evgeny saldin - arxiv.org

SPECIAL RELATIVITY

Applications to astronomy and the acceleratorphysics

Evgeny Saldin

Preprint submitted to 8 December 2021

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aFifth draft from December 2021

Current draft was completed by a new Chapter 8 where weanalyze the potential for exploiting earth-based sources inorder to confirm the predictions of the aberration of light

(particle) theory. We demonstrate that aberration image shiftis regulated by the Wigner rotation. According to the specialtheory of relativity, two observers with different trajectorieshave different 3-spaces. In contrast, the accepted in previousliterature incorrect assumption that an earth-based observerand an inertial ( e.g. sun - based ) observer have common

3-space has always been considered obvious. We propose anexperiment using commercial ( STEM ) electron microscopeas a ( relativistic ) particle source. The proposed method ofmeasuring the angle of aberration involves the use of earthbased sources and have a big advantage. The rotation of theearth on its axis should produce a corresponding shift of theimage. The aberration shift depends only upon the value of

v⊥, the component of the orbital velocity perpendicular to theearth rotation axis. In principle, records could be taken over

a period of one day. Published experimental results ofunavoidable image drift (with constant speed and direction)during a period of one hour is quite close to the theoretical

prediction.If you have any comments, please send an email to

[email protected].

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*Preface

There are many books on the classical subject of special relativity. However,after having spent a number of years, both in relativistic engineering andresearch with relativity, I have come to the conclusion that there exist a placefor a new book. I do believe that the present book is not quite the same asthe others, mainly due to attempt to cast light on dark corners.

I should make it clear what this little book is not. It is not a textbook onrelativity theory. What the book is about is the nature of special relativistickinematics, its relation to space and time, and the operational interpretationof coordinate transformations. Every theory contains a number of quantitiesthat can be measured by experiment and an expressions that cannot possiblybe observed. Whenever we have a theory containing an arbitrary conven-tion, we should examine what parts of the theory depend on the choice ofthat convention and what parts do not. The distinction is not always madeand many authors claim some data to be observable, according to arbitraryconventions, which do not correspond to any physical experiment. Thisleads to inconsistencies and paradoxes that should be avoided at all cost.

The practical approach used in the book should be acceptable to astronomers,space engineers, accelerator engineers, and more generally, relativistic engi-neers. This approach, unusual in the relativistic literature, may be clarifiedby quoting one of the problems discussed in the text: the new light beamkinematics for rotating frame of references. Since we live on such a rotating(earth-based) frame of reference, difference in relativistic kinematics be-tween rotating and non-rotating frames of reference is of great practical aswell as theoretical significance. A correct solution of this problem requiresthe use of relativistic principles even at low velocities since the first-orderterms in (v/c) play a fundamental role in the non-inertial relativistic kine-matics of light propagation.

All the results presented here are derived from the ”first principles”, and allsteps involving physical principles are given. To preserve a self-consistentstyle, I place the derivation of auxiliary results in appendices. To help readersform their own opinion on the topics discussed, the end of each chapterhas a suggested bibliography together with relevant remarks. The list ofreferences includes only the papers I have consulted directly. A lot of papersremain unmentioned, and for this I apologize.

I am grateful to my longtime friends Gianluca Geloni and Vitaly Kocharyanfor discussions over many years about much of the material in this book.I should also like to express my thanks to DESY (Deutsches Electronen-Synchrotron) for enabling me to work in this interesting field.

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Contents

1 Introduction 10

2 A Critical Survey of the Present Approaches to Special Relativity 21

2.1 What is Special Relativity? 21

2.2 Different Approaches to Special Relativity 22

2.3 Myth About the Incorrectness of Galilean Transformations 25

2.4 Myth About the Non-Relativistic Limit of the LorentzTransformations 26

2.5 Myth about the Constancy of the Speed of Light 27

2.6 Convention-Dependent and Convention-Invariant Parts of theTheory 28

2.7 Myth about the Reality of Relativistic Time Dilation and LengthContraction 29

2.8 Relativistic Particle Dynamics 29

2.9 Mistake in Commonly Used Method of Coupling Fields andParticles 31

2.10 Clarification of the True Content of the ”Single Frame” Theory 33

2.11 Bibliography and Notes 34

3 Space-Time and Its Coordinatization 38

3.1 Introductory Remarks 38

3.2 Choice of Coordinates System in an Inertial Frame 38

3.3 Inertial Frame where a Source of Light is at Rest 39

3.4 Motion of a Light Source With Respect to the Inertial Frame 41

3.5 Discussion 42

3.6 A Clock Resynchronization Procedure 43


4 Aberration of Light Phenomenon. Inertial Frame of Reference 47

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4.2 The ”Plane Wave” Emitter 47

4.3 A Moving Emitter. The ”Single Frame” Description 48

4.4 A Moving Emitter. Explanation on the Basis of the LorentzTransformations 52

4.5 Reflection from a Mirror Moving Transversely. Mistake in ExistingTheory 53

4.6 Way to Solve the Emitter-Mirror Problem in the (3+1) Space andTime 56

4.7 Covariant Way to Solve the Emitter-Mirror Problem 59

4.8 Discussion 61

4.9 Large Aperture Mirror 63

4.10 Analysis of Transmission through a Hole in a Opaque Screen 64

4.11 Spatiotemporal Transformation of the Transmitted Light Beam 65

4.12 Applicability of the Ray Optics 66

4.13 Measure of Aberration 68

4.14 Moving Large Aperture Emitter 69

4.15 A Point Source in an Inertial Frame of Reference 71


5 Aberration of Light Phenomenon. Noninertial Frame of Reference 76


5.2 Absolute Time Coordinatization in the Accelerated Systems 77

5.3 The Asymmetry between the Inertial and Accelerated Frames 78

5.4 A Resynchronization of the Accelerated Clocks 81

5.5 A Single Moving Emitter in a Noninertial Frame of Reference 82

5.6 A Useful Analogy 86

5.7 A Point Source in a Nonertial Frame of Reference 87

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5.8 Concept of Reciprocity in the Aberration of Light Theory 89

5.9 Explanation of the Aberration on the Basis of Electrodynamics 91

5.10 Explanation of the Aberration on the Basis of the Ether Theory 93


6 Stellar Aberration 97

6.1 A New Approach to the Stellar Aberration 97

6.2 Heliocentric Inertial Frame of Reference 100

6.3 Earth-Based Non-Inertial Frame of Reference 101

6.4 Region of Applicability 103


7 Notion of Ordinary Space in Special Relativity 107

7.1 Inertial Frame View of Observations of the Noninertial Observer 107

7.2 The Non-Existence of an Instantaneous 3-Space 112

7.3 Acceleration of a Rigid Body and Special Relativity 114

7.4 Discussion 117

7.5 Aberration of Relativistic Particles 118

8 Earth-Based Setups to Detect the Aberration Phenomena 122

8.1 Potential of Earth-Based Electron Microscopes 122

8.2 Explanation of the Aberration of Electrons on the Basis ofElectrodynamics 124

8.3 Absent of the Aberration of Light from the Earth-based Laser Source 125

8.4 Thought Experiment using Point-Like Source at Optical Frequencies 130

8.5 Working Way of the Setup Using Incoherent Source at OpticalFrequencies 132


9 Special Relativity. Absence of Reciprocity 137

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9.1 Time Dilation 137

9.2 Light-Clock. Observations of the Noninertial Observer 138

9.3 Experimental Evidence for the Absence of Reciprocity 141

9.4 Pseudo-gravitation Field and the Langevin Metric 142


10 Space and Time Coordinates in an Inertial Frame 148

10.1 Active and Passive Transformations 148

10.2 Two Metrics 151

10.3 Metrics Connected with Optics of Moving Media and Light Sources 151

10.4 The Notion of Time in Special Relativity Theory 154

10.5 The Aberration of Light from a Moving Laser Source 155

10.6 The Aberration of Particles from a Moving Source in an InertialFrame 159

10.7 Inertial Frame View of Accelerated Light Clock 163

10.8 Aberration of Light and the Principle of Relativity 169


11 Relativistic Particle Dynamics 172

11.1 Manifestly Covariant Particle Dynamics 172

11.2 Conventional Particle Tracking. Hidden Absolute TimeCoordinatization 173

11.3 Incorrect Expansion of the Relation dτ = dt/γ to an ArbitraryMotion 176

11.4 The Usual Integration of the 4D Covariant Equation of Motion 177

11.5 Covariant Particle Tracking 179

11.6 Convention-Invariant Particle Tracking 181

11.7 The Nature of Mass 183


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12 Relativity and Electrodynamics 188

12.1 Why did the Error in Radiation Theory Remain so Long Undetected? 189

12.2 An Illustrative Example 191


13 Synchrotron Radiation 204


13.2 Paraxial Approximation for the Radiation Field 205

13.3 Undulator Radiation 206

13.4 Synchrotron Radiation from Bending Magnets 217

13.5 How to Solve Problems Involving Many Trajectory Kicks 225

13.6 Synchrotron Radiation in the Case of Particle Motion on a Helix 229


14 Relativity and X-Ray Free Electron Lasers 237


14.2 Modulation Wavefront Orientation 240

14.3 XFEL Radiation Setup 242

14.4 Modulation Wavefront Tilt in Maxwell’s Theory. LogicalInconsistency 248


15 Kinematics of the Wigner Rotation 254


15.2 The Commutativity of Collinear Lorentz Boosts 254

15.3 The Noncommutativity of Two Lorentz Boosts in NonparallelDirections 255

15.4 Wigner Rotation 256

15.5 How to Measure a Wigner rotation? 257

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15.6 Wigner Rotation and the Reciprocity 259


16 Relativistic Spin Dynamics 262


16.2 Magnetic Dipole at Rest in an Electromagnetic Field 262

16.3 Derivation of the Covariant (BMT) Equation of Motion of Spin 263

16.4 Change Spin Variables 265

16.5 An Alternative Approach to the BMT Theory 266

16.6 Spin Tracking 268

16.7 Spin Rotation in the Limit g→ 0 as Dynamics Effect 271

16.8 Incorrect Expression for Wigner Rotation. Myth About ExperimentalTest 273

16.9 Reasons that Underlie the Derivation of the Incorrect Expressions 276

16.10Bibliography and Notes 278

References 279

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1 Introduction

The standard books on special relativity do not usually address the questionsof the physical meaning of relativistic effects and the nature of space-time.The presentation of the subject in the present book differs somewhat fromthe usual one in that the four-dimensional geometric formulation of thetheory plays a dominant role than in most of the current textbooks.

The book begins with a critical survey of the present approaches to spe-cial relativity. The established way of looking at special relativity is basedon Einstein postulates: the principle of relativity and the constancy of thevelocity of light. In the most general geometric approach to the theory ofspecial relativity, the principle of relativity, in contrast to Einstein formu-lation, is only a consequence of the geometry of space-time. We point thatthe essence of the special theory of relativity consists in the following pos-tulate: all physical processes proceed in four-dimensional space-time, thegeometry of which is pseudo-Euclidean.

The space-time geometric approach to special relativity deals with all pos-sible choices of coordinates of the chosen reference frames, and thereforethe second Einstein postulate, referred to as the constancy of the coordinatespeed of light, does not have a place in this more general formulation. Onlyin Lorentz coordinates, when Einstein’s synchronization of distant clocksand Cartesian space coordinates are used, the coordinate speed of light isisotropic and constant. Thus, the basic elements of the space-time geometricformulation of the special relativity and the usual Einstein’s formulation,are quite different.

It should be emphasized that in practical applications there are two choicesof clock synchronization convention useful to consider:

(a) Einstein’s convention, leading to the Lorentz transformations betweenframes.

(b) Absolute time convention, leading to the Galilean transformations be-tween frames.

Absolute time (or simultaneity) can be introduced in special relativity with-out affecting neither the logical structure, no the (convention-independent)predictions of the theory. In the theory of relativity, this choice may seemquite unusual, but it is usually most convenient when one wants to connectto laboratory reality.

There is a widespread view that only philosophers of physics discuss theissue of distant clock synchronization. Indeed, a typical physical laboratory

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contains no space-time grid. It should be clear that a rule-clock structureexist only in our mind and manipulations with non existing clocks in thespecial relativity are an indispensable prerequisite for the application of dy-namics and electrodynamics theory in the coordinate representation. Suchsituation usually forces physicist to believe that the application of the theoryof relativity to the study of physical processes is possible without detailedknowledge of the clocks synchronization procedure.

However, many problems in special relativity can be adequately treated onlyby an approach which uses the non-standard absolute time synchronization.One of the features that is unique to this book is its treatment of the absolutetime coordinatization. No other books deals with Galilean transformationsin the framework of special relativity.

Third chapter presents an ”operational interpretation” of the Lorentz andabsolute time coordinatizations. This is probably the most important chapterof this book. Today the statement about correctness of Galilean transforma-tions is a ”shocking heresy”, which offends the ”relativistic” intuition andthe generally accepted way of looking at special relativity of most physi-cists. The difference between absolute time synchronization and Einstein’stime synchronization from the operational point of view will be an im-portant discovery for every special relativity expert. To our knowledge,neither operational interpretation of the absolute time coordinatization northe difference between absolute time synchronization and Einstein’s timesynchronization from the operational point of view, are given elsewhere inthe literature.

We have tried to keep the mathematical complexity of the discussion toa minimum. The derivation of relativistic kinematics in the first part ofthe book (Chapter 3-10) is fairly elementary from a mathematical point ofview, but it is conceptually subtle. We use synthetic approach to present thematerial: some simple models are studied first, and more complicated onesare introduced gradually.

We start with aberration of light phenomena. Light, being a special case ofelectromagnetic waves, is described by the electrodynamics theory. It is wellknown that the electrodynamics theory meets all requirements of the theoryof relativity and therefore must accurately describe the properties of such atypical relativistic object as light.

In the Chapter 4-5, we present a critical reexamination of the existing aber-ration of light theory. The phenomenon of aberration of light is by no meanssimple to describe, even in the first order in v/c: a large number of incorrectresults can be found in the literature. The utilization of the electrodynam-ics in the absolute time coordinatization becomes indispensable when we

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consider optical phenomena associated with a relative motion of two (ormore) light sources. There is no problem to introduce Lorentz coordinateswhen these sources are independent. The possibility to introduce Lorentzcoordinates in this situation consists in introducing individual coordinatesystem (i.e. individual set of clocks) for each sources. Suppose now thatthe second (e.g. secondary) source interacts with the light beam radiatedfrom the first source. The peculiarity of this problem with the viewpoint ofthe special relativity is that here we cannot prepare for sources a commonLorentz coordinatization in the single inertial frame.

Suppose that we have a point source (and observer with his measuringdevices) is at rest in the inertial lab frame and a (finite aperture) mirroris moving with a constant speed and interacts with the radiated light. Itis generally believed that for a mirror moving tangentially to its surfacethe law of reflection which holds for the stationary mirror is preserved.This statement, presented in textbooks, is incorrect. We demonstrate thatthere is a deviation of the energy transport for reflected light. The resultis extremely important from the applied point of view. According to theBabinet’s principle, this remarkable prediction of our aberration of lighttheory is correct also for light transmitted through a hole in the movingopaque screen or, consequently, through a moving open end of the telescopebarrel.

We demonstrate that the aberration of light is a complex phenomenon whichmust be branched out into a number of varieties according to their origin.These branching out takes place depending on what is the cause of aberra-tion - whether it is a jump in the velocity (with respect to the fixed stars)of the observer or of the light source. Aberration could undergo furthersplitting - depending on the physical influence of the optical instrument onthe measurement.

Questions related to transmission through the transversely moving pupildetection (e.g. a moving end of the telescope barrel) lead to serious mis-understanding, which is actually due to an inadequate understanding ofseveral complicated aspects of the statistical optics. Today one is told thatthe phenomenon of aberration of light could be interpreted, using ray modelof light. One case of rather great interest is that which corresponds to trans-mission through a moving telescope tube. The rays of light coming from thestar are falling on the telescope tube and, according to literature, not interactwith its sides. One could naively expect that the region of applicability of rayoptics, following from the textbooks reasoning, should be identified withany spatially incoherent radiation. This is misconception. In particular, aspatially completely incoherent source (e.g. an incandescent lamp or a star)is actually a system of elementary (statistically independent) point sourceswith different offsets. Radiation field generated by a completely incoherent

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source can be seen as a linear superposition of fields of individually inco-herent sources. An elementary source produces in front of pupil detectioneffectively a plane wave and the measuring instrument always influencesthe measured radiation (due to unavoidable diffraction of a plane wave byan aperture). It should be remarked that any linear superposition of radi-ation fields from elementary point sources conserves single point sourcecharacteristics like independence on source motion.

The Chapter 6 deals with astronomical applications. The effect of stellaraberration seems to be one of the simplest phenomena in astronomicalobservations. In spite of its apparent simplicity, aberration seems to beone of the most intricate effects in special relativity. It is widely believedthat stellar aberration depends on the relative velocity of the source (star)and observer. Observations show clearly that stellar aberration does notdepend on the relative motion between star and telescope on the earth. Thelack of symmetry, between the cases when either the source or telescope ismoving is shown clearly on the basis of the separation of binary stars. Therelative motion of these stars with respect to each other (and hence, withrespect to the earth) is never followed by any aberration, although motionof these stars is, sometimes, much faster than that of the earth aroundthe sun. It should be stressed that it is the telescope and not the star thatmust change its velocity (relative to the fixed stars) to cause aberration.Contradiction is so plane that some physicists use this fact to argue thatstellar aberration contradicts the special theory of relativity. There is noavailable explanation for the fact that, while the observational data on stellaraberration are compatible with moving earth, the symmetric description,when the star possesses the relative transverse motion, does not apparentlylead to observations compatible with predictions. We demonstrate that thefact that we do not see myriads of widely separated binaries in wild gyrationdoes not require any fundamental change of outlook, but it does requirethat aberration of ”distant” stars is treated in the framework of space-timegeometric approach.

It is generally believed that the phenomenon of aberration of light could beinterpreted, using the corpuscular model of light, as being analogous of theobservation of the oblique fall of raindrops by a moving observer. This isa classical kinematics method to the computation of the stellar aberrationused in astronomy for about three hundred years. This book develops arelativistic theory of aberration of light in rotating frames. To see the con-sequences of relativistic modification of stellar aberration, we start with theclassical aberration increment, or θa = v/V. Here v is the velocity of observerand V is the velocity of particles. It is assumed that v V. According to theconventional approach, if we neglect v2/c2 comparison with 1, stellar aber-ration increment is simplified to elementary formula θa = v/c. This result isparticularly remarkable because the study of stellar aberration is intimately

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connected with the old kinematics: the Galilean vectorial law of additionof velocities is actually used. We come to the conclusion that the standardanalysis of the stellar aberration does not take into account the fundamentaldifference between velocity of light and velocity of raindrops.

The satisfactory treatment of the relativistic modification should be basedon the two relativistic parameters. For an observer on the earth, it is, withrespect to the solar referential frame, of about 30 km/s, corresponding tothe earth motion around the sun. Clearly, in the theory of stellar aberration,we consider the small expansion parameter v/c ' 10−4 neglecting terms oforder of v2/c2. In the theory of stellar aberration, we, however, must alsoconsider the relativistic parameter V/c. For light V/c = 1 and the special the-ory of relativity says that stellar aberration cannot be discussed as a topic inthe classical theory. Light is always a (”ultra”) relativistic object, no matterhow small the ratio v/c may be. According to the special theory of relativ-ity, physical processes takes place in a metric space-time geometry, so thegeodesic interval must be used to describe aberration of light phenomenon.

The problem of the earth-based measurements is solved with the discov-ery of the essential asymmetry between the earth-based and the sun-basedobservers, namely, the acceleration of the traveling earth-based observerrelative to the fixed stars. We demonstrate that for explanation of the aber-ration of light effect in the rotating frame of reference one does not needneither modify the special theory of relativity, nor apply general theory ofrelativity. It is only necessary to strictly follow the special relativity.

All phenomena in non-inertial reference system should be considered in theframework of the space-time geometric approach using metric tensor. UsingLangevin metric in a frame of reference attending rotation enables us toexplain all earth-based experiments. A correct solution of this problem in therotating frame requires the use of metric tensor even in experiments of firstorder in v/c since the crossed term in the Langevin metric (which involvesthe first-order deviation of metric tensor from its Minkowski form) plays afundamental role in the non-inertial kinematics of a light beam propagation.The historical rate of the studies of the optical effects in rotating frames isvery unusual. The Sagnac effect was discovered in 1913 and described byLangevin in 1921. It is interesting that the Langevin metric has in the pasthardly ever been applied in the stellar aberration theory.

The main facts which a theory of stellar aberration in the earth-based frameof reference must explain are (1) the annual apparent motion of the fixedstars about their locations and (2) the null apparent aberration of rotatingbinary systems (they exhibit aberrations not different from other stars). Wepresent here a theory which accounts for all these, and in addition givesnew results. All earth-based experiments can be explained on the basis of

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the effect of the measuring instrument (i.e. physical unavoidable influenceof the telescope on the measurement) and the acceleration of traveling earth-based observer relative to the fixed stars.

The second part of the book (Chapter 11 - 16) deals with accelerator physics.This part is technically more challenging, but perhaps also more practical.In the Chapter 11 we describe the particle dynamics method in the Lorentzlab frame using lab time t as evolution parameter. The study of relativisticparticle motion in a constant magnetic field according to usual acceleratorengineering, is intimately connected with the old (Newtonian) kinematics:the Galilean vectorial law of addition of velocities is actually used. A non-covariant approach to relativistic particle dynamics is based on the absolutetime coordinatization, but this is actually a hidden coordinatization. Theabsolute time synchronization convention is self-evident and this is thereason why this subject is not discussed in accelerator physics.

There is a reason to prefer the non-covariant way within the framework ofdynamics only. In this approach we have no mixture of positions and time.This (3+1) dimensional non-covariant particle tracking method is simple,self-evident, and adequate to the laboratory reality. It can be demonstratedthat there is no principle difficulty with the non-covariant approach in me-chanics and electrodynamics. It is perfectly satisfactory. It does not matterwhich transformation is used to describe the same reality. What matter isthat, once fixed, such convention should be applied and kept in a consistentway for both dynamics and electrodynamics.

The common mistake made in accelerator physics is connected with theincorrect algorithm for solving the electromagnetic field equations. If onewants to use the usual Maxwell’s equations, only the solution of the dy-namics equations in covariant form (i.e. in Lorentz coordinates) gives thecorrect coupling between the Maxwell’s equations and particle trajectoriesin the lab frame.

First, we examine the reasoning presented in textbooks. It is generally be-lieved that the magnetic field, in the Lorentz coordinatization, is only ca-pable altering the direction of motion, but not the speed of the electron.Nevertheless, there is argument against this commonly accepted derivationof the composition of velocities. The standard presentation of the velocitytransformation is based on the hidden assumption that (x′, y′, z′) axes ofthe moving observer and (x, y, z) axes of the lab observer are parallel. Inother words, it is assumed that observers have common 3-space. This ismisconception. The fact that two observers with different trajectories in theLorentz coordinatization have different 3-spaces is not considered in text-books. First of all, there is no absolute notion of simultaneity in the theoryof special relativity. In the case of the relativity of simultaneity we have a

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mixture - of positions and time. In other words, in the space measurementof one observer there is mixing of space and time, as seen by the other.Therefore, there is no notion of an instantaneous 3-space.

We demonstrate that velocities addition in the Lorentz coordinatizationis regulated by the Wigner rotation. One of the consequences of non-commutativity of non-collinear Lorentz boosts is not the existence of acommon ordinary space. Our result is at odds with the prediction fromtextbooks. The commonly accepted derivation of the composition of non-collinear velocities based on the use the relation dt′ = dt/γ and does not ac-count for the Wigner rotation (i.e. mixture of transverse positions and time).We illustrate error in standard coupling fields and particles by consideringthe relatively simple example, wherein the essential physical features arenot obscured by unnecessary mathematical difficulties. We hope that thepresent example will draw wide attention to the central role that Wignerrotation plays in the transformation of non-collinear velocities.

In the Chapter 11 we examine what parts of the dynamics theory dependon the choice of synchronization convention and what parts do not. Thepresent analysis shows the difference between the notions of path and tra-jectory. The path is rather a purely geometrical notion. The trajectory of aparticle conveys more information about its motion because every positionis described additionally by the corresponding time instant. The path hasexact objective meaning. In fact, the curvature radius of the path in themagnetic field (and consequently the three-momentum) has obviously anobjective meaning, i.e. is convention-invariant. In contrast to this, and con-sistently with the conventionality intrinsic in the velocity, the trajectory ~x(t)of the particle is convention dependent and has no exact objective meaning.Dynamics theory contains a particle trajectory that we do not need to checkdirectly, but which is used in the analysis of electrodynamics problem.

In the Chapter 12 we present a critical reexamination of radiation theory.To evaluate radiation fields arising from an external sources, we need toknow the particle velocity ~v and the position ~x as a function of the lab frametime t. As discussed above, one should solve the usual Maxwell’s equa-tions in the lab frame with current and charge density created by particlesmoving along covariant trajectory. For an arbitrary parameter v/c covariantcalculations of the radiation process is very difficult. There are, however,circumstances in which calculations can be greatly simplified. As exampleof such circumstance is a non-relativistic radiation setup. The reason is thatthe non-relativistic assumption implies the dipole approximation which isof great practical significance. In accounting only for the dipole part of the ra-diation we neglect all information about the electron trajectory. That meansthat the dipole radiation does not show any sensitivity to the differencebetween covariant and non-covariant particle trajectories.

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But that is only the first and most practically important term. The otherterms (higher multi-poles of radiation) tell us that there are higher ordercorrections to the dipole radiation approximation. The calculation of thiscorrection requires detailed information about the electron trajectory. Obvi-ously, in order to calculate the correction to the dipole radiation, we will haveto use the covariant trajectory and not be satisfied with the non-covariantapproach.

Accelerator physicists, who try to understand the situation related to theuse of the theory of relativity in the synchrotron radiation phenomena,are often troubled by the fact that the difference between covariant andnon-covariant particle trajectories was never understood, and that nobodyrealized that there was a contribution to the synchrotron radiation from rela-tivistic kinematics effects. Accelerator physics was always thought in termsof the old (Newtonian) kinematics that is not compatible with Maxwell’sequations. At this point, a reasonable question arises: since storage rings aredesigned without accounting for the relativistic kinematics effects, how canthey actually operate? In fact, electron dynamics in storage ring is greatlyinfluenced by the emission of radiation. We answer this question in great de-tail in the Chapter 13. The main emphasis of this chapter is on spontaneoussynchrotron radiation from bending magnets and undulators.

Similar to the non-relativistic asymptote, the ultrarelativistic asymptote alsoprovides the essential simplicity of the covariant calculation. The reason isthat the ultrarelativistic assumption implies the paraxial approximation.Note that the geometry of the electron motion in a bending magnet has acylindrical symmetry. There are a number of remarkable effects which are aconsequence of the cylindrical symmetry and the paraxial approximation.We demonstrate that there is no needs to use covariant particle tracking forderivation of the bending magnet radiation. However, there is one situa-tion where the conventional theory fails. The covariant approach predictsa non-zero red shift of the critical frequency, which arises when there areperturbations of the electron motion along the magnetic field.

One way to demonstrate incompatibility between the standard approachto relativistic electrodynamics, which deals with the usual Maxwell’s equa-tions, and particle trajectories calculated by using non-covariant particletracking, is to make a direct laboratory test of synchrotron radiation theory.In other words, we are stating that, despite the many measurements doneduring decades, synchrotron radiation theory is not an experimentally well-confirmed theory. We analyze the potential for exploiting synchrotron radi-ation sources in order to confirm the predictions of corrected synchrotronradiation theory. Proposed experiments have never been performed at syn-chrotron facilities.

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The theory of relativity shows that the relativity of simultaneity, which isa most fundamental relativistic kinematics effect, is related with extendedrelativistic objects. But up to 21 st century there were no macroscopic objectspossessing relativistic velocities, and there was a general belief that onlymicroscopic particles in experiments can travel at velocities close to that oflight. The 2010s saw a rapid development of new laser light sources in theX-ray wavelength range. An X-ray free electron laser (XFEL) is an examplewhere improvements in accelerator technology makes it possible to developultrarelativistic macroscopic objects with an internal structure (modulatedelectron bunches), and the relativistic kinematics plays an essential role intheir description.

In Chapter 14 we present a critical reexamination of existing XFEL theory.It is mainly addressed to readers with limiting knowledge of acceleratorphysics. We study the production of coherent undulator radiation by mod-ulated ultrarelativistic electron beam. Fortunately, the principle of produc-tion of coherent undulator radiation does not require specific knowledgeof undulator radiation theory presented in the Chapter 13 and can be ex-plained in a very simple way. Relativistic kinematics enters XFEL physicsin a most fundamental way through the rotation of the modulation wave-front, which, in ultrarelativistic approximation, is closely associated to therelativity of simultaneity. When the trajectories of particles calculated in theLorentz reference frame (i.e. an inertial frame where Einstein synchroniza-tion procedure is used to assign values to the time coordinate) they mustinclude relativistic kinematics effects such as relativity of simultaneity. Inthe ultrarelativistic asymptote, the orientation of the modulation wavefront,i.e the orientation of the plane of simultaneity, is always perpendicular tothe electron beam velocity when the evolution of the modulated electronbeam is treated using Lorentz coordinates.

We should remark that Maxwell’s equations are valid only in Lorentz refer-ence frames. Einstein’s time order should obviously be applied and kept inconsistent way both in dynamics and electrodynamics. It is important at thispoint to emphasize that the theory of relativity dictates that a modulatedelectron beam in the ultrarelativistic asymptote has the same kinematics,in Lorentz coordinates, as a laser beam. According to Maxwell’s equations,the wavefront of a laser beam is always orthogonal to the propagation di-rection. Experiments show that this prediction is, in fact, true. The theoryof relativity as a theory of space-time with pseudo-Euclidean geometry hashad more than hundred years of development, and rather suddenly it hasbegun to be fully exploited in practical ways in XFEL physics.

As known, a composition of noncollinear Lorentz boosts does not resultin a different boost but in a Lorentz transformation involving a boost anda spatial rotation, the Wigner rotation (it is often called the ”Thomas rota-

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tion”). We discuss the Wigner rotation in Chapter 7 and Chapter 11. Ourearlier discussion is really about aberration of light and particles. The re-sults for the Wigner rotation in the Lorentz lab frame obtained by manyexperts on special relativity are incorrect. They overestimate the angle ofthe Wigner rotation by a relativistic factor γ compared to its real value, andthe direction of the rotation is also determined incorrectly. It should be notethat the correct result was obtained in the works of several authors, whichwere published more than half century ago but remained unnoticed againstthe background of numerous incorrect works. To deduce the expression forWigner rotation is the main purpose of Chapter 15. We demonstrate thatwhat are usually considered the advanced parts of the theory of relativityare, in fact, quite simple.

The Wigner rotation is also relates the angular rotation velocity (Thomasprecession) of the spin of an elementary particles following a curvalinearorbit. The kinematics tools needed to study the motion of charged elemen-tary spinning particles in the storage ring are the main topic of final chapterof the book. In 1959, Bargman, Michel, and Telegdi (BMT) proposed a con-sistent relativistic theory for the dynamics of the spin as observed in thelab frame, which was successfully tested in experiments. It is commonly be-lieved that the BMT equation contains the standard (and incorrect) result forthe Wigner rotation in the Lorentz lab frame. There are many physicists whohave already received knowledge about the Wigner (Thomas) rotation fromwell-known textbooks and who would say, ”The extremely precise mea-surements of the magnetic-moment anomaly of the electron made on highlyrelativistic electrons are based on the BMT equation, of which the Wigner ro-tation is an integral part, and can be taken as experimental confirmations ofthe standard expression for the Wigner rotation” This misconception aboutexperimental test of the incorrect expression for the Wigner rotation in thelab frame is common and pernicious. We should remark that the results inthe Bargmann-Michel-Telegdi paper were obtained by the method of semi-classical approximation of the Dirac equation. The Wigner rotation was notconsidered at all, because the Dirac equation allow obtaining the solutionfor the total particle’s spin motion without an explicit splitting it into theLarmor and Wigner parts. We demonstrate that the notion that the standard(incorrect) result for the Wigner rotation as an integral part of the BMTequation in most texts is based, in turn, on an incorrect physical argument.

A large number of incorrect expressions for the Wigner (Thomas) rotationcan be found in the literature. Let us discuss the reasons that underlie thederivation of the incorrect expressions. The situation relating to the use ofthe Wigner (Thomas) rotation theory in the relativistic spin dynamics is com-plex. The starting point of Bargman, Michel and Telegdi was the particle restframe and the equation of motion for particle angular momentum, whichthey generalized to the Lorentz lab frame and then transformed back to the

19

rest frame. This brings up an interesting question: Why it is convenient totransform this equation to the rest frame as of that instant? The explanationis deep down in relativistic quantum mechanics. The approach in which wedeal with the proper spin is much preferred in the experimental practice dueto clear physical meaning of the spin vector ~s. Unlike momentum, whichhas definite components in each reference frame, angular momentum isdefined only in one particular reference frame. It does not transform. Anystatement about it refers to the rest frame as of that instant. If we say thatin the lab frame the spin of a particle makes the angle φ with its velocity,we mean that in the particle’s rest frame the spin makes this angle withthe line motion of the lab frame. The analysis of physical phenomena innon-inertial frames of reference can be described in an inertial frame withinstandard (Einstein) special relativity taking advantage of well known rela-tivistic kinematic effects. In the case of spin dynamics one has no occasionto make measurements with non-inertial devices. Common textbooks pre-sentations of the spin dynamics use the approach which deals only withobservations of the inertial (lab) observer. The conventional method usedto explain the spin dynamics in the lab frame is very unusual. First, wealready pointed that the laws of physics in the lab frame are able to accountfor all physical phenomena, including the observation made by non-inertialobserver. In particular, it is possible to determine the result of spin orienta-tion measurements with respect to the lab frame axes in the proper frame,which is moving with velocity −v and acceleration in the proper frame. Inthis case, when viewed from the lab frame, the interpretation of the spinrotation experiments in the lab frame is simple and may be derived from theprediction of the proper observer as concerns the measurement made by thelab observer. Therefore, the spin rotation measurement by a polarimeter inthe lab frame is interpreted with viewpoint of the proper observer as view-ing this of the lab observer. This complexity is one of the reason why authorsof textbooks obtained an incorrect expression for the Wigner rotation.

It is also important to elucidate the reason why did the error in the Wignerrotation theory remain so long undetected. The main conclusion drawnfrom the review of the literature is as follows. Wigner (Thomas) rotationis generally presented only in the context of spin kinematics as a peculiareffect of special relativity. Regretfully, such limiting point of view still pre-vails in textbooks and monographs expounding relativity theory. This issupposedly the reason why until recently no researches have noticed soserious a discrepancy between the results of different works. We emphasizethat the Wigner rotation is very basic relativistic phenomenon. It is as basicas the time dilation and length contraction. For example, we demonstrateby elementary means that relativistic velocities addition is regulated by theWigner rotation. Therefore, it is valid to say that velocity addition and otherapplications (e.g. the aberration of light effect in the rotating frame) demon-strate the absurdity of the textbook expression for the Wigner rotation.

20

2 A Critical Survey of the Present Approaches to Special Relativity

2.1 What is Special Relativity?

The laws of physics are invariant with respect to Lorentz transformations.This is a restrictive principle and does not determine the exact form ofthe dynamics in question. Understanding the postulates of the theory ofrelativity is similar to understanding energy conservation: at first we learnthis as a principle and later on we study microscopic interpretations thatmust be consistent with this principle. For any system to which the energyconservation principle can be applied, a deeper theory should exist whichyields insight into the detailed physical processes involved. Of course, thisdeeper theory must lead to energy conservation.

The principle of conservation of energy is very useful in making analyseswithout knowing all the formulas of the fundamental theory. A method-ological analogy with the postulates of the special relativity emerges byitself. Suppose we do not know why a muon disintegrates, but we knowthe law of decay in the Lorentz rest frame. This law would then be a phe-nomenological law. The relativistic generalization of this law to any Lorentzframe allows us to make a prediction on the average distance traveled bya muon. In particular, when a Lorentz transformation of the decay law istried, one obtains the prediction that after the travel distance γvτ0, the popu-lation in the lab frame would be reduced to 1/2 of the origin population. Wemay interpret this result by saying that, in the lab frame, the characteristiclifetime of a particle has increased from τ0 to γτ0.

However, the theory of relativity is necessary incomplete. Constructive (mi-croscopic) theories like electrodynamics or quantum field theory providemore insight into the nature of things than restrictive theories like specialrelativity. Relativistic kinematics is only an interpretation of the behaviorof the dynamical matter fields in the view of different observers. The pointis that one can, in principle compute any relativistic quantity directly fromthe underlying theories of matter without involving relativity at all. Forexample, muons in motion behave relativistically because the field forcesthat are responsible for the muon disintegration satisfy quantum field equa-tions that are Lorentz covariant. Of course, in the ”microscopic” approachto relativistic phenomena, Lorentz covariance of all the fundamental lawsof physics remains, similarly to energy conservation, an unexplained fact,but all explanation must stop somewhere.

21

2.2 Different Approaches to Special Relativity

In literature, three approaches to special relativity are discussed: Einstein’sapproach, the usual covariant approach, and the space-time geometric ap-proach.

Einstein formulation is based on postulates: the principle of relativity andthe constancy of the velocity of light. The usual covariant formulation of thetheory of relativity deals with the pseudo-Eucledian space-time geometryand with the invariance of interval ds, but it is understood only in a limitedsense when the metric is strictly diagonal. Assuming diagonality of themetric we also automatically assume Lorentz coordinates, and that differentinertial frames are related by Lorentz transformations.

In space-time geometric approach, primary importance is attributed to thegeometry of space-time; it is supposed that the geometry of space-time isa pseudo-Euclidean geometry in which only 4-tensors quantities do havereal physical meaning. In this most general approach the principle of rela-tivity in contrast to Einstein formulation of the special relativity is a simpleconsequence of the space-time geometry. Since the space-time geometricapproach deals with all possible choices of coordinates of the chosen refer-ence frames, the second Einstein postulate referred to the constancy of thecoordinate velocity of light does not hold in this formulation of the theoryof relativity. Only in Lorentz coordinates, when Einstein’s synchronizationof distant clocks and Cartesian space coordinates are used, the coordinatespeed of light is isotropic and constant.

2.2.1 The Usual Einstein’s Approach

Traditionally, the special theory of relativity is built on the principle ofrelativity and on a second additional postulate concerning the velocity oflight:

1. Principle of relativity. The laws of nature are the same (or take the sameform) in all inertial frames

2. Constancy of the speed of light. Light propagates with constant velocityc independently of the direction of propagation, and of the velocity of itssource.

The constancy of the light velocity in all inertial systems of reference is nota fundamental statement of the theory of relativity. The central principleof special relativity is the Lorentz covariance of all the fundamental lawsof physics. It it important to stress at this point that the second ”postu-

22

late”, contrary to the view presented in textbooks, is not a separate physicalassumption, but a convention that cannot be the subject of experimentaltests.

Assuming postulate 2 on the constancy of the speed of light in all inertialframes we also automatically assume Lorentz coordinates, and that differ-ent inertial frames are related by Lorentz transformations. In other words,according to such limiting understanding of the theory of relativity it isassumed that only Lorentz transformations must be used to map the coor-dinates of events between inertial observers.

2.2.2 The Usual Covariant Approach

In the usual covariant approach the special of relativity is understood asthe theory of space-time with pseudo-Euclidean geometry. Quantities ofphysical interest are represented by tensors in a four-dimensional space-time, i.e. by covariant quantities, and the laws of physics are written inmanifestly covariant way as four-tensor equations.

In order to develop space-time geometry, it is necessary to introduce a metricor a measure ds of space-time intervals. The type of measure determinesthe nature of the geometry. Any event in the usual covariant approach ismathematically represented by a point in space-time, called world-point.The evolution of a particle is, instead, represented by a curve in space-time,called world-line. If ds is the infinitesimal displacement along a particleworld-line, then

ds2 = c2dT2− dX2

− dY2− dZ2 , (1)

where we have selected a special type of coordinate system (a Lorentzcoordinate system), defined by the requirement that Eq. (1) holds.

To simplify our writing we will use, instead of variables T,X,Y,Z, variablesX0 = cT, X1 = X, X2 = Y, X3 = Z. Then, by adopting the tensor notation, Eq.(1) becomes ds2 = ηi jdXidX j, where Einstein summation is understood. Hereηi j are the Cartesian components of the metric tensor and by definition, inany Lorentz system, they are given by ηi j = diag[1,−1,−1,−1], which is themetric canonical, diagonal form. As a consequence of the space-time geome-try, Lorentz coordinates systems are connected by Lorentz transformations.

Physical quantities are represented by space-time geometric (tensor) quan-tities. When some basis is introduced, the representation of a tensor asgeometric quantity comprise both components and basis. In usual covari-ant approach, one only deals with the basis components of tensors in the

23

Lorentz coordinates i.e. with the case when the basis four-vectors are or-thogonal. As a result one deals only with four-tensor equations of physicswritten out in the component form.

However, the concept of a tensor in the usual covariant approach is givenin terms of the transformation properties of its components. For examplein the usual covariant approach the electromagnetic ”tensor” Fµν is actu-ally not a tensor since Fµν are only components implicitly taken in standard(orthogonal) basis. The components are coordinate quantities and they donot contain the whole information about the physical quantity, since a basisof the space-time is not included. This is no problem only in the limitingcase when transformations from one orthogonal basis to another orthogo-nal basis are selected i.e. only assuming that Lorentz transformations mustbe used to map the coordinates of events. According to the usual covari-ant approach, another transformations from standard to non standard (notorthogonal) basis, like Galilean transformations, are ”incorrect”.

2.2.3 The Space-Time Geometric Approach

We emphasize the great freedom one has in the choice of a Minkovskispace-time coordinatization. The space-time continuum, determined by theinterval Eq. (1) can be described in arbitrary coordinates and not only inLorentz coordinates. In the transition to arbitrary coordinates, the geome-try of four-dimensional space-time obviously does not change, and in thespecial theory of relativity we are not limited in any way in the choice ofa coordinates system. The space coordinates x1, x2, x3 can be any quantitiesdefining the position of particles in space, and the time coordinate x0 can bedefined by an arbitrary running clock. The components of the metric tensorin the coordinate system xi can be determined by performing the transfor-mation from the Lorentz coordinates Xi to the arbitrary variables x j, whichare fixed as Xi = f i(x j). One then obtains

ds2 = ηi jdXidX j = ηi j∂Xi

∂xk

∂X j

∂xm dxkdxm = gkmdxkdxm , (2)

This expression represents the general form of the pseudo-Euclidean metric.In textbooks and monographs, the special theory of relativity is generallypresented in relation to an interval ds in the Minkowski form Eq.(1), whileEq.(2) is ascribed to the theory of general relativity.

However, in the space-time geometric approach, special relativity is under-stood as a theory of four-dimensional space-time with pseudo-Euclideangeometry. In this formulation of the theory of relativity the space-time con-

24

tinuum can be described equally well from the point of view of any co-ordinate system, which cannot possibly change ds. At variance, the usualformulation of the theory of relativity also deals with the invariance ofds, but it is understood only in a limited sense when the metric is strictlydiagonal.

Common textbook presentations of the special theory of relativity use theEinstein approach or, as generalization, the usual covariant approach whichdeals, as discussed above, only with components of the 4-tensors in spe-cific (orthogonal) Lorentz basis. The fact that in the process of transition toarbitrary coordinates the geometry of the space-time does not change, isnot considered in textbooks. As a consequence there is a widespread beliefamong experts that a transformation from an orthogonal Lorentz basis toa non orthogonal basis is incorrect, while a Lorentz transformation, whichis a transformation from an orthogonal Lorentz basis to another orthogonalLorentz basis, is correct. This is not true. We can describe physics in anyarbitrary coordinates system. The different transformations of coordinatesonly correspond to a change in the way of components of 4-tensors arewritten, but not influence of 4-tensors themselves. Although the Einsteinsynchronization i.e. Lorentz coordinates choice, is preferred by physicistsdue to its simplicity and symmetry, it is nothing more ”physical” than anyother. A particularly very unusual choice of coordinates, the absolute timecoordinate choice, will be considered and exploited in this book.

2.3 Myth About the Incorrectness of Galilean Transformations

The use of Galilean transformations within the theory of relativity requiressome special discussion. Many physicists still tend to think of Galileantransformations (which is actually a transformations from an orthogonalLorentz basis to a non-orthogonal basis) as old, incorrect transformationsbetween spatial coordinates and time. A widespread argument used tosupport the incorrectness of Galilean transformations is that they do notpreserve the form-invariance of Maxwell’s equations under a change ofinertial frame. This idea is a part of the material in well-known books andmonographs (1).

Authors of textbooks are mistaken in their belief about the incorrectness ofGalilean transformations. The special theory of relativity is the theory offour-dimensional space-time with pseudo-Euclidean geometry. From thisviewpoint, the principle of relativity is a simple consequence of the space-time geometry, and the space-time continuum can be described in arbitrarycoordinates (2). Therefore, contrary to the view presented in most textbooks,Galilean transformations are actually compatible with the principle of rela-

25

tivity although, of course, they alter the form of Maxwell’s equations.

2.4 Myth About the Non-Relativistic Limit of the Lorentz Transformations

It is generally believed that a Lorentz transformation reduces to a Galileantransformation in the non-relativistic limit. We state that this typical text-book statement is incorrect and misleading. Kinematics is a comparativestudy which requires two coordinate systems, and one needs to assign timecoordinates to the two systems. Different types of clock synchronizationprovide different time coordinates. The convention on the clock synchro-nization amounts to nothing more than a definite choice of the coordinatesystem in an inertial frame of reference in Minkowski space-time. Pragmaticarguments for choosing one coordinate system over another may thereforelead to different choices in different situations. Usually, in relativistic engi-neering, we have a choice between absolute time coordinate and Lorentztime coordinate. The space-time continuum can be described equally wellin both coordinate systems. This means that for arbitrary particle speed, theGalilean coordinate transformations well characterize a change in the ref-erence frame from the lab inertial observer to a comoving inertial observerin the context of the theory of relativity. Let us consider the non relativisticlimit. The Lorentz transformation, for v/c so small that v2/c2 is neglectedcan be written as x′ = x − vt, t′ = t − xv/c2. This infinitesimal Lorentz trans-formation differs from the infinitesimal Galilean transformation x′ = x − vt,t′ = t. The difference is in the term xv/c2 in the Lorentz transformation fortime, which is a first order term.

We only wish to emphasize here the following point. An infinitesimalLorentz transformation differs from Galilean transformation only by theinclusion of the relativity of simultaneity, which is the only relativistic effectthat appears in the first order in v/c. All other higher order effects, thatare Lorentz-Fitzgerald contraction, time dilation, and relativistic correctionin the law of composition of velocities, can be derived mathematically, byiterating this infinitesimal transformation (3).

The main difference between the Lorentz coordinatization and the absolutetime coordinatization is that the transformation laws connecting coordinatesand times between relatively moving systems are different. It is impossible toagree with the textbook statement that there is reduction of t′ = γ(t−vx/c2) toGalilean relation t′ = t in the non-relativistic limit. This would mean that inthe non-relativistic limit infinitesimal Lorentz transformations are identicalto infinitesimal Galilean transformations. This statement is absurd conclu-sion from a mathematical standpoint. The essence of Lorentz (or Galilean)transformations consists in their infinitesimal form: relativistic kinematics

26

effects cannot be found by the mathematical procedure of iterating the in-finitesimal Galilean transformations.

2.5 Myth about the Constancy of the Speed of Light

It is generally believed that experiments show that the speed of light in vac-uum is independent of the source or observer (4). This statement presentedin most textbooks and is obviously incorrect. The constancy of the speed oflight is related to the choice of synchronization convention, and cannot besubject to experimental tests (5).

In fact, in order to measure the one-way speed of light one has first to syn-chronize the infinity of clocks assumed attached to every position in space,which allows us to perform time measurements. Obviously, an unavoidabledeadlock appears if one synchronizes the clocks by assuming a-priori thatthe one-way speed of light is c. In fact, in that case, the one-way speed oflight measured with these clocks (that is the Einstein speed of light) cannotbe anything else but c: this is because the clocks have been set assuming thatparticular one-way speed in advance.

Therefore, it can be said that the value of the one-way speed of light is justa matter of convention without physical meaning. In contrast to this, thetwo-way speed of light, directly measurable along a round-trip, has physi-cal meaning, because round-trip experiments rely upon the observation ofsimultaneity or non-simultaneity of events at a single point in space andnot depends on clock synchronization convention. All well known methodsto measure the speed of light are, indeed, round-trip measurements. Thecardinal example is given by the Michelson-Morley experiment: this exper-iment uses, indeed, an interferometer where light beams are compared in atwo-way fashion.

Because of our usage of Galilean transformations within electrodynamicswe have some apparent paradoxes. One finds many books which say thata Galilean transformations of the velocity of light is not consistent with theelectron-theoretical explanation of refraction and reflection (6). It is widelyaccepted that if we consider a moving source and a stationary glass, theincident light wave and wave scattered by the dipoles of the glass cannotinterfere as required by the electron theory of dispersion since their velocityare different. This is misconception. It is clear that an incident wave witha certain frequency, no matter what its velocity, excites the electrons of aglass into oscillations of the same frequency. They then emit radiation withthe same frequency. Thus, the incident and scattered wave at any givenpoint have the same frequency and can interfere. The effect of the different

27

velocities is to produce a relative phase which varies with position in space.This, according to well known ideas, affects the velocity and amplitudeenvelope of the single wave which results from the superposition of the twoseparate waves.

2.6 Convention-Dependent and Convention-Invariant Parts of the Theory

Consider the motion of charged particle in a given magnetic field. The theoryof relativity says that the particle trajectory ~x(t) in the lab frame dependson the choice of a convention, namely the synchronization convention ofclocks in the lab frame. Whenever we have a theory containing an arbitraryconvention, we should examine what parts of the theory depend on thechoice of that convention and what parts do not. We may call the formerconvention-dependent, and the latter convention-invariant parts. Clearly,physically meaningful measurement results must be convention-invariant.

Consider the motion of two charged particles in a given magnetic field,which is used to control the particle trajectories. Suppose there are twoapertures at point A and at point A′. From the solution of the dynamicsequation of motion we may conclude that the first particle gets throughthe aperture at A and the second particle gets through the aperture at A′

simultaneity. The two events, i.e. the passage of particles at point A andpoint A′ have exact objective meaning i.e. convention-invariant. However,the simultaneity of these two events is convention-dependent and has noexact objective meaning. It is important at this point to emphasize that,consistently with the conventionality of simultaneity, also the value of thespeed of particle is a matter of convention and has no definite objectivemeaning.

In order to examine what parts of the dynamics theory depend on thechoice of that convention and what parts do not, we want to show thedifference between the notions of path and trajectory. Let us consider themotion of a particle in three-dimensional space using the vector-valuedfunction ~x(t). We have a prescribed curve (path) along which the particlemoves. The motion along the path is described by l(t), where l is a certainparameter (in our case of interest the length of the arc). The trajectory of aparticle conveys more information about its motion because every positionis described additionally by the corresponding time instant. The path israther a purely geometrical notion. If we take the origin of the (Cartesian)coordinate system and we connect the point to the point laying on the pathand describing the motion of the particle, then the creating vector will be aposition vector ~x(l) (7).

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The difference between trajectory ~x(t) and path ~x(l) is very interesting. Thepath has exact objective meaning i.e. it is convention-invariant. In contrastto this, and consistently with the conventionality intrinsic in the velocity,the trajectory ~x(t) of the particle is convention dependent and has no exactobjective meaning.

In order to avoid being to abstract for to long we have given some examples:just think of the experiments related with accelerator physics. Suppose wewant to perform a particle momentum measurement. A uniform magneticfield can be used in making a ”momentum analyzer” for high-energy chargeparticles, and it must be recognized that this method for determining the par-ticle’s momentum is convention-independent. In fact, the curvature radiusof the path in the magnetic field (and consequently the three-momentum)has obviously an objective meaning, i.e. is convention-invariant. Dynamicstheory contains a particle trajectory that we do not need to check directly,but which is used in the analysis of electrodynamics problem.

2.7 Myth about the Reality of Relativistic Time Dilation and Length Contraction

Generally, experts on the theory of relativity erroneously identify the proper-ties of Minkovski space-time with the familiar form that certain convention-dependent quantities assume under the standard Lorentz coordinatization.These quantities usually are called ”relativistic kinematics effects”. There isa widespread belief that the convention-dependent quantities like the timedilation, length contraction, and Einstein’s addition of velocities have di-rect physical meaning. We found that statement like ” moving clocks runslow” is not true under the adopted absolute time clock synchronization,and, hence, are by no means intrinsic features of Minkowski space-time.Relativistic kinematic effects are coordinate (i.e. convention-dependent) ef-fects and have no exact objective meaning (8). In the case of Lorentz co-ordinatization, one will experience e.g. the time dilation phenomenon. Incontrast to this, in the case of absolute time coordinatization there are norelativistic kinematics effects and no time dilation will be found. However,all coordinate-independent quantities like the particle path ~x(l) and momen-tum |~p| remain independent of such a change in clock synchronization.

2.8 Relativistic Particle Dynamics

2.8.1 Covariant Approach

The accelerated motion is described by a covariant equation of motion for arelativistic charged particle under the action of the four-force in the Lorentz

29

lab frame. The trajectory of a particle ~xcov(t) is viewed from the Lorentz labframe as a result of successive infinitesimal Lorentz transformations. Thelab frame time t in the equation of motion cannot be independent from thespace variables. This is because Lorentz transformations lead to a mixtureof positions and time, and the relativistic kinematics effects are consideredto be a manifestation of the relativity of simultaneity.

2.8.2 Noncovariant (”Single Frame”) Approach

Let us consider the conventional particle tracking approach. It is generallyaccepted that in order to describe dynamics of relativistic particles in thelab reference frame, which we assume inertial, one only needs to take intoaccount the relativistic dependence of the particle momentum on the veloc-ity. The treatment of relativistic particle dynamics involves only correctedNewton’s second law. In a given lab frame, there is an electric field ~E andmagnetic field ~B. They push on a particle in accordance with

d~pdt

= e(~E +

~vc× ~B

),

~p = m~v(1 −

v2

c2

)−1/2

, (3)

where here the particle’s mass, charge, and velocity are denoted by m, e, and~v respectively. The Lorentz force law, plus measurements on the componentsof acceleration of test particles, can be viewed as defining the components ofthe electric and magnetic fields. Once field components are known from theacceleration of test particles, they can be used to predict the accelerations ofother particles.

This solution of the dynamics problem in the lab frame makes no referenceto Lorentz transformations. Conventional particle tracking treats the space-time continuum in a non-relativistic format, as a (3+1) manifold. In otherwords, in this approach, introducing as only modification to the classicalcase the relativistic mass, time differ from space. In fact, we have no mixtureof positions and time (9).

Most of the interesting phenomena in which charges move under the actionof electromagnetic fields occur in very complicated situations. But here wejust want to discuss the simple problem of the accelerated motion of parti-cles in a constant magnetic field. According to the non-covariant treatment,the magnetic field is only capable of altering the direction of motion, but notthe speed (i.e. mass) of a particle. This study of relativistic particle motion in

30

a constant magnetic field, usual for accelerator engineering, looks preciselythe same as in nonrelativistic Newtonian dynamics and kinematics. Thetrajectory of a particle ~x(t), which follows from the solution of the correctedNewton’s second law, does not include relativistic kinematics effects as rela-tivity of simultaneity and the Galilean vectorial law of addition of velocitiesis actually used.

Let us discuss the important problem of the addition of velocities in relativ-ity. Suppose that in the case of accelerated motion one introduces an infinitesequence of co-moving frames. At each instant, the rest frame is a Lorentzframe centered on the particle and moving with it. Suppose that in inertialframe where particle is at rest at a given time, the traveler was observinglight itself. In other words measured speed of light v = c, and yet the frameis moving relative the lab frame. How will it look to the observer in thelab frame? According to relativistic law of addition of velocities the answerwill be c. Maxwell’s equations remain in the same form when Lorentz trans-formations are applied to them, but Lorentz transformations give rise tonon-Galilean transformation rules for velocities, and therefore the theory ofrelativity shows that, if Maxwell’s equations is to be valid in the lab frame,the trajectories of the particles must include relativistic kinematics effects.In other words, Maxwell,s equations can be applied in the lab frame onlyin the case when particle trajectories are viewed, from the lab frame as theresult of successive infinitesimal Lorentz transformations.

The absence of relativistic kinematics effects is the prediction of conven-tional non-covariant theory and is obviously absurd from the viewpoint ofMaxwell’s electrodynamics. Therefore, something is fundamentally, power-fully, and absolutely wrong in coupling fields and particles within a ”singleinertial frame”.

2.9 Mistake in Commonly Used Method of Coupling Fields and Particles

It is generally believed that the electrodynamics problem can be treatedwithin the same ”single inertial frame” description without reference toLorentz transformations. In all standard derivations it is assumed that usualMaxwell’s equations and corrected Newton’s second law can explain allexperiments that are performed in a single inertial frame, for instance thelab reference frame.

Going to electrodynamics problem, the differential form of Maxwell’s equa-tions describing electromagnetic phenomena in the same inertial lab frame(in cgs units) is given by the following expressions:

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~∇ · ~E = 4πρ ,~∇ · ~B = 0 ,

~∇ × ~E = −1c∂~B∂t,

~∇ × ~B =4πc~j +

1c∂~E∂t. (4)

Here the charge density ρ and current density ~j are written as

ρ(~x, t) =∑

n

enδ(~x − ~xn(t)) ,

~j(~x, t) =∑

n

en~vn(t)δ(~x − ~xn(t)) , (5)

where δ(~x − ~xn(t)) is three-dimensional delta function, mn, en, ~xn(t), and ~vn =d~xn(t)/dt denote the mass, charge, position, and the velocity of the nth parti-cle, respectively. To evaluate radiation fields arising from an external sourcesEq. (4) we need the velocity ~vn and the position ~xn as a function of lab frametime t. It is generally accepted by physics community that equation of mo-tion, which describes how coordinates of the particle carrying the chargechange with time t, is described by corrected Newton’s second law Eq. (3).

This coupling of Maxwell’s equations and corrected Newton’s equation iscommonly accepted as useful method in accelerator physics and, in partic-ular, in analytical and numerical calculations of radiation properties. Suchapproach to relativistic dynamics and electrodynamics usually forces the ac-celerator physicist to believe that the design of particle accelerators possiblewithout detailed knowledge of the theory of relativity.

However, there is a common mistake made in accelerator physics connectedwith the difference between ~x(t) and ~xcov(t) trajectories. Let us look at this dif-ference from the point of view of electrodynamics of relativistically movingcharges. To evaluate fields arising from external sources we need to knowtheir velocity and positions as a function of the lab frame time t. Suppose onewants to calculate properties of radiation. Given our previous discussionthe question arises, whether one should solve the usual Maxwell’s equa-tions in the lab frame with current and charge density created by particlemoving along non-covariant trajectories like ~x(t). We claim that the answerto this question is negative. This algorithm for solving usual Maxwell’sequations in the lab frame, which is considered in all standard treatmentsas relativistically correct, is at odds with the principle of relativity. This es-sential point has never received attention in the physical community. Only

32

the solution of the dynamics equations in covariant form gives the correctcoupling between the usual Maxwell’s equations and particle trajectories inthe lab frame.

2.10 Clarification of the True Content of the ”Single Frame” Theory

Let us now examine the logical content of the concept of a ”single inertialframe”. If a traveler in a co moving frame, similar to an observer in the labframe, introduces a definite coordinate-time grid, there is always a definitetransformation between these two four-dimensional coordinate systems.Thus, particle trajectories are always viewed from the lab frame as a result ofsuccessive transformations, and the form of these transformations dependson the choice of coordinate systems in the comoving and the lab frame.

One might well wonder why it is necessary to discuss how different inertialframes are related to one another. The point is that all natural phenomenafollow the principle of relativity, which is a restrictive principle: it says thatthe laws of nature are the same (or take the same form) in all inertial frames.In agreement with this principle, usual Maxwell’s equations can always beexploited in any inertial frame where electromagnetic sources are at restusing Einstein synchronization procedure in the rest frame of the source.The fact that one can deduce electromagnetic field equations for arbitrarymoving sources by studying the form taken by Maxwell’s equations underthe transformation between rest frame of the source and the frame wherethe source is moving is a practical application of the principle of relativity.The question now arises how to assign a time coordinate to the lab frame.

Coordinates serve the purpose of labeling events in an unambiguous way,and this can be done in infinitely many different ways. The principle of rela-tivity dictates that Maxwell’s equations can be applied in the lab frame onlyin the case when Lorentz coordinates are assigned and particle trajectoriesare viewing from the lab frame as a result of successive infinitesimal Lorentztransformations between the lab and comoving inertial frames.

A ”single frame” (non-covariant 3+1) approach to relativistic particle dy-namics has been used in particle tracking calculations for about seventyyears. However, the type of clock synchronization which provides the timecoordinate t in the corrected Newton’s equation has never been discussedin literature. It is clear that without an answer to the question about themethod of synchronization used, not only the concept of velocity, but alsothe dynamics law has no physical meaning. A ”single frame” approachto relativistic particle dynamics is forcefully based on a definite synchro-nization assumption but this is actually a hidden assumption. According to

33

conventional particle tracking, the dynamical evolution in the lab frame isbased on the use of the lab frame time t as an independent variable. Suchapproach to relativistic particle dynamics is actually based on the use ofa not standard (not Einstein) clock synchronization assumption in the labframe.

In fact, the usual for accelerator engineering study of relativistic particlemotion in a constant magnetic field looks precisely the same as in non-relativistic Newtonian mechanics and the trajectories of the electrons doesnot include relativistic kinematics effects. According to textbooks, this is noproblem. If no more than one frame is involved, one does not need to use(and does not need to know) the theory of relativity. Only when one passesfrom one reference frame to another the relativistic context is important.Conventional particle tracking in a constant magnetic field is actually basedon classical Newton mechanics. It is generally believed that the electrody-namics problem, similar to conventional particle tracking, can be treatedwithin a description involving a single inertial frame and one should solvethe usual Maxwell’s equations in the lab frame with current and chargedensity created by particles moving along the non-covariant (single frame)trajectories.

This is misconception. The situation when only one frame is involved andthe relativistic context is unimportant cannot be realized. The lab observermay argue, ”I don’t care about other frames.” Perhaps the lab observerdoesn’t, but nature knows that, according to the principle of relativity,Maxwell’s equations are always valid in the Lorentz comoving frame. Elec-trodynamics equations can be written down in the lab frame only whena space-time coordinate system has been specified. An observer in the labframe has only one freedom. This is the choice of a coordinate system (i.e.the choice of clock synchronization convention) in the lab frame. After this,the theory of relativity states that the electrodynamics equations in the labframe are the result of transformation of Maxwell’s equations from theLorentz comoving frame to the lab frame.

2.11 Bibliography and Notes

1. Many physicists tend to think of Galilean transformations as pre-relativistictransformations between spatial coordinates and time that are not compat-ible with the special theory of relativity. To quote e.g. Bohm [1] ”... theGalilean law of addition of velocities implies that the speed of light shouldvary with the speed of the observing equipment. Since this predicted varia-tion is contrary to the fact, the Galilean transformations evidently cannot bethe correct one.”. Similar statements can also be found in recently published

34

pedagogical papers. To quote e.g. Drake and Purvis [2] ”One of the greatinsights to come relativity theory was the realization that Galilean transfor-mations are wrong. The correct way to translate the space-time measure ofevents between inertial frames is with the Lorentz transformations” How-ever, this is not true. Galilean transformations are simply transformationsrelating a given coordinate set to another coordinate set. The space-timecontinuum can be described in arbitrary coordinates, and choice of this setof coordinates cannot change the geometry of space-time.

2. The mathematical argument that in the process of transition to arbitrarycoordinates the geometry of the space-time does not change, is consideredin textbooks as erroneous. To quote L. Landau and E. Lifshitz [3]: ”Thisformula is called the Galileo transformation. It is easily to verify that thistransformation, as was to be expected, does not satisfy the requirementsof the theory of relativity; it does not leave the interval between eventsinvariant.”. This fact is ascribed to a lack of understanding of the differ-ence between convention-dependent and convention-invariant parts of thetheory. In pseudo-Euclidean geometry the interval between events is an in-variant in arbitrary coordinates. A comparison with three-dimensional Eu-clidean space might help here. In the usual 3D Euclidean space, one can con-sider a Cartesian coordinate system (x, y, z), a cylindrical coordinate system(r, φ, z), a spherical coordinate system (ρ, θ, φ), or any other. Depending onthe choice of the coordinate system one respectively has ds2 = dx2 +dy2 +dz2,ds2 = dr2 + r2dφ2 + dz2, ds2 = dρ2 + ρ2dθ2 + ρ2 sinθ2dφ2. The metric actuallydoes not change, but the components of the metric do, depending only onthe choice of coordinates. In general, in fact, we write ds2 = gikdxidxk. Consid-ering Cartesian coordinates we will always have gi j = diag(1, 1, 1). Similarly,Lorentz transformations between inertial frames with Einstein coordinatesleave the components of the metric tensor unvaried.

3. There is a widespread belief that a Lorentz transformation reduces to aGalilean transformation in the non-relativistic limit. To quote e.g. French [4]:”The reduction of t′ = γ(t−vx/c2) to Galilean relation t′ = t requires x ct aswell as v/c 1”. Similar statements can also be found in recently publishedtextbooks. To quote Rafelski [5]: ”A wealth of daily experience shows thatthe Galilean coordinate transformation (GT) is correct in the nonrelativisticlimit in which the speed of light is so large that it plays no physical role.Any coordinate transformation replacing the GT must also agree with thisexperience, and thus must contain the GT in the nonrelativistic limit.” Thisstatement is absurd conclusion even from a mathematical standpoint. Thishas been recognized by some expects, perhaps most explicitly by Baierleinwho states [6] that ”If the Lorentz transformation for infinitesimal v/c wereto reduce to the Galilean transformation, then iterative process could nevergenerate a finite Lorentz transformation that is radically different from theGalilean transformation. But the finite transformations are radically differ-

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ent, and so -however subtly-the infinitesimal Lorentz transformation mustdiffer significantly from the Galilean transformation.”

4. Many authors of textbooks still attribute a measurable status to the con-ventional quantities. To quote e.g. Cristodoulides [7] ”The fact that Galileantransformation does not leave Maxwell’s equations has already been men-tioned [...] On the other hand, experiments show that the speed of light invacuum is independent of the source or observer.”.

5. Because we have empirical access only to the round-trip average speedof light, statements about the magnitude and isotrophy of the one-wayspeed of light must reflect the assumptions made in the choice of timecoordinatization, and such entities change as the theory is re-synchronized.To quote C. Anderson, I. Vetharaniam, and G. Stedman [8]: ”No experiment,then, is a ”one-way” experiment. An empirical test of any property of theone-way speed of light is not possible. Such quantities as the one-way speedof light are irreducibly conventional in nature, and recognizing this aspectis to recognize a profound feature of nature”.

6. The peculiarity of the kinematic consequence of using Galilean transfor-mations is that the speed of light emitted by a moving source depends onthe relative velocity between source and observer. A widespread theoreticalargument used to support the incorrectness of Galilean transformations isthe conclusion that a Galilean transformation of the velocity of light is notconsistent with the explanation of reflection and refraction. This idea is apart of the material in well-known books. To quote Pauli [9] ”[...] it is es-sential that the spherical waves emitted by the dipoles in the body shouldinterfere with the incident wave. If we now think of the body as at rest,and the light source moving relative to it, then [...] the wave emitted by thedipoles will have velocity different from that of the incident wave. Interfer-ence is therefore not possible.” This conclusion is incorrect. It is clear thatthe incident and scattered wave at any given point have the same frequencyand can interfere.

7. For a general discussion of the difference between path and trajectory wesuggest reading the book [10].

8. The standard textbooks erroneously identify the properties of space-timewith the familiar form that certain synchronization-dependent quantitiesassume under the Einstein’s clock synchronization. These quantities are, forexample, the formulae for ”time dilation” and ”length contraction”. Fortu-nately, it has also been occasionally stressed in the literature that the formsof the above ”relativistic effects” are coordinate-dependent, their forms de-pend in turn on the kind of synchronization procedure adopted. To quoteLeubner, K. Auflinger, and P. Krumm [11]: ”... there is a widespread belief

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among students that the familiar form of coordinate-dependent quantitieslike the measured velocity of light, the Lorentz transformation between twoobservers, ”addition of velocities”, ”time dilation”, ”length contraction”which they assume under the standard clock synchronization, is relativityproper. In order to demonstrate that this is by no means so, this paper studiesthe consequences of a non-standard synchronization, and it is shown thatdrastic changes in the appearance of all these quantities are thus induced.”

9. Clarification of the true content of the non-covariant theory can be foundin various advanced textbooks. To quote e.g. Ferrarese and Bini [12]: ”...within a single inertial frame, the time is an absolute quantity in specialrelativity also. As a consequence, if no more than one frame is involved, onewould not expect differences between classical and relativistic kinematics.But in the relativistic context there are differences in the transformationlaws of the various relative quantities (of kinematics or dynamics), whenpassing from one reference frame to another.” We see that authors give aspecial role to concept of a ”single inertial frame”. The name ”single inertialframe” tends to suggest that a distinctive trait of non-covariant theory isthe absence of relativistic kinematics in the description of particle motion.This point was expressed by Friedman [15]: ”Within any single inertialframe, things looks precisely the same as in Newtonian kinematics: there isan enduring Euclidean three-space, a global (i.e. absolute) time t, and lawof motion. But different inertial frames are related to one another in a non-Newtonian fashion.” According to conventional particle tracking, within the”single” frame there is no relativistic kinematics effects. This, as we alreadymentioned, contradicts the Maxwell’s electrodynamics. We cannot take oldkinematics for mechanics and Einstein’s kinematics for electrodynamics.If one wants to use the usual Maxwell’s equations, only solution of thedynamics equations in covariant form gives the correct coupling betweenMaxwell’s equations and particle trajectories in the ”single frame”.

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3 Space-Time and Its Coordinatization

3.1 Introductory Remarks

Let us discuss an ”operational interpretation” of the Lorentz and absolutetime coordinatizations. We should underline that we claim the non covari-ant approach to relativistic particle dynamics is actually based on the use ofa not standard and unusual clock synchronization assumption within thetheory of relativity. It is important to know how to operationally interpretthe absolute time convention i.e. how one should perform the clock syn-chronization in the lab frame. The result is very interesting, since it tell usabout difference between absolute time synchronization and Einstein’s timesynchronization from the operational point of view.

3.2 Choice of Coordinates System in an Inertial Frame

Each physical phenomenon occurs in space and time. A concrete methodfor representing space and time is a frame of reference. One-and-the samespace and time can be represented by various coordinate-time grids, i.e.,by various frames of references. Even the simplest space-time coordinatesystems require carefully description.

Clocks reveal the motion of a particle through the coordinate-time grid.The general approach to the determination of the motion of a particle isthe following: at any instant a particle has a well-defined velocity ~v asmeasured in a laboratory frame of reference. How is a velocity of a particlefound? The velocity is determined once the coordinates in the lab frameare chosen, and is then measured at appropriate time intervals along theparticle’s trajectory. But how to measure a time interval between eventsoccurring at different points in space? In order to do so, and hence measurethe velocity of a particle within a single inertial lab frame, one first has tosynchronize distant clocks. The concept of synchronization is a key conceptin the understanding of special relativity. It is possible to think of variousmethods to synchronize distant clocks (1). The choice of a convention onclock synchronization is nothing more than a definite choice of coordinatessystem in an inertial frame of reference of the Minkowski space-time.

The space-time continuum can be described in arbitrary coordinates. Bychanging these arbitrary coordinates, the geometry of the four-dimensionalspace-time obviously does not change, and in the special theory of relativitywe are not limited in any way in the choice of a coordinates system. Relyingon the geometric structure of Minkowski space-time, one defines the class of

38

inertial frames and adopts a Lorentz frame with orthonormal basis vectors.Within the chosen Lorentz frame, Einstein’s synchronization procedure ofdistant clocks (which based on the constancy of the speed of light in allinertial framers) and Cartesian space coordinates are enforced.

3.3 Inertial Frame where a Source of Light is at Rest

Let us give an ”operational interpretation” of the Lorentz coordinatizations.The fundamental laws of electrodynamics are expressed by Maxwell’s equa-tions, according to which, as well-known, light propagates with the samevelocity c in all directions. This is because Maxwell’s theory has no intrinsicanisotropy. It has been stated that in their original form Maxwell’s equa-tions are only valid in inertial frames. However, Maxwell’s equations can bewritten down in coordinate representation only if the space-time coordinatesystem has already been specified.

The problem of assigning Lorentz coordinates to the lab frame in the case ofaccelerated motion is complicated. We would like to start with the simplerquestion of how to assign space-time coordinates to an inertial frame, wherea source of light is at rest. We need to give a ”practical”, ”operational” answerto this question. The most natural method of synchronization consists inputting all the ideal clocks together at the same point in space, where theycan be synchronized. Then, they can be transported slowly to their originalplaces (slow clock transport) (2).

The usual Maxwell’s equations are valid in any inertial frame where sourcesare at rest and the procedure of slow clock transport is used to assign val-ues to the time coordinate. The same considerations apply when chargedparticles are moving in non-relativistic manner. In particular, when oscillat-ing, charged particles emit radiation, and in the non-relativistic case, whencharges oscillate with velocities much smaller than c, dipole radiation isgenerated and described with the help of the Maxwell’s equations in theirusual form.

Let’s examine in a more detail how the dipole radiation term comes about.The retardation time in the integrands of the expression for the radiationfield amplitude, can be neglected in the cases where the trajectory of thecharge changes little during this time. It is easy to find the conditions forsatisfying this requirement. Let us denote by a the order of magnitude ofthe dimensions of the system. Then the retardation time ∼ a/c. In order toensure that the distribution of the charges in the system does not undergo asignificant change during this time, it is necessary that a λ, where λ is theradiation wavelength. Thus, the dimensions of the system must be small

39

compared to radiation wavelength. This condition can be written in stillanother form v c, where v is of the order of magnitude of the velocitiesof the charges. In accounting only for the dipole part of the radiation weneglect all information about the electron trajectory. Therefore, one shouldnot be surprised to find that dipole radiation theory gives fields very muchlike the instantaneous theory.

The theory of relativity offers an alternative procedure of clocks synchro-nization based on the constancy of the speed of light in all inertial frames.This is usually considered a postulate but, as we have seen, it is just a con-vention. The synchronization procedure that follows is the usual Einsteinsynchronization procedure. Suppose we have a dipole radiation source.When the dipole light source is at rest, the field equations are constituted bythe usual Maxwell’s equations. Indeed, in dipole radiation theory we con-sider the small expansion parameter v/c 1 neglecting terms of order v/c.In other words, in dipole radiation theory we use zero order non relativisticapproximation. Einstein synchronization is defined in terms of light signalsemitted by the dipole source at rest, assuming that light propagate withthe same velocity c in all directions. Using Einstein synchronization proce-dure in the rest frame of the dipole source, we actually select the Lorentzcoordinate system.

Slow transport synchronization is equivalent to Einstein synchronization ininertial system where the dipole light source is at rest (3). In other words,suppose we have two sets of synchronized clocks spaced along the x axis.Suppose that one set of clocks is synchronized by using the slow clocktransport procedure and the other by light signals. If we would ride to-gether with any clock in either set, we could see that it has the same time asthe adjacent clocks, with which its reading is compared. This is because inour case of interest, when light source is at rest, field equations are the usualMaxwell’s equations and Einstein synchronization is defined in terms oflight signals emitted by a source at rest assuming that light propagates withthe same velocity c in all directions. Using any of these synchronizationprocedures in the rest frame we actually select a Lorentz coordinate sys-tem. In this coordinate system the metric of the light source has Minkowskiform ds2 = c2dt2

− dx2− dy2

− dz2. We want now to turn to the subjectof electromagnetic waves. In free space, the electric field ~E of an electro-magnetic wave emitted by the dipole source at rest satisfies the equation2~E = ∇2~E − ∂2~E/∂(ct)2 = 0.

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3.4 Motion of a Light Source With Respect to the Inertial Frame

We now consider the case when the light source in the lab frame is acceler-ated from rest up to velocity v along the x-axis. In other words, we considerso-called active (physical) boost. When we state that the source undergoesan acceleration, and the inertial observer (with measuring devices) doesnot, the acceleration means acceleration relative to the inertial frame. Anyacceleration relative to the inertial frame (i.e. any active boost of velocity)has an absolute meaning.

In the case of an active boost we are dealing with motion of the samephysical system, evolving in time and treated from the point of view ofthe same reference system. A fundamental question to ask is whether ourlab clock synchronization method depends on the state of motion of thelight source or not. The answer simply fixes a convention. The simplestmethod of synchronization consists in keeping, without changes, the sameset of uniformly synchronized clocks used in the case when the light sourcewas at rest, i.e. we still enforce the clock transport synchronization ( orEinstein synchronization which is defined in terms of light signals emittedby the dipole source at rest). This choice is usually the most convenient onefrom the viewpoint of connection to laboratory reality. This synchronizationconvention preserves simultaneity and is actually based on the absolute time(or absolute simultaneity) convention.

It is always possible to create a new frame of reference by relabeling co-ordinates, and then discussing physical phenomena in terms of the newcoordinate labels - a passive transformation. For example, it is always pos-sible to create a so-called comoving coordinate system in the lab frame,and then discussing radiation from the moving source in therm of the new(comoving) coordinate labels. In the comoving coordinate system, fields areexpressed as a function of the independent variables x′, y′, z′, and t′. Thevariables x′, y′, z′, t′ can be expressed in terms of the independent variablesx, y, z, t by means of a passive Galilean transformation, so that fields can bewritten in terms of x′, y′, z′, t′. After the passive transformation, the Carte-sian coordinates of the source transform as x′ = x − vt, y′ = y, z′ = z. Thistransformation completes with the invariance of simultaneity, t′ = t. Thetransformation of time and spatial coordinates of any event has the form ofa Galilean transformation.

The principle of covariance implies the equivalence of active and passivetransformations within a single inertial frame. The equivalence of the ac-tive and passive pictures is due to the fact that moving system one wayis equivalent to moving the coordinate system the other way by an equalamount. According to this principle of relativity, in our case of interest the

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Maxwell’s equations always valid in the comoving frame. In the comov-ing frame, fields are expressed as a function of the independent variablesx′, y′, z′, and t′. The electric field ~E′ of an electromagnetic wave satisfies theequation ′2~E′ = ∇′2~E′ − ∂2~E′/∂(ct′)2 = 0. However, the variables x′, y′, z′, t′

can be expressed in terms of the independent variables x, y, z, t by means ofa Galilean transformation, so that fields can be written in terms of x, y, z, t.From the Galilean transformation x′ = x− vt, y′ = y, z′ = z, t′ = t, after par-tial differentiation, one obtains ∂/∂t = ∂/∂t′ − v∂/∂x′, ∂/∂x = ∂/∂x′. Hencethe wave equation transforms into

2~E =

(1 −

v2

c2

)∂2~E∂x2 − 2

(vc

)∂2~E∂ct∂x

+∂2~E∂y2 +

∂2~E∂z2 −

1c2

∂2~E∂t2 = 0 , (6)

The solution of this equation F[x− (c+v)t]+G[x+ (−c+v)t] is the sum of twoarbitrary functions, one of argument x − (c + v)t and the other of argumentx + (−c + v)t. Here we obtained the solution for waves which move in the xdirection by supposing that the field does not depend on y and z. The firstterm represents a wave traveling forward in the positive x direction, andthe second term a wave traveling backwards in the negative direction.

We conclude that the speed of light emitted by a moving source measured inthe lab frame (t, x) depends on the relative velocity of source and observer,in our example v. In other words, the speed of light is compatible withthe Galilean law of addition of velocities. In fact, the coordinate velocity oflight parallel to the x-axis is given by dx/dt = c + v in the positive direction,and dx/dt = −c + v in the negative direction. The reason why it is differentfrom the electrodynamics constant c is due to the fact that the clocks aresynchronized following the absolute time convention.

3.5 Discussion

No one can ”explain” any more than what just explained. We have no ideasabout a more basic mechanism from which these results can be deduced. Itis important to stress at this point that there is no machinery behind the lawsof electrodynamics. It is obvious that electromagnetic fields are fundamentaland the electrodynamics theory meets all requirements of the theory of rel-ativity. Consequently, one can use any (e.g. absolute time) coordinatizationto describe the electromagnetic field dynamics.

We would like to emphasize a very important difference between funda-mental and phenomenological theory. For example, we can characterizeNewton’s equation in the comoving frame as a phenomenological law. The

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microscopic interpretation of the inertial mass of a particle is not given andthe rest mass is introduced in an ad hoc manner. Any phenomenologicallaw, which is valid in the Lorentz rest frame, can be embedded in the fourdimensional space-time only by using Lorentz coordinatization. It can besaid that the particle has some kind of internal works - some inner variables- that we do not yet know about.

However, the remarkable progress in experimental and theoretical physicsmade during the past few decades has considerably deepened our knowl-edge concerning the nature of mass. From a microscopic viewpoint, todayaccepted explanation of how particles acquire mass is based on the quantumfield theory. This mechanism can be invoked to explain Newton’s secondlaw from microscopic viewpoint even for structure-less particles like lep-tons. It is enough to assume Lorentz covariance of the quantum field theoryinvolved in particle mass calculations in order to obtain the same resultfrom the two synchronization conventions discussed here, and it does notmatter which transformation (Galilean or Lorentz) is used. We will discussthis subject further in the Chapter 11.

3.6 A Clock Resynchronization Procedure

After the active boost in the case of the absolute time coordinatization wecan see that the homogeneous wave equation for the field in the lab framehas nearly but not quite the usual, standard form that takes when there isno acceleration up to velocity v. The main difference consists in the crossedterm ∂2/∂t∂x, which complicates the solution of the equation. To get aroundthis difficulty, we observe that simplification is always possible. The trickneeded here is to further make a change of the time variable. Suppose that wedescribe the effect of radiation by working only up to the first order in v/c. Inthis approximation, it is always possible to introduce a time transformationt→ t + xvx/c2. After this time shift we obtain the usual wave equation.

Let us now consider the case of an arbitrary velocity. We have, then, a generalmethod for finding solution of electrodynamics problem in the case of theabsolute time coordinatization. The new independent variables (xL, tL) canbe expressed in terms of the old independent variables (x, t):

ctL = t√

1 − v2x/c2 + vxx/[c2

√1 − v2

x/c2] , xL = x/√

1 − v2x/c2 , (7)

Since the change variables completed by the Galilean transformation ismathematically equivalent to the Lorentz transformation, it obviously fol-

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lows that transforming to new variables xL, tL leads to the usual Maxwell’sequations. In particular, the wave equation Eq. (6) transforms into

2LE = ∇2

LE − ∂2E/∂(ctL)2 = 0 . (8)

In the new variables the velocity of light emitted by a moving source isconstant in all directions, and equal to the electrodynamics constant c.

The overall combination of the active boost and variable changes actuallyyields the Lorentz transformation in the case of absolute time coordina-tization in the lab frame, but in this context this transformation are onlyto be understood as useful mathematical device, which allow one to solvethe electrodynamics problem in the choice of absolute time synchronizationwith minimal effort.

We can now rise an interesting question: do we need to transform the resultsof the electrodynamics problem solution into the original variables? Westate that the variable changes performed above have no intrinsic meaning- their meaning only being assigned by a convention. In particular, one cansee the connection between the time shift t → t + xv/c2 and the issue ofclock synchrony. Note that the final change in the scale of time and spatialcoordinates is unrecognizable also from a physical viewpoint. It is clear thatthe convention-independent results of calculations are precisely the same inthe new variables. As a consequence, we should not care to transform theresults of the electrodynamics problem solution into the original variables.

Consider now two light sources say ”1” and ”2”. Suppose that in the labframe the velocities of ”1” and ”2” are ~v1, ~v2 and ~v1 , ~v2. The questionnow arises how to assign a time coordinate to the lab reference frame. Wehave a choice between an absolute time coordinate and a Lorentz timecoordinate. The most natural choice, from the point of view of connectingto the laboratory reality, is the absolute time synchronization. In this casesimultaneity is absolute, and for this we should prepare, for two sources,only one set of synchronized clocks in the lab frame. On the other hand,Maxwell’s equations are not form-invariant under Galilean transformations,that is, their form is different on the lab frame. In fact, the use of the absolutetime convention, implies the use of much more complicated field equations,and these equations are different for each source. Now we are in the positionto assign Lorentz coordinates. The only possibility to introduce Lorentzcoordinates in this situation consists in introducing individual coordinatesystems (i.e. individual set of clocks) for each source. It is clear that ifoperational methods are at hand to fix the coordinates for the first source,the same methods can be used to assign values to the coordinates for thesecond source and these will be two different Lorentz coordinate systems.

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It should be note, however, that there is another satisfactory way of ex-plaining the effect of an active Galilean boost in the inertial frame. Theexplanation consists in using non-diagonal metric of the accelerated lightsources, ds2 = (1−v2

x/c2)c2dt2 +2vxdxdt−dx2. By changing a four-dimensionalcoordinate system, one can obtain a more suitable description of a physicalsystem. For example, it is always possible to chose such variables, in whichmetric of the accelerated source will be diagonal, Eq. (7). We discuss themetric associated with the wave equation Eq. (6) in the Chapter 10.


1. According to the thesis of conventionality of simultaneity [13–17], simul-taneity of distant events is a conventional matter, as it can be legitimatelyfixed in different manners in any given inertial reference frame. To quote e.g.Moeller [13]: ”All methods for the regulation of clocks meet with the samefundamental difficulty. The concept of simultaneity between two events indifferent places obviously has no exact objective meaning at all, since wecannot specify any experimental method by which this simultaneity couldbe ascertained. The same is therefore true also for concept of velocity.” Letus illustrate a particular special example of the simultaneity convention.Let the synchronization of clocks in different spatial points be provided bylight signals having, respectively, velocity c1 in the direction parallel to thepositive axis x, and velocity c2 in the opposite direction. Then, a light signalsent from point A at time tA will arrive at point B at time tB = tA +xAB/c1. Thereflected signal will arrive back at point A at time t′A = tB +xAB/c2. From thesetwo expressions we get t′A − tA = xAB(c2 + c1)/(c1c2). Summing up we havetB = tA +c2(t′A−tA)/(c1 +c2). So we come to the synchronization first proposedin [14]: tB = tA + ε(t′A − tA), where ε = c2/(c2 − c1). After substituting in thisexpression Einstein choice c1 = −c2 = c we get ε = 1/2. This time order, asfixed by the standard synchronization, is frame dependent. This is the well-known thesis of relativity of simultaneity. The conventional nature of distantsimultaneity in special relativity is not to be confused with the relativity ofsimultaneity. Clearly, the conventionality of simultaneity within a single in-ertial frame is quite distinct from the relativity of simultaneity in Einstein’ssynchronization. If ε , 1/2, then speed of light from A to B differs from thespeed of light from B to A. Although the Einstein synchronization choice,is preferred by physicists, it is nothing more ”physical” than any other. Anice pedagogical (but artificial) example of a non-standard synchronizationis clock synchronization [11] in which ε = 0. To any convention on the si-multaneity there will correspond a definite choice of the coordinate systemin an inertial coordinate system of reference of the Minkowski space-time.A particular very unusual choice of coordinates, the absolute time coordi-nate choice, is exploited in this book. In our case, when time is absolute, we

45

have c1 = c + v, c2 = −c + v. Substituting into the expression for ε we obtainε = (1/2)(1 − v/c).

2. In the text first published in 1923, Eddington discussed, apparently for thefirst time, a procedure for synchronization using slow transport of clocks[18]. The details can be found in review [8] ( see also [19]).

3. We already pointed that we have empirical access only to the round-tripaverage speed of light. An empirical test of any property of the one-wayspeed of light is not possible. Many authors of textbooks still attribute areality status of the one-way speed of light. To quote Hrasko [20]: ”It issometimes claimed that Einstein synchronization of distant clocks A ans Bis circular. The argument is very simple: Einstein synchronization is basedon the equality of light velocity on the path from A to B and back from Bto A, but measurement of light velocity in one direction between to distantpoints is impossible unless the clocks at these points have already beensynchronized. This argument is, however, fallacious. It is true that one-waymeasurement of light velocity can be performed only if clocks at the end-points are synchronized correctly. But since they need not show the correctcoordinate time, they can be synchronized without light signals by trans-porting them from common site in a symmetrical manner. The procedureconsists of following steps: ... . As we see, the thought experiment describedis capable to prove constancy of light speed if it is true, or to disprove it ifit is false. It provided, therefore, solid logical foundation for Einstein’s syn-chronization prescription.” This logical argument is incorrect. Slow clocktransport synchronization is equivalent to Einstein’s synchronization in in-ertial system where the light source is at rest.

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4 Aberration of Light Phenomenon. Inertial Frame of Reference


The effect of light aberration in an inertial frame of reference is usuallyunderstood as a change in the direction of light propagation ascribed toboosted light sources. The phenomenon of aberration of light in an inertialframe of reference by no means simple to describe, even in the first orderin v/c: a large number of incorrect results can be found in the textbooks.Thus, in order to keep the mathematical complexity of the discussion to aminimum, we will describe the effect of aberration by working only up tothe first order. The appearance of relativistic effects in optical phenomenadoes not depend on a large speed of the radiation sources. Light is alwaysa relativistic object, no matter how small the ratio v/c may be.

The subject of this chapter is the physical influence of the optical instrumenton the aberration of light measurement. First, we will demonstrate that whenone has finite-aperture mirror moving transversely and the plane wave oflight is falling normally on the mirror, there is the aberration (deviation ofthe energy transport) for light reflected from the mirror. The next problemto be discussed in this chapter is of more practical importance. According tothe Babinet’s principle, the problem of reflection from a transversely mov-ing (finite-aperture) mirror complementary to the problem of transmissionthrough a hole in the moving opaque screen. One case of rather great interestis that which corresponds to transmission through a moving end of the tele-scope barrel. It is shown that the (electromagnetic) wave theory approach tothe aberration of light in an inertial frame leads to a substantially differentresult for energy transport than the predicted by conventional theory.

4.2 The ”Plane Wave” Emitter

We shall try to understand the effect in a very simple case. The explanationof the effect of aberration is actually based on the use of the model of singleplane-wave emitter. As a simple model of a plane-wave emitter, we use atwo-dimensional array of identical coherent elementary sources (dipoles),uniformly distributed on a given (x − y) plane P. We take the elementarysources to start radiating waves simultaneously with respect to the labreference frame where the plane P is at rest. Therefore we have a planefull of sources, oscillating together, with their motion in the plane and allhaving the same amplitude and phase. Let us suppose that the elementarysources are oscillating at frequency ω. By letting the distance between each

47

two adjacent elementary sources approach zero (i.e. much smaller withrespect to the radiation wavelength λ = 2πc/ω), we may consider this two-dimensional arrangement as an ideal plane-wave emitter.

The concept of an (infinite) plane wave is widely used in physics. It is ananalytically well-behaved solution of Maxwell’s equations. However, it isnot a physically realizable solution, because the total energy content of sucha wave is infinite. In any physically realizable situation, one will have toconsider a finite source aperture. It is always hiddenly assumed that thedetector for the direction of the radiation is an energy propagation detectorand the size of the detector aperture is sufficiently large compared with theradiation beam size. Indeed, what is usually considered as an aberration is,in fact, an apparent deviation of the energy transport direction.

4.3 A Moving Emitter. The ”Single Frame” Description

Let us consider the case when the (x − y) plane emitter in the lab inertialframe is accelerated from rest up to velocity v along the x-axis. Suppose thatan observer, which is at rest with respect to the inertial frame of referenceperforms the direction of the energy transport measurement.

There could be two approaches to the analysis of the aberration o light ra-diated by a single moving emitter. The first one is the covariant approach.The explanation of the effect of aberration of light in the case of a singlemoving emitter presented in the literature is actually based on the use of aLorentz boost to describe how the direction of a beam of light depends onthe velocity of the light source relative to the inertial frame of reference. An-other non-covariant approach consists in using a ”single frame” descriptionwithout reference to Lorentz transformations.

The two approaches, treated according to Einstein’s or absolute time syn-chronization conventions give the same result in the case of a single movingemitter. The choice between these two different approaches in this case is amatter of pragmatics.

In this section we demonstrate both approaches. Let us start with non-covariant approach. We must emphasize that there is no principle difficultywith the a non-covariant (3+1) approach in relativistic electrodynamics. It isperfectly satisfactory. The aberration of light problem can be treated withinthe same ”single inertial frame” description without reference to Lorentztransformations. Within a single inertial frame, the time is an absolute quan-tity in special relativity also. What does ”absolute” time mean? It means thatsimultaneity is absolute and there is no mixture of positions and time whensources change their velocities in the inertial frame. A distinctive trait of

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Fig. 1. The case when the (x− y) plane emitter in an inertial lab frame is acceleratedfrom rest up to velocity v along the x-axis. It is assumed that there is no physicalinfluence of the detector for the direction of the radiation on the measurement. Theaberration increment is connected with the physical parameters by the relation:θa = v/c. The explanation of the effect of aberration is based on the use of a Galilean(left) and Lorentz (right) boost to describe how the direction of a light beam dependson the velocity of light source relative to the lab frame. If make a Lorentz boost, weautomatically introduce a time transformation t′ = t − xv/c2 and the effect of thistransformation is just a rotation of the radiation phase front in the lab frame.

our non-covariant theory is the absence of relativistic kinematics in the de-scription of aberration of light phenomena. When one has a transverselymoving emitter there is the deviation of the energy transport for radiatedlight. According to the ”single frame” approach, this effect is a consequenceof the fact that the Doppler effect is responsible for the angular frequencydispersion of the radiated light waves (Fig. 1 left).

Let us consider the electrodynamics of the moving source. The explanationof the phenomenon of radiation in our case of interest consists in using a(active) Galileo boost to describe the uniform translational motion of thesource in the inertial lab frame. Maxwell’s equations are not preservedin form by the Galilean transformation, i.e. Maxwell’s equations are notinvariant under Galilean transformation. The new terms that have to be putinto the field equations due to use of Galilean transformation lead to theprediction of the Doppler effect.

One should not to be surprised to find that electrodynamics problem ofmoving emitter has intrinsic anisotropy. If fact, anisontropy results directlyfrom the time-dependence of the transverse position of the moving emitter

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with finite aperture. What must be recognized is that in the time-dependentemitter problem, the results will depend on the direction of the velocityvector.

The present approach to moving emitter problem uses the Fourier transformmethods. When we are dealing with linear systems it is useful to decomposea complicated input into a number of more simple inputs, to calculate theresponse of the system to each of these elementary functions, and to super-impose the individual responses to find the total response. Fourier analysisprovides the basic means for performing such a decomposition (1).

Consider the inverse transform relationship g(x) =∫∞

−∞G(K) exp(iKx)dK ex-

pressing the profile function in terms of its wavenumber spectrum. We mayregard this expression as a decomposition of the function g(x) into a linearcombination (in our case into an integral) of elementary functions, eachwith specific form exp(iKx). From this it is clear that the number G(K) issimply a weighting factor that must be applied to elementary function ofwavenumber K to synthesize the desired g(x).

The emitter with finite aperture is a kind of active medium which breaks upthe radiated beam into a number of diffracted beams of plane waves. Eachof these beams corresponds to one of the Fourier components into whichan active medium can be resolved. Let us assume that the dipole densityof the elementary source varies according to the law ρdip = g(K⊥) cos (K⊥x).We conclude, then, that the active medium of the elementary source issinusoidally space-modulated.

From the Galilean transformation, after partial differentiation, one obtainswave equation Eq.(6). Let us demonstrate that the new terms that have tobe put into the field equations due to use of Galilean transformation leadto the prediction of the Doppler effect. Consider as a possible solution aradiated plane wave exp(i~k ·~r− iωt). With a plane wave exp(i~k ·~r− iωt) withthe wavenumber vector ~k and the frequency ω equation Eq.(6) becomes:(1 − v2/c2)k2

x + 2vkxω/c + k2z − ω

2/c2 = 0. The wavenumber vector of the

radiated plane wave is fixed by initial conditions. In fact, kz =√ω2

i /c2 − k2

x,kx = K⊥, where K⊥ is the wavenumber of sinusoidally space-modulateddipole density. From initial conditions we will find that it necessary touse ~k as independent variable and we will consider ω as a function of kx:ω = ωi +∆ω(kx), whereωi is the frequency of the emitter radiation before theacceleration. From this dispersion equation, we find the requirement that thewavenumber K⊥ and the frequency change ∆ω are related by ∆ω = K⊥v. Itis worth noting that we consider an aberration angle that is relatively largecompared to the divergence of the radiated light beam. In other words,o/De v/c (i.e cK⊥/ωi v/c), where De is the transverse size of the emitter.

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There is a different physical viewpoint on the Doppler effect of the light beamradiated from a moving emitter that is equivalent to the presented above.The phase of the wave at the world point (~r, t) is invariant at a changeof the reference system. This invariance is independent of the coordinatetransformation, by which we describe the change of the reference system.Therefore, the phase~k·~r−ωt must be invariant of the Galilean transformation.Consequently,~k′ ·~r′−ω′t′ = ~k·~r−ωt. Substituting the Galilean transformationformulae x′ = x − vt, t′ = t into the phase equality formula we obtain∆ω = K⊥v. This frequency change coincides with the result derived directlyfrom the dispersion equation, as must be.

Let us first remind the reader of the fact that the usual velocity of waves isdefined as given the phase difference between the oscillations observed attwo different points in a free plane wave. It is primary used for computinginterference fringes that makes phase differences visible. In a plane wavewe observe the phase velocity ω/k. Another (group, or energy propagation)velocity can be defined, if we consider the propagation of a peculiarity, thatis change in amplitude impressed on a train of waves. A simple combinationof groups obtains when two waves ω1 = ω + ∆ω, k1 = k + ∆k and ω2 = ω −∆ω, k2 = k − ∆k are superimposed. This represents a carrier with frequencyω and a modulation with frequency ∆ω. The wave may be described as asuccession of moving beats (or groups). The carrier’s velocity is ω/k, whilethe group velocity is given by vg = ∆ω/∆k→ dω/dk.

Many textbooks on electromagnetic theory discuss the aberration of lightphenomena in the context of plane wave. However, in dealing with planewave one will have an incorrect model of the aberration of light. When aninfinite sinusoidal wave travels, there is a uniform average energy densitythroughout the space. Does this energy remain where it is, or does it prop-agate through the space? It is impossible to know this. All experimentalmethods for measuring the aberration of light operate with light signals,and hence do not measure the phase velocity but the signal velocity andthis velocity coincides with group velocity.

In our example, the plane waves with different wavenumber vectors prop-agate out from the moving emitter with the different frequencies. Thenequation ∆ω/∆kx = v holds for each scattered waves independently on thesign and the magnitude of the radiated angle. In fact, the ∆ω in our case ofinterest is the Doppler shift ∆ω = ~K⊥ · ~v and the ∆kx is simply the transversecomponent of the radiated wavenumber vector ∆kx = K⊥. The last equationsstate that radiated light beam with finite transverse size moves along the xdirection with group velocity dω/dkx = v.

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4.4 A Moving Emitter. Explanation on the Basis of the Lorentz Transformations

It should be note, however, that there is another satisfactory way of explain-ing the effect of aberration of light from the moving source. The explanationconsists in using a Lorentz boost to describe the uniform translation motionof the light source in the lab frame. On the one hand, the Maxwell’s equa-tions remain invariant with respect to Lorentz transformations. On the otherhand, if make a Lorentz boost, we automatically introduce a time transfor-mation t′ = t− xv/c2 and the effect of this transformation is just a rotation ofthe radiation phase front in the lab frame. This is because the effect of thistime transformation is just a dislocation in the timing of processes, whichhas the effect of rotating the plane of simultaneity on the angle v/c in thefirst order approximation.

According to Maxwell’s electrodynamics, coherent radiation is always emit-ted in the direction normal to the radiation phase front. This is becauseMaxwell’s equations have no intrinsic anisotropy. In other words, when auniform translational motion of the source is treated according to Lorentztransformations, the aberration of light effect is described in the languageof relativistic kinematics. According to the relativistic kinematics, the extraphase chirp dφ/dx = kx = vωi/c2 is introduced and the array of identicalelementary sources of the moving emitter now have different phases. Asa consequence of this, the plane wave wavefront rotates after the Lorentztransformation. Then, the radiated light beam is propagated at the anglev/c, yielding the phenomenon of the aberration of light (Fig. 1 right).

We now ask about the group velocity of the radiated beam. With a planewave exp(i~k · ~r − iωt) dispersion equation in the case of Maxwell’s electro-dynamics is reduced to k2

z + k2x − ω

2/c2 = 0. From the initial conditions and

the Lorentz transformation we find that ω = γ(ωi + vK⊥), kz =√ω2

i /c2 − K2

⊥,

kx = γ(vωi/c2 + K⊥), where K⊥ is the wavenumber of sinusoidally space-modulated dipole density,ωi is the frequency of the emitter radiation beforethe acceleration. Substituting these expressions in dispersion equation wefind that the latter is satisfied, as must be.

As one of the consequences of the Doppler effect in the Lorentz coordinati-zation, we find an angular frequency dispersion of the light waves radiatedfrom the moving emitter with finite aperture. The Doppler shift, ∆ω, of ra-diated light wave (in the first order approximation) is given by ∆ω = ~K⊥ · ~v,where K⊥ is the transverse component of the radiated wavenumber vector.The last equation state that radiated light beam with finite transverse sizemoves along the x direction with group velocity dω/dkx = v.

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It is interesting to discuss what it means that there are two different (covari-ant and non-covariant) approaches that produce the same group velocity.The point is that both approaches describe correctly the same physical re-ality and since the group velocity has obviously an objective meaning (i.e.convention-invariant), both approaches yield the same physical results.

The difference between the absolute time coordinatization and the Lorentzcoordinatization is very interesting. In the Chapter 3 we already discussedhow one can transform the absolute time coordinatization to Lorentz co-ordinatization. We can interpret manipulations with rule-clock structure inthe lab frame simply as a change of the time variable according to the trans-formation t→ t + xvx/c2. The overall combination of Galileo transformationand variable changes actually yields the Lorentz transformation in the caseof absolute time coordinatization in the lab frame. This variable change hasno intrinsic meaning. One can see the connection between the time shiftand the issue of clock synchrony. The convention-independent results ofcalculations are precisely the same in the new variables. As a consequence,we should not care to transform the results of the electrodynamics problemsolution into the original (3+1) variables.

An idea of studying the relativistic electrodynamics using technique involv-ing a change of variables is useful from a pedagogical point of view. It isworth remarking that the absent of a dynamical explanation for wavefrontrotation in the Lorentz coordinatization has disturbed some physicists. Itshould be clear from the discussion in the Chapter 3 that a good way tothink of the wavefront rotation is to regard it as a result of transformationto a new time variable in the framework of the Galilean (”single frame”)electrodynamics.

4.5 Reflection from a Mirror Moving Transversely. Mistake in Existing Theory

It is generally believed that for a mirror moving tangentially to its surfacethe law of reflection which holds for the stationary mirror is preserved,as shown in Fig.2. In other words, the velocity of the energy transport isequal to the phase velocity. This statement, presented in most textbooks, isincorrect.

First, we examine the reasoning presented in textbooks (2). The reflectionfrom the mirror is analyzed in two Lorentz reference frames. The fixed (lab)frame S is at rest with respect to the plane-wave emitter. The moving frameS′ has velocity v. In this frame, the mirror is at rest. In both frames, we use aCartesian coordinate system in which x− y plane is tangent to the reflectionsurface. The x direction coincides with the direction of v. For simplicity,

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Fig. 2. Transversely moving mirror with small aperture at normal incidence. Ac-cording to textbooks, there is no deviation of the energy transport for the reflectedlight beam. A monochromatic plane wave of light is falling normally on the smallmoving aperture mirror, and generates a reflected oblique beam.

Fig. 3. The effect of aberration of light is described in the language of relativistickinematics, in terms of the wavenumber four vector. Geometry of the reflectionas seen from (a) lab frame, (b) inertial frame moving with the same velocity asthe mirror. According to textbooks, there is no aberration for light reflected from atangentially moving mirror.

we consider the case in which light is incident from the z direction in thelab frame. Incident light is described by its four-dimensional wave vector,whose time like component is the angular frequencyω and whose space likecomponents define the direction of propagation. In the lab frame, (t, x, y, z),this vector has components k1 = (ω, 0, 0,−ω/c), where the negative signindicates propagation towards the mirror (Fig. 3a). Our task is to determinethe wave vector for the reflected beam.

The argument that there is no aberration for light reflected from a trans-versely moving mirror runs something like this. It is easiest to consider the

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reflection in the moving frame (Fig. 3b). In this frame, the surface is at rest, sothe usual laws of optical reflection apply. We will describe the effect of aber-ration of light by working only up to the first order v/c. An observer mov-ing with the mirror surface sees the wave vector k′1 = (ω,−vω/c2, 0,−ω/c).The effect of reflection is to reverse the sign of the z′ component of thewave vector, k′2 = (ω,−vω/c2, 0, ω/c). We now obtain the reflected wavevector in the lab frame by applying the inverse Lorentz transformation:k2 = (ω, 0, 0, ω/c).This vector represents a light beam traveling away fromthe mirror, having the same frequency as the incoming beam. This showsthat the beam is reflected according to the usual geometrical optics laws,and the beam suffers no aberration.

The error in the last argument follows from the fact that the concept of aplane wave and an infinite plane mirror is used in the case of the tangentialmotion. The absurdity of this, i.e. the impossibility of such a motion everbeen detected is put forward as an obvious argument against the reasoningpresented in textbooks. Surprisingly, it was not recognized in the literaturethat in this case the ”mirror motion” is not a real observable effect. Indeed, ifthe mirror is infinite the problem is not time dependent (we must concludethat when we are dealing with hidden assumption that problem is nottime dependent we have no aberration and that is not too surprising). It isobvious that only the motion of a finite mirror is meaningful. In view of theelectrodynamics, only a velocity of the finite mirror has a physical meaning.The reasoning in textbooks ignores completely the interaction of the lightand the moving mirror edges.

We would also noticed that the concept of plane wave is used in textbooks.There is a common mistake made in the electrodynamics connected withthe energy transport direction in the case of an (infinite) plane wave. Whenan infinite plane wave travels, there is a uniform average energy densitythroughout the space. It is impossible to know the energy transport direc-tion when one has deal with a plane wave. All experimental methods formeasuring the aberration increment operate with light signals, and hencedo not measure the phase velocity (i.e. frequency and wavenumber vector)but the group velocity. It can be defined only if we consider the propagationof a peculiarity, that is change in amplitude impressed on a train of waves.What authors of textbooks generally overlooked is the fact that the energytransport problem is well-defined only if the source and mirror apertureshave already been specified.

We shall discuss the situation where there is a finite aperture mirror movingtangentially to its surface. For simplicity, we shall assume that the transversesize of the moving mirror is very small relative to the transverse size ofthe ”plane-wave” emitter, as sketched in Fig. 2. It is worth noting thatwe consider an aberration angle that is relatively large compared to the

55

divergence of the reflected radiation. In other words, o/De o/Dm

v/c, where De and Dm are the transverse size of the emitter and mirror,respectively.

We will demonstrate that when one has a transversely moving mirror withfinite aperture and a plane wave of light is falling normally onto the mirror,there is a deviation of the energy transport for the reflected light. In this case,the effect of aberration of light results directly from the time-dependence ofthe position of the moving mirror, since a finite aperture has to be consid-ered. What must be recognized is that in the time-dependent emitter-mirrorproblem, the solution involves light beams with different frequencies. Itcomes out quite natural that the result will depend on the velocity vector:in other words, textbooks overlook the influence of the Doppler effect. Fora transversely moving mirror with finite aperture we cannot neglect theangular frequency dispersion, which is an effect of the first order in v/c.

4.6 Way to Solve the Emitter-Mirror Problem in the (3+1) Space and Time

The aberration of light problem can be treated within the same ”single iner-tial frame” description without reference to Lorentz transformations. Whenthe illumination of the object originates from a monochromatic spatially co-herent source there exists a method for calculating the reflected intensity thathas the special appeal of conceptual simplicity. It uses Fourier transformsof spatial filtering theory that is the Abbe diffraction theory.

The essence of Abbe’s approach, in our case of interest, is that one regardsthe mirror as a kind of a diffraction grating which breaks up the incidentbeam of the plane wave into a number of diffracted beams constituted byplane waves. Each of these beams corresponds to one of the Fourier com-ponents into which the reflected power of the mirror can be resolved. Thefinite-aperture mirror is a non-periodic object. It gives an infinite number ofdiffracted beams forming a continuum.

A simple example of a diffraction grating is shown in Fig.4. Let us as-sume that the reflectance of the grating varies according to the law R =

g(K⊥) cos (~K⊥ · ~r). The reflectance is sinusoidally space-modulated. It shouldbe noted that the permanent reflectance distribution grating discussed hereis only our mathematical model and we do not need to discuss how it canbe created.

The ~k vectors shown in Fig.4 represent the propagation vector of the inci-dent plane wave ~ki, which is assumed to be directed perpendicularly to the

surface. The vectors ~ks(+)

and ~ks(−)

are added to indicate the scattered light.

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Fig. 4. The Bragg diffraction grating at normal incidence. The reflectance is sinu-soidally space-modulated.

The Bragg condition ~ks = ~ki ± ~K⊥ shows how the direction of the incidentand scattered wave are related. The first-order maxima dominate due to thefact that light is being scattered from a sinusoidal grating, rather than a setof discrete planes (grooves).

We assume that the ~K⊥ vector is directed parallel to the side of the diffractiongrating with the incident wave impinging on the grating perpendicularly,as shown in Fig.4. The length of the vectors ~ks and ~ki must, of course,be the same, but the vector diagram does not quite match up. The Braggconditions are then not satisfied precisely. For small angles,~ks = ~ki ± ~K⊥ stillholds approximately, so that we obtain the scattering angle θ = K⊥/ki.

When the scatterer wave is a progressive wave rather than a fixed modula-tion, the frequency of the scattered wave is different from that of the incidentwave. This fact is interpreted as a Doppler effect, since the reflection is froma moving, rather than a stationary, set of waves. In the case of a transverselymoving grating, light in the diffraction maxima undergoes a Doppler shiftresulting from the fact that it has been reflected from moving waves withwavenumber vectors ~K⊥.

Let us demonstrate that the new terms that have to put into the field equa-tions due to use of absolute time coordinatization lead to the prediction ofthe Doppler effect. We recall that with a plane wave exp(i~k ·~r− iωt) with thewavenumber vector~k and the frequency ω dispersion equation in the abso-lute time coordinatization becomes: (1 − v2/c2)k2

x + 2vkxω/c + k2z −ω

2/c2 = 0.The wavenumber vector of the radiated plane wave is fixed by initial con-

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ditions. In fact, kz =√ω2

i /c2 − k2

x, kx = K⊥, where K⊥ is the wavenumber ofsinusoidally space-modulated reflectance. From this dispersion equation,we find the requirement that the wavenumber K⊥ and the frequency change∆ω are related by ∆ω = K⊥v.

The following important detail of such ”single inertial frame” descriptioncan hardly be emphasize enough. If the source of light is at rest and themirror is in motion, it is obvious that the electrodynamics equations mustbe identical for all electromagnetic waves. In other words, the dispersionequation in the absolute time coordinatization should be applied and keptin a consistent way for both incoming and scattered waves (Fig. 4). In ourprevious discussion of absolute time coordinatization we learned that theemitter at rest must be in the same time described by Maxwell’s electrody-namics. A dispersion equation in the case of Maxwell’s electrodynamics isreduced to k2

i − ω2i = 0. From the initial conditions we find that ~ki = ~ezkz,

ωi = ckz. The contradiction, however disappears if we perform geometri-cal analysis of light reflection. The peculiarity of the discussed geometry isthat even after the Galilean transformation along the x-axis the dispersionequation in the absolute time coordinatization will have the same (diagonal)form k2

z − ω2 = 0 for the incident wave.

As one of the consequences of the Doppler effect, we find an angular fre-quency dispersion of the light waves reflected off the moving mirror withfinite aperture. If ~n = ~k/|~k| denotes a unit vector in the direction of the wavenormal, and ~v is the mirror velocity vector relative to the lab frame, we getthe equation ωs = ωi(1 + ~n · ~v/c) = ωi + (ωiv/c) cosθ. The Doppler effect isresponsible for angular frequency dispersion to the first order of v/c evenwhen ~n ·~v = 0 (i.e when cosθ = 0). In fact, dωs/dθ = −(ωiv/c) sinθ = −ωiv/cat θ = π/2. We can rewrite this equation in a different way. The differentialof the scattered angle is given by dθ = −dkx/ki. With the help of this relationand account for that ki = ωi/c we have dωs/dkx = v.

One of the most important conclusions of the foregoing discussion is aremarkable prediction on the theory of the aberration of light, concerningthe deviation of the energy transport for light reflected from a mirror movingtransversely. Namely, when a plane wave of light is falling normally on themirror, there is a deviation of the energy transport for reflected light beam(see Fig. 5). This phenomenon can be regarded as a simple consequence ofthe Doppler effect.

The argument that in the process of reflection from a transversely mov-ing mirror the direction of propagation is not given by the normal to thewavefront is considered erroneous in literature (4). This fact is ascribed to alack of understanding of the difference between convention-dependent and

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Fig. 5. Transversely moving mirror with a small aperture at normal incidence. Whena plane wave of light is falling normally on the mirror, there is a deviation of theenergy transport for the reflected light beam. This effect is a consequence of the factthat the Doppler effect is responsible for angular frequency dispersion of the lightwaves reflected from the mirror. As a result, the velocity of the energy transport isnot equal to the phase velocity.

convention-invariant parts of the theory. The direction of the energy trans-port has an exact objective meaning i.e. is convention-invariant. However,the phase front orientation (i.e. the plane of simultaneity in the judgementof an observer) has no exact objective meaning since, due to the finitenessof the speed of light, we cannot specify any experimental method by whichthis orientation could be ascertained.

Since the phase front orientation does not exist as physical reality withinthe angular range v/c, a question arises: why do we need to account forthe exact phase front orientation in our electrodynamics calculations? Theanswer is that when the evolution of the radiation beam is treated accordingto the single inertial frame of reference, one will experience that phasefront orientation remains unvaried: this has no objective meaning but isused in the analysis of the electrodynamics problem. A comparison with agauge transformation in Maxwell’s electrodynamics might help here. Evenif the phenomena are quite different, the common mathematical formulationpermits us to draw this analogue.

4.7 Covariant Way to Solve the Emitter-Mirror Problem

Physicists who try to understand the situation related to the use of thecovariant approach in the aberration of light phenomena, are often troubledby the fact that in the situation where there is a mirror moving normally

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to its surface the reasoning presented in textbooks is correct. The reflectionfrom the mirror is analyzed in two Lorentz reference frames. It is interestingto note that, for this case, relativistic kinematics correctly predicts the lightfrequency variation on reflection from a moving mirror. At this point areasonable question arises: why the same method gives incorrect result inthe case of reflection of light from a mirror moving tangentially to its surface?

It is not difficult to see that the peculiarity of the situation where there isan infinite mirror moving normally to its surface is that here the ”mirrormotion” is a real observable effect. Indeed, even if the mirror is infinitethe problem is time dependent. In this situation, a velocity of the infinitemirror has physical meaning. Thus, for normal motion, the infinite mirrormay be a useful concept for analysis of the Doppler effect. This method wasincorrectly extended to the case of tangential motion.

The authors of textbooks did not make a computational mistake in theirtreatment of the aberration of light phenomena in an inertial frame of ref-erence, but rather a conceptual one. We must say that there is no objectionsto the moving frame transformation. It is easiest to consider the reflectionin the inertial frame moving with the same velocity as the mirror (Fig. 3b).An observer moving with the mirror surface sees the incoming wave vectork′1 = (ω,−vω/c2, 0,−ω/c). Then, where does the mistake comes from? Thepresented above commonly accepted covariant treatment of reflection froma mirror moving transversely includes one delicate point. We state that thetypical textbook statement ”The effect of reflection is to reverse the sign ofthe z′ component of the wave vector, k′2 = (ω,−vω/c2, 0, ω/c)” is incorrect. Infact, as we have already discussed in this section, the infinite plane mirrorcannot be used when we deal with aberration of light phenomena.

The finite aperture mirror, that is treated as the source of reflected radiation,is usually modeled with the help of a physical optics approach. This iswell-known high-frequency approximation technique, often used in theanalysis of the electromagnetic waves scattered from large (relative to thewavelength) objects. The present approach to mirror reflection problemuses the Fourier transform methods. The beam is reflected according tousual physical optics laws and has the angular spectrum width ∆θ ' o/Dm,where Dm is the characteristic mirror size. A reflected light beam in thecomoving frame traveling away from the mirror has the same frequency asthe incoming plane wave. So we must conclude that the Doppler effect isabsent and the velocity of the energy transport in the x direction is equal tozero.

From the initial conditions and the Lorentz transformation we find that in thelab frameω = γ(ωi +vK⊥), kz =

√ω2

i /c2 − K2

⊥, kx = γ(vωi/c2 +K⊥), where K⊥ is

the transverse wavenumber of the plane wave in the Fourier decomposition

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of the reflected beam. As one of the consequences of the Doppler effect inthe Lorentz coordinatization, we find an angular frequency dispersion ofthe light waves reflected from the moving mirror with finite aperture. TheDoppler shift, ∆ω, of reflected light wave (in the first order approximation)is given by ∆ω = ~K⊥ · ~v. The last equation states that reflected light beamwith finite transverse size moves along the x direction with group velocitydω/dkx = v. That is the reflection appears as shown in Fig. 5.

Let us examine in a little more detail how group velocity comes about fromcovariant and non-covariant point of view. After the Galilean transforma-tion x′ = x − vt, t′ = t we would obtain the same group velocity as after theLorentz transformation x′ = x − vt, t′ = t − vx/c2. The two approaches givethe same result for real observable effect. First we want to rise the followinginteresting and important point. An acceleration of the mirror with respectto the inertial frame is absolute (i.e. is physical reality) and described in bothapproaches by the same coordinate transformation x′ = x − vt. This trans-formation (boost) in the x direction leads to angular frequency dispersionof the light waves reflected from the moving mirror with finite aperture, in-dependently of the coordinatization. On the other hand, if make a Lorentztransformation, we introduce a time transformation t′ = t − vx/c2 and theeffect of this transformation is just a rotation of the radiation wavefront.This rotation is not a real observable effect.

4.8 Discussion

This is a good point to make a general remark about the emitter-mirrorproblem. The peculiarity of this problem with the viewpoint of relativistickinematics is that here the emitter (and observer with his measuring devices)is at rest in the lab inertial frame and the mirror is moving with the constantspeed with respect to the lab frame and interacts with the radiated lightbeam. How can we solve a problem involving the emitter-mirror relativevelocity? Each physical phenomenon occurs in space and time. A concretemethod for representing space and time is a frame of reference (coordinate-time grid), which requires careful description.

The Maxwell’s equations can be applied in the lab inertial frame only in thecase when Lorentz coordinates are assigned. It is incorrectly believed thatthe emitter-mirror electrodynamics problem can be treated within the same”single Lorentz frame” description. In other words, it is incorrectly believedthat the common Lorentz time coordinate axis for emitter and mirror canbe assigned. This is misconception. We can prepare, for mirror and emitter,a common set of synchronized clocks in the lab frame only in the case ofabsolute time coordinatization i.e in the case when simultaneity is absolute.

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As we already mentioned, the utilization of the electrodynamics in theabsolute time coordinatization becomes indispensable when we consideroptical phenomena associated with a relative motion of two or more (in-teracting) bodies. Suppose that we assign the Lorentz time coordinate forthe description of the emitter radiation. But this will be the absolute timecoordinatization for the boosted mirror and this boost will be describedin such coordinatization by the Galilean transformation. Suppose that were-synchronize clocks in the lab frame in order to assign the Lorentz timecoordinate for the boosted mirror. In this coordinatization, we describe thereflection of light using the usual Maxwell’s equations. But this new time co-ordinate in the lab frame is interpreted by saying that Maxwell’s equationsare not applicable to the emitter radiation description.

So far we have considered the covariant way to solve the emitter-mirrorproblem. It is interesting to note that, the use of relativistic kinematics forthe calculation of the reflection from a transversely moving mirror doesnot necessarily leads to mistake. We would like to discuss the followingquestion: since the common Lorentz time coordinate axis for emitter andmirror cannot be assigned, how the relativistic kinematic method leads to thecorrect result if applied to computation of the reflection from a transverselymoving mirror? Above we demonstrated that the both (covariant and non-covariant) approaches give the same result for group velocity of the reflectedlight beam. The reason is that an acceleration of the mirror with respect tothe inertial frame is physical reality and described in both approaches bythe same boost x′ = x − vt. This transformation leads to the Doppler effectof the light waves reflected from the moving mirror with finite apertureindependently of the coordinatization.

Let us now discuss more about consequences of the Lorentz transforma-tions. If we rely on the relativistic kinematic method, the reflection resultsin a difference between the direction reflected beam motion and the normalto the radiation wavefront. This is already a conflict result, because we nowconclude that, according to covariant approach, the direction of propagationafter the reflection is not perpendicular to the radiation wavefront. This iswhat we would get for the case when our analysis is based on the relativistickinematics and is obviously absurd from the viewpoint of Maxwell’s elec-trodynamics. In fact, we demonstrated that our assumption about existenceof common Lorentz time axis for emitter and mirror leads to logical incon-sistency. We conclude that this assumption is incorrect. Only the solutionof emitter-mirror problem in the absolute time coordinatization gives theconsistent description of the reflection from a mirror moving transversely.

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Fig. 6. Transversely moving mirror with a large aperture at normal incident. Whena beam of light is falling normally on the mirror, there is no deviation of the energytransport for the reflected light beam. The velocity of the energy transport is equalto the phase velocity.

4.9 Large Aperture Mirror

In this chapter our discussion is limited to the region of problem parameters,in which we forget about the emitter edges. Although this aberration of lighttheory is just an approximation, it is a very great importance practically. Weshall also discuss the situation where the transverse size of the emitter isvery small relative to the transverse size of the moving mirror. It should benote that this situation is not realized in the stellar aberration measurements.We shall work out this case in order to understand all the physical principlesvery clearly. It is easy to show that the deviation of the energy transport isabsent in this case, Fig. 6. The direct approach to moving mirror problemuses the Fourier transform methods. We now consider another method ofcalculating the aberration of light effect - we want to illustrate the greatvariety of possibilities. The way of thinking that made the law about thebehavior of light reflected from a large aperture mirror evident is based onBabinet’s principle. It is well known that, when light comes through a holeof a given shape, made in an opaque screen, the distribution of intensityafter the hole (i.e. the diffraction pattern) is the same as in the case when thehole is replaced by sources (dipoles) uniformly distributed over the hole. Inother words, the diffracted plane wave from a hole, or from a source withthe same shape of the hole are the same.

This is a particular case of Babinet’s principle, which states that the sum ofdiffraction fields behind two complementary opaque screens is the incidentwave. We know from this principle, that the solution we have found using

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Abbe’s approach also corresponds to that for large aperture hole in a movingopaque screen. We see clearly that there is no electromagnetic interactionof a light beam with screen. Indeed light beam is not scattered by the holeedges. Does this discussion about large aperture hole have any meaning?To see whether it does, we should remember about the Babinet’s principle.Here we only wish to show how easy the law of reflection from a largeaperture mirror can be found with the help of the Babinet’s principle.

4.10 Analysis of Transmission through a Hole in a Opaque Screen

Above we demonstrated that when one has a small aperture mirror movingtransversely and the plane wave of light is falling normally on the mirror,there is the aberration (deviation of the energy transport) for light reflectedfrom the mirror. The problem to be considered in this section is of morepractical importance. We now consider the case of a screen, in the lab frame,moving with velocity v along its surface. It is generally believed that thereis no deviation of the energy transport for light transmitted through a holein the moving opaque screen, Fig. 7.

However, there is a common mistake made in relativistic optics, connectedwith aberration effects from a transversely moving screen containing a hole.We describe the system using, again, a Fourier transform method similar tothat considered above. The screen containing a hole is a kind of diffractiongrating which breaks up the incident beam of the plane wave into a numberof diffracted beams of plane waves. Each of these beams corresponds toone of the Fourier components into which a transmitted light beam can beresolved.

The gratings discussed so far modulate the amplitude of the incident planewave by a periodic reflection function. However, we can immediately ex-tend the range of validity of our analysis to gratings that modulate theamplitude of the incident light by a periodic transmission function. Letus assume that the transmittance of the grating varies according to thelaw T = g(K⊥) cos (~K⊥ · ~r), Fig.8. The transmittance is sinusoidally space-modulated. All the equations that we derived so far hold immediately forthe forward scattered beams.

According to our approach, there is a remarkable prediction of the theoryof aberration of light concerning the deviation of the energy transport forlight transmitted through a hole in a moving screen. Namely, when one hasa transversely moving screen with a hole in it and a plane wave of light isfalling normally on the screen, there is a deviation of the energy transportfor light transmitted through the hole (see Fig. 9).

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Fig. 7. Aberration of light in an inertial frame of reference. Transversely movingscreen which has a hole in it. According to textbooks, a monochromatic plane waveof light is falling normally on the screen and generates a transmitted oblique beam.There is no deviation of the energy transport for the transmitted oblique light beam.The velocity of the energy transport is equal to the phase velocity.

Fig. 8. The Bragg diffraction grating at the normal incident. The transmittance issinusoidally space-modulated.

4.11 Spatiotemporal Transformation of the Transmitted Light Beam

Let us suppose that transmitted light pulse propagates in the x − z plane.Now we are interested in the space-time intensity distribution in this plane.Spatiotemporal coupling arises naturally in transmitted radiation behindthe screen, because the transmission process involves the introduction ofan angular-frequency dispersion of the transmitted radiation. The emittedlight beam is represented with sufficient accuracy as the product of factorsseparately depending on space and time. However, when the manipulation

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Fig. 9. Aberration of light in an inertial frame of reference. Transversely movingscreen which has a hole in it. A monochromatic plane wave of light is fallingnormally on the screen and generates a transmitted light beam. The Doppler effectis responsible for the angular frequency dispersion of the light waves transmittedthrough the hole. As a result, the velocity of the energy transport is not equal to thephase velocity.

of the emitted light requires the transmission through a hole in a movingopaque screen, such assumption fails.

We start by writing the field of an emitted pulse as E(x, t) = bi(x) exp[iωi(z/c−t)]. The initial amplitude distribution bi(x) in front of the moving screen is theoptical replica of the emitter aperture. The electric field of the transmittedpulse expressed in the reciprocal domain as E(∆kx,∆ω) = E(K⊥,K⊥dω/dkx),which is the Fourier component of the electric field of a beam with angularfrequency dispersion and dω/dkx = v. The inverse Fourier transform tothe space-time domain can be expressed as E = b(x − vt) exp[iωi(z/c − t)].This is the field immediately behind the moving screen. Profile of the beamb(x − vt) is the optical replica of the moving aperture. Consider a screen atrest position on the distance l behind the moving aperture. For simplicitywe assume here that Fresnel number is large, NF = D2

h/(ol) 1, and we canneglect the diffraction effects. Here Dh is the characteristic aperture size. Weconclude that the light spot on the observer screen moves with the samevelocity v as the moving screen (5).

4.12 Applicability of the Ray Optics

Let us move on to consider the predictions of the existing aberration oflight theory in the case of a transversely moving screen which has a hole

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Fig. 10. Transverse moving screen which has a hole in it. Light corpuscles are fallingnormally on the screen and, according to literature, generates an oblique light beam.

in it (Fig. 7). According to conventional theory, the spot of the transmittedlight beam on the observer screen also moves with the same velocity as themoving screen. It is important at this point to emphasize that the electro-dynamics dictates that this would also lead to a consequent introduction ofan angular-frequency dispersion. However, such angular spectrum changewould mean a correction to a deviation of the energy transport direction ofthe transmitted light beam so that there is a glaring conflict with the predic-tion of the energy transport direction according to conventional aberrationof light theory. The absence of the group velocity along the moving directionis the prediction of conventional aberration of light theory and is obviouslyabsurd from the viewpoint of electrodynamics.

This incorrect statement is a straightforward consequence of the generallyaccepted way of looking at the aberration of light phenomena of most au-thors of the texbooks. Today one is told that the phenomenon of aberrationof light could be interpreted, using corpuscular model of light. Light corpus-cles are falling normally on the moving screen and, according to literature,generate oblique light beam as Fig. 10 shows. This wrong argument persiststo this day. If the optical system is spatially coherently illuminated, thena satisfactory treatment of the aberration of light should be based on theelectromagnetic wave theory.

Some experts believe that the applicability of corpuscular model in thetheory of light should be reinterpreted as the applicability of ray optics. Letus see what happens, according to the ray optics, in our case of interest.Light rays are falling normally on the moving screen and generate obliqueray beam as Fig. 10 shows (6). In this situation, we just treat light rayslike little particles and the effect is entirely familiar. We would like now todiscuss the region of applicability of ray optics. The situation relating to usethe ray optics in the theory of aberration of light is complicated. One could

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naively expect that the region of applicability of ray optics, following fromthe textbooks reasoning, should be identified with any spatially incoherentradiation. However, incorrect results are obtained by doing so. In particular,a spatially completely incoherent source (e.g. an incandescent lamp or astar) is actually a system of elementary (statistically independent) pointsources with different offsets. Radiation field generated by a completelyincoherent source can be seen as a linear superposition of fields of individualelementary point sources. An elementary source produces in front of ahole aperture effectively a plane wave. In other words, the transmissionprocess involves the introduction of an angular frequency dispersion of thetransmitted radiation. It should be remarked that any linear superpositionof radiation fields from elementary point sources conserves single pointsource characteristics like a deviation of the energy transport direction. Thisargument gives reason why ray optics is not applicable in the theory ofaberration of light from (spatially) completely incoherent sources.

We will illustrate the applicability of ray optics in the theory of aberrationof light for a particular class of spatially incoherent light beams. In fact, theray beam shown in Fig. 10 can be realized as follows. Such beam may beproduced by many commonly used lasers with a random spread of phases.One of the very useful properties of laser sources is their ability to producefields that are highly directional.The intensity of such fields is concentratedin a very narrow solid angle. A method can be proposed for generating raybeam from primary sources by the use of array of randomly phased lasers.Such planar source generates rays which are falling normally on the movingscreen (within the laser Rayleigh range) and generate oblique transmittedray beam, Fig. 10. Intuitively, a transversely moving screen containing a holeacts like a switcher for lasers. Surely, a luminous spot moving at velocityv can be realized simpler, so to speak, ”manually”. We can arrange thelaser-like sources along the x axis and switch them on one after another(independently) from left to right with a given time lag. Naturally, we canget a luminous spot moving at any velocity (even at v > c). From thisexample it is seen that in this process no information can be transmitted(along the x axis) since each source radiates independently.

4.13 Measure of Aberration

Aberration of light theory describes the deviation of the energy transportfor transmitted light beam. But how to measure this deviation? A movingwith group velovity v transmitted light beam changes its position alongthe x axis in time. The question arises whether it is possible to give anexperimental interpretation of the aberration effects. We illustrate the prob-lem of how to represent the deviation of the energy transport in the case

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of a time-dependent aberration of light problem with a simple example.Let us imaging the practical situation in which emitter radiated pulse. Weconsider a pulse of nearly monochromatic radiation having a duration andbandwidth equal to Tp and ∆ω, respectively. The present approach to lighttransmission problem uses the Fourier transform methods. The essenceof Fourier approach, in our case of interest, is that the incident radiationpulse expanded into the superposition of incoming beams constituted byplane monochromatic waves. Each of these beams corresponds to one of theFourier components into which the incoming light pulse can be resolved.It is useful to calculate the transmission to each of these elementary beams,and to superimpose the individual responses to find the total response. Oneof the important conclusions of the Fourier analysis is follows. When a lightpulse is falling normally on the moving screen there is a deviation of theenergy transport for the transmitted light pulse. Consider a light positiondetector in the rest position. The detector is placed at the distance l fromthe screen. It worth noting that we consider the aberration shift is relativelylarge compared to the hole size Dh. In other words, Dh vl/c. Also note that,in order to resolve aberration shift, we must require that cTp l. In smalldiffraction angle approximation o/Dh v/c we also have a second smallproblem parameter Dh/l v/c. Let us discuss interdependence of these twosmall parameters. A combination of these two parameters NF = D2

h/(ol) canbe refereed to as the Fresnel number. It is worth noting that, in our caseof interest, there is no restriction on the parameter NF. At first glance, onecan determine the aberration shift v/c to any desired degree of accuracy byincreasing distance l. However, the measuring device produces the uncer-tainty. In fact, the direction of light pulse propagation cannot be ascertainedmore accurately than up to the finite angle of the hole aperture o/Dh.

4.14 Moving Large Aperture Emitter

Let us now consider the case when a ”plane-wave” emitter in the lab inertialframe is accelerated from rest up to velocity v along the x axis. An emitterwith finite aperture is a kind of active medium which breaks up the radiatedbeam into a number of diffracted beams of plane waves. Each of thesebeams corresponds to one of the Fourier components into which an activemedium can be resolved. We already know from our discussion from verybeginning of this chapter that there is a deviation of the energy transportfor the coherent light radiated by the transversely moving emitter, which isnothing else but well-known result: there is the aberration of light from thetransversely moving emitter in the inertial frame of reference (Fig. 1).

The specific of our case of interest with the viewpoint of kinematics isthat here the screen is at rest in the lab frame and the emitter is moving

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Fig. 11. Aberration of light in an inertial frame of reference. The large aperture”plane-wave” emitter moving tangentially to its surface. As one of the consequencesof the Doppler effect, we find group velocity of the light waves radiated off a largeaperture moving emitter. The screen is at rest and we have actually the problem ofsteady-state transmission. The transmitted light beam is going vertically because ithas lost its horizontal (group) velocity component.

tangentially to its surface with constant speed with respect to the lab frame.For simplicity we shall assume that the transverse size of the moving ”plane-wave” emitter is very large relative to the transverse size of the hole in thescreen. Suppose that an observer, which is at rest with respect to the screenperforms the direction of the energy transport measurement.

The way of thinking that made the law about the behaviour of transmittedlight evident is called ”Abbe’s approach”. We call attention to the fact thatif the transverse size of the incoming light beam De is much large thanthe transverse size of the hole Dh, the group velocity of the transmittedbeam is dramatically reduced. This suppression is not surprising, if oneanalyzes the expression for the group velocity (vx)g = ∆ω/∆kx. In fact, theDoppler shift of a light wave radiated from the moving emitter is given by∆ω = ~Ke

⊥· ~v, where Ke

⊥∼ 1/De is the characteristic emitter wave number.

But the transverse component of the transmitted wavenumber vector ∆kx

in our case of interest can be written as ∆kx = (kx)i + Kh⊥

, where (kx)i ∼ 1/De

is the transverse component of the incoming wavenumber vector, Kh⊥

is thecharacteristic hole wavenumber. In the large aperture emitter case we have∆ω/∆kx ∼ (Dh/De)v v.

At close look at the physics of this subject shows that in the inertial lab frame,where the screen is at rest, we have actually the problem of steady-statetransmission. The Doppler effect is absent and the transmitted beam is going

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vertically because it has lost its horizontal (group velocity) component. Thatis the transmission appears as shown in Fig.11. We only wish to emphasizehere the following point. When the light passes through the small aperturehole we have a light beam whose fields have been perturbed by diffraction,and now not include information about emitter motion.

4.15 A Point Source in an Inertial Frame of Reference

Above we considered a single moving ”plane wave” emitter in an inertialframe of reference. In the description of the aberration of light in an inertialframe there are two choices of sources useful to consider:

(a) A ”plane wave”-like emitter

(b) A point-like ( or, more generally, spatially completely incoherent) source

Source field diffraction can be divided into categories - the Fresnel (near-zone) diffraction and Fraunhofer (far-zone) diffraction. In Fraunhofer diffrac-tion, the phase of the wave is assumed to vary linearly across the detectoraperture. This would occur if, for example, a plane wave were incident onthe aperture at an angle with respect to the optical axis. In the Fresnel diffrac-tion, we replace the assumption of a linear phase variation with quadraticphase variation. In the far zone, both types of sources produce in front ofpupil detection effectively a plane wave. In other words, there is always thephysical influence of the instrument on the measurement of the aberrationof light in the Fraunhofer zone.

One of the specific properties of a ”plane wave” emitter is ability to producefields that are highly directional. Within the near-zone at z z0 = D2

e/o anearly planar wave propagates, and the wavefront changes only marginally.At z z0 we have possibility to discuss about the aberration of light radiatedby a single ”plane wave” emitter. Indeed, at large detector size there is noinfluence of the detector on the measurement. In contrast, the peculiarityof point-like sources is that radiation emitted at one instant form a spherearound the source and the measuring instrument always influences themeasured radiation.

One of the important conclusions of the discussion presented above (seeFig. 11 ) is that the aberration of point source is absent in an inertial frame,Fig. 12. There are a number of remarkable effects which are a consequenceof the fact that the information about a point source motion is not includedinto the light beam transmitted through the hole. In fact, this is the key tothe binary star paradox discussed in the Chapter 6.

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Fig. 12. Aberration of light in an inertial frame of reference. A point source producesin front of a hole aperture effectively a plane wave. If the motion of the point sourceis parallel to the screen, transmitted beam is going vertically. The aberration of lightphenomenon is absent in this situation.

Fig. 13. A drawing of the modified source of the ”plane wave” emitter type. A pointsource of light is placed in the front focal plane of the lens.

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For the rest of our discussion of the aberration of light phenomena in aninertial frame, it will be more convenient if we consider a somewhat mod-ified source of the ”plane wave” emitter type. We have chosen the modeldescribed in the Section 4.2 because it is relatively simple. It is sufficientlycomplicated that it can stand as a prototype which can be generalized forthe description of the aberration of light phenomena. No one has ever doneall of ”thought experiments” we described in just this way, but we knowwhat would happen from the laws of special relativity which are, of course,based on other experiments.

Fig. 13 shows a drawing of the modified source of the ”plane wave” emittertype. The setup looks more complicated at first, but it is practically possibleto make. A point source of light is placed in the front focal plane of the lens.It is assumed that the order of magnitude of the dimensions of the ”point”source is about λ, where λ is the optical radiation wavelength. Note that,with reference to the system shown in Fig. 13, the field distribution in thefocal plane is uniform.

At this point, a reasonable question arises: why we are not discuss aboutthe aberration of light radiated by a single laser. At first glance, the outputlight beam shown in Fig. 13 may be produced by a laser. Indeed, one of thespecific properties of a laser is ability to produce coherent light beams thatare highly directional. We will not be able actually to get into the subjectuntil later. We will analyze the aberration of light transmitted from the laserresonator which is accelerated in an inertial frame in the Section 5.7 and 8.3.

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1. For a general discussion of the Fourier transform methods of spatialfiltering theory or Abbe diffraction theory we suggest reading the book [21].

2. It is generally believed that there is no aberration for light reflected frommirrors moving transversely. To quote e.g. Sommerfeld [22]: ”Thus, for amirror moving tangentially to its surface the law of reflection which holdsfor the stationary mirror is preserved.” Similar statements can also be foundin other textbooks. To quote e.g. Ugarov [23]: ” Hence, when the mirrormoves parallel to itself the frequency of incident light is equal to that ofreflected light, and the angle of incidence is equal to the angle of reflection.”

3. There is another interpretation that we can give to the scattering process.We note the very close correspondence between angular and frequencyresponse of moving diffraction gratings and the Raman scattering [24]. Sincethe theory of Bragg diffraction applies to light scattering by sound wavesin liquids and solids, it is not surprising that we can obtain our resultsfrom the quantum theory of light scattering by phonon. This scatteringprocess is also known as Brillouin scattering. The quantum theory leads toequation ~ks = ~ki ± ~K⊥ by the requirement that the momentum is conservedbetween interacting photon and phonon. The process of Brillouin scatteringis a special case of Raman scattering. The momentum balance for scatteringinvolving the emission or absorption of phonon leads to the Bragg condition.The requirement of conservation of energy leads to the equation ωs = ωi ±

ωphon with ωs, ωi, and ωphon being the radian frequencies, of the scatteredphoton, the incident photon, and the phonon. The frequency shift that occursfor light scattering from sound waves comes about as a result of the Dopplereffect. Quantum mechanically, it is a consequence of the conservation ofenergy between the participating particles. We come to the conclusion that,according to the hidden choice of absolute time coordinatization in non-relativistic quantum mechanics, the lab observer actually sees the radiationafter reflection as a result of a Galilean boost rather than a Lorentz boost.

4. There is a common misconception that the radiation wavefront orienta-tion has objective meaning. To quote Norton [25]: ”One might try to escapethe problem by supposing that the direction of propagation is not alwaysgiven by the normal to the wavefront. We might identify the direction ofpropagation with the direction of energy propagation, supposing the latterto transform differently from the wave normal under Galilean transforma-tion. Whatever may be the merits of such proposals, they are unavailableto some trying to implement a principle of relativity. If the direction ofpropagation of a plane wave is normal to the wavefronts in one inertialframe then that must be true in all inertial frame.” This incorrect statement

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is a straightforward consequence of the generally accepted way of look-ing at special relativity of most physicists. Accepting the postulate on theconstancy of the speed of light one also automatically assumes Lorentz coor-dinates. According to such limiting understanding of the theory of relativity,it is assumed that only Lorentz coordinatization must be used to map thecoordinates of events.

5. Note that spatiotemporal coupling is discussed in literature usually in re-lation with ultrashort laser pulse propagation through a grating monochro-mator. Ultrashort laser pulses are usually represented as a products of elec-tric field factors separately dependent on space and time. However, whenthe manipulation of ultrashort laser pulses requires the propagation througha grating monochromator, such assumption fails. In this situation one hasto consider in addition (to phase fronts), planes of constant intensity, thatis pulse fronts. A pulse-front tilt can be present in the beam due to prop-agation through an optical setup incorporating dispersive optical element.In the grating monochromator case, the different spectral components ofthe out-coming pulse travel in different directions. The electric field of apulse including angular dispersion can be expressed in the Fourier domain[kx, ω] as E(kx − pω,ω), while the inverse Fourier transform from the [kx, ω]to the space-time domain [x, t] can be expressed as E(x, t + px), which is theelectric field of pulse with a pulse-front tilt. The tilt angle θtilt is given bytanθtilt = cp. More specifically p = dkx/dω = kdθD/dω = λ/(cθDd), whereλ = 2πc/ω, θD is the diffracted angle, and d is the groove spacing. Thediffracted angle θD is a function of frequency, according to the well-knownplane grating equation. Assuming diffraction into the first order, one hasλ = (cosθi − cosθD)d, where θi is the incident angle. By differentiating thisequation one obtains dθD/dλ = 1/(θDd), where we assume for simplisitygrazing incidence geometry, θi 1 and θD 1. The physical meaning ofthe former equation is that different spectral components of the out-comingpulse travel in different directions. Therefore one concludes that the pulse-front tilt is invariably accompanied by angular dispersion. It follows thatany device like a grating monochromator, that produces an angular disper-sion, also introduces significant pulse-front tilt. In our case of interest wehave deal with light diffracted by a (transversely) moving set of gratings andthe Doppler effect is responsible for frequency dispersion dω/dkx = v. Thusthe spatitemporal coupling due to the light beam transmission through ahole in a moving screen and the usual pulse-front tilt distortion are quitedifferent.

6. To quote Brillouin [26]: ” Fig. 5 explain the situation assuming a simplifieddevice consisting of parallel plate moving with uniform velocity v in thehorizontal direction. Monochromatic light is falling normally on the plateand generates an oblique ray.” This oblique effect is demonstrated in ourFig. 10.

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5 Aberration of Light Phenomenon. Noninertial Frame of Reference


In this section we reexamine the issue of light transmission in non-inertialframes with particular reference to the aberration of light phenomena. Wederive the aberration for a pulse of light traveling in the accelerated systemsusing the Langevin metric in general relativity (1). It comes out naturallyif one writes the equation of the time transfer, from the inertial frame tothe accelerated frame, in a generally covariant context. We only wish toemphasize here the following point. From a mathematical standpoint, thereis no difference between calculations in the framework of the general theoryof relativity and of the special theory of relativity in the absence of space-time curvature.

The aberration of light problem is solved with discovery of the essentialasymmetry between the non-inertial and the inertial observers. Actually, inresent years it seems to be almost normally accepted in scientific communitythat the ”theory of relativity” is just a name, not to be taken literally. Onecan conclude that not all is relative in relativity, because this theory alsocontains some features which are absolute.

Has all our talk about asymmetry violated the relativity principle? At a firstglance it might seem so, since the relativity principle is often interpreted asimplying perfect symmetry among moving frames. It is generally incorrectlybelieved that key feature distinguishing special relativity from the classicaltheory is its distinctive dependence on the unqualified motional symmetryimplied by the relativity of motion: therefore seemingly not restricted torelative velocity but applicable to all form of relative motion, includingthe higher time derivatives of separation distance. In that case accelerationdoes not spoil the motional symmetry between the non-inertial referenceframe and inertial reference frame, and the asymmetry ”paradox” (actuallya disagreement with experimental facts) persists.

The principle of relativity denied the possibility for an observer partakingin a uniform motion relative to an inertial frame of discovering by anymeasurement such a motion, of course, that one does not look outside.The arguments concerning the relativity of motion in our case of interestcannot be applied, since the inertial and non-inertial reference systems arenot equitable.

A typical resolution of the asymmetry paradox identifies acceleration as theagency of asymmetry. It should be note that the duration of the accelerationperiod has a negligible effect on the anisotropy in the accelerated frame, so

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the acceleration need not be considered explicitly in working the problem.Nevertheless, the acceleration completely determines the problem.

However, there remains an intriguing puzzle to solve: how can the ob-servers tell which observer took the acceleration? The surprising fact is, thedetermination of which observer took the acceleration can be made only byobservation of the ”fixed stars”. The acceleration is in principle defined interms of motion relative to the fixed stars, and they must be consulted inorder to determine whether an acceleration occurred. Thus when we statethat the earth-base observer undergoes an acceleration, and the sun-basedobserver does not, there is a hidden assumption concerning the distribu-tion of mass in the universe. The implicit ”absolute” acceleration meansacceleration relative to the fixed stars.

5.2 Absolute Time Coordinatization in the Accelerated Systems

We will consider the problem of aberration of light in the accelerated systemson the basis of the theory of special relativity. Let us demonstrate that forexplanation of the optical effects in the rotating frame of reference one doesnot need neither modify the special theory of relativity, nor apply the generaltheory of relativity. It is only necessary to strictly follow the special theoryof relativity.

Suppose that an inertial system S is at rest with respect to the fixed stars (weuse this particular initial conditions for sketch simplification) and a systemSn (and an observer with his measuring instruments) in the lab frame S isaccelerated from the rest with respect to the fixed stars up to velocity v alongthe x-axis. In accelerated systems, only the theory maintaining an absolutesimultaneity is logically consistent with the natural behavior of clocks. Themethod of synchronization consists in keeping, without changes, the sameset of uniformly synchronized clocks used in the case when the system Sn

was at rest. It is well known that during the motion with acceleration (withrespect to the fixed stars) the procedure of Einstein’s clock synchronizationcannot be performed and the interval in the accelerated reference systemSn will, by the moment when the system Sn starts moving with constantvelocity, have the non-diagonal form.

Absolute simultaneity can be introduced in special relativity without affect-ing neither the logical structure, no the (convention-independent) predic-tions of the theory. We begin with the metric as the true measure of space-time intervals for an non-inertial observer Sn with coordinates (tn, xn, yn, zn).We transform coordinates (t, x, y, z) that would be coordinates of an inertialobserver S moving with velocity −v with respect to the observer Sn, using

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a Galilean transformation: we substitute xn = x − vt, while leaving timeunchanged tn = t into the Minkowski metric ds2 = c2dt2

− dx2− dy2

− dz2 toobtain (1)

ds2 = c2(1 − v2/c2)dt2n − 2vdxndtn − dx2

n − dy2n − dz2

n . (9)

Inspecting Eq. (9) we can find the components of the metric tensor gµν inthe coordinate system of Sn. We obtain g00 = 1 − v2/c2, g01 = −v/c, g11 = −1.Note that the metric in Eq. (9) is not diagonal, since, g01 , 0, and this impliesthat time is not orthogonal to space.

The velocity of light emitted by a source at rest in the coordinate system (t, x)for S is c. In that case the Minkowski metric Eq.(1) associated with inertialframe S predict a symmetry in the one-way speed of light. In the coordinatesystem (tn, xn), however, the speed of light emitted by the accelerated source(i.e. source which is at rest with respect to the accelerated frame Sn) cannotbe equal c anymore because (tn, xn) is related to (t, x) via a Galilean transfor-mation. This is readily verified if one recalls that the velocity of light in thereference system S is equal to c. If ds is the infinitesimal displacement alongthe world line of a ray of light, then ds2 = 0 and we obtain c2 = (dx/dt)2. Inthe accelerated reference system, since xn = x− vt and t = tn, this expressiontakes the form c2 = (dxn/dtn + v)2, which can be seen by setting ds2 = 0 inEq. (9). This means that in the accelerated reference system of coordinates(ctn, xn) the velocity of light parallel to the x-axis, is dxn/dtn = c − v in thepositive direction, and dxn/dtn = −c − v in the negative direction as statedabove. The reason why it is different from the electrodynamics constant c isdue to the fact that the clocks are synchronized following the absolute timeconvention, which is fixed because (tn, xn) is related to (t, x) via a Galileantransformation.

5.3 The Asymmetry between the Inertial and Accelerated Frames

We discovery of the essential asymmetry between the inertial and acceler-ated frames, namely, the Maxwell’s equations are not applicable from theviewpoint of an observer at rest with respect to accelerated frame Sn. In fact,the metric Eq.(9) associated with accelerated reference frame Sn predicts anasymmetry in the one-way speed of light in the relative velocity direction.Accelerations (with respect to the fixed stars) have an effect on the propa-gation of light. On accelerated system Sn, the velocity of light emitted bya source at rest must be added to (or subtracted from) the speed due toacceleration and the velocity of light is different in opposite directions. Incontrast, the Maxwell’s equations continue to hold from the viewpoint ofan observer at rest with respect to inertial frame S. On inertial system S, the

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velocity of light emitted by a source at rest is c.

Suppose that an observer in the accelerated frame Sn performs an aberrationmeasurement. How shall we describe the aberration of light from the ”planewave” emitter which is at rest in the inertial frame S? In order to predictthe result of the aberration measurement the accelerated observer shoulduse the non-diagonal metric Eq.(9) (i.e. anisotropic electrodynamics fieldequations). It is important to note that the source of asymmetry is not therelative acceleration of two frames; rather, it is the relative acceleration ofeach frame separately with respect to the ”third frame” - the inertial frame(all frames in uniform motion with respect to the fixed stars).

According to the asymmetry between the inertial and accelerated frames,there is a remarkable prediction on the theory of the aberration of light.Namely, if the opaque screen with hole was at rest relative to the fixedstars and the screen started from rest to motion relative to the fixed stars,then the apparent angular position of the ”plane-wave” emitter seen in theaccelerated frame through the aperture would jump by angle v/c. That isthe transmission through the hole in the opaque screen in the frame Sn

appears as shown in Fig 14. It could be said that the crossed term in metricEq.(9) generates anisotropy in the accelerated frame that is responsible forthe change of radiation direction (aberration).

Above (see Chapter 4) we considered the emitter-screen problem in theframe of reference S which is at rest (or an uniform motion) with respectto the fixed stars. It is found that there is no aberration proceeding fromthe emitter, independently of their motion in the case of the small aperturehole. It is important to emphasize that the aberration of the light beamtransmitted through the small aperture hole also does not dependent on theemitter motion in the accelerated system Sn, Fig. 15. The point is that thecrossed term in metric Eq.(9), which generates aberration in this particularcase, depends only on the velocity change of the accelerated frame relativeto the inertial frame.

One finds many books which say that we cannot distinguish by any ex-periment which observer remains ”at rest” during the acceleration, becauseaccording to the principle of relativity only relative motion has any physicalmeaning. It follows that any observed asymmetry would lead to a contra-diction with the principle of relativity. This argument is wrong. We discussthe apparent conflict between aberration of light and the principle of rela-tivity in the Chapter 10. It is demonstrated that there is no conflict betweenthe fundamental structure of special relativity on the one hand, and theaberration of light phenomena. Principle of special relativity is irrelevancyof velocity with regard to physical laws, not with regard to anything.

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Fig. 14. Aberration of light in an accelerated frame of reference Sn. Radiation wave-front orientation and the anisotropy of speed of light presented in the absolute timecoordinatization (tn = t) for the screen. The crossed term in metric Eq.(9) generatesanisotropy in the accelerated frame that is responsible for the change of radiationdirection (aberration).

Fig. 15. Aberration of light in an accelerated frame of reference Sn. Radiation wave-front orientation and the anisotropy of speed of light presented in the absolutetime coordinatization (tn = t) for the screen. The aberration of the transmittedlight is considered independent of the source speed and to have just a local originexclusively based on the observer acceleration relative to the fixed stars.

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Fig. 16. Aberration of light in the accelerated frame of reference Sn. The plane wave-front rotates in the accelerated frame after the accelerated clock re-synchronization.The Maxwell’s equations can now be used to describe the transmitted (throughthe aperture) light beam. According to the Maxwell’s electrodynamics, the trans-mitted light beam is propagated at the angle v/c, yelding the phenomenon of theaberration of light.

5.4 A Resynchronization of the Accelerated Clocks

It should be noted, however, that there is another satisfactory way of ex-plaining the effect of aberration of light in the accelerated frame Sn. Theexplanation consists in using a clock re-synchronization procedure. Wellknown that in their original form Maxwell’s equations are valid in the iner-tial frames. But Maxwell’s equations can be written down only if the Lorentzcoordinates has already been specified.

When the system Sn starts moving with constant velocity the standard pro-cedure of Einstein’s clock synchronization can be performed. The Einsteinsynchronization is defined in terms of light signals emitted by a source atrest assuming that light propagates with the same velocity c in all direc-tion. Using such synchronization procedure we actually select a Lorentzcoordinate system for the screen. In this synchronization, we describe thetransmission through the aperture using usual Maxwell’s equations. Theinterval in the accelerated reference system Sn will have the diagonal formEq.(1) for the transmitted light beam.

The time t′n under the Einstein’s synchronization in the Sn frame is readily ob-tained by introducing the offset factor xnv/c2 and substituting t′n = tn−xnv/c2

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in the first order approximation. This time shift has the effect of rotation theplane of simultaneity (that is emitter radiation wavefront) on the angle v/cin the first order approximation. As a consequence of this, the plane wave-front rotates in the accelerated frame after re-synchronization. The new timecoordinate in the accelerated frame is interpreted by saying that Maxwell’sequations are applicable to the light transmission (through the aperture)description. Then, the transmitted light beam is propagated at the anglev/c, yielding the phenomenon of the aberration of light: the two approachesgive the same result. The choice between these two different clock syn-chronizations is a matter of pragmatics. By changing the (four-dimensional)coordinate system, one cannot obtain a physics in which new physical phe-nomena appear. But we can obtain a more consistent description of thesephenomena.

In relation with this discussion, we can now describe the origin of theasymmetry between the inertial frame of reference S and accelerated frameSn. In Fig. 11 we consider the case when the ”plane-wave” emitter in theinertial frame is accelerated from rest up to velocity v along the x axis. InFig. 16 an observer in the accelerated frame Sn performs a measurement ofthe aberration of light from the ”plane wave” emitter which is at rest in theinertial frame S. When the system Sn starts moving with constant velocity itwill be inertial frame of reference. At first glance, after the re-synchronizationof the accelerated clocks, we have the symmetry between the inertial frameS and inertial frame Sn. Where does the asymmetry come from? The mostimportant difference, however, is that the type of clock synchronization,which provides the wavefront orientation of the source in both situations,is based on the use of the synchronization in the inertial frame S which isat rest with respect to the fixed stars. The electrodynamics equation needsto be integrated with initial condition for the radiation wavefront. From thespecial theory of relativity (Galilean boost xn = x − vt, tn = t) we know thatacceleration has no effect on the wavefront orientation and the wavefrontof the emitted light beam is perpendicular to the vertical direction zn afterthe acceleration, Fig. 14.

5.5 A Single Moving Emitter in a Noninertial Frame of Reference

Let us analyze the aberration of light radiated by a single emitter movingin the accelerated system. Before we go on to analyze an observations of anon-inertial observer, we should make one more remark about observationsof an inertial observer. In the Chapter 4 we already emphasized that in thedescription of the aberration of light in an inertial frame of reference thereare two choices of (four-dimensional) coordinatizations useful to consider:

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(a) Non-standard (absolute time) coordinatization for a moving source

(b) Standard Lorentz coordinatization for a moving source

We should underline that we claim the ”single frame” approach to relativis-tic electrodynamics is actually based on the use of a not standard (absolutetime) clock synchronization assumption within the theory of relativity.

When the light source in the inertial frame is accelerated from rest up tovelocity v along the x axis, the simplest ( absolute time) method of synchro-nization consists in keeping, without changes, the same set of uniformlysynchronized clocks used in the case when the light source was at rest, i.e.we still enforce the clock transport synchronization ( or Einstein synchro-nization which is defined in terms of light signals emitted by the light sourceat rest). This choice is usually the most convenient one from the viewpointof connection to laboratory reality.

Let us, then, discuss (according to the Chapter 3) how to assign Lorentzcoordinates in the case when a light source in an inertial frame is acceleratedfrom rest up to velocity v along the x-axis. In order to assign a Lorentzcoordinate system in the inertial frame after the Galilean boost x′ = x−vt, t =t, one needs to perform the distant clock resynchronization t′ = t−xv/c2. Thisnew space-time coordinates in the lab frame are interpreted, mathematically,by saying that the metric is now diagonal and the speed of light from themoving source is isotropic and equal to c.

Now let us return to observations of a non-inertial observer. In the descrip-tion of the aberration of light radiated by a single moving emitter in anaccelerated frame of reference there are also two choices of coordinatiza-tions:

(a) Non-standard (absolute time) coordinatization for an accelerated source

(b) Standard Lorentz coordinatization for an accelerated source

(a) When the system Sn in the stationary inertial lab frame S is acceleratedfrom the rest up to velocity v along the x-axis, the simplest ( absolute time)method of synchronization consists in keeping, without changes, the sameset of uniformly synchronized clocks used in the case when the system Sn

was at rest. Acceleration with respect to the fixed stars have an effect on thepropagation of light. The velocity of light from the source which is at restin the accelerated frame Sn will, by the moment when the system Sn startsmoving with constant velocity, have anisotropy along the xn axis (see Eq.(9)).

(b) When the system Sn starts moving with constant velocity the standard

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Fig. 17. Aberration of light in an accelerated frame of reference Sn. Emitter is atrest with respect to the fixed stars. Wave fronts orientation and the group velocitypresented in the Einstein synchronization. The Einstein synchronization is definedin terms of light signals emitted by a source at rest (in the accelerated frame) assum-ing that light propagates with the same velocity c in all direction. The aberrationincrement θa is connected with the physical parameters by the relation: θa = 2v/c,where v is the velocity of the fixed stars in the accelerated frame Sn.

procedure of Einstein’s clock synchronization can be performed. The Ein-stein synchronization is defined in terms of light signals emitted by a sourceat rest (in the accelerated frame) assuming that light propagates with thesame velocity c in all direction. Using such (Einstein time) synchroniza-tion procedure we actually describe the light from the source at rest (in theaccelerated frame) using usual Maxwell’s equations.

Above we considered the single emitter problem in the frame of referenceS which is at rest (or uniform motion) with respect to the fixed stars. It isfound that there is a deviation of the energy transport for the light radiatedby the transversely moving emitter, which is nothing else but well-known(from textbooks) result: there is the aberration of light from the transverselymoving emitter in the inertial frame of reference (Fig. 1).

We now consider the case when the emitter is at rest in the lab inertial frameS (i.e. at rest with respect to the fixed stars) and the observer, which is at restwith respect to the accelerated frame of reference Sn performs the directionof the energy transport measurement. It is important to emphasize that theaberration of light radiated by the single emitter is also dependent of theemitter motion in the accelerated system Sn. According to the asymmetrybetween the inertial and accelerated frames, there is a remarkable prediction

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Fig. 18. Aberration of light in an accelerated frame of reference Sn. Emitter is atrest with respect to the fixed stars. The aberration of light might be calculatedfrom classical theory, in which the light beam may be treated, say, as a corpuscular(raindrops) beam. On the basis of classical theory one gets a light aberration anglethat is only one-half as big as that predicted by special relativity.

on the theory of the aberration of light. Namely, if the emitter is at rest relativeto the fixed stars and the observer started from rest to motion relative tothe fixed stars, then the apparent angular position of the ”plane-wave”emitter seen in the accelerated frame would jump by angle 2v/c. That is theaberration of light in the frame Sn appears as shown in Fig 17.

Let us now consider the most general case when emitter in the inertialframe S is accelerated from rest up to velocity u along the x axis and thesystem Sn in the inertial frame S accelerated from the rest up to velocity valong the same x axis. Suppose that an observer in the accelerated frame Sn

performs an aberration measurement. At close look at the physics of thissubject shows that the aberration increment is connected with the problemparameters by the relation θa = 2v/c−u/c. The point is that the crossed termin metric Eq.(9), which generates the anisotropy in the accelerated frame,depends only on the velocity v of the accelerated frame relative to the fixedstars.

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Fig. 19. The deflection of starlight by the field of the sun. The light deflection mightbe calculated from classical theory, in which the light beam may be treated, say, asa comet approaches with the velocity of light. On the basis of classical theory onegets a deflection of light that is only one-half as big as that predicted by generalrelativity.

5.6 A Useful Analogy

We have presented theoretical evidence of an aberration increment 2v/cconcerning the single emitter problem in a non-inertial frame of reference.First, we notice that there is an extra factor 2. On the basis of classical theoryone gets aberration increment that is only one-half as big as that predictedby special relativity, Fig. 18. According to classical approach, there is noprinciple difference between the aberration of light and the aberration ofraindrops.

There is a methodological analogy between the aberration of light effectand the deflection of light in general relativity. A ray of light which, comingfrom fixed star, passes close by the sun will thus be attracted to it and willdescribe a somewhat concave orbit with respect to the sun. This deflectionmight be calculated from Newton’s theory, in which the ray of light maybe treated, say, as a comet which approaches with the velocity of light.We then get a formula similar to that of Einstein, but giving only halfthe value of the deflection, Fig. 19. Errors inherent in the classical methodapplied to the computation of the deflection of light are due to the use of

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Fig. 20. Aberration of light in an accelerated frame of reference. Point source is atrest in the accelerated frame. Wavefront orientation presented in the absolute timecoordinatization for the screen.

the velocity of light simply as a classical velocity. According to classicalapproach, there is no principal difference between the deflection of lightand the deflection of comet. A methodological analogy with the calculationof the aberration increment from classical (raindrops) theory emerges. Doesthis analogy have any physical meaning? To see whether it does, we shouldcompare the motion of relativistic particles in the gravitation field with theaberration of particle effect. We shall do this in the Section 7.5.

5.7 A Point Source in a Nonertial Frame of Reference

Now let us return to observations of a non-inertial observer. The peculiarityof point-like sources is that radiation emitted at one instant form a spherearound the source and the measuring instrument always influences the mea-sured radiation, Fig. 20-Fig. 21 . The aberration of point source is consideredindependent of the source speed and to have just a local origin exclusivelybased on the observer (with his measuring instruments) speed relative tothe fixed stars. The crossed term in metric Eq.(9) generates anisotropy in theaccelerated frame that is responsible for the change of transmitted radiationdirection. The point source follows the same pattern as fixed stars underthe same aberration angle i.e. its apparent position changes with an angulardisplacement common to fixed stars.

In the framework of the conventional theory of the aberration of light, there

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Fig. 21. Aberration of light in an accelerated frame of reference. Point source is atrest with respect to the fixed stars.

is an outstanding puzzle concerning the stellar aberration. There are doublestar systems the components of which change their velocity on a time scaleranging from days to years. The components of such binary systems atsome times can have velocities relative to the earth very different fromone another; nevertheless it is well known that these components exhibitsalways the same aberration angle. Rotating binary systems follow the samepattern as all fixed stars and are observed within a period of a year underthe same universal aberration angle, i.e. their apparent position changeswith an annual period common to all distant stars. This argument suggeststhat results of the astronomical observations confirm our prediction foraberration of light from a point source in an non-inertial frame. In the nextchapter we discuss this experimental test in more detail.

In regard to light aberration one should differentiate between that from thepoint source and that from the laser source. Let us analyze the aberration oflight transmitted from the laser resonator which is at rest in the acceleratedframe. A close examination of of all experiments inside the acceleratedframe shows that all optical phenomena in an optical resonator appearedto be independent of the acceleration relative to the fixed stars. It shouldbe note that there is no influence of the difference between Langevin andMinkowski metrics on the parameter of a laser source. It is not hard tounderstand this result. It is clear that the crossed term in the Langevin metric,which generates aberration, cancels during the (round-trip) evolution of theradiation in the laser resonator. We will analyze the aberration of lighttransmitted from the laser resonator in great detail in the Section 8.3

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Fig. 22. The aberration of light from stationary and moving sources. According tothe asymmetry between the inertial and accelerated frames, there is a remarkableprediction on the theory of the aberration of light. Namely, if the emitter is at restrelative to the fixed stars and the observer with measuring devices started from restto uniform motion relative to the fixed stars, then the apparent angular position ofthe emitter seen in the accelerated frame would jump by angle θa = −2v/c. Thissituation is not symmetrical with respect to the change of the reference frames. Ifthe observer is at rest relative to the fixed stars and the emitter started from restto uniform motion relative to the fixed stars, then the apparent angular position ofthe emitter seen in the inertial frame would jump by angle θa = v/c.

5.8 Concept of Reciprocity in the Aberration of Light Theory

In the present section we shall continue our discussion of the aberrationof light radiated by a single moving emitter. Imaging that there are twoidentical emitters. Let us consider the case when the first emitter is at restin an inertial frame and the second emitter is accelerated from rest up tovelocity v along the x axis. Suppose that an observer, which is at rest withrespect to the inertial frame of reference performs the direction of the energytransport measurement. Now we must be careful about initial phasing ofthese emitters. As example, we consider the case in which initially thevelocity component of the light beam along the x-axis is equal to zero.Then how does the light beam from the moving emitter looks? The inertialobserver would find that angular displacement is equal to θa = v/c. That isthe radiation appears as shown in Fig. 22. This is example what called thephenomenon of aberration of light and it is well known.

When the accelerated system starts moving with constant velocity the stan-

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Fig. 23. Reciprocity in a theory of the aberration of light. One must use the samesource phasing to demonstrate the reciprocity. After the re-phasing procedure theinertial observer would find that angular displacement is θa = 2v/c.

dard procedure of Einstein’s clock synchronization can be performed. In thissynchronization, the accelerated observer describes the light beam from anemitter at rest using usual Maxwell’s equations. One may ask: ”Where isthe information about the observer acceleration recorded in the case of theLorentz clock synchronization?” When the accelerated observer performedthe standard procedure of clock synchronization, the time shift has the ef-fect of rotation the plane of simultaneity, that is source radiation wavefront,on the angle v/c. It is easy to show that information about the observeracceleration is recorded in the phase front orientation (with respect to thecoordinate axes of the accelerated frame) of the light beam radiated fromthe source at rest in the accelerated frame.

Let us describe what happens when accelerated observer performed there-directing of the accelerated emitter. We consider also the case in whichfinally the velocity component of the light beam along the x-axis is equal tozero. After this re-directing procedure the inertial observer would find thatangular displacement is θa = 2v/c. Suppose that the accelerated observeralso performs an aberration measurement. Fig. 23 shows that the aberrationincrement is connected with the problem parameters by the relation θa =−2v/c.

The situation can be described quite naturally in the following way. Wedemonstrated that symmetry is a correct concept in the aberration of ”planewave” emitters. It should be note, however, that our formulas can be con-

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sidered only as a first order approximation in v/c. We have never in allour discussions of the past chapters, said anything about the high ordereffects. Of course we can compute a better approximation with the help ofLangevin metric. The most important thing is that new effects have to dowith the asymmetry between the accelerated frame and the inertial frame.We discuss this asymmetry in the Chapter 9.

To conclude this discussion about reciprocity, let us consider the situationwhen the accelerated source automatically re-phasing. Let us analyze theaberration of light transmitted from the laser resonator which is in therest in the accelerated frame. The accelerated observer will find that thedeviation of the energy transport is absent. As we already discussed, alloptical phenomena in a laser resonator appeared to be independent of theacceleration with respect to the fixed stars. We would like to emphasize avery important difference between point-like and laser sources. In fact, thepoint-like sources are not re-directed. What we want to make clear is thatreciprocity can be an incorrect concept in the aberration of light phenomenaeven if we neglect terms of order higher than the first in v/c.

It is generally accepted that aberration of light phenomena depends onthe relative motion of the source and observer. However, the statementconcerning reciprocity is refuted by the experimental facts showing withoutambiguity that one always can make a distinction if either point source orthe observer is accelerating relative to the fixed stars.

5.9 Explanation of the Aberration on the Basis of Electrodynamics

It is well known that the electrodynamics theory meets all requirements ofthe theory of relativity and therefore must accurately describe the propertiesof such a relativistic object as light. For example, we are able to demonstratethat velocity of light in the accelerated frame can be derived independentlyof a technique involving metric Eq. (9). In the inertial frame, fields areexpressed as a function of the independent variables x, y, z, and t. Accordingto the principle of relativity, the Maxwell’s equations always valid in theLorentz reference frame. The electric field ~E of an electromagnetic waveradiated by the emitter at rest in the inertial frame satisfies the equation2~E = ∇2~E−∂2~E/∂(ct)2 = 0. However, the variables x, y, z, t can be expressedin terms of the variables xn, yn, zn, tn by means of a Galilean transformation,so that fields can be written in terms of xn, yn, zn, tn. Hence the wave equationtransforms into

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2~En =

(1 −

v2

c2

)∂2~En

∂x2n

+ 2(v

c

)∂2~En

∂tn∂xn+∂2~En

∂y2n

+∂2~En

∂z2n−

1c2

∂2~En

∂t2n

= 0 , (10)

where coordinates and time are transformed according to a Galilean trans-formation. The solution of this equation F[xn − (c − v)tn] + G[xn + (−c − v)tn]is the sum of two arbitrary functions, one of argument x − (c − v)t and theother of argument x + (−c − v)t. The first term represents a wave travelingforward in the positive xn direction, and the second term a wave travelingbackwards in the negative xn direction. Acceleration has an effect on thepropagation of light. In fact, the coordinate velocity of light parallel to thex-axis is given by dx/dt = c − v in the positive direction, and dx/dt = −c − vin the negative direction.

We describe the aberration of light radiated by a single ”plane wave” emitterbased on the observations made by an observer in the same acceleratedframe as the emitter. It is assumed that the detector for the direction ofthe radiation is an energy propagation detector and the size of the detectoraperture is sufficiently large compared with the radiation beam size. In otherwords, it is assumed that there is no physical influence of the detector (e.g.aperture) on the measurement.

The present approach to accelerated emitter uses the Fourier transformmethods. The emitter with finite aperture is a kind of active medium whichbreaks up the radiated beam into a number of diffracted beams of planewaves. Each of these beams corresponds to one of the Fourier compo-nents into which an active medium can be resolved. Let us assume thatthe dipole density of the elementary source varies according to the lawρdip = g(K⊥) cos (K⊥x).

Acceleration has an effect on the field equations. In fact, the wave equationtransforms into Eq.(10). Consider as a possible solution a radiated planewave exp(i~k ·~r− iωt). With a plane wave exp(i~k ·~r− iωt) with the wavenum-ber vector ~k and the frequency ω equation Eq.(10) becomes: (1 − v2/c2)k2

x −

2vkxω/c+k2z−ω

2/c2 = 0. The wavenumber vector of the radiated plane waveis fixed by initial conditions. Let us assume that the wavefront of the emittedlight beam is perpendicular to the vertical direction before the acceleration.

Then we have kz =√ω2

i /c2 − k2

x, kx = K⊥, where K⊥ is the wavenumberof sinusoidally space-modulated dipole density. From initial conditions wewill find it necessary to use~k as independent variable and we will considerω as a function of kx: ω = ωi + ∆ω(kx), where ωi is the frequency of theemitter radiation before the acceleration. From this dispersion equation, wefind the requirement that the wavenumber K⊥ and the frequency change∆ω are related by ∆ω = −K⊥v. Because of the assumption cK⊥/ωi v/c (i.e.

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Fig. 24. Aberration of light in an accelerated frame of reference Sn. The wavefrontof the emitted light beam is perpendicular to the vertical direction before theacceleration. When one has an accelerated emitter there is a deviation of the energytransport for radiated light. In order to predict result of the aberration measurement,the accelerated observer should use the field equation Eq.(10) (metric Eq. (9) ). Thecrossed term in the field equation Eq.(10) ( crossed term in metric Eq. (9)) generatesanisotropy in the accelerated frame that is responsible for the change of radiationdirection (aberration)

o/De v/c, where De is the transverse size of the emitter) there is no secondorder terms in cK⊥/ωi when we find ∆ω.

In our example, the plane waves with different wavenumber vectors prop-agate out from the accelerated emitter with the different frequencies. Thenequation ∆ω/∆kx = −v holds for each scattered waves independently on thesign and the magnitude of the radiated angle. The last equations state thatradiated light beam with finite transverse size moves along the x directionwith group velocity dω/dkx = −v, Fig. 24.

5.10 Explanation of the Aberration on the Basis of the Ether Theory

5.10.1 Introductory Remarks

Now we wish to continue in our analysis a little further. We will look fora different way of calculating the aberration effect. Another solution to theaccelerated emitter problem will be discussed going back to ether theory.We are going to demonstrate that the ether-related solution is simple and

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straightforward.

While the results presented above are fundamental, there is nothing un-expected about them, except perhaps that they can be derived using pre-relativistic theory only, and thus that they could have been proven long ago.Indeed, the effect of light aberration in an accelerated frame of referencecan easily be explained on the basis of the pre-relativistic ether theory if wemay assume that terms of the second-order are below the accuracy of theexperiments.

We note that the denial of the ether by the special relativity cannot betaken seriously anymore. We remark that, as shown by Lorentz, there isan agreement between the pre-relativistic ether theory and the theory ofrelativity as regards all optical effects of the first order in v/c. How shall wechange the ether theory so that it will be completely equivalent to the theoryof relativity? As it turn out, the only requirement is that the length of anyobject moving in ether must be contracted. When this change is made, theether theory and the theory of special relativity will harmonize (2).

5.10.2 Noninertial Frame of Reference

We accept the ether theory in the original form: there is ether, which rulesthe speed of light. Because the ether is immovable, it is causes anisotropyin every frames moving relative to the ether. Because the non-inertial frameis accelerated with respect to the fixed stars; consequently we must feelan ”ether wind” when measuring light propagation. Firstly we discuss theeffect of the ether wind on the light speed.

According to the hypothesis of an ether at rest in the unaccelerated labora-tory, the velocity of light judged from an accelerated reference system wouldbe c + u for the beam propagating in the same direction as the acceleratedsystem and−c+u for the beam propagating in the opposite direction, whereu = −v is the ether velocity in the accelerated reference system.

The aberration effect is an effect of the ether-wind on the speed of lightthat must be corrected for in transformations of accelerating coordinatesystems (Fig. 25). We remark again that there is an agreement between theether theory and the theory of relativity. In fact, the metric Eq.(9) associatedwith accelerated reference frame Sn predicts an asymmetry in the one-wayspeed of light in the relative velocity direction. Accelerations (with respectto the fixed stars) have an effect on the propagation of light in the theory ofrelativity. On accelerated system Sn, the velocity of light must be added to(or subtracted from) the speed due to acceleration and the velocity of lightis different in opposite directions (3).

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Fig. 25. Aberration of light in an accelerated frame of reference Sn. According to theether theory, the ether wind generates anisotropy in the accelerated ( relative to theether) frame that is responsible for change of radiation direction.

The fact that an ether theory is consistent with accelerated motion providesstrong evidence that an ether exists, but does not inevitably imply that uni-form motion relative to the ether is measurable. It should be note that allwell known methods to test the special relativity are round-trip measure-ments. It is important to remark that a close examination in the frameworkof the ether theory of all round-trip experiments inside the uniformly mov-ing (relative to the fixed stars) frame, however, shows that in reality allphenomena appeared to be independent of the uniform motion relative tothe fixed stars. However, the source-observer asymmetry associated withthe aberration of light phenomena at first glance contradicts the principleof relativity. In the Chapter 10, we explain that there is no conflict betweenthe fundamental structure of special theory of relativity and the aberrationof light phenomena. According to special principle of relativity all inertialframes are equivalent with regard to physical laws, not with regards, toanything.

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1. The aberration of light problem in the accelerated systems is solved withdiscovery of the essential asymmetry between the non-inertial and the in-ertial observers. This asymmetry is of the same nature as that of the well-known Sagnac effect [27–29]. For instance Langevin’s 1921 explanation ofthe Sagnac effect rested upon the assertion that ”any change of velocity, orany acceleration has an absolute meaning.” [30].

2. As a way out of disagreement between the ether theory and the principleof relativity in the second and higher order in v/c Fitzgerald (1889) andindependently Lorentz (1892) proposed that the length of bodies movingin the ether is reduced in the direction of their motion; the amount of thislength reduction was assumed to be such as to explain the absence of anyeffect due to the motion of the earth in Michelson’s experiment. The abolitionof the ether concept is often credited to Einstein. On the contrary, Einsteinhas stated the absolute necessity of the ether. To quote Einstein [31]: ”Thenegation of ether is not necessarily required by the principle of relativity.We can admit the existence of ether but we have to give up attributing it toa particular motion. The hypothesis of the ether as such does not contradictthe theory of special relativity.”

3. The presented explanation of the aberration of light effect in a rotatingframe of reference is based on the concept of the immovable ether. There isa certain degree of analogy between the aberration of light and the Sagnaceffect. The latter was first proposed and knowingly measured by GeorgeSagnac, and was then interpreted as the proof of existence of the immovableether and as a measurement of rotation relative to it [27]. It has been shownthat the Sagnac effect can be understood to result from a rotational ethermotion, reveling a close relationship between the light transmission in arotating frame of reference and the Sagnac effect. A close look at the physicsof these two subjects shows things which are common to these phenomena:In both situations these are experiments in the first order in v/c and caneasily be described in classical (pre-relativistic physics) terms. It is foundthat the non-standard (absolute) time coordinate in special relativity bettersuited for the description of both effects in the rotating system. If we lookmore closely at the physics, we would see aspects that are not commonto these phenomena. The difference is given by the global nature of theSagnac effect. When comparing the local effects with the global ones, wefound that the time coordinate defined by the standard (Einstein’s) isotropicsynchronization convention can not be used as global coordinate becauseof a time-lag associated with the round travel.

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6 Stellar Aberration

6.1 A New Approach to the Stellar Aberration

6.1.1 The Corpuscular Model of Light and the Stellar Aberration

It is generally believed that the phenomenon of aberration of light could beinterpreted, using the corpuscular model of light, as being analogous of theobservation of the oblique fall of raindrops by a moving observer. This is aclassical kinematics method to the computation of the stellar aberration usedin astronomy for about three hundred years (1). What had to be added in the20th century, was that the dynamical laws of Newton for light were foundto be all wrong, and electromagnetic wave theory had to be introduced tocorrect them.

As well-known, according to textbooks the physical basis of the stellar aber-ration is the fact that the velocity of light is finite and changes its directionwhen seen from another reference frame. It is a consequence of the for-mula for addition of velocities applied to a light beam when the observer ischanging its reference frame.

It is sufficient to describe the effect of stellar aberration by working only upto the first order v/c. For an observer on the earth, it is, with respect to thesolar referential frame, of about 30 km/s, corresponding to the earth motionaround the sun. Clearly, in the theory of stellar aberration, we consider thesmall expansion parameter v/c ' 10−4 neglecting terms of order of v2/c2.According to the conventional approach, the study of stellar aberration isintimately connected with the old (Newtonian) kinematics: the Galileanvectorial law of addition of velocities is actually used.

6.1.2 Wave Theory of the Stellar Aberration. Effect of the Measuring Instruments

It is generally accepted that it is much more difficult to describe the aber-ration phenomenon based on wave optics then based on the corpusculartheory of light. It can be easily demonstrated that the light produced by adistant star is approximately coherent over a circular area whose diameter,in all practical cases, is much larger than the telescope diameter. Hence, wesample such a tiny portion of the coherent area of the starlight with ourtelescopes that the waveforms are effectively (flat) plane waves (2). Mostauthors treat light propagating through the telescope barrel as raindrops,and not as a plane wave. Questioning the validity of standard reasoning weargue that a satisfactory treatment of stellar aberration should be based onthe coherent wave optics.

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The conventional method used to explain the aberration phenomenon inthe framework of wave theory is based on the belief that there is no aberra-tion for light reflected from mirrors moving transversely. According to theBabinet’s principle, this prediction of standard theory should be correct alsofor light transmitted through a hole punched in a moving opaque screen or,consequently, through the moving open end of a telescope barrel.

In Chaper 4, we presented a critical reexamination of the textbook statementthat wavefronts and raindrops are to have the same aberration. We usedthe theory of relativity to show that when one has a transversely movingmirror and a plane wave of light is falling normally on the mirror, thereis a deviation of the energy transport for light reflected from the mirror.This effect is a consequence of the fact that the Doppler effect is responsiblefor angular frequency dispersion of light waves reflected from the movingmirror with finite aperture. As a result, the velocity of the energy transportis not equal to the phase velocity. This remarkable prediction of our theory iscorrect also for light transmitted through the moving open end of a telescopebarrel.

It is generally believed that the theory of relativity appears to conform to thephenomenon of stellar aberration discovered by Bradley by claiming it is aconsequence of the motion of observers relative to light sources, but binarystars do not exhibit such shifts when traversing our direction of view (3).

Spectroscopic binaries have velocities exceeding the earth’s velocity roundthe sun. They revolve around their common center of gravity within days,a period during which the motion of the earth is practically constant. Thecomponents of the binary system should be easily separable, when theirchanging velocities are comparable to the earth’s velocity round the sun.This is, however, not observed (Fig.26-Fig.27) (4).

In principle the binary star paradox is resolved by noting that when lightpasses through the end of a telescope barrel we have a light beam whosefields have been perturbed by diffraction, and, therefore, do not includeinformation about star motion relative to the fixed stars. If the telescopewere at rest relative to the fixed stars and the star started to move from rest,then the apparent position of the star seen in the telescope would neverjump by any angle.

However, other difficulties arise in the explanation of the earth-based ob-servations, i.e the change in an apparent positions of the fixed stars, whichhappens when the earth-based telescope changes of its motion relative tothe fixed stars. It should be stressed that it is the telescope and not the starthat must change its velocity (relative to the fixed stars) to cause aberration.Though stellar aberration behaves asymmetrically, it does not contradict

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Fig. 26. Aberration shift as inferred from astronomical observations. The plane wavefronts of starlight entering the telescope are imaged by the lens to a diffraction spotwhich lies in the focal plane. Two cases are selected with different velocities of aearth-based telescope and a star. In the first case (left) the star is at rest with respectto the fixed stars. In the second case (right) the star moves with the same velocityas the earth.

Fig. 27. Aberration shift as predicted by the conventional theory of the stellaraberration. Two cases are selected with different velocities of a earth-based telescopeand a star. In the first case (left) the star is at rest with respect to the fixed stars.In the second case (right) the star moves with the same velocity as the earth. Theconventional theory predicts no aberration of the image in this case.

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special relativity, because the heliocentric (i.e. sun-based) reference systemis inertial and the geocentric reference system is non-inertial.

6.1.3 Examination of Stellar aberration in the Context of Special Relativity

We will come to the conclusion that the standard analysis of the stellaraberration does not take into account the fundamental difference betweenvelocity of light and velocity of the raindrops. The main difference is thelimiting character of the velocity of light. No ”causal” signal can propagatewith velocity greater than that of light. Light is always a relativistic object,no matter how small the ratio v/c may be.

Where does the relativistic kinematics in the aberration of light physicscomes from? It is immediately understood that, in contrast to the classicaltheory, the light wavefront (i.e. the plane of simultaneity) orientation doesnot exist as a physical reality within the angle interval v/c since, due to thelimiting character of the speed of light, we cannot specify any experimentalmethod by which this orientation could be ascertained. The relativistic kine-matics enters aberration of light physics in a most fundamental way throughthe rotating of the wavefront which, in the case of light, is associated to thelowest order (v/c) relativistic kinematics effect (relativity of simultaneity).

An objective of this chapter is to consider non-relativistic interpretations ofthe stellar aberration, and clearly demonstrate that they are incorrect. Classi-cal kinematics effects leads to serious mistakes if applied to the computationof stellar aberration as seen on the earth rotating around the sun (i.e. in anon-inertial geocentric frame of reference). The problem of the earth-basedmeasurements is solved with the discovery of the essential asymmetry be-tween the earth-based and the sun-based observers, namely, the accelerationof the traveling earth-based observer relative to the fixed stars (5). We derivethe aberration for a pulse of light traveling on the surface of the earth usingthe Langevin metric.

6.2 Heliocentric Inertial Frame of Reference

Above, we demonstrated that when one has some hole in the opaque screenat rest with respect to the fixed stars and the transversely moving pointsource, there is no aberration (deviation of the energy transport) for lighttransmitted through the hole. The absence of the effects of moving (relativeto the fixed stars) source in this setup automatically implies the same prob-lem for stellar aberration theory in the heliocentric frame of reference. Howshall we solve it? It is like a hole-point source problem with the end of thetelescope barrel as a hole.

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Suppose that an observer, which is at rest relative to the telescope, performsthe direction of the energy transport measurement. At close look at thephysics of this subject shows that in the heliocentric frame of reference,where the telescope is at rest, we have actually the problem of steady-statetransmission. Then how does the transmitted light beam looks? It looks asthough the transmitted beam is going along the telescope axis because ithas lost its horizontal group velocity component. That is the transmissionappears as shown in Fig 12. It takes the case of a telescope positionedperpendicular to the plane phase front. In other words, the telescope pointeddirectly at the star. If the motion of the star is parallel to the phase front(i.e. perpendicular to the telescope axis), starlight entering the end of thetelescope would be able to pass its full length.

One of the most important conclusions of the discussion presented aboveis that the aberration of starlight phenomenon is absent in this situation.In particular, the binary components remains unresolved which means thattheir velocity has no influence on aberration.

6.3 Earth-Based Non-Inertial Frame of Reference

6.3.1 Experimental Results and Its Interpretation Based on Our Theory

The analogy between the obliquity of raindrops and the stellar aberration isincorrect. Only on the basis of the theory of relativity and the wave optics,we are able to describe all earth-based experimental observations of stellaraberration. Our theory predicts an effect of stellar aberration in completeagreement to the Bradley’s results, Fig. 28. According to the asymmetrybetween the inertial and rotating frames, there is a remarkable predictionon the theory of the aberration of light. Namely, if the telescope is at restrelative to the earth and the earth rotating relative to the fixed stars, thenthe direction of a star as seen from the earth is not the same as the directionwhen viewed by a hypothetical sun-based observer. Apparent angle is lessthan the actual angle. The difference between the actual angle and apparentangle θa is connected with the physical parameters by the relation: θa = v/c,where v is the velocity of the earth in its orbit around the sun. It could be saidthat the crossed term in metric Eq.(9) generates anisotropy in the rotatingframe that is responsible for the change of radiation direction (aberration).That is the transmission through the telescope aperture in the rotating frameappears as shown in Fig. 20. The stellar aberration in the geocentric frameof reference is considered independent of the star speed and to have justa local origin exclusively based on the observer speed with respect to thefixed stars.

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Fig. 28. The direction of a star as seen from the earth is not the same as the directionwhen viewed by a hypothetical observer at the sun center. Apparent angle φ is lessthan the actual angle θ. The difference between the actual angle and apparent angleis connected with the physical parameters by the relation: θ − φ = v/c, where v isthe velocity of the earth in its orbit around the sun.

The main facts which a theory of stellar aberration in the earth-based frameof reference must explain are (1) the annual apparent motion of the fixedstars about their locations and (2) the null apparent aberration of rotatingbinary systems. We presented here a theory which accounts for all these,and in addition gives new results. We demonstrated that the aberration oflight is a complex phenomenon which must be branched out into a num-ber of varieties according to their origin. These branching out takes placedepending on what is the cause of aberration - whether it is a jump in thevelocity (relative to the fixed stars) of the observer or of the light source.Aberration could undergo further splitting - depending on the physical in-fluence of the optical instrument on the measurement. According to ourinterpretation, there are many kinds of aberration and the stellar aberrationin the earth-based frame is only one of these. We demonstrated that allearth-based experiments can be explained on the basis of the effect of themeasuring instrument (i.e. physical influence of the telescope on the mea-surement) and the acceleration of traveling earth-based observer relative tothe fixed stars.

6.3.2 Physical Coordinate System in Space

The aberration of light is the geometric phenomenon. In order to detect theaberration effect inside the earth-based frame, it is obvious that some coor-dinate system with reference direction is needed. A conventional approachto the aberration of light effect is forcefully based on a definite assumption ofreference direction, but this is actually a hidden assumption. Traditionallythe physical interpretation of the aberration of light effect in terms of mea-

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surements performed with rods that are at rest in the observer’s frame. This(local frame of reference) convention is self-evident and this is the reasonwhy it is never discussed in the aberration of light theory

Fortunately, it is possible to find the space description on a more fundamen-tal level than that of measuring rods. The reference axis in the earth-basedframe is formed by gravitation field vector. The plumb-line direction isknown as the nadir, leading to the earth’s center. This is the most funda-mental local earth-based coordinate system. For example, Bradley used thevertically-mounted telescope. The star he chose was Draconis because ittransited almost exactly in zenith. The traditional plumb line provided asufficiently accurate zenith-point for observations of the stellar aberration.

6.4 Region of Applicability

The region of applicability of our stellar aberration theory is more widerthan one might think. Above we assumed for simplicity that a star image isactually a point spread function in the image plane of a telescope. In otherwords, it is assumed that the input signal is effectively a plane wave (6).

Suppose that a telescope is able to distinguish detail in the star image. Acomplete understanding of the relation between object and image can beobtained if the effect of diffraction are included. The effect of diffraction isto convolve that ideal image with the Fraunhofer diffraction pattern of thetelescope pupil.

The star is a spatially completely incoherent source (7). This means thatsuch source is actually a system of elementary (statistically independent)point sources with different offsets. An elementary source with a givenoffset produces in front of a telescope pupil effectively a plane wave. Anelementary source offset tilts the far zone field. Radiation field generatedby an completely incoherent source can be seen as a linear superposition offields of individual elementary point sources. The image of an elementarypoint source is a point spread function. In other words, there is alwaysthe physical influence of the telescope on the measurement of a completelyincoherent source. It should be remarked that any linear superposition ofradiation fields from elementary point sources conserves single point sourcecharacteristics like independence on source motion. This argument givesreason why our theory of stellar aberration is correct also for imaging ofarbitrary completely incoherent sources (6) .

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1. The phenomenon of the annual apparent motion of celestial objects abouttheir locations, named stellar aberration, was discovered by Bradley in 1727,who also explained it employing the corpuscular model of light [32].

2. A star is a completely incoherent source. The character of the mutualintensity function produced by an incoherent source is fully described bythe Van Cittert-Zernike theorem [33]. Any star can be considered as very faraway from the sun. In all cases of practical interest, telescopes are situated inthe far-zone of the (distant star) source. Consider a star like Sirius, which isone of the the nearest stars. The coherent area of light observed from Siriushas a diameter of about 6 m. This correlation was observed by Brown andTwiss in 1956 [34]. The star Bradley chose was Draconis, which is also one ofthe nearest stars. In this case the diameter of the coherent area on the earthsurface can be estimated to be about 100 m.

3. It is widely believed that stellar aberration depends on the relative velocityof the source (star) and observer. In the paper on the theory of relativityEinstein deduced the aberration formula from the idea that the velocity ofv is the relative velocity of the star-earth system. The idea was representedby many authors of textbooks. To quote Moeller [13]: ”This phenomenon,which is called aberration, was observed ... by Bradley who noticed that thestars seem to perform a collective annual motion in the sky. This apparentmotion is simply due to the fact that the observed direction of a light raycoming from a star depends on the velocity of the earth relative to the star.”

4. In 1950 Ives [35] stressed for the first time that the presence of binariesin the sky gave rise to an important difficulty for the theory of relativity. Itis stated that the idea that aberration may be described in terms of relativemotions of the bodies concerned is immediately refuted by the existence ofspectroscopic binaries with velocities comparable with that of the Earth inits orbit. Still this exhibit aberrations not different from other stars. For ex-ample, a spectroscopic binary, Mizar A, has well-known orbital parameters,from which can be calculated an observable angular separation of 1’10” ifaberration were due to relative velocity. The empirical value is less than0.01”, clearly incompatible with authors of textbooks point of view [36,37].There is no available explanation for the fact that, while the observationaldata on stellar aberration are compatible with moving earth, the symmetricdescription, when the star possesses the relative transverse motion, doesnot apparently lead to observations compatible with predictions.

5. Aberration exist as observable phenomenon only in the presence of chang-ing states of motion (i.e. acceleration). The problem of the earth-based mea-

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surements is related with the essential asymmetry between the earth-basedand the sun-based observers, namely, the acceleration of the traveling earth-based observer relative to the fixed stars. This has been recognized by someexpects, perhaps most explicitly by Selleri who states [37] that ”Thus a com-plete explanation of the aberration effect is given in terms of variations ofthe earth absolute velocity due to orbital motion, while the star/earth rela-tive velocity is irrelevant. Thus acceleration (of planet, this time) enters oncemore into a game.”

6. There is a widespread belief that the rays of light coming from the star arefalling on the telescope tube and not interact with its sides. To quote French[4]: ”Regarding light as being composed of rain of photons, we can easilycalculate the change in apparent direction of a distant object such a star.”Similar statements can also be found in recently published textbooks. Toquote Rafelski [5]: ”We consider a light ray originating in a distant star. ... Thefollowing discussion is addressing the observation of well focusing light-rays and not (spherical) plane wave light. ... This picture of the experimentalsituation is accurate since the light emitted by a star consists of an incoherentflux of photons produced in independent atomic processes.” This conclusionis incorrect. We emphasize that the design of a telescope optical system isbased on the classical theory of diffraction (originated directly from theclassical electromagnetic theory). The resolution of a telescope is limitedby the diffraction of light waves and the telescope always influences themeasured radiation (due to unavoidable diffraction of a star light by thetelescope aperture). In a well corrected optical system and in a circularpupil, the size Airy disc of diffraction pattern (i.e. size of the image of apoint source) is inversely proportional to the diameter of the pupil.

7. From everything that authors of textbooks have written about the starlight (”rain of photons”) it clearly follows that they did not understand theseveral aspects of the statistical optics. In the statistical optics for star emis-sion, the fields are described classically at the level of Maxwell’s equationsand the emitting medium is treated (as a ensemble of atoms) by quantummechanics. The atomic polarization induced by the radiation field appears asa driving term in Maxwell’s equations and sustains the oscillations. The staris actually a system of elementary (statistically independent) point sourceswith different offsets. The order of magnitude of the dimensions of the ele-mentary statistically independent source is about λ, where λ is the (visible)radiation wavelength. Within an elementary source volume (λ3) there is anenormous number of atoms. Semi-classical theory is treating these atomsas coherent, interacting radiating dipoles. The induced macroscopic dipolemoment in an elementary source leads to the classical electromagnetic ra-diation. The distinguishing characteristic of the statistical optics is the factthat electromagnetic fields are treated in a completely classical manner untilthey interact with the atoms of the photosensitive material on which they

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are incident. Thus there is no necessity to deal with quantization of theelectromagnetic field; only the interaction of the classical field and matter isquantized. Light interact with matter in a fundamentally random way. As aconsequence, any measurement of (star) light will be accompanied by cer-tain unavoidable fluctuations in the detector. Stochastic fluctuations of theclassical intensity can influence the statistical properties of the photoeventsthat are observed. Note in particular that the variance σ2

K of photoevents Kconsists of two distinct terms, each of which has a physical interpretation:σ2

K = [K2− < K >2]/ < K >2= 1/ < K > +σ2

W. The first term is simply thevariation of the counts that would be observed if classical intensities wereconstant and the photocounts were purely Poisson. We refer to this con-tribution to the count fluctuations as ”quantum noise”. The second therm,σ2

W, is clearly zero if there are no fluctuations of the classical intensity. Forinstance, in the case of laser light, this component would be identically zero,and the count variance would be simply that arising from the photon ”shotnoise”. When star (thermal) light is incident on the photodetector, the clas-sical fluctuations are nonzero, and the variance of the photocounts is largerthan that expected for a Poisson distribution.

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7 Notion of Ordinary Space in Special Relativity

We introduce new approach to the aberration of light theory in non-inertialframes of reference, finding another way in which this complicated prob-lem can be solved. For example, when viewed from the inertial frame, theaberration of light effect in the accelerated frame (Fig. 16) is easy to calculatein the framework of the special theory of relativity taking advantage of therelativity of simultaneity. In contrast, in order to compute the aberration oflight effect in non-inertial systems we used a metric tensor.

7.1 Inertial Frame View of Observations of the Noninertial Observer

First we want to rise the following interesting and important point. Thelaws of physics in any inertial reference frame should be able to account forall physical phenomena, including the observations made by non-inertialobservers. It is widely believed that all phenomena in non-inertial (e.g.rotating) reference system should be considered only in the framework ofthe space-time geometric approach using Langevin metric. However, theanalysis of physical phenomena in non-inertial frames of reference can bedescribed in an inertial frame within standard (Einstein) special relativitytaking advantage of well known relativistic kinematic effects.

For example, when viewed from the inertial lab system, the interpretationof the Sagnac effect is simple and the phase difference (attributed to theSagnac effect) between counter-propagating waves may be derived fromthe relativistic law of velocity composition. In other words, Sagnac effectin the rotating frame, as viewed from the inertial lab frame, presents akinematic effect (Einstein’s velocity addition) of special theory of relativity.It is implicitly assumed that the speed of light is c in the lab frame which, inaccordance with Einstein special relativity, is independent of motion of thesource. In contrast, the Sagnac effect is not easy to calculate in a frame ofreference attending rotation. In this case, authors of textbook used a metrictensor (Langevin metric) in a plane four-dimensional Minkowski space-time to calculate the propagation time difference between counter-runningwaves. Can we not look at the aberration of light effect in the same way?Here also we used the Langevin metric in a non-inertial frame of reference.

Let us first consider the application of the classical kinematic method tothe computation of the aberration of light effect in a non-inertial frame, asviewed from an inertial frame. In first order approximation, the Galileanlaw of velocity composition may be used. However, the classical kinematicsmethod leads to a serious mistake if applied to the computation of the

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aberration of light effect. Let us consider the Fig. 9. Based only on the(Galilean or Einstein’s) velocity addition, we arrive at the conclusion thatthere is no aberration of the transmitted light beam in the accelerated frame.

Errors inherent in the classical kinematics method applied to the computa-tion of the aberration of light effect are due to the use of the velocity of lightsimply as a classical velocity. Relativistic effects do not have a place in thisdescription. According to classical approach, there is no principal differencebetween the aberration of light and the aberration of raindrops. We firstnotice that one of the postulate of special relativity states that if two distinctevents cannot be connected by a causal signal that travels no faster than alight signal, then they cannot be connected by a causal signal at all. We willshow that the aberration of light effect is a corollary to the relativistic kine-matic effects. The appearance of relativistic effects in radiation phenomenadoes not depends on a large speed of the radiation sources. Light is alwaysa relativistic object. In particular the relativity of simultaneity is responsiblefor aberrations to the first order in v/c.

Now, it is very interesting to show that the geometric effects in our ordinaryspace world are closely associated with the relativity of simultaneity. In Fig.16 the transmitted light beam is propagated at the angle v/c, yielding thephenomenon of the aberration of light. The question cannot be avoided rel-ative to what a light beam propagated in the accelerated frame with angulardisplacement v/c? Suppose that an observer in the accelerated frame per-forms the direction of the light beam measurement and the plane wavefrontof transmitted light beam is imaged by a lens to a diffraction spot which liesin the focal plane on the optical axis. Measurement of the direction of theoptical axis with respect to the frame axes is equivalent to the determinationthe angular displacement. In order to detect the aberration of light effectinside the accelerated frame, it is obvious that some coordinate system withreference direction is needed. We must inquire in detail by what method weassign coordinates. This method involves some sort of physical procedure;eventually it must be such that it will be give us coordinates in both (inertialand accelerated) frame of reference.

In ordinary space we find that the accelerated frame moves with respect tothe inertial frame along the line motion and the inertial frame moves withrespect to the accelerated frame along the line motion. The angle betweenthe axis of the observer’s coordinate system and the line motion is a simpleordinary space geometric parameter. Using the line motion as a referencexn-axis, the accelerated observer can then define the second reference axis.We need to give a ”practical”, ”operational” answer to the question ofhow to assign an axis perpendicular to the xn-axis. Clearly, it is possible todefine a reference direction using a light beam. We will define the secondreference direction in the following way. Let us suppose that the aberration

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Fig. 29. Aberration of light in the accelerated frame of reference. In the Einstein timesynchronization for the screen, the accelerated observer describes the transmittedlight beam and the reference beam from an emitter at rest using usual Maxwell’sequations. The xn-axis is parallel to the wavefront of the reference beam. The aber-ration increment may be defined as an angle between the reference and transmittedlight beams.

direction inside the accelerated frame is determined with reference to thefixed direction of light beam from a ”plane-wave” emitter which is at rest inthe accelerated frame. In other words, a coordinate system is formed hereby electromagnetic axis and line motion. The motion of the aberrated lightbeam are assumed for simplicity, to lie in the same plane and the angularposition of the aberrated beam is described by one angle.

When the accelerated system starts moving with constant velocity the stan-dard procedure of Einstein’s clock synchronization can be performed. TheEinstein synchronization is defined in terms of light signals emitted by asource at rest assuming that light propagates with the same velocity c inall direction. In this synchronization, the accelerated observer describes thelight beam from an emitter at rest using usual Maxwell’s equations. Accord-ing to Maxwell’s electrodynamics, light is always emitted in the directionnormal to the radiation wavefront. We consider the case in which the ve-locity component of the (reference) light beam along the x-axis is equal tozero. In other words, the x-axis is parallel to the wavefront of the reference”plane wave”, Fig. 29. The number that specifies the aberration incrementof the light beam transmitted through the hole may be defined as an anglebetween the reference and transmitted light beams.

Let us try to get an understanding of the relationship between the reference

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Fig. 30. Aberration of light in the inertial frame of reference. In the absolute timesynchronization for the screen, the inertial observer describes the reference beamfrom an emitter at rest using usual Maxwell’s equations. The x-axis is parallel tothe wavefront of the reference beam. The aberration increment may be defined asan angle between the reference and transmitted light beams.

directions inside the inertial frame and the reference directions inside theaccelerated frame. Such approach in our case of interest is based on theuse of local (reference) light sources. We determine the reference directionsperpendicular to the line motion in the inertial frame and the acceleratedframe using two local light sources. In effect, one source should be stay atrest in the accelerated frame (Fig. 29) while the second should be at rest inthe accelerated frame (Fig. 30). In other words, the reference electromagneticaxis in each frame is formed by an individual light beam. Of course, it ispossible to find the space description on a more practical level than that oflight beams. For instance, the reference axis in the earth-based frame may beformed by the gravitation field vector (e.g. a standard direction of plumb-line). The equivalence of all local physical frames of reference underlies thetheory of relativity, so when the aberration angle emerges, it emerges in thesame manner in all local physical reference systems. A comparison with alight clock might help here. When inertial observer looks at the light clockinside the accelerated frame, he sees that clock run slowly. Not only doesthis particular kind of clock run more slowly, but if the theory of relativityis correct, any other clock, operating on any principle whatsoever, wouldalso appear to run slower.

Now let us return to observations of an accelerated observer, as viewedfrom an inertial frame. From what has preceded it is easy to see that in the

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Fig. 31. Inertial frame view of observations of the accelerated observer. Accordingto textbooks, the line motion of observers is the same in the accelerated frame asin the inertial frame (in other words, it is hiddenly assumed that accelerated andinertial observers have common ordinary space). In the Lorentz coordinatization,the reference light beam in the accelerated frame is emitted in the direction normalto the wavefront. If make a Lorentz boost, we introduce a time transformationtn = t − xv/c2 and the effect of this transformation is just an inclination of theelectromagnetic axis in the accelerated frame.

accelerated system, from the point of view of the inertial observer, the elec-tromagnetic axis, assigned by the accelerated observer, will not orthogonalto the x-axis (i.e. line motion) of the inertial observer, Fig. 31. Based on theGalilean velocity addition, we arrive at the conclusion that in the acceler-ated frame transmitted light beam propagates along the z-axis of the inertialframe of reference. But in the Lorentz coordinatization there is an angulardisplacement, v/c, between the inertial and accelerated electromagnetic ref-erence directions. In other words, aberration of light effect in the acceleratedframe, as viewed from the inertial frame, presents a kinematic effect (rela-tivity of simultaneity) of special theory of relativity. The transformation ofobservations from the inertial frame with Lorentz coordinates to the accel-erated frame is described by a Lorentz boost. On the one hand, the waveequation remains invariant with respect to Lorentz transformations. On theother hand, if make a Lorentz boost, we automatically introduce a timetransformation tn = t − xv/c2 and the effect of this transformation is just arotation of the radiation phase front (and,consequently, the electromagneticaxis) in the accelerated frame on the angle v/c.

To continue our discussion of inertial frame view of observations of the non-

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inertial observer, let us consider the opposite situation. We determine thereference directions in the inertial frame and the accelerated frame using twolocal light sources. This has interesting consequences. For example, supposethat an observer in the inertial frame performs the direction of transmittedlight measurement relative to the reference light beam from the emitterwhich is at rest with respect to the inertial frame (Fig. 30). Now we must becareful about sign of an angular displacement. The inertial observer wouldfind that the angular displacement is positive and equal to θa = v/c. Thedefinition of the sing that we use for the angular displacement is analogousto this: we suppose that the both light beams are imaged by a lens to thetwo diffraction spots which lie in the focal plane and the spot of transmittedlight is shifted in positive direction along the x-axis relative to the spot ofreference beam.

Now we ask about the angular displacement of transmitted light withrespect to the reference beam inside the accelerated frame. On the onehand, we know that angular displacement of the transmitted light withrespect to the reference light beam from the emitter which is at rest withrespect to the accelerated frame is −v/c. On the other hand, the angulardisplacement of the inertial reference light beam is −2v/c. We come tothe conclusion that the accelerated observer can directly measure the an-gular displacement of the transmitted light with respect to the referencelight beam of the inertial frame. In fact, the relative angular displacementis θa(transmitted beam) − θa(inertial reference beam) = −v/c + 2v/c = v/c.This result is consistent with the inertial observer measurements (Fig. 30),as must be.

7.2 The Non-Existence of an Instantaneous 3-Space

We derived the results for observations of the non-inertial observer withthe help of Lorentz transformations. Let us assume, according to textbooks,that the line motion of observers is the same in the accelerated frame as inthe inertial frame. At first site, if reference axes were orthogonal to the linemotion they must be parallel to each other. Since there exist the angulardisplacement v/c, the situation seems paradoxical.

First of all, as noted above, the key idea in resolution of this paradox isthe realization of the fact that there is no absolute notion of simultaneity inthe theory of special relativity. In the case of the relativity of simultaneitywe have a mixture - of positions and time. In other words, in the spacemeasurement of one observer there is mixed a little bit of the time as seenby the other. Therefore, there is no notion of an instantaneous 3-space.The point, then, is not the existence of a common ordinary space. The fact

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Fig. 32. It is assumed that the line motion of the reference light beam in the ac-celerated frame is the same in the inertial frame. In this case the lab observer seesthe rotation of the moving axes with respect to the lab frame axes, which is closelyassociated with the Wigner rotation.

that two observers with different trajectories have different 3-spaces is notconsidered in textbooks.

If we look at the situation carefully we see that the accepted in textbooks as-sumption of common (x) line motion of observers is based on the belief thatobservers have common 3-space. This is misconception. Right way wouldbe to use the light beam as the common reference direction. We have alreadyseen that the inertial observer can directly measure the angular displace-ment between the inertial (z) and accelerated (zn) electromagnetic referencedirections and this result is consistent with the accelerated observer mea-surement. In other words, the displacement between electromagnetic axesis a physical reality. Let us accept that the electromagnetic zn axis (i.e the ref-erence light beam in the accelerated frame) is the same in the inertial frame.Now the lab observer sees the rotation of the moving frame axes with re-spect to the lab frame axes, Fig. 32. This rotation is a relativistic kinematiceffect. One can verify directly that the axes of accelerated frame are actuallyrotated with respect to the lab frame axes by the angle equal to ΦR = v/c.

Let us clear up the meaning of the parameter ΦR. It can be associated with theangle of the so-called Wigner rotation (see the Section 7.5 for more details).Our results show that we cannot remain with the framework of parallel axeszn, z when considering the inertial frame view of light beam observationsof the non-inertial observer in perpendicular geometry. A Lorentz transfor-

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mation makes axes (z and zn) oblique-angled. This inclination angle v/c canbe interpreted as a rotation angle ΦR = v/c of the moving axes with respectto the lab frame axes.

7.3 Acceleration of a Rigid Body and Special Relativity

Now let us look at some further consequences of relativistic mixture of po-sitions and time. The concept of rigid motion, as known from Newtonianmechanics, cannot be taken over unchanged to the special theory of rela-tivity. This follows from the fact that the distance between two points is nolonger an invariant quantity. Thus, even if the distance between two pointsis constant in time in one system of reference, this will in general not thetrue in another system.

First, let us study the relativistic kinematics of rotation of the moving frameaxes with respect to the inertial frame. Consider an emitter-detector setup inthe lab inertial frame, Fig. 33. Suppose that the setup is accelerated from therest with respect to the lab frame up to velocity v along the x-axis. The maindifference between the relativistic kinematics and the kinematics of Newtonis that in the lab frame, the accelerated emitter-detector setup exhibits ashearing motion. This counterintuitive fact is not a paradox. In fact, thisshearing contribution adds up to the Wigner rotation. The shearing (3-spacegeometric) effect is closely associated with the relativity of simultaneity.

Above we demonstrated that aberration of light effect in the acceleratedframe, as viewed from the inertial lab frame, may be derived from theLorentz transformations. In other words, aberration of light effect in the ac-celerated frame, as viewed from the inertial lab frame, presents a kinematicseffect of special theory of relativity, Fig. 34. It is shown that the relativityof simultaneity is responsible for aberrations in the first order of v/c. Wediscovered of the essential asymmetry between the inertial and acceleratedframe, namely, an acceleration (with respect to the inertial frame) have aneffect on the propagation of light in the accelerated frame. On acceleratedsystem, the accelerated detector does not receive light radiated by the accel-erated emitter, Fig. 34. In contrast, the inertial detector at rest continues toreceive light radiated by the inertial emitter at rest, Fig. 33 left.

Common textbook presentations of the special relativity use the (3+1) ap-proach which deals with hidden assumption that distant clocks are synchro-nized according to the absolute simultaneity convention. In fact, accordingto textbooks, when a reference frame at rest is put into motion, then allpoints in the 3D reference grid have a start to move at same time. Conse-quently, rigidity in the sense introduced by textbooks cannot be considered

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Fig. 33. Aberration of light in an inertial frame of reference. The aberration of lighteffect can be described within the standard special relativity taking advantage ofthe Wigner rotation theory. When one has a transversely moving emitter there is thedeviation of the energy transport for radiated light. This effect is a consequence ofthe fact that the Doppler effect is responsible for the angular frequency dispersionof the radiated light waves (see Fig. 1). As viewed from the Lorentz lab frame,the coordinate axes of the accelerated frame is rotated through the angle v/c withrespect to the coordinate axes of the lab frame. In the first order approximation inv/c, the accelerated emitter-detector setup exhibits a shearing motion (during accel-eration). This shearing contribution adds up to the Wigner rotation. The wavefrontorientation has no exact objective meaning. Because of this, only the wavefront ori-entation in the absolute time coordinatization for the moving source is presentedhere for sketch simplification.

as a relativistic kinematics property. The standard presentation of the aber-ration of light effect is based on the hidden assumption that (xx, yn, zn) axesof the accelerated frame and (x, y, z) axes of the inertial frame are parallel. Inother words, it is incorrectly assumed that accelerated and inertial observershave common 3-space, Fig. 35,36. According to the conventional theory, anacceleration does not spoil the motional symmetry between the acceleratedreference frame and inertial reference frame. Our result (Fig. 33,34 ) is atodds with the prediction from textbooks. The commonly accepted deriva-tion of the aberration of light effect does not account for the Wigner rotation,which in our case closely associated with the relativity of simultaneity.

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Fig. 34. Aberration of light in an accelerated frame of reference. In order to predictresult of the aberration measurement, the accelerated observer should use theLangevin metric. On accelerated system, the accelerated detector does not receivelight radiated by the accelerated emitter.

Fig. 35. Aberration of light in an inertial frame of reference as predicted by theconventional theory (see Fig. 1).

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Fig. 36. Aberration of light in an accelerated frame of reference as predicted bythe conventional theory. An acceleration does not spoil the motional symmetrybetween the accelerated reference frame and inertial reference frame.

7.4 Discussion

Let us return to observations of an inertial observer. Special theory of rel-ativity says that the orthogonal emitter-detector setup exhibits a shearingmotion during acceleration, Fig. 33. This is a purely kinematic effect, involv-ing no forces. How can that be? That’s not an easy question, but here is oneway of thinking about this.

It is clear that our ordinary intuition are quite wrong when we have dealwith two space-like separated events. We see that the accepted in textbooksassumption of rigidity of accelerated frame is based on the belief that simul-taneous acceleration of the reference space grid has direct physical meaning.This is misconception. In fact, when we have two distant events, we havedeal with the conventionality of distant simultaneity within the time inter-val L/c, where L is the space separation of these two events. The positionalong the x-axis of the accelerated emitter with respect to the accelerateddetector has no exact objective meaning since, due to the finiteness of thespeed of light, we cannot specify any experimental method by which thisposition could be ascertained . There is an uncertainty (blurring) of the rel-ative position in the x-direction of amount Lv/c (due to the uncertainty inthe moments of the acceleration).

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7.5 Aberration of Relativistic Particles

We want now to discuss the phenomenon of the aberration of relativisticparticles. We begin by recalling a very interesting (methodological) analogybetween the aberration of light effect and the deflection of light in gen-eral relativity. Consider a ”point” mass m moving with a velocity V in thegravitation field of a point mass M. Both the motion of particles and thepath of light in the gravitation field are described by the same equation ofmotion. The deflection angle for a particle is θd = (1 + V2/c2)θc, where θc

is the particle deflection angle according to classical approach. The factor(1 + V2/c2) is new from relativistic (Schwarzschild) metric, but reduced tofamiliar Newtonian limit as V2/c2

→ 0. For light V/c = 1 and there is anextra factor 2. On the basis of classical theory one gets a deflection of lightthat is only one-half as big as predicted by general relativity.

Suppose we have a particle source. All particles which come out of thesource will have the same velocity V. Particles trace paths along the z-axis.We consider the case when the source is at rest in the lab inertial frame andthe observer (with his measuring devices) is accelerated from rest up to ve-locity v along the x axis. It is assumed that v V. Suppose that an observerin the accelerated frame performs an aberration measurement. How shallwe describe the aberration of particle beam from the source which is at restin the inertial frame? In order to predict the result of the aberration measure-ment the accelerated observer should use the Langevin metric Eq.(9). Theaberration increment is connected with the problem parameters by the rela-tion θa = −[1 + (1 − 1/γp)]θc, where θc = v/V is the particle deflection angleaccording to classical approach. Here γp = 1/

√1 − V2/c2 is the relativistic

factor. The term [1 + (1 − 1/γp)] is new from relativistic (Langevin) metric,but reduced to familiar classical limit as V2/c2

→ 0. For light V/c = 1 andthere is an extra factor 2. A correct solution of this problem in the rotatingframe requires the use of metric tensor even in first-order experiments sincethe crossed term in the Langevin metric (which involves the first-order de-viation of metric tensor from its Minkowski form) plays a fundamental rolein the non-inertial kinematics of a relativistic particle beam propagation.

Imaging that there are two identical sources. For the sake of simplicity weassume that the relativistic factor of the beam γp 1. Let us considerthe case when the first source is at rest in an inertial frame and the secondsource is accelerated from rest up to velocity v along the x-axis. Suppose thatan observer, which is at rest with respect to the inertial frame of referenceperforms the direction of the particle transport measurement. Now we mustbe careful about the initial direction of the velocity of a particle beam.As example, we consider the case in which particles are initially directedalong the z-axis. Then how does the particle beam from the moving source

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Fig. 37. The aberration of ultrarelativistic (γp 1) particles from stationary andmoving sources. If the particle source is at rest relative to the fixed stars and theobserver with measuring devices started from rest to uniform motion with velocityv c, then the apparent angular position of the source seen in the acceleratedframe would jump by angle θa = −2v/c. This situation is not symmetrical withrespect to the change of the reference frames. If observer is at rest relative to thefixed stars and the source started from rest to uniform motion relative to the fixedstars, then the apparent angular position of the source seen in the inertial framewould jump by angle θa = v/c

looks? The inertial observer would find that angular displacement is equalto θa = v/c. This is example what called aberration of relativistic particlesand it is well known.

The difference between measurements made in both reference frames is veryinteresting. According to the asymmetry between the inertial and acceler-ated frames, there is a remarkable prediction on the theory of the aberrationof particles. Namely, if the source is at rest relative to the fixed stars andthe observer with measuring devices started from rest to uniform motionrelative to the fixed stars, then the apparent angular position of the emitterseen in the accelerated frame would jump by angle θa = −2v/c. This situa-tion is not symmetrical with respect to the change of the reference frames,Fig. 37. It could be said that the crossed term in metric Eq.(9) generatesanisotropy in the accelerated frame that is responsible for the differencebetween aberration increments.

When the accelerated system starts moving with constant velocity the pro-cedure of symmetrization can be performed. Let us describe what happenswhen accelerated observer performed the re-directing of the accelerated

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Fig. 38. Reciprocity in the theory of the aberration of particles. One must use thesame beam directing to demonstrate the reciprocity. After the redirecting proce-dure of the accelerated source the inertial observer would find that the angulardisplacement is θa = 2v/c (for the sake of simplicity it is assumed that γp 1).

source. We consider the case in which finally the particles trace paths alongthe zn-axis. After this re-directing procedure the inertial observer wouldfind that angular displacement is θa = 2v/c. Suppose that the acceleratedobserver also performs an aberration measurement. Fig. 38 shows that theaberration increment is connected with the problem parameters by the re-lation θa = −2v/c.

Above in this chapter we demonstrated that aberration of light effect in theaccelerated frame, as viewed from the inertial lab frame, presents a wellknown kinematic effect of special theory of relativity. What about particles?Can we not look at the aberration of particles in the accelerated frame in thesame way? As we shall see, the aberration of particles effect in non-inertialframes of reference can be described in an inertial frame within standardspecial relativity taking advantage of the Wigner rotation theory.

The Wigner rotation is a relativistic kinematic effect, which consists in thata coordinate axes of a reference frame, moves along a curvilinear trajectoryrotating about the axes of a Lorentz lab frame. What is the formula forWigner rotation? It turns out that this is complicated, and it takes a greatdeal of study and sophistication to appreciate it. [We shall do this in theChapter 15.] We only write it down to give the impression. The expressionfor the rotation angle in the lab frame can be presented in vector form~θw = (1 − 1/γp)~V × ~v/V2, where ~V is the vector of particle velocity in the

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Lorent lab frame before acceleration, ~v is the vector of small velocity change(v V) due to acceleration, θw is the Wigner rotation angle of the spatialcoordinate axes of the system comoving with a particle relative to Lorentzlab frame. We shall discuss this expression further in the Chapter 15. In themeantime, we shall accept it as true.

To find the rotation magnitude in the stated problem, we introduce a com-position of boosts. Let S be a lab frame of reference, Sn an accelerated framewith velocity ~v relative to the lab frame, and S′ is a particle comoving framewhich moves relative to the Sn with velocity ~V along the zn-axis. Two se-quential boosts from the lab frame S to Sn and then to S′ are equivalent tothe boost from S to S′ and the subsequent rotation. The comoving frame S′

is rotated through the angle θw with respect to the lab frame S. Now wemust be careful about direction of rotation. There is a good mnemonic rule.The rule says that the direction of the Wigner rotation in the Lorentz labframe is the same as the direction of the velocity rotation in the Lorentzlab frame. We can easily understand that the coordinate axes of the parti-cle comoving frame (x′, y′, z′) will be parallel to the coordinate axes of theaccelerated frame (xn, yn, zn). As viewed from the lab frame, the coordinateaxes of the accelerated frame (xn, yn, zn) is also rotated through the angleθw with respect to the coordinate axes of the lab frame (x, y, z), Fig. 38. Ourresults show that we cannot remain with the framework of parallel axes zn, zwhen considering the inertial frame view of particle beam observations ofthe accelerated observer in perpendicular geometry. For light V/c → 1 andthe axes of accelerated frame are then rotated with respect to the lab frameaxes by angle θw = v/c.

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8 Earth-Based Setups to Detect the Aberration Phenomena

8.1 Potential of Earth-Based Electron Microscopes

Above we discussed the new relativistic particle kinematics for rotatingframe of references. Since we live on rotating earth frame, difference in rel-ativistic kinematics between rotating and non-rotating frames of referencesis of practical as well as theoretical significance. We propose an aberrationof particle experiment using an earth-based particle source. Experimentscan be explained on the basis of special relativity. A correct solution of thisproblem in the earth-based frame requires the use of metric tensor even infirst-order experiments since the crossed term in the Langevin metric (whichinvolves the first-order deviation of metric tensor from its Minkowski form)plays a fundamental role in the non-inertial kinematics of a relativistic par-ticle beam propagation.

We propose an experiment using commercial 200-kV scanning transmissionelectron microscope (STEM) as a particle source. One could use a verticallyoriented optical setup. We suppose that the electron beam is imaged by alens to a spot which lies in the image plane. Measurement of the spot shiftwith respect to the optical axis (which is parallel to the zenith-nadir axis)is equivalent to the determination of the angular displacement. Due to thehigh stability, the aberration increment could be observed (1).

Consider the inertial sun-based reference system. In this system there is theearth which rotates around the sun with orbital velocity v. Developing intopowers of v/c we can classify effects for velocities v c as of the first order,of the second order or of higher orders. The essentially relativistic effects areof the second (or higher) orders. Clearly, in the case of the rotating (aroundthe sun) earth-based frame, we consider the small expansion parameterv/c ' 10−4 neglecting terms of order of v2/c2.

Suppose that an earth-based observer performs an aberration of particlemeasurement. The aberration angle varies with one-year temporal period.An approximate formula to express the aberration angle can be found byusing Langevin metric in the earth-based frame or by using Wigner rotationtheory in the sun-based frame. In the aberration of particles, the aberrationangle is the apparent angular deviation of the position of the particle sourcerelative to the real location of the source. The reference axis in the earth-based frame can be formed by plumb line, which is the most fundamentallocal earth-based coordinate system.

When measured with an earth-based particle source, the aberration angle is

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Fig. 39. Definitions related observer’s position on the earth.

~θa = −~θw = −(1 − 1/γp)~V × ~v/V2, where v is the orbital velocity of the earthrelative to the sun, V is the particle velocity, and γp = 1/

√1 − V2/c2 is the

particle relativistic factor. This means that the angle of Wigner rotation de-pends on V2/c2. It should be note that presented formula can be consideredonly as a first order approximation in v/c. In the theory of (infinitesimal)Wigner rotation we must also consider the relativistic parameter V/c. If aparticle motion velocity is non-relativistic, the binomial expansion yields~θa = −[V2/(2c2)]~V × ~v/V2.

Let us calculate the parameters for the STEM setup. The Wigner rotationangle parameter (constant of aberration) defines the scale of the aberrationincrement, and is equal to θw = (1 − 1/1/γp)v/V ' 40µrad. Here γp ' 1.39,V/c = 0.7, v/c ' 100µrad. Earth moves around the sun, with a consequentchange of the direction of ~v. Therefore the angle of aberration changes. Thetotal change of the order of 40µrad. It is oscillatory with a period of one year.

A specific aspect of our case study needs further investigations. The pro-posed method of measuring the angle of aberration involves the use ofearth-based sources and have a big advantage. The rotation of the earth onits axis should produce a corresponding shift of the image. Obviously, it isimportant that observation should be recorded in the shortest possible time.In principle, records could be taken over a period of one day. The aberrationshift depends only upon the value of v⊥, the component of the orbital veloc-ity perpendicular to the earth rotation axis (2). In practice, the measurement

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procedure is complicated by a number of factors, Fig. 39. A corrections hasto be made for the observer position on the earth’s surface and the obliquityof the ecliptic δ = 23o. The image appears to move in an ellipse. The shape ofthe aberration ellipse obviously depends on the observer’s latitude φ. Foran observer on the equator the ellipse degenerates into a line segment andfor an observer at the pole of the earth, the ellipse is a circle. At the valuel = 2 mm of the focal length (3), the major axis of the aberration ellipse of theorder of 80 (cos δ) nm. The value of minor axis depends on the observer’sposition and is given by 80 (cos δ sinφ)) nm. When the latitude is φ = 35o,the value of the scale of image shift per hour depends on the local time andvarious from a value about 10 nm/60 min to about 20 nm/60 min throughthe day (4).

8.2 Explanation of the Aberration of Electrons on the Basis of Electrodynamics

A large number of incorrect expression for the Wigner rotation (it is oftencalled the ”Thomas rotation”) can be found in the literature. For those whohave already received knowledge about the Wigner rotation from the well-known textbooks, it may be expedient to begin its aberration of particle(theoretical and experimental) development with the microscopic approach.The point is that one can compute any relativistic quantity directly from theunderlying theories of matter without involving relativity at all. Wignerrotation associated with the transformation from the inertial (sun-based)coordinate system to the rotation (earth-base) coordinate system may beregarded as a result of the action of certain force. This problem has norelation to the problem of the ether and is already formulated within theframework of special relativity. The electromagnetic forces which govern theproperties of an emitted electron beam must be affected by (earth-based)source acceleration around the sun in a such a way that they lead to adeviation of the electron transport direction. In fact, there is a machinerybehind the electron momentum ~p changes when the source and observer(with his measuring instruments) are at rest in the rotating frame. Its originis explained in the framework of the electrodynamics theory.

We can obtain the electrodynamics equations in rotating reference frameusing the Galilean transformation. When we consider the electrodynamicsequations under Galilean transformation, there is a question: what is thetransformation law for the electromagnetic fields ~E and ~B? The fields ~En and~Bn observed in the accelerated frame where the particle source is at rest arenot the same as those ~E and ~B in the inertial frame before the active Galileanboost.

We need to find the relation between ~En and ~Bn and ~E and ~B. To do this,

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we transform coordinates (t, x) that would be coordinates of an inertialobserver S moving with velocity −v with respect to the observer Sn, usinga Galilean transformation: we substitute xn = x − vt, while leaving timeunchanged tn = t into the Minkowski metric ds2 = c2dt2

− dx2− dy2

− dz2 toobtain Langevin metric ds2 = c2(1 − v2/c2)dt2

n − 2vdxndtn − dx2n − dy2

n − dz2n.

We can obtain the electrodynamics equations in rotating frame using theGalilean transformation (with velocity−v) of the Maxwell’s equations. Withthe Galilean transformation, the field transformation, for the case ~B = 0, is~En = ~E, ~Bn = −~v × ~E/c. This expression describe first order (in v/c) effectsonly. Here ~v is the velocity vector of the earth in the sun-based frame.

The accelerated observer has experimentally obtained the Lorentz force ~Fn

on a charge e moving with velocity ~V in the region of electron (microscope)gun in which ~En and ~Bn are presented, ~Fn = e~E − e~V × ~v × ~E/c2. In thecase - where ~v · ~V = 0, ~E · ~V = EV - the expression for magnetic forceis equivalent to −e~V × ~v × ~E/c2 = −~v(e~V · ~E/c2). Using the relativisticallycorrect equation of motion, ~Fn = d~p/dt, we expect after the time dt emittedelectron will have a transverse momentum in the accelerated frame givenby d~p⊥ = −~v(e~V · ~E/c2) dt. The differential change in the kinetic energy ofthe accelerated electrons is dT = ~F · ~Vdt = ~F · d~s. If we now integrate, we getT = e

∫~E ·d~s = mc2/

√1 − V2/c2−mc2. Since we have chosen to make ~v · ~V = 0,

aberration increment becomes θa = p⊥/p, where p = mV/√

1 − V2/c2 is themomentum of the accelerated electron. So we have θa = −(1 − 1/γp)v/V.We can put our expression for the aberration increment in vector form:~θa = −(1 − 1/γp)~V × ~v/V2.

This formula correspond to the result we derived for observation of thenon-inertial observer with the help of Lorentz transformations. The the-ory of special relativity shows that the aberration of particles effect in therotating earth-based frame, as viewed from the inertial sun-based frame,presents a kinematic effect (Wigner rotation) of special theory of relativity.It is important to stress at this point that the dynamical line of argumentsdiscussed here explains what the Wigner rotation physically means in therotating frame. The relativistic kinematics effects in the rotating frame, asviewed from the inertial observer, is only an interpretation of the behaviorof the electromagnetic fields.

8.3 Absent of the Aberration of Light from the Earth-based Laser Source

Let us make some comments on the aberration of light experiments usingan earth-based source. In regard to light aberration one should differentiate

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between that from the incoherent source and that from the laser source. Onecould naively expect that the rotation of the earth around the sun producesaberration in an amount large enough to be taken into account in preciseobservation work using laser as a earth-based light source. Let us stress thatthe aberration of light cannot be measured using laser.

In order to understand this phenomenon, we consider the simple case whena laser resonator is equipped with plane mirrors. For oscillation to occur, thetotal loss in power due to diffraction and reflection loss must be less than thepower gained by travel through the active medium. Thus diffraction loss isexpected to be an important factor, both in determining the start-oscillationcondition, and in determining the distribution of energy in the interferom-eter during oscillation. The purpose of our study is to investigate the effectof diffraction on the electromagnetic field in a Fabry-Perot interferometerin free space. The conclusion can be applied equally well to gaseous lasersprovided the interferometer immersed in the active medium.

The electric field ~E of an electromagnetic wave in a passive resonator satis-fies the equation ∇2~E − ∂2~E/∂(ct)2 = 0. For simplicity, we shall assume thatthe field in the resonator to be circularly polarized. Such an assumption doesnot violate the generality of the analysis because the polarization degener-acy takes place in the system under study. The electric field vector in theplane resonator is presented as an oscillation superposition with differentlongitudinal wave numbers, Ex + iEy =

∑m Em(x, y, t) exp(−iωmt) sin(Kmz+δ),

where Km = mπ/l − i/(nl), ωm = mπc/l, δ = i/n, n is the (complex) refractiveindex of the mirror matter (|n| 1), l is the distance between the mirrors,and m is an integer number (m 1). It is evident that this expression satis-fies Leontovich’s boundary conduction (5) on the mirror surface when z = 0and z = l. We assume the field change per one resonator pass to be small i.e.|∂Em/∂t|l/c |Em|, which means that the laser frequency is close to the natu-ral frequenciesωm of the longitudinal resonator modes. Let us find a solutionfor the field amplitude Em in the form Em = Φ(x, y) exp(Λt). Substituting theansatz in the wave equation we get ∇⊥Φ + 2iωm[Λ/c2 + 2/(nl)]Φ = 0.

In a plane resonator the diffraction effects at the edges of the mirrors canbe taken into account if one considers the space between the mirrors as awaveguide, and uses diffraction theory at the open end of waveguide. Ifalmost an integer number of half waves is laid along the length between themirrors then at open resonator end we may put boundary conditions on theamplitude function Φ(x, y), equivalent by its effect to the open waveguideend action (so-called impedance boundary conditions of resonance type (6)):Φ+(1+ i)β0

√cl/(4ωm)∂Φ/∂ζ = 0, where ζ is the direction of the normal to the

imaginary side surface of the resonator, and the parameter β0 = 0.824 (7). Theboundary condition do not depend on the polarization which means that

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polarization degeneracy takes place in the system under study. Therefore,the problem concerning the open plane resonator excitation by means ofequivalent boundary condition can be reduced to the more usual classicalformulation of the problem of closed resonator excitation.

Let us consider a resonator equipped with plane circular mirrors with radiusR. The distance between the mirrors is equal to l. The system is azimuthallysymmetric relatively to the resonator axis which coincides with the z axisof the polar coordinates. We shall seek the solution of the wave equationin the following form: Φ(~r) = Φν(r) cos(νφ),Φν(r) sin(νφ), where ν is an inte-ger.The eigenfunctions of the resonator are the solution of the homogeneousequation r2d2Φν j/dr2 +rdΦν j/dr+ (k2

ν jr2−ν2)Φν j = 0, satisfying the impedance

boundary conditions. The eigenfunctions Φν j are orthogonal and form acomplete system. In the first approximation for small parameter M = 1/

√N,

where N = ωmR2/(cl) 1 can be referred to as the Fresnel number, the func-tions Φν j have the form Φν(r) = Jν(kν jr)/Jν+1(µν j), where µν j is the jth root ofthe Bessel function of order ν, kν j = µν j(1 − ∆)/R, ∆ = (1 + i)β0M/2. Eachmode for ν > 0 is four time degenerate. Firstly, for each value of the az-imuthal number ν > 0 there are two linear independent functions ∝ cos(νφ)and ∝ sin(νφ). Secondly, because the boundary conditions do not dependson the polarization, two different field polarizations exist for each linearindependent eigenfunction.

The rigorous results of the three-dimensional theory of a plane Fabry-Perotresonator have shown that a radiation in the resonator may be presentedby a set of modes. Each mode is characterized by the decrement and aneigenfunction of the field amplitude distribution along the mirror surface.For the eigenmode with transverse wave number kν j, we obtain the follow-ing expression for eigenvalue: Λ = −2c/(nl) − iµ2

ν j(1 − 2∆)/(2R2). Due to theexponential dependence on µ2

ν j, only the lower order modes tend to survivein the passive open resonator. We designate a normal mode of a circularplane mirrors as a TEMν j mode, with ν denoting the order of angular varia-tions ( the angular variation are sinusoidal in form) and j denoting the orderof radial variation. The lowest order, of TEM00 mode, we designate as thedominate mode for circular plane mirrors.

Suppose that an earth-based observer performs an aberration of light mea-surement using laser as a light source. One could use a vertically orientedoptical setup. Above, we demonstrate that acceleration has an effect on thefield equations. The electric field ~E of an electromagnetic wave into theresonator satisfies the wave equation transformed into Eq. (10), where co-ordinates and time are transformed according to a Galilean transformation.At first glance, measurement of the diffraction losses of the dominate modeis equivalent to the determination of the angular displacement.

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Fig. 40. Absent ot the aberration of light from the earth-based laser source. Theelectromagnetic wave travels in the resonator forward and back reflecting fromthe mirrors. The crossed term in the Langevin metric, which generates aberration,cancels during the evolution of the radiation in the optical resonator.

It should be note, however, that there is no influence of the difference be-tween the Langevin and Minkowski metrics on the parameters of a laserwith optical resonator. In our previous discussion of the Langevin metric, welearned that after the Galilean transformation of the wave equation we ob-tain the complicated anisotropic equation Eq. (10). How shall we solve thiselectrodynamics equation together with boundary conditions? It is enoughto assume the Lorentz covariance of the electrodynamics theory involved inthe optical resonator calculations. We can make a mathematical trick for thesolution of the electrodynamics field equations with anisotropic terms: inorder to eliminate these terms in the transformed field equations, we makea change variables. The overall combination of the Langevin metric andvariable changes actually yields the Minkowski metric in the acceleratedframe, but in this context this transformation are only to be understood asuseful mathematical device, which allow one to solve the electrodynamicsproblem in the case of the Langevin metric with minimal effort.

We state that the variable changes performed above have no intrinsic mean-ing - their meaning only being assigned by a convention. It is clear thatthe convention-independent results of calculations are precisely the same inthe new variables. By changing the (four-dimensional) coordinate system,one cannot obtain a physics in which new physical phenomena appear. It

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is important to remark that a close examination of all experiments insidethe uniformly moving accelerated Sn frame, however, shows that all opti-cal phenomena in an optical resonator appeared to be independent of theacceleration relative to the fixed stars. The point is that all methods to mea-sure the radiation in an optical resonator, indeed, the standing wave (i.e.round-trip or two beams) measurements.

Let us analyze the aberration of light transmitted from the laser resonatorwhich is at rest in the accelerated frame. When the measuring data isanalyzed, the accelerated observer finds that the deviation of the energytransport direction is absent. It could be said that the crossed term in theLangevin metric, which generates aberration, cancels during the (round-trip-to-round-trip) evolution of the radiation in the optical resonator.

It is generally believed that the special relativity is a reciprocal theory. Anillustration of the absence of reciprocity is provided by asymmetry in theexperiments involved the time dilation. Imaging that there are two identicallaser sources. We set up the lasers such that light is travels between mirrorsparallel to the z-axis. Let us consider the case when the first source is at restin an inertial frame and the second source is accelerated from rest up tovelocity v along the x-axis (i.e. transverse to the direction of optical beam).Suppose that an observer in the inertial frame performs a measurementof the frequency ωm = mπc/l of the TEM00 mode. In order to measure thefrequency, the observer has to specify frequency standard. Suppose he tookan ordinary atom, which had natural frequency ωo at rest. Let us considerthe opposite situation. Suppose that an observer in the accelerated frameperforms a measurement of the TEM00 mode. It is also assumed that thesame atom at rest in the accelerated frame is used as frequency standard. Inthe case of Langevin metric Eq.(9), the speed of light propagated between themirrors in the transverse zn-axis direction cannot be equal c anymore. UsingEq.(9) we obtain (dzn/dtn)2 = c2(1 − v2/c2). Therefore, the frequency of theTEM00 mode is ωm = mπc

√1 − v2/c2/l. Because tn = t (note that coordinate

clock in the accelerated frame , although fixed in this frame , read the sametime as the clocks in the inertial frame) and ωm = mπc/l has the physicalmeaning of the frequency of the dominate mode at rest in the inertial frame,this implies that the frequency of the TEM00 mode at rest in the acceleratedframe is red shifted compared to ωm in the original inertial frame. Takinginto account the Langevin metric we obtain the time dilation of a physicalclock at rest in the accelerated frame as compared to physical clock in theinertial frame. In the next chapter we shall discuss the asymmetry in theexperiments involved the measurement of quantities of the second order inv/c.

An observer on the accelerated frame cannot observe the time dilation effect,of course, that one does not look outside. In order to detect the red shift of the

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TEM00 mode at rest in the accelerated frame, it is obvious that some referencefrequency is needed. The point is that the natural frequency of the atomobserved in the accelerated frame is ω0

√1 − v2/c2. We have found that the

accelerated observer will measure the same mode frequency. It is not hard tounderstand this result. It is clear that an observer performs the measurementof standing wave mode of the optical resonator. If we analyze the geometryof the situation, we find that from the standing wave measurements in theoptical resonator we can only extract information about two-way speedof light. In other words, the measurement of the two-way speed of lightis universal. Although the optical cavity we have been describing seemsto be quite different from a simple electromagnetic model for the light-clock of special relativity, the two systems are, of course closely related. Thediscussion of the behavior of an accelerated light clock is presented in thenext chapter.

8.4 Thought Experiment using Point-Like Source at Optical Frequencies

It is generally believed that the theory of relativity appears to conform tothe phenomenon of stellar aberration claiming it is a consequence of themotion of observers relative to light sources. In particular, the conventionalmethod used to explain the aberration phenomena is based on the belief thatthere is no aberration for the starlight radiated from the star moving withthe same velocity as the earth. However, aberration shift, as inferred fromastronomical observations, behaves asymmetrically. It should be stressedthat it is the telescope and not the star that must change its velocity (relativeto the fixed stars) to cause aberration.

We want now to discuss a simple scaling model for the stellar aberration, Fig.41. Let us consider the scale transformation of the source linear dimensionand the distance between the source and observer. We will obtain a conditionfor optical similarity between the aberration of light from a distant star andfrom the earth-based source. This scale transformation may help in theinitial design of the earth-based setup using incandescent lamp as a point-like source instead of a star.

The peculiarity of point-like sources is that radiation emitted at one instantform a sphere around the source and the measuring instrument alwaysinfluences the measured radiation. Source field diffraction can be dividedinto categories - the Fresnel (near-zone) diffraction and Fraunhofer ( ortelescopic) diffraction. If the Fraunhofer (or equivalently, far zone) approx-imation z > D2/λ is satisfied, then quadratic phase factor is approximatelyunity over the detection aperture. Here D is the aperture width, λ is thewavelength of the radiation, and z is the distance between source and ob-

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Fig. 41. Scaling model for the stellar aberration. Earth-based setup using incan-descent lamp. We know (without using our knowledge of special relativity) whatwould happen from the astronomical observations. Without looking at anythingexternal to the earth, one could determine the speed of the earth around the sun bymeans of aberration of light measurements.

server. In the far zone a point source produces in front of pupil detectioneffectively a plane wave. The plane wave fronts of starlight entering thetelescope are imaged by the lens to a diffraction spot which lies in the focalplane. It worth noting that we consider the divergence of the transmittedradiation, λ/D, that is relatively small compared to the aberration angle,θa = v/c.

An incandescent lamp is a completely incoherent source. The character ofthe mutual intensity function produced by an incoherent source is fullydescribed by the Van Cittert-Zernike theorem (8). In our case of interest,imaging system is situated in the far-zone of the source (9). Under thesecircumstances, the Van Cittert-Zernike theorem takes its simplest form. It isthe source linear dimension d (source diameter) that determines the coherentarea of the observed radiation (zλ/d)2. The condition for neglecting thetransverse size of the source, d, formulated as the requirement that thechange of the correlation function along the aperture be less than unity, thatis d D.

Let us estimate the parameters for the earth-based setup using incandescentlamp. We want to describe some thought - but possible - experiment. At

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optical frequencies, the conditions required for validity of the Fraunhoferapproximation can be sever ones. For example, at a wavelength ofλ = 0.3µmand an aperture width of D = 3 cm, the observation distance must satisfyz > 3000m. Obviously, the required conditions are met in a number ofimportant problems. No one has ever done such experiment, but we know(without using our knowledge of special relativity) what would happenfrom the astronomical observations. The purpose of this thought experimentis to suggest that without looking at anything external to the earth, one coulddetermine the speed of the earth around the sun by means of aberration oflight measurements.

The aberration of light problem in the accelerated frame demonstrates theessential asymmetry between the non-inertial and inertial observers. Thereis a certain degree of analogy between the aberration of light and the Sagnaceffect. Clearly, it is possible to detect a rotating velocity without lookingoutside using Sagnac phenomenon. A close look at the physics of these twosubjects shows things which are common to these phenomena: In both situ-ations these are experiments in the first order in v/c. Indeed, the aberrationof light is the geometric effect in our ordinary space which is (similar to theSagnac phenomenon) closely associated with the first order (crossed) termin the Langevin metric.

8.5 Working Way of the Setup Using Incoherent Source at Optical Frequencies

It should be note that Fraunhofer diffraction patterns can be observed atdistances much closer than implied by equation z > D2/λ provided theaperture is illuminated by a spherical wave converging towards the ob-server, or if a positive lens is properly situated between the observer andthe aperture.

A working way of the earth-based optical setup to detect the aberrationof light is schematically shown in Fig. 42. This is the well-known two-lensimage formation scheme. This scheme allows for magnification by changingthe focal distance of the second lens but for simplicity, in the following wewill assume that the two focal distances are the same (i.e. we consider 1:1imaging). This two-lens setup is usually employed for image-processingpurposes, as it can be better used for image modification compared to thesingle-lens system. Given the two-lens setup discussed above, we considerthe relatively simple problem of characterization of point source radiationin the image plane. We assume that the two lenses in Fig. 42 are identical.

To perform an aberration of light measurement, the source linear dimensiond should be radically decreased. We assume that the order of magnitude of

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the dimensions of the ”point” source is about λ, where λ is the optical ra-diation wavelength. Within an elementary source volume (λ3) there is anenormous number of atoms. Semi-classical theory is treating these atomsas coherent, interacting radiating dipoles. The induced macroscopic dipolemoment in an elementary source leads to the classical electromagnetic radi-ation. In the optics for such ”point” source emission, the fields are describedclassically at the level of Maxwell’s equations and the emitting mediumis treated (as a ensemble of atoms) by quantum mechanics. The atomicpolarization induced by the radiation field appears as a driving term inMaxwell’s equations and sustains the oscillations. With the help of mod-ern lithography it is not difficult to produce an unresolved point source atoptical wavelengths and let sufficiently bright radiation from it.

Consider a diffracting aperture that is circular, and let the radius of theaperture be a. The physical properties of the lens can be combined in a singlenumber f called the focal length. We assume that the image produced by adiffraction-limited optical system (i.e. a system that is free from aberrations).Once the wavelength is fixed, the resolution only depends on the numericalaperture NA = a/ f 1 of the system. The response of the system at point(xi, yi) of the image plane to a δ function input at coordinates (xo, yo)ofthe object plane is called the point spread function of the system. In ourcase of interest, one obtains the following amplitude point spread function:A(xi, yi) = J1(α)/α, which is the diffraction pattern of a circular aperture,where α = 2πari/( fλ), x0 = 0, y0 = 0, ri = (x2

i + y2i )1/2. The first zero of the

Airy’s pattern is at ri = 0.6λ/NA.

Let us estimate the parameters for the optical setup shown in Fig.42. Supposea = 1.5 cm and λ = 0.3µm. We will use NA = 0.1. The resolution analysis isreduced to the theoretical framework of standard imaging theory. One cantake advantage of well-known resolution criteria like Rayleigh criteria i.e.0.6λ/NA = 2µm. One could use a vertically oriented optical setup. Constantof aberration defines the scale of the aberration increment, and is equal tov/c = 100µrad. Measurement of the spot shift with respect to the optical axis(which is parallel to the nadir direction) is equivalent to the determination ofthe angular displacement. As already pointed out, the proposed method ofmeasuring the angle of aberration involves the use of earth-based sourcesand have a big advantage. The rotation of the earth on its axis shouldproduce a corresponding shift of the image. In principle, records could betaken over a period of one day. The image appears to move in an ellipse. Themajor axis of the ellipse is about 14 µm. The value of minor axis depends onthe observer’s position and is given by 14(sinφ) µm.

We derived our results under the assumption that the order of magnitude ofthe dimensions of the source is about λ. It should be note that the resolutionof proposed optical techniques is not limited by size of incoherent source.

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Fig. 42. A working way of the earth-based optical setup to detect the aberration oflight. Two-lens image formation setup using the earth-based incoherent source atoptical frequency.

Radiation field generated by an incoherent source can be seen as a linearsuperposition of fields of individual elementary point sources. The image ofan elementary point source is a point spread function. Optical system usingincoherent light could be completely described in therm of convolutionintegral relating the object distribution to the image distribution. In sucha description, the optical system is characterized by its impulse response,i.e. the point image. The rounding or softening of the source’s edges ischaracteristic of the convolution (”smoothing”) integral.

It should be remarked that any linear superposition of radiation fields fromelementary point sources conserves single point source characteristics likeoscillatory image shift with a period of one day. This argument gives reasonwhy our technique can be extended to allow for the reconstruction of theangle of aberration from a measurement of image shift of incoherent source.The edge object has become useful in aberration shift detection. An alterna-tive application of a ”point” source is an incoherent source with (physically)sharp edges. The edge width of the source is strictly related to the practicalresolution achievable. For our a particular example, the edge width shouldbe about 1µm .

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1. The dedicated STEM with high stability was made [38]. It was developedbased on a 200-kV commercial instrument. The electron source was installedin the anti-seismic room. Temperature fluctuations in the room were within0.2 oC. The magnetic field in the room was less than 0.2mG near the sourcecolumn. To prevent thermal drift, a shield box was placed over the high-voltage generator. The column was covered with rubber sheets to reducethe temperature fluctuations. The image drift measured during a period of61 minutes was ' 6 nm. The drift speed and direction during this periodwere approximately constant. The average drift speed was 0.12 nm/min. Itis substantially smaller than that conventional commercial microscope, inwhich the drift speed is about 2 nm/min.

2. The rotation of the earth on its axis produces aberration in an amountlarge enough to be taken into account in precise observation work usingelectron microscope as a particle source. The Wigner rotation angle pa-rameter (constant of diurnal aberration) defines the scale of the aberrationincrement, and is equal to θw = (1 − 1/1/γp)ve/V ' 0.6µrad. Here γp ' 1.39,V/c = 0.7, ve/c ' 1, 55µrad where ve = 0.45km/s is the linear velocity of theearth rotation at equator.

3. Electron-optical system of the STEM contains an electron gun and severalmagnetic lenses, stacked vertically to form a lens column. The illuminationsystem of the instrument comprises the electron gun, together with two con-denser lenses that focus the electrons onto the spacemen. Electrons enteringthe lens column appear to come from a virtual source. The first condenserlens (C1) is a strong magnetic lens, with a focal length f1 of typically 2mm. Using a virtual electron source as its object, C1 produces a real image.Because the lens is located D1 = 15 cm bellow the object (virtual source),its object distance is 15 cm f1 and the image distance ' f1. The secondcondenser lens (C2) is a weak magnetic lens ( f2 ' 10 cm) that provideslittle magnification but allows the diameter of illumination at the specimento varied continuously over a wide range. Distance between lens centerstypically 10 cm. Spacemen is located D3 = 25 cm bellow the C2.

4. Experimental results (see Fig. 44 p.167 in [38]) shows that the image shiftis quite close to the theoretical prediction ' 10 nm. It is important to notethat we used the typical focal length (2 mm) in our estimations. It is alsoassumed that the second condenser lens provides no magnification (M ' 1).The details of the optical setup may influence the image shift.

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5. We have presented an example of introducing an approximate boundaryconditions. This approximate boundary conditions were derived by Leon-tovich. They are valid on the surface of a material having a large value ofthe modulus of the refractive index. An extended theoretical treatment ofLeontovich’s boundary conditions appears in the monograph [39].

6. Analytical method is based on the introduction of particular, complexboundary condition, which are called impedance boundary conditions [40].Once these boundary conditions for the field are formulated on the virtualside surface of the open plane resonator, the original problem, which is anopen resonator excitation, is simplified to the that of a closed resonator.

7. β0 turns out to be related with one of the most famous mathematicalfunctions, the Riemann zeta function ζ. In fact, it is given by −ζ(1/2)/

√π,

see [40].

8. The original optical source (such as a star or an incandescent lamp) consistsof an extended collection of independent radiators. Accordingly, it is of somespecial interest to know precisely how mutual intensity propagates awayfrom an incoherent source. The character of the mutual intensity functionproduced by an incoherent source is fully described by the Van Cittert-Zernike theorem [33].

9. In all cases of practical interest, telescopes are situated in the far zone of thedistant stars. If the far zone approximation z > 2d2/λ is satisfied, then the VanCittert-Zernike theorem takes its simplest form. For example, the coherencearea of the light emitted by the circular incoherent source of diameter d atdistance z is Ac = 4λ2z2/(πd2). The maximal coherent area of light observedon the earth’s surface from a nearest star has diameter of about 10 m. Inother words, any star can be considered as a point source and a star image isactually a point spread function in the image plane of a telescope. The starconsists of an extended collection of independent ”point” sources. The orderof magnitude of the dimensions of the elementary statistically independentsource is about λ. It is obvious that the star diameter d is much larger thana telescope aperture D and that far zone condition z > D2/λ for elementary”point” source is automatically satisfied. Our scaling model is based on thefar zone approximation. We have only to change the distance and the sourcediameter. With the limitation of the small far zone parameters (d2/(zλ) < 1,D2/(zλ) < 1) assumption, the proposed scaling method gives d D.

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9 Special Relativity. Absence of Reciprocity

In this chapter we shall discuss the observations of a non-inertial observer.Common textbook presentations of the special relativity use the approachwhich deals only with observations of an inertial observer and, conse-quently, only with a coordinate transformations within a single inertialframe. However, one often has occasion to make measurements with non-inertial devices; for example, aberration of light measurements made ina laboratory rotating with the earth. The situation with measurements inspecial relativity is not symmetrical with respect to the change of the ref-erence frames. The fact that in the real process of observer transmission toa comoving reference frame (i.e in the process of an observer acceleratingwith respect to the fixed stars) the metric of the observer is changed, is notconsidered in textbooks.

9.1 Time Dilation

It is generally believed that the special relativity is a reciprocal theory. Anexcellent illustration of the absence of reciprocity is provided by the asym-metry in the experiments involved the measurement of quantities of thesecond order in v/c. Let us discuss the time dilation effect. Consider thecourse of time in two inertial reference frames, one of which, S, will be con-sidered to be at rest, while another one, Sn, will accelerate with respect tothe first one up to velocity v along the x- axis. As we already discussed, thetransformation from the inertial frame to the accelerated frame apparentlyfirst used by Langevin. The transformation to the accelerated frame, withcoordinates denoted by ”n”, has the Galilean form, t = tn, x = xn + vtn. Interms of these new coordinates the invariant interval given by Eq. (1) isEq.(9). Note that coordinate clocks on the accelerated frame, although fixedin that frame, read the same time as the clocks in the inertial frame. With thecross-term, dtndxn, clocks in the accelerated frame are not synchronized inthe standard way (Einstein synchronization) of sending a light signal backand forth between the clocks.

Suppose we have two identical physical clocks at one and the same pointof the inertial reference system S. Consider their readings to coincide atthe initial moment t = 0. Let first of these clocks always be at rest in theframe S. At moment t = 0 the second clock accelerates and starts to movewith a constant velocity v along the x axis. Now consider the reading ofthe clocks in the accelerated reference system, where the second clock isalways at rest. The system Sn is not inertial, since it accelerated with respectto the fixed stars. In the accelerated frame the first clock moves with velocity

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dxn/dtn = −v. Taking into account the Langevin metric Eq.(9), ds2 = c2(1 −v2/c2)dt2

n − 2vdxndtn − dx2n, we obtain dτ1 = ds/c = dtn, i.e. the time, shown by

the first physical clock coincidence with the time t = tn of coordinate clock.Since the second physical clock is at rest, its reading of its proper time isdτ2 =

√1 − v2/c2dtn. This demonstrates the time dilation of a physical clock

at rest in the accelerated frame as compared to physical clocks in the inertialframe. We have found that the slowly down of the accelerated physical clockdoes not depend on the choice of reference system, in which this effect ismeasured as it must be.

9.2 Light-Clock. Observations of the Noninertial Observer

9.2.1 Introductory remarks

In order to understand the slowing of the clock in an accelerated system,we have to watch the machinery of the clock and see what happens whenit is moving. We shall try to understand the effect in a very simple case. Aclock which we shall call ”light-clock” is a rod (meter stick) with a mirrorat each end. The tick of time is the reflection of light in the mirror. Let usnow consider the parallel and orthogonal to motion orientations of the lightclock. The equivalence of all clocks underlines the theory of relativity andwe shall show that irrespective of the orientation of the light path withrespect to the motion of the light-clock, the clock scores the same time. Twolight-clocks are shown in Fig. 43

9.2.2 Parallel Light-Clock. Langevin Metric

Now we have to watch the machinery of the physical clock and see whathappens to the accelerated light-clock. We describe the clock based on theobservations made by an observer in the same accelerated frame as theclock. In a first step, we set up the light clock such that light travels betweenmirrors parallel to the direction of fixed stars motion. In the case of Langevinmetric, the speed of light radiated by the accelerated emitter cannot be equalc anymore. If ds is the infinitesimal displacement along the world line of abeam of light, then ds2 = 0 and using Eq.(9) we obtain c2 = (dxn/dtn + v)2.This means that in the accelerated reference system of coordinates (ctn, xn)the velocity of light radiated by the accelerated emitter parallel to the xn-axis, is dxn/dtn = c − v in the positive direction, and dxn/dtn = −c − v in thenegative direction. Designating the rod (meter stick) length, unknown sofar, by ln, we obtain the time in which light gets from the left mirror to theright mirror, t1 = ln/(c − v), and the time in which light gets from the rightmirror to the left mirror, t2 = ln/(c + v). Therefore, the time interval between

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Fig. 43. Parallel and orthogonal light-clock setups.

the sending and reception of the light signal is ∆t = t1 +t2 = 2ln/[c(1−v2/c2)].

One must first determine the length of physical rods. For the measurementof distances between fixed points on the accelerated frame we shall usephysical measuring rods of the same kind as those used in the inertialframe, but now at rest relative to the accelerated frame. In connection withthe process of measuring distances in accelerated frames a problem ariseswhich did not occur in inertial frames. For according to the special relativity,no rigid bodies can exist, since they would provide a means of transmittingsignals with velocity larger than velocity of light.

With regard to spatial distances, the interpretive principle is that√

−ds2

gives the length of an infinitesimal rod whose endpoints are simultaneousaccording to standard simultaneity in the rod’s rest frame. Taking into ac-count the Langevin metric Eq.(9), we obtain at dtn = 0 that dxn = dx, i.e. thelength of physical rod in the inertial frame coincidence with the coordinatedistance in the accelerated frame. When we apply this rule to physical rodsthat are at rest in the accelerated frame, we encounter the complication thatdtn = 0 does not automatically correspond to standard simultaneity in theaccelerated frame. The length will surely be different for a physical rod atrest in the accelerated frame compared to a physical rod at rest in the inertialframe.

The spatial geometry in the accelerated frame is characterized by the dif-ferential spatial line element on it. It should be note that spatial geometry

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has a conventional character. However, the proper spatial line element dln isof special significance, since it defines a coordinate-independent spatial ge-ometry in the accelerated frame. The relation dl2

n = (−gαβ + g0αg0β/g00)dxαdxβgives the connection between the 3-dimensional spatial line element and themetric of the four-dimensional space-time. It can be shown that the elementdln is invariant under a transformation connecting two different coordinatesystems inside the same frame of reference. We note that the spatial lineelement dl obtained by putting dtn = dt = 0 in Eq.(9), dl2 = dx2

n, is differ fromdln.

With the help of Eq.(9), we find dl2n = dx2

n/(1 − v2/c2), which determinesthe spatial geometry in the accelerated frame. This equation is asymmetricwith respect to the lengths dxn = dl and dln, since it relates the physicallength dl of the rod in the inertial frame S to the physical length dln in theaccelerated frame Sn. Our analysis based on using standard measuring rodsin an accelerated frame to measure it geometrical properties shows thatcoordinate distance dxn has physical length dxn/

√1 − v2/c2. With reference

to the inertial frame this is explained as due to Lorentz contraction of themeasuring rods. With reference to the accelerated frame it is interpreted asan effect of acceleration with respect to the fixed stars, i.e. the geometry ofspace is non-Euclidean in an accelerated frame.

This shows that the physical meter stick on which the mirrors are mountedundergoes compress on in the direction of fixed stars motion. Thus wehave ascertained the asymmetry of meter stick’s length directly from themetric. Therefore, the time interval between the sending and reception ofthe light signal in the parallel light-clock in the accelerated frame is ∆tn =

(2l/c)/√

1 − v2/c2. Because tn = t and ∆t = 2l/c has the physical meaning ofthe time indicated by a physical clock at rest in the inertial frame, this impliesthat physical clocks at rest in the accelerated frame are slow compared tophysical clocks in the original inertial frame.

9.2.3 Orthogonal Light-Clock. Langevin Metric

Let us consider a light-clock accelerated up to velocity v transverse to thedirection of optical pulse. In other words, we set up light-clock such that theeffect of a meter stick contraction is not present. We describe the orthogonallight-clock based on the observations made by an observer in the sameaccelerated frame as the light-clock. If ds is the infinitesimal displacementalong the world line of a beam of light, then ds2 = 0 and using Eq.(9) weobtain (dzn/dtn)2 = c2(1 − v2/c2). Therefore, the time interval between thesending and reception of the light signal is ∆tn = 2l/[c

√1 − v2/c2]. Because

tn = t and ∆t = 2l/c has the physical meaning of the time indicated by aphysical clock at rest in the inertial frame, this implies that physical clocks

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at rest in the accelerated frame are slow compared to physical clocks in theoriginal inertial frame. The theory of relativity shows us that irrespective ofthe orientation of the light path with respect to the motion of the light-clock,the clock scores the same time.

9.3 Experimental Evidence for the Absence of Reciprocity

It should be emphasized that the decisive experiments for the reciprocalspecial theory of relativity had to be such as involved the measurement ofquantities of the second order in v/c. Let us consider the transverse Dopplereffect. The measurements of the resonance absorption of gamma rays in aMossbauer rotating disk experiments for the same relative velocity betweenthe source and the observer give a blue shift when the source is at the center,and a red shift when it is at the tip. This experiment demonstrates that thereis no reciprocity in time dilation (1).

The source of the clock paradox is an assertion that when there is relativemotion of two clocks - they experience reciprocal time dilation. Imagingtwo observers, first observer is at rest in the center of a circle, and secondobserver is moving in a circle. Observer on the circle cannot observe theeffect of slowness of the clock and red shifting of observer’s source locatedin the center of a circle. He has no right to explain the blue shift of the sourcein terms of the transverse Doppler effect caused by the central observer’scounter motion. The accelerated observer on the circle can account for theblue shift by the fact that, according to the Langevin metric, he experiencestime dilation and, being in delayed time conditions, perceives light as blueshifted. The four-geometric approach does not need any tricks to explainthe rotating disk experiment. The blue-red shift asymmetry in the geometricapproach is an elementary consequence of the relative acceleration of theobservers with respect to an important third frame - the fixed stars (i.e.remaining matter in the universe).

The historical rate of the studies of the optical effects in rotating framesis very unusual. We are dealing with classical effects such as Sagnac effect,transverse Doppler effect, and aberration of light effect, and which are quan-titative described by Langevin metric. The Sagnac effect was discovered in1913 and described by Langevin in 1921. The blue shift in a Mossbauer ro-tating disk experiments turned out to be observed and explained in 1965.It is interesting that the Langevin metric has in the past hardly ever beenapplied in the aberration of light theory.

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9.4 Pseudo-gravitation Field and the Langevin Metric

This section concentrates on the nature of special relativistic effects. Langevinmetric associated with the transformation from the inertial coordinate sys-tem to the rotating coordinate system may be regarded as a result of theaction of certain force. This problem formulated within the framework ofgeneral relativity. Note that it is not correct to say that this is explana-tion of the effects discussed in the last four chapters. One of the aims inthis section is to show that one can describe the non-inertial frames in thepseudo-Euclidean space-time using the language of general relativity.

Let (ct, x, y, z) again be Lorentz coordinates in a system of inertia S and asystem Sn in the frame S is uniformly accelerated from the rest with respectto the system S up to velocity v along the x-axis. The active Galilean boostxn = x−vt, tn = t then defines a new system of reference which is the systemof inertia Sn moving in the direction of the x-axis with the velocity v relativeto S.

We can treat a uniformly accelerating frame as if it were an inertial framewith the addition of a uniform pseudo-gravitational field. By a ”pseudo-gravitational field”, we mean an apparent field (not a real gravitationalfield) that acts on all objects proportionally to their mass; by ”uniform” wemean that the force felt by each object is independent of its position. This isbasic content of equivalence principle. The principle of equivalence can beapplied to solve non-inertial kinematic problems with dynamics methods.

In order to keep the mathematical complexity of the discussion to a mini-mum, we will describe the effects by working only up to the second orderin v/c. Suppose that each reference point in the system S has a constantacceleration g in the negative direction along the xn- axis during the timeT = v/|g|. In other words, the system S is accelerating (freely falling) ina pseudo-gravitational field. It is now immediately clear that the pseudo-gravitational acceleration is simply equal to the gradient of the scalar poten-tial: g = −∂φ/∂xn. A simple calculation gives xn = gT2/2 + x at t = T. Whenthe system S starts moving with constant velocity the pseudo-gravitationalfield is zero. For the pseudo-gravitational potential difference between thetwo frames we get after some calculation φ = −(∂φ/∂xn)gT2/2 = −v2/2. It isassumed that φ = 0 at x = xn.

The equivalence principle analysis of the accelerated frame does not use anyreal gravitational field, and so does not use the general relativity. Neverthe-less, what general relativity does say about real gravitational fields doeshold in the restrictive sense for pseudo-gravitational fields. One thing weneed here is that time runs slower as you descend into the potential well of

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a pseudo force field. We can use that fact to our advantage when analyzingthe metric in the non-inertial frame.

The equivalence principle implies that gravity can shift the frequency of anelectromagnetic wave, and cause clocks to run slow. The light frequencyincreases with increase absolute value of the potential of the gravitationalfield. If a ray of light emitted at a point where the gravitational potential isφ1, has at that point the frequency ω1, then upon arriving at a point wherethe potential is φ2, it will have a frequency measured in units of the propertime at that point equal to ω2 = ω1[1 + (φ1 − φ2)/c2].

Frequency is proportional to the inverse of the local proper time rate; thegravitational frequency shift formula can be converted to a time dilationformula. The rate of a coordinate clock thus depends on the gravitationalpotential at the place where the clock is situated. The relation betweencoordinate time and proper time can be written in the form dτ = dtn(1 +φ/c2). Thus proper time elapses the more slowly the smaller the gravitationpotential at a given point in a space, i.e. the lager the absolute value. If of twoidentical clocks is placed in a gravitation field for some time, the clock whichhas been in the field will thereafter appear to be slow. For clock at rest inthe accelerated frame we have dτ = dtn(1 +φ/c2) = dtn[1− v2/(2c2)]. The factthat the inertial observer always turns out to have aged more than the non-inertial one, in the equivalence principle analysis is elementary consequenceof the slowly down of the rate of the processes in the pseudo-gravitationalpotential, which can be shown by means of simple algebraic calculations.

Furthermore it is important to point out that the phenomenon of the relativ-ity o simultaneity can be understood in terms of dynamical consideration.The time-offset relation can be interpreted as the accumulated time dif-ference between two spatially separated clocks because of pseudo-gravityexperienced by the accelerated observer. This suggests that the clock athigher gravitational potential (placed along the direction of acceleration)runs faster. For the pseudo-gravitational potential difference between twopoints along the xn-axis we get ∆φ = φ1−φ2 = (∂φ/∂xn)[xn(1)−xn(2)]. Whenthe system S starts moving with constant velocity the gradient of poten-tial in the system Sn is zero. For accumulated time difference between twospatially separated clocks we have tn(1)− tn(2) = −g[xn(1)− xn(2)]v/(c2

|g|) =v[xn(1) − xn(2)]/c2.

The next thing we must investigate is the origin of the Lorentz deformation.Here we may point out that the Lorentz deformation (changes of the relativeposition in the xn direction) is closely related to the shift in the momentsof the acceleration. With the help of time-offset relation, integrating withrespect to the infinitesimal value dv we find dln = dxn[1 + v2/(2c2)], whichdetermines the spatial geometry in the accelerated frame. This equation re-

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lates the length dl = dxn of the rod in the inertial frame S to the length dln inthe accelerated frame Sn. Our analysis based on using standard measuringrods in an accelerated frame to measure it geometrical properties showsthat coordinate distance dxn has length dxn[1 + v2/(2c2)]. The coordinate ap-proach shows clearly inevitable consequence: In accelerating process therod in the accelerated coordinate system Sn must be understood to developa contraction. This assertion may sound bizarre, but is a purely kinematiceffect, involving no forces. We come to the conclusion that the various rel-ativistic effects are not independent of each other, but that these effects canall be understood through a general equivalence principle.

We have now all quantities we wanted. Let us put them all together into theexpression for the new space-time variables (2). The new independent vari-ables (x(d)

n , t(d)n ) can be expressed in terms of the old independent variables

(xn, tn): ct(d)n = tn[1 − v2/(2c2)] − (vxn/c2)[1 + v2/(2c2)], x(d)

n = xn[1 + v2/(2c2)].Above we made a simplification in calculating time and space transfor-mations by considering only low velocities. The physical principles thatproduced the space-time transformation were, however, made clear. Thenext question is: What is the transformation in the case of an arbitrary ve-locity? That is easy: the answer is t(d)

n = tn√

1 − v2/c2 − (vxn/c2)/√

1 − v2/c2,x(d)

n = xn/√

1 − v2/c2. Quantity dt(d)n characterizes physical time, which is

independent on the choice of coordinate time. Let us note that tn in theexpression for t(d)

n is the coordinate time for the accelerated frame. Physicaltime determines the flow of time in a physical process. The |dx(d)

n | is nothing,but the physical distance between two points of 3-space. In new physical co-ordinates we obtain interval in the following (Langevin) form ds2 = (cdt(d)

n )2

- (dx(d)n )2 = (1 − v2/c2)c2dt2

n − 2vdxndtn − dx2n.

The previous derivation of the Langevin metric in the accelerated framefrom pseudo-gravitation field includes one delicate point. Metric appliesto physical laws, not to physical facts. We interpret the Langevin metric tomean that the law of electrodynamics is expressed by equation that havethe form Eq.(10) in the accelerated frame. By a physical facts in this contextwe mean the aberration of light radiated by a single ”plane wave” emitterin the accelerated frame. The electrodynamics equation needs to be inte-grated with initial condition for the radiation wavefront. From the specialtheory of relativity we know that acceleration has no effect on the wave-front orientation. In fact, the variables (t, x, y, z) can be expressed in terms ofthe variables (tn, xn, yn, zn) by means of Galilean transformation xn = x − vt,tn = t, so that the wavefront of the emitted light beam is perpendicular tothe vertical direction zn after the acceleration, Fig. 24.

The following simple analysis in the framework of general relativity con-firms this initial condition. We treat accelerated frame as if it were an inertial

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frame with addition of a pseudo-gravity. First, one could expect that the ac-cumulated time difference between spatially separated clocks (because ofpseudo-gravity experienced by the accelerated emitter) has the effect ofrotation the radiation wavefront on the angle v/c in the counter clockwisedirection. Second, the overall combination of the (reciprocal) active Galileanboost of the inertial system S with velocity −v and gravitational space-timetransformation actually yields the Lorentz boost as viewing this of the ac-celerated observer.

Now the accelerated observer sees the Wigner rotation of the moving frameaxes with respect to the accelerated frame axes in the clockwise direction onthe angle −v/c. It comes quite naturally that the wavefront of the light fromthe ”plane wave” emitter which is at rest in the inertial frame S is perpendic-ular to the vertical direction zn in the accelerated frame Sn. According to theLangevin metric, there is a remarkable prediction on the theory of the aber-ration of light. Namely, the apparent angular position of the ”plane wave”emitter seen in the accelerated frame through the aperture would jump byangle −v/c. That is transmission through the aperture in the acceleratedframe Sn appears as shown in Fig. 14.

Above we considered a single moving ”plane wave” emitter in an accel-erated frame. Now let us return to observations of a point source. Theaberration of point source is considered independent of the source speedand to have just a local origin exclusively based on the observer (with hismeasuring instruments) speed relative to the fixed stars Fig. 20, Fig. 21. As-tronomical observations show that this prediction is, in fact true. The pointsource can have velocities relative to earth very different from one another,nevertheless it follows the same pattern as fixed stars under the same aber-ration angle i.e. its apparent position changes with an angular displacementcommon to fixed stars.

An idea of treating the accelerated frame using the equivalence principle isuseful from a pedagogical point of view. It is worth remarking that the absentof a dynamical explanation for the Langevin metric in special relativity hasdisturbed some physicists. A good way to think of the asymmetry betweenthe inertial and accelerated frames is to regard it as a result of pseudo-gravityexperienced by the accelerated observer.

When the system Sn starts moving with constant velocity it will be inertialframe of reference. At first glance, we have the symmetry (reciprocity) be-tween the inertial frame S and inertial frame Sn. Where does the asymmetrycomes from? Equivalence principle analysis does not need any tricks to ac-count for the asymmetry between an inertial frame and a non-inertial frame.A resolution of the asymmetry paradox identifies a pseudo-gravitationalfield within the system Sn as the agency of asymmetry.

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Although both observers consider themselves as at rest, they both predictthat the Maxwell’s equations continue to hold in the inertial frame andthe Minkowski metric is not applicable in the accelerated frame. In fact,the Langevin metric associated with accelerated reference frame predictsan asymmetry in the one-way speed of light. But observers have totallydifferent explanation for this difference of metrics. Inertial observer saysthat physical processes in the accelerated frame describes by the Langevinmetric due to the active Galilean boost. In other words, the analysis ofphysical phenomena in non-inertial frames of reference can be described inan inertial frame within standard special relativity. Accelerated observer, onthe other hand, says that Langevin metric in his frame can be understoodin terms of dynamical consideration. He can apply the idea of gravitationaltime dilation, Lorentz deformation, and mixture of positions and time. Theexplanation of the Minkowski metric associated with the inertial frame Scan be described in the accelerated frame within the same approach.

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1. The measurements of the resonance absorption of gamma rays in a Moss-bouer rotating disk experiment [41] for the same relative velocity betweenthe source and the observer give a blue shift when the source is at the center,and a red shift when it is at the tip.

2. We should make one further remark about the equivalence principle. Thespecial theory of relativity, using equivalence principle, can be made into auseful tool for a discussion of a variety of effects in relativity theory. Withthe special relativity we can discuss the case of earth-based experimentswhere pseudo-gravity and the real gravity are inextricably mixed. In thiscase, the introduction of sun gravitation can be accomplished by usingthe equivalence principle, and the components of the metric tensor areg00 = (1 + 2φ/c2

− v2/c2), g01 = −v/c, g11 = −1.

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10 Space and Time Coordinates in an Inertial Frame

10.1 Active and Passive Transformations

Now let us return to observations of an inertial observer. To continue ourdiscussion of the space-time transformations and relativistic effects, we con-sider so-called active and passive transformations. First, we notice that eachphysical state, motion, or effect admits many different descriptions. The va-riety of methods of description physical systems makes it possible to selectfor each problem the representation most suitable for its solution. It is al-ways possible to create a new frame of reference by relabeling coordinates,and then discussing physical phenomena in terms of the new coordinatelabels - a passive transformation. An active transformation of a physicalsystem is its motion, i.e., a variation in its characteristics under the effect ofsome internal or external interactions. We are dealing with motion of thesame physical system, evolving in time and treated from the point of viewof the same reference system.

Imaging that there are two identical emitters. The first emitter is at rest in theobserver frame and the second emitter is accelerated up to velocity v alongthe x-axis. This circumstance is called ”active boost”. In the case of an active(physical) boost of velocity we consider the effect of interaction on motionwhich defined in terms of accelerating motion relative to the fixed stars.Thus when we state that the second emitter undergoes an acceleration, andthe inertial observer (with measuring devices) does not, the accelerationmeans acceleration relative to the fixed stars. Any acceleration relative tothe fixed stars (i.e. any active boost of velocity) has an absolute meaning.

Let us consider the case when the velocity of light emitted by a source at restin the inertial coordinate system (t, x) for S is c. In that case the Minkowskimetric ds2 = c2dt2

− dx2 associated with inertial frame S predict a symmetryin the one-way speed of light radiated by the first emitter. As a result of theactive Galilean boost, we obtain the metric of the accelerated emitter

ds2 = c2(1 − v2/c2)dt2 + 2vdxdt − dx2 . (11)

Inspecting Eq. (11), we can find the components of the metric tensor gµνin the coordinate system (ct, x) of S. We obtain g00 = 1 − v2/c2, g01 = v/c,g11 = −1.

In the chosen coordinatization the velocity of light radiated by the firstemitter is c. In this coordinate system, however, the speed of light radiatedby the second emitter cannot be equal to c anymore. If ds is the infinitesimal

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displacement along the world line of a beam of light, then ds2 = 0 andusing Eq. (11) we obtain c2 = (dx/dt − v)2. This means that in the lab inertialreference system of coordinates (ct, x) the velocity of light radiated by theaccelerated emitter parallel to the x-axis, is dx/dt = c + v in the positivedirection, and dx/dt = −c + v in the negative direction.

We conclude that the speed of light emitted by a moving source measuredin the inertial frame (t, x) depends on the relative velocity of source andobserver, in our example v. In other words, the speed of light is compatiblewith the Galilean law of addition of velocities. The reason why it is differentfrom the electrodynamics constant c is due to the fact that we chose the sim-plest method of synchronization. This method consists in keeping, withoutchanges, the same set of uniformly synchronized clocks used in the casewhen the second light source was at rest. When the second emitter startsmoving with constant velocity the clock synchronization is still defined interms of light signals emitted by a source at rest assuming that light prop-agates with velocity c. This simplest synchronization convention preservessimultaneity and using such synchronization procedure we actually selectthe absolute time synchronization for the moving emitter. Note that fromwhat we just discussed follows the statement that the difference between thespeed of light and the electrodynamics constant c is convention-dependentand has no direct physical meaning.

In order to examine what parts of the dynamics and electrodynamics theorydepend on the choice of that convention and what parts do not, we want toshow the difference between the notions of coordinate time and proper time.The proper frame can be fixed to moving object. The object is at rest in thisframe, so that events happening with this object are registered by one clock.Since the first clock is at rest in the inertial frame S, its reading of its propertime is dτ1 = ds/c = dt, i.e. the time, shown by the first clock coincidencewith the coordinate time t. In the inertial frame S the second emitter moveswith velocity dx/dt = v. Taking into account the metric ds2 = c2dt2

− dx2, weobtain dτ2 =

√1 − v2/c2dt. This demonstrates the time dilation of a physical

clock at rest in the accelerated frame as compared to physical clock in theinertial frame. The slowly down of the second clock, as compared to thefirst, is an absolute effect and does not depend on the choice of referencesystem, in which this effect is computed.

Let us consider now a passive boost of velocity. At passive boost a four-vector of event is thought to be fixed and one system of coordinates changeswith respect to the other coordinate system. It should be clear that a goodway to think of coordinate transformations is to regard it as a result ofchange variables. By changing a four-dimensional coordinate system, onecannot obtain a physics in which new physical phenomena appear. But wecan obtain a more suitable description of a physical system.

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For example, suppose we transform the metric Eq. (11) from the variables(x, t) to variables convenient to describe the radiation of the moving emitter.It is possible to chose such variables, in which metric of the acceleratedemitter will be diagonal. We chose the new coordinate x′ = x − vt. Thistransformation completes with the invariance of time, t′ = t. We begin withmetric in the coordinate system (x, t) and substitute x′ = x − vt, t′ = t intothe Eq.(11) to obtain Minkovski metric ds2 = c2dt′2 − dx′2. In other words,we describe a complicated situation by finding a coordinate system wherethe analysis is already done (radiation in the case where a source of lightis at rest) and transforms radiation back to the old coordinates. We discussthe methodology of solving synchrotron radiation problem involving activeand passive boosts in Subsection 13.6.2

To continue our discussion a passive boost of velocity, let us consider themetric associated with the first emitter. We begin with metric in the coordi-nate system (x, t) and substitute (x′, t′) into the Minkowski metric to obtainds2 = c2(1− v2/c2)dt′2 − 2vdx′dt′ − dx′2. This metric is reciprocal to the metricof the accelerated emitter in the coordinate system S, Eq.(11). There is aproblem here. According to this result, time dilation is symmetric and thereis a disagreement with experiments.

It is generally accepted that the principle of covariance implies the equiv-alence of active and passive transformations. There is a widespread beliefamong experts that the equivalence of the active and passive pictures is dueto the fact that moving system one way is equivalent to moving the coordi-nate system the other way by an equal amount. In Chapter 3 we discussedthe equivalence of active and passive transformations, but we had to restrictourselves to apply this principle only for a single inertial frame.

However, it should be note that passive boost is only a mathematical trick.The contradiction described above is resolved by noting that the passivetransformations within a single inertial frame S is quite distinct from theacceleration with respect to the fixed stars of an observer with his measuringdevices from inertial frame to the accelerated frame Sn. According to passivetransformation, the emitter (and the observer) motion relative to the fixedstars does not changes. In fact, in the new variables (x′, t′) the observer andthe system of fixed stars move with velocity dx′/dt′ = −v. A passive trans-formation within a single inertial frame is simply another parametrizationof the observations of the inertial observer (1).

The ignorance of the distinction between the active and passive transfor-mations is the source of much confusion in the literature. There is even aspecial relativity expert which says that the reciprocity is a correct conceptand the apparent frequency of emitted radiation depends on the relativemotion of source and observer (2). We argue that reciprocity of the Doppler

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wavelength shift, which is considered in standard treatments as relativisti-cally correct, is at odds with special relativity. An excellent illustration of theabsence of reciprocity is provided by the blue-red shift asymmetry in themeasurements of the resonance absorption of gamma rays in a Mossbauerrotating disk experiments.

10.2 Two Metrics

As a result of active Galilean boost, we obtained the metric of the acceleratedemitter in the inertial frame Eq. (11). In this coordinatization, we describethe non-accelerated light source using diagonal metric, ds2 = c2dt2

− dx2.Now we see that the Langeving metric (9) of the accelerated emitter in theaccelerated frame is the same as the metric Eq. (11) of the accelerated emitterin the inertial frame. In fact, the difference is only that the sign of velocityv. Space-time geometrical analysis of an accelerated emitter leads to thesymmetry between the metric Eq. (11) and the the Langeving metric (9).These two metrics are actually two sides of the same coin. In fact, we dealwith the same active Galilean boost. The metric Eq. (11) is presented here forthe first time. It is difficult to see why this should be, since Langevin metricis widely used in the applications to the optical phenomena in a rotatingframe.

10.3 Metrics Connected with Optics of Moving Media and Light Sources

In the present section we shall continue our discussion of inertial measure-ments. Let us now consider a reflection and refraction in the context ofspecial relativity. We are now in a position to understand what is happeningwhen light radiated by a moving emitter goes through a stationary materiallike glass. We see that what we have to do is to calculate the interferencebetween the incident radiation and the radiation due to the material. Atfirst glance, if one wants to calculate the refraction and the reflection in asingle inertial frame one should take into account that the metrics, and con-sequently the electrodynamics equations, are different for a moving emitterand a stationary glass. So the possibility of calculating interference effects isnot clear.

We must now discuss a certain feature of the phenomenon of interference.One finds many books which say that a Galilean transformation of thevelocity of light is not consistent with the electron-theoretical explanationof refraction and reflection. It is widely accepted that if we consider a movingsource and a stationary glass, the incident light wave and wave scattered bythe dipoles of the glass cannot interfere as required by the electron theory

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Fig. 44. Light radiated by a moving emitter reflects from a stationary mirror inthe collinear geometry. There is interference between the incident radiation andthe radiation due to the material. Galilean transformation of the velocity of lightis consistent with the electron-theoretical explanation of refraction and reflection.The Minkowski metric and the metric Eq. (11) give the same prediction for thewavelength of the standing wave.

of dispersion since their velocity are different. This is simply not true inphysics. It is clear that an incident wave with a certain frequency, no matterwhat its velocity, excites the electrons of a glass into oscillations of thesame frequency. They then emit radiation with the same frequency. Thus,the incident and scattered wave at any given point have the same frequencyand can interfere. The effect of the different velocities is to produce a relativephase which varies with position in space. This affects the velocity andamplitude envelope of the single wave which results from the superpositionof the two separate waves.

If the mirror is at rest and the emitter is in motion, it is obvious that theelectrodynamics equations must be identical for all electromagnetic waves.In other words, the same metric should be applied and kept in a consis-tent way for both incident and scattered waves. The requirement that theelectrodynamics equations must be identical for both incident and reflectedwaves appear to be a paradox here. In fact, we have two different metrics.We demonstrated that the contradiction disappear in the aberration of lightproblem. The peculiarity of the aberration of light (perpendicular) geometryis that even after the Galilean transformation the electrodynamics equationsin the absolute time coordinatization will have Maxwell’s form for the inci-

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dent plane wave. Let us now see what happens in the collinear geometry,Fig. 44.

Nature apparently doesn’t see any paradox, however, because we discussinterference effects. We will show that Minkowski metric and metric Eq.(11) give the same prediction. The point is that all methods to measure theinterference, indeed, the standing wave (i.e. round-trip) measurements. Incontrast, the deviation of the energy transport direction is a geometricaleffect. Because we have empirical access only to the round-trip averagespeed of light the value of the one-way speed of light is just a matter ofconvention without physical meaning. In contrast to this, the two-way speedof light, directly measurable along a round-trip, has physical meaning.

The following simple analysis confirms these ideas. Let exp i(ωt − kx) rep-resent an incoming wave whose velocity is ω/k = c + v. Similarly, letexp i(ωt+k′x+φ) represent another out-coming (scattered radiation) wave ofthe same amplitude and the same frequency, a different velocityω/k′ = c−vand different phase. The superposition of these two waves is representedby exp i(ωt− kx) + exp i(ωt + k′x +φ) = 2[cos[(k + k′)x/2 +φ/2]] exp i[ωt− (k−k′)x/2 +φ/2]. There is the cosine factor representing an amplitude envelopewhich is stationary in space and whose periodicity is inversely proportionalto the difference in the propagating constants k and k′ of the two compo-nent waves. This can be written in a simpler form 2[cos[ωx/[c(1 − v2/c2)] +φ/2]] exp i[ωt − xvω/[c2(1 − v2/c2)] + φ/2].

Suppose that the source at rest is emitting waves at frequency ω0. In the labframe after the Galilean transformation the velocity of incoming wave is c+v.Thus if ω0 is the natural frequency, the observed in the lab frame frequencywould be ω = ω0/[1 − v/(c + v)]. Therefore, the observed in the lab framefrequency is ω = ω0(1 + v/c). The shift in frequency observed in the abovesituation is the well known Doppler effect. Our equation for superpositionof two waves now looks like 2[cos[ω0x/[c(1−v/c)]+φ/2]] exp i[ω0(1+v/c)t−xvω0/[c2(1 − v/c)] + φ/2].

Suppose that an observer in the laboratory performs the standing wavemeasurement. We should examine what parts of the measured data dependson the choice of synchronization convention and what parts do not. Westate that time oscillation has no intrinsic meaning - its meaning only beingassigned by a convention. In particular, one can see the connection betweenthe time shift xvω0(1+v/c)/c2 in exp i[ω0(1+v/c)t−xvω0/[c2(1−v/c)]+φ/2] andthe issue of distant clock synchrony. Note that the scale of time (frequency)is also unrecognizable from physical viewpoint.

Suppose that the laboratory observer performs a measurement of the wave-length of the standing wave. The relation dl2

i = (−g11 + g01g01/g00)dx2 gives

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the connection between the spatial line element and the metric. With the helpof Eq.(11), we find dl2

i = dx2/(1− v2/c2), which determines the spatial geom-etry in the inertial frame. This is explained as due to Lorentz contractionof the measuring rods in the inertial frame. Then, when the measured datais analyzed, the laboratory observer finds that the wave number is equalto√

1 − v2/c2ω0/[c(1−v/c)] = ω0√

1 + v/c/√

1 − v/c. The observer finds thatthe wavelength of radiation from moving source (the source moves towardsthe observer) is decreased by the factor

√1 − v/c/

√1 + v/c. We see that it is

the same factor that we can obtain by assuming that the metric is diagonal.In other words, the laboratory observer will measure the same two-wayspeed of light, irrespective of the choice of metric.

10.4 The Notion of Time in Special Relativity Theory

There are many changes in what concepts are important when we go fromclassical to relativistic theory. In particular, the velocity concept gradu-ally fades away, while the concepts of energy and momentum become ofparamount importance. Now we wish to continue our analysis a little fur-ther. It is necessary to take a careful look at the term ”time”, which playsa fundamental role in the relativistic description of the world. The theoryof relativity show us that the notion of time is not what we would haveexpected on the basis of our intuitive ideas.

In our approach to physics in general and of the theory of relativity in par-ticular we think it very important always to remember that we are dealingwith objective physical quantities and that we attempt to describe the latterin terms of measures. We attempt to give a definition of time on the basisof a principle ”Each physical magnitude is defined by the method it is mea-sured.” Let us see how this definition works. In the case of length we knowthat there exist a length standard and we know that any body has its partic-ular length. It should be note, however, that we are dealing with objectivephysical length only in the case of a stationary object. In contrast to this, thelength of a moving object (e.g. moving rod) is convention-dependent (i.e.synchronization-dependent) and has no exact objective meaning. Now weare ready to consider the time problem in special relativity theory.

Here is one example which shows in a circumstance that is easy to under-stand. Suppose we know the law of muon decay in the Lorentz rest frame.When a Lorentz transformation of the decay law is tried, one obtains theprediction that, in the lab frame, the characteristic lifetime of a particle hasincreased from τ0 to γτ0. We can interpreted this result by saying that afterthe travel distance γvτ0, the population in the lab frame would reducedto 1/2 of the origin population. The time measured in this case is ”length”

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which is metrized as described above. That leads us to interesting question.Is the same statement perhaps also true for all cases of time measurementsin the framework of special relativity? We claim that the answer to thisquestion is positive.

Our next example has to do with frequency. We noted in the previous sec-tion that the incident and scattered wave have the same frequency and caninterfere. But what about the frequency measurements? Let us suppose thatwe have a (Fabry-Perot) interferometer. The frequency measured in this caseis (standing wave) ”length”. Frequency measurements automatically obeythe relationship between the frequency and interference. Another exampleof the relation between interference and frequency measurements is a grat-ing spectrometer. The frequency measurement in this case is a position ofthe light spot along the dispersion direction. There is no specific, importantphysical difference between diffraction grating and interferometer. Gratingspectrometer is also based on the interference of incident and reflected lightwaves.

10.5 The Aberration of Light from a Moving Laser Source

We want now to discuss the aberration of light from a moving laser source inan inertial frame of reference. The aberration of light effect can be describedwithin the standard special relativity taking advantage of the Wigner rota-tion theory. When one has a transversely (i.e. parallel to the mirrors) movinglaser there is the deviation of the energy transport for light transmitted fromthe optical resonator. In the first order approximation in v/c, the inertial ob-server would find that angular displacement is θa = 2v/c, Fig. 45.

How shall we describe the aberration of light beam from the laser sourcewhich is accelerated from rest up to velocity v in the inertial frame? First,we notice that there is an extra factor 2. On the basis of conventional theoryone gets aberration increment only one-half as big as that predicted byspecial relativity. One-half of this increment is a consequence of the factthat the Doppler effect is responsible for the angular frequency dispersionof the radiated light waves. Another source of the aberration of light is theWigner rotation. As viewed from the Lorentz lab frame, the coordinate axesof the accelerated frame is rotated through the angle v/c with respect to thecoordinate axes of the inertial frame.

In regard to light aberration one should differentiate between that from the”plane wave” emitter and that from the laser source. Suppose we have a”plane wave” emitter. We consider the case when the ”plane wave” emitteris accelerated from rest up to velocity v in the direction perpendicular to

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its optical axis. Suppose that an observer, which is at rest with respect tothe inertial frame of reference performs the direction of the light transportmeasurement. Then how does the light beam from the moving ”plane wave”emitter looks? The inertial observer would find that angular displacementis equal to θa = v/c, Fig. 46.

The first thing we would say about inertial frame measurements is that thereis an intuitively plausible way to understanding the aberration of light froma moving ”plane wave” emitter. An elementary explanation of this effect iswell-known. This phenomenon is fully understandable in terms of transfor-mation of velocities between different reference frames. The aberration oflight can also be easily explained on the basis of corpuscular theory of light.This is plausible if one keeps in mind that a light signal represents a cer-tain amount of electromagnetic energy. Energy, like mass, is a quantity thatis conserved, so that a light signal resembles, in many aspects, a materialparticles. Therefore, we should expect that group velocities of light signalsobey the same addition theorem for particle velocities. A closer treatmentbased on wave theory of light confirm this expectation.

In the case of the moving laser source, intuition would seem to tell us thataberration increment would be the same. But the special relativity saysthat there is an extra factor 2. The commonly accepted derivation of theaberration of light effect does not account for the Wigner rotation. It isincorrectly assumed that accelerated and inertial observers have common3-space.

Now we ask about the angular displacement of light transmitted from thelaser resonator inside the accelerated frame. The accelerated observer wouldfind that the deviation of the energy transport direction is absent, Fig. 47.According to asymmetry between the ”plane wave” emitter and the lasersource, there is a remarkable prediction on the theory of the aberration oflight. Namely, if the ”plane wave” emitter was at rest relative to the fixedstars and this source started from rest to motion relative to the fixed stars,then the apparent angular position of the ”plane wave” emitter seen in theaccelerated frame would jump by angle −v/c, Fig. 48. Obviously, in order tounderstand the aberration of light, we will have to use the special theory ofrelativity and not be satisfied with classical theory.

How can such difference between a ”plane wave” emitter and a laser sourcecome about? Let us first ask: ”What happens to accelerated optical res-onator?” When we were dealing with optical resonators we used the elec-trodynamics boundary conditions in 3-space, and they defined the energypropagation direction along the optical axis (i.e. perpendicular to the mirrorsurface). According to special relativity, an acceleration does not changethe direction of the optical axis (i.e mirrors orientation) with respect to the

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Fig. 45. Aberration of light from a moving laser source in an inertial frame ofreference. The motion of the laser resonator is parallel to the mirrors. If an observeris at rest relative to the fixed stars and the laser source started from rest to uniformmotion relative to the fixed stars, then the apparent angular position of the lasersource seen in the inertial frame would jump by angle θa = 2v/c. It is assumed thatthe accelerated laser operates in the steady-state regime.

Fig. 46. Aberration of light from a moving ”plane wave” emitter in an inertialframe of reference. The motion of the source is perpendicular to the optical axis. Ifan observer is at rest relative to the fixed stars and the ”plane wave” emitter startedfrom rest to uniform motion relative to the fixed stars, then the apparent angularposition of the source seen in the inertial frame would jump by angle θa = v/c.

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Fig. 47. Aberration of light in an accelerated frame of reference. It is assumedthat the accelerated laser operates in the steady-state regime. The aberration oflight from the accelerated laser source is absent. The electromagnetic wave travelsin the resonator forward and back reflecting from mirrors. The crossed term inthe Langevin metric, which generates aberration, cancels during evolution of theradiation in the optical resonator.

Fig. 48. Aberration of light from a ”plane wave” emitter in an accelerated frame ofreference. If the ”plane wave” emitter is at rest in the accelerated frame, then theapparent angular position of the source seen in the accelerated frame would jumpby angle θa = −v/c.

accelerated frame (3-space) axes. An observer in the rest frame of a lasersource finds that the direction of the energy transport is independent of themotion with respect to the fixed stars, Fig. 47. We turn now to the radiationemitted by the accelerated laser source in the inertial frame. As viewed from

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the inertial frame, the coordinate axes of the accelerated frame (xn, yn, zn) isrotated through the Wigner angle θw with respect to the axes of the inertialframe (x, y, z). For light V/c→ 1 and the zn-axis of accelerated frame is thenrotated with respect to the z-axis by angle θw = v/c. As a result, an extrafactor of 2 appears in the aberration increment, in Fig. 45.

Now consider a ”plane wave” emitter which is at rest in an acceleratedframe. How shall we find the general wave solution in the case of a pointsource? The answer is that electrodynamics equation needs to be integratedwith initial condition for the radiation wavefront. According to the specialrelativity, physical process of light creation takes place in a metric space-timegeometry, so the metric tensor must be used to describe the light creation andpropagation in empty space. We only wish to emphasize here the followingpoint. According to special relativity, accelerated and inertial observers havecommon space-time but have different metrics and, consequently, different3-spaces. When we are dealing with optical resonator the (3-space) boundaryconditions define the energy propagation direction. In contrast, when weare dealing with a point source the combination of initial condition (i.eorientation of the radiation wavefront ) and metric tensor is important.From the special theory of relativity we know that the acceleration has noeffect on the wavefront orientation and the wavefront of the emitted lightbeam is perpendicular to the direction zn after the acceleration. Using thisinitial condition and the Langevin metric in a frame of reference attendingacceleration enables us to explain the aberration of light from a ”plane wave”emitter, Fig. 48.

10.6 The Aberration of Particles from a Moving Source in an Inertial Frame

The next interesting problem to consider what happens if the electron gun,an electron source for a relativistic electron beam, in the lab frame is accel-erated from rest up to velocity v along the x-axis, Fig. 49. In other words, weconsider so-called active (physical) boost. In the case of an active boost weare dealing with motion of the same physical system, evolving in time andtreated from the point of view of the same reference system. The simplestmethod of synchronization consists in keeping, without changes, the sameset of uniformly synchronized clocks used in the case when the particlesource was at rest, i.e. we still enforce the clock transport synchronization (or Einstein synchronization which is defined in terms of light signals emit-ted by the dipole source at rest). This choice is usually the most convenientone from the viewpoint of connection to laboratory reality. This synchro-nization convention preserves simultaneity and is actually based on theabsolute time (or absolute simultaneity) convention.

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Fig. 49. Aberration of electrons from a moving source in an inertial frame. Themotion of the electron gun is perpendicular to the optical axis. If an observer isat rest relative to the fixed stars and the electron gun started from rest to uniformmotion relative to the fixed stars, then the apparent angular position of the sourceseen in the inertial frame would jump by angle θa = v/V, where V is the velocityof the accelerated electrons.

It is always possible to create a new frame of reference by relabeling co-ordinates, and then discussing physical phenomena in terms of the newcoordinate labels - a passive transformation. For example, it is always pos-sible to create a so-called comoving coordinate system in the lab frame, andthen discussing accelerated particle beam from the moving source in thermof the new (comoving) coordinate labels. In the comoving coordinate sys-tem, fields are expressed as a function of the independent variables x′, y′, z′,and t′. The variables x′, y′, z′, t′ can be expressed in terms of the indepen-dent variables x, y, z, t by means of a passive Galilean transformation, so thatfields can be written in terms of x′, y′, z′, t′. After the passive transformation,the Cartesian coordinates of the source transform as x′ = x−vt, y′ = y, z′ = z.This transformation completes with the invariance of simultaneity, t′ = t.The transformation of time and spatial coordinates of any event has theform of a Galilean transformation.

The principle of covariance implies the equivalence of active and passivetransformations within a single inertial frame. The equivalence of the ac-tive and passive pictures is due to the fact that moving system one wayis equivalent to moving the coordinate system the other way by an equalamount. According to the principle of relativity, the Maxwell’s equationsalways valid in the Lorentz comoving frame. Let us consider the electro-dynamics of the moving source. The explanation of the phenomenon of

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particle acceleration in our case of interest consists in using a Galileo boostto describe the uniform translation motion of the source in the inertial labframe.

This result is interesting and important enough that we should deduce it bya purely microscopic analysis instead of by argument about the kinematictransformations. When we consider the electrodynamics equations underGalilean transformation, there is a question: what is the transformation lawfor the electromagnetic fields ~E and ~B? The fields ~E and ~B observed in theinertial frame where the particle source is moving with velocity v are notthe same as those ~E′ and ~B′ in the inertial frame before the active Galileanboost. With the Galilean transformation, the field transformation, for thecase ~B′ = 0, is ~E = ~E′, ~B = ~v × ~E′/c. This expression describe first order (inv/c) effects only. Here ~v is the velocity vector of the source in the lab frame.

The inertial observer has experimentally obtained the Lorentz force ~F ona charge e moving with velocity ~V in the region of electron gun in which~E and ~B are presented, ~F = e~E′ + e~V × ~v × ~E′/c2. In the case - where ~v ·~V = 0, ~E · ~V = EV - the expression for magnetic force is equivalent toe~V × ~v × ~E′/c2 = ~v(e~V · ~E′/c2). Using the (relativistically) corrected equationof motion, ~F = d~p/dt, we expect after the time dt emitted electron willhave the differential change in the transverse momentum given by d~p⊥ =

~v(e~V · ~E′/c2) dt. The differential change in the kinetic energy of the acceleratedelectrons is dT = ~F · ~Vdt = ~F · d~s. If we now integrate, we get T = e

∫~E · d~s =

mc2/√

1 − V2/c2 − mc2. The total transverse momentum, p⊥, of acceleratedelectron beam (in the first order approximation in v/c) is given by p⊥ = p0 +

∆p⊥ = mv + vT/c2 = mv/√

1 − V2/c2. Here p0 = mv is the initial momentumof the emitted electron. Since we have chosen to make ~v · ~V = 0, aberrationincrement becomes θa = p⊥/p, where p = mV/

√1 − V2/c2 is the momentum

of the accelerated electron. So we have θa = v/V. We can put our expressionfor the aberration increment in vector form: ~θa = ~V × ~v/V2.

We described the effect of aberration of particles by working only up to thefirst order v/c. An elementary explanation of this effect is well-known. Theaberration of particles in the inertial lab frame can be easily explained interms of Galilean transformation of velocities between different referenceframes. It is important to stress at this point that the dynamical line of ar-guments discussed here explains what the aberration of particles physicallymeans in the inertial frame. The kinematics effects is only an interpretationof the behavior of the electromagnetic fields.

In Chapter 8 we already discussed the aberration of electrons. We had torestrict ourselves to discussing the aberration only for the case when the

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Fig. 50. Aberration of electrons when the electron source and observer are at rest inthe accelerated frame. The aberration increment is connected with the problem pa-rameters by the relation θa = (1−1/γ)v/V. Here γ = 1/

√1 − V2/c2 is the relativistic

factor.

electron source and observer are at rest in the accelerated frame. The physi-cal principles that produces the aberration increment were, however, madeclear. We have seen in Chapter 8 that the Wigner rotation associated with thetransformation from inertial coordinate system to the accelerated coordinatesystem may be regarded as a result of the action of certain electromagnetic(Lorentz) force. In fact, the electromagnetic forces which govern the prop-erties of an emitted electron beam must be affected by acceleration withrespect to the fixed stars in such a way that they lead to a deviation of theelectron transport direction, Fig. 50.

The microscopic approach for the motion of electrons in the acceleratedframe gives a similar (for the motion of electrons in the inertial frame)answer for the total transverse momentum, p⊥, of the accelerated electronbeam except with the following modification. Because the electron sourceis at rest in the accelerated frame, the initial condition is modified. We canrewrite our formula for the transverse momentum ( in the inertial frame)as p⊥ = p0 + ∆p⊥ = ∆p⊥ = vT/c2 = mv/

√1 − V2/c2 − mv, which says that

quantity p0 is equal to zero. Therefore, when the electron source is at rest inthe accelerated frame, there is an aberration increment given by θa = p⊥/p =(1 − 1/γ)v/V. So we see that we could analyze this complicated situationeither by the idea that there is an electromagnetic forces and that they leadto a deviation of the energy transport, or else by the Lorentz transformationsand that the aberration of particles effect in the accelerated frame (as viewedfrom the inertial frame) presents a kinematics effect (Wigner rotation) ofspecial theory of relativity.

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10.7 Inertial Frame View of Accelerated Light Clock

Slowing of the clock in a moving system is a very peculiar phenomenon.We have already discussed accelerated clocks in the previous chapter. Wedescribed the parallel and orthogonal light clocks based on the observationsmade by an observer in the same accelerated frame as the clock. Now letus return to observations of an inertial observer. We must say immediatelythat there is no objection to the standard description of the parallel lightclock in the inertial lab frame. What must be recognized is that the conceptof Wigner rotation is only introduced in the description of an orthogonallight clock.

Let us consider a light clock accelerated up to velocity v transverse to thedirection of optical pulse. First, we examine the reasoning presented intextbooks. Suppose that an inertial system S (and an observer with hismeasuring instruments) is at rest with respect to the fixed stars (we use thisparticular initial conditions for sketch simplification) and a system Sn in thelab frame S is accelerated from the rest with respect to the fixed stars upto velocity v along the x-axis. A clock which we call ”light-clock” is a rodwith mirror at each end. According to textbooks, when inertial observerlooks at the orthogonal light clock going by, he sees that the light, in goingfrom mirror to mirror is taking a zigzag path. If in a giving time the clockmoves forward a distance proportional to v in Fig. 51, the distance the lighttravels in the same time is proportional to c, and the vertical distance istherefore proportional to

√

c2 − v2. That is it takes a longer time for light togo from end to end in the moving clock than in stationary clock. Thereforethe apparent time between clicks is longer for the moving clock in theproportion 1/

√1 − v2/c2.

Nevertheless, there is argument against this commonly accepted derivationof the slowing down of an orthogonal light clock in a moving system. Thestandard explanation is based on the hidden assumption that (xn, yn, zn) axesof the moving observer and (x, y, z) axes of the lab observer are parallel.In other words, it is assumed that observers have common 3-space. Thisis misconception. In fact, two observers with different trajectories in theLorentz coordinatization have different 3-spaces.

The accepted in textbooks assumption of rigidity of accelerated frame isbased on the belief that simultaneous acceleration of the reference spacegrid (xn, yn, zn) has direct physical meaning. However, the position alongthe x-axis of the accelerated first mirror with respect to the acceleratedsecond mirror has no exact objective meaning since, due to the finiteness ofthe speed of light, we cannot specify any experimental method by whichthis position could be ascertained. There is an uncertainty (blurring) of

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Fig. 51. The ”light clock” thought experiment. The calculation of the time intervalbetween the sending and reception of the light signal in the reference frame Srelative to which a light clock moves. According to textbooks, it is assumed that(xn, yn, zn) axes of the moving frame and (x, y, z) axes of the lab observer are parallel.

the relative position in the x-direction of amount lv/c, where l is the rodlength. This uncertainty in the relative position exist due to the uncertaintyin the moments of the acceleration. The theory of relativity shows us that therelationship of positions and times are not what we would expect intuitively.When we have two distant events, we have deal with the conventionalityof distant simultaneity within the time interval l/c, where l is the spaceseparation of these two events.

Now let us see how we can compare the clock rates in the frame S and Sn. Itis generally believed that observing clock rates in the two frames S and Sn

moving relative to each other, one can only compare readings of one clockfrom one frame with readings of several clocks from another frame, becausetwo clocks from different frames get together at the same point in spaceonly once. In one of the frames there must be at least two clocks which aresupposed to be synchronized.

We have already pointed out that in all cases of time measurements in theframework of special relativity the time measured in this case is actually”length”. In the case of length measurement we are dealing with objective(i.e. convention independent) physical quantity. Because we have empiricalaccess only to the length measurements, it is impossible to agree with text-book statement that one can only compare readings of one clock from one

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frame with readings of several synchronized clocks from another frame. Letus try out our operational interpretation of time measurements on the lightclock example, to see how it works.

We shall discuss the situation where time marks can be imprinted on themoving object (screen) by a clock, as sketched in Fig. 52. First, we describethe orthogonal light-clock based on the observations made by an observer inthe same accelerated frame as the light-clock. In the case of Langevin metricEq.(9), the speed of light radiated by the accelerated emitter in the transversedirection is c

√1 − v2/c2. Therefore, the time interval between the sending

and reception of the light signal is ∆tn = 2l/[c√

1 − v2/c2]. The spacing of thetime marks, i.e the distance ∆xn through which the object (screen in the labframe) falls in the time is ∆xn = ∆x = 2lv/[c

√1 − v2/c2].

Taking into account the Langevin metric Eq.(9), we obtain at dtn = dt = 0that dxn = dx, i.e. the length of physical rod in the inertial frame coincidencewith the coordinate distance in the accelerated frame. Because tn = t and∆t = 2l/c has the physical meaning of the time indicated by a physical clockat rest in the inertial frame, this implies that, as physical clocks at rest inthe accelerated frame are slow compared to physical clocks in the originalinertial frame. The accelerated clock rate measured by the lab observer isactually spacing of the time marks on the lab screen.

Second, we describe the orthogonal light-clock based on the observationsmade by an observer in the same inertial frame as the light-clock, Fig. 53.Therefore, the time interval between the sending and reception of the lightsignal is ∆t = 2l/c. The spacing of the time marks, i.e the distance ∆x throughwhich the screen in the accelerated frame falls in the time is ∆x = ∆xn = 2lv/c.

Analysis based on using standard measuring rods in an accelerated frameto measure it geometrical properties shows that coordinate distance dxn hasphysical length dxn/

√1 − v2/c2. We have ascertained the asymmetry of me-

ter stick’s length directly from the Langevin metric. With reference to theinertial frame this is explained as due to Lorentz contraction of the mea-suring rods. This shows that physical spacing of the time marks measuredin the accelerated frame undergoes compress on in the direction of fixedstars motion. Because ∆xn = ∆x and ∆x = 2vl/c has the physical meaningof the time rate indicated by a physical clock at rest in the inertial frame,this implies that the accelerated observer can observe the shortness of markspacing ∆ln = 2lv

√1 − v2/c2]/c. Here ∆ln is the physical mark spacing mea-

sured in the accelerated frame. The asymmetry in spacing of time marks andthe blue-red shift asymmetry in the transverse Doppler effect are actuallytwo sides of the same coin. In fact, we deal with the same time dilationeffect. This thought experiment illustrates that there is no reciprocity in timedilation.

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Fig. 52. The ”light clock” thought experiment. Time marks can be imprinted on themoving screen by clock. Observations made by an observer in the same acceleratedframe as the light clock. The anisotropy of speed of light presented in the absolutetime coordinatization. The distance ∆xn through which the screen in the lab framefalls in the time (i.e. spacing of the time marks) is ∆xn = ∆x = 2vlγ/c.

Fig. 53. The ”light clock” thought experiment. Time marks can be imprinted on themoving screen by clock. Observations made by an observer in the same inertialframe as the light clock. The distance ∆xn through which the screen in the acceler-ated frame falls in the time (i.e. spacing of the time marks) is ∆xn = ∆x = 2vl/c.

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Fig. 54. The ”light clock” thought experiment. Initialization of the orthogonal lightclock in the accelerated frame. Observation made by an observer in the inertial labframe. The initial condition of the accelerated light clock cannot be measured.

Fig. 55. The ”light clock” thought experiment. Observation made by an observer inthe inertial lab frame. The position along x- axis of the accelerated first mirror withrespect to the second mirror has no exact objective meaning. As viewed from theLorentz lab frame, the orientation of coordinate axes of the accelerated frame withrespect to coordinate axes of the lab frame is regulated by the Wigner rotation. Theaccepted in textbooks assumption of rigidity of accelerated orthogonal light clockis based on the incorrect belief that axes of the accelerated frame and axes of the labframe are parallel. In order to keep the complexity of the discussion to a minimum,we describe the Wigner rotation effect by working only up to the first order in v/c.

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Fig. 56. The ”light clock” thought experiment. Initialization of the orthogonal lightclock in the accelerated frame. Observation made by an observer in the inertiallab frame. According to textbooks, it is incorrectly assumed that accelerated andinertial observer have common 3-space. The possibility of measurement of theinitial condition is prediction of conventional light clock theory and is obviouslyabsurd from the viewpoint of special relativity.

We demonstrated that apparent time between clicks is longer for acceleratedclock. Now, continuing our discussion of the light clock operation in theframe S and Sn, let us consider the phase of light clocks. The word ”phase”in this case has meaning initialization (i.e. initial conditions) of these lightclocks. We have already (Fig. 52 - 53) discussed how a clock rate can bemeasured when the clock is moving. Above we found that the rate of movingclock (i.e. the time interval between the sending and reception of the lightsignal) measured by the lab observer is actually spacing of the time markson the lab screen. Is the same statement also true for phase measurements?Let us try to demonstrate that it is.

Suppose that the lab observer has two screens and we shall discuss theup-and-down imprinting method (i.e. the situation where time marks canbe imprinting on the both screens by a moving clock). We learned in theChapter 7 that the orientation of coordinate axes of the accelerated framewith respect to coordinate axes of the Lorentz lab frame is regulated by theWigner rotation. We can show that, as consequence of Wigner rotation, theinitial condition (phase) of the accelerated orthogonal light clock cannot bemeasured in the lab frame, Fig. 54.

The standard presentation of the orthogonal light clock operation is based onthe hidden assumption that accelerated and inertial observers have common3-space. According to the conventional theory, when a reference frame at rest

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is put into motion, then all points in the 3D reference grid have a start to moveat same time. Consequently, rigidity of the orthogonal light clock in the senseintroduced by textbooks cannot be considered as relativistic kinematicsproperty, Fig. 55. Our result (Fig. 54) is at odds with the prediction fromtextbooks, Fig. 56. The commonly accepted description of the orthogonallight clock operation does not account for the Wigner rotation, which in ourcase closely associated with the relativity of simultaneity. The possibilityof measurement of the initial condition is prediction of conventional lightclock theory and is obviously absurd from the viewpoint of special relativity.Surprisingly, it was not recognized in the literature that in this case, due tothe finiteness of the speed of light, we cannot specify any experimentalmethod by which this initial condition could be ascertained.

10.8 Aberration of Light and the Principle of Relativity

In the last few chapters we have treated the essential ideas necessary for anunderstanding of the aberration of light phenomena. We have not had toworry yet about the fact that the source-observer asymmetry associated withthe stellar aberration phenomenon at first glance contradicts the principleof relativity.

The principle of relativity which we owe to Poincare, who first coined theterm, denied the possibility for an observer partaking in a uniform motionrelative to the fixed stars of discovering, by any measurement, such a motion,under the assumption, of course, that one does not look outside.

Now let us to analyze the stellar aberration experiment. Perhaps we shouldsay: ” In the case of stellar aberration, the peculiarity of the aberrationof light measurements is that the accelerated (earth-based) observer looksoutside to the stars.” But no! The motion of the stars with respect to theearth is never followed by any aberration. The aberration shift (as inferredfrom astronomical observations) exists even in the case when star moveswith the same velocity as the earth. We cannot remain with the framework of”looking outside to the star” when considering the earth-based observationsof the stellar aberration.

In the Section 8.4 we discussed a simple scaling model for the stellar aberra-tion. We obtained a condition for optical similarity between the aberrationof light from a distant star and from the earth-based incoherent source. Wefind that without looking outside, it is possible to determine the speed ofthe earth around the sun by means of aberration of light measurements. Noone has ever done such experiment with earth-based incoherent source, butwe know would happen from the astronomical observations.

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It should be note that all well known methods to test the special relativity areround-trip measurements. The cardinal example is given by the Michelson-Morley experiment. A close examination of all round-trip experiments in-side an inertial frame shows that phenomena appear to be independentof the uniform motion relative to the fixed stars. We conclude the follow-ing: The usual formulation of the theory of relativity deals with formulatedabove principle of irrelevancy of velocity, but it should be understood onlyin a limiting sense when we are dealing with round-trip measurements.

Another illustration of the irrelevancy of velocity is provided by absenceof the aberration of light in the experiments involved the light transmittedfrom the laser resonator in the accelerated frame. The accelerated observerfinds that the direction of the energy transport is independent of the motionwith respect to the fixed stars. In fact, the electromagnetic wave travels inthe laser resonator forward and back reflecting from the mirrors. It couldbe said that the asymmetry cancels during the (round-trip-to-round-trip)evolution of the radiation in the optical resonator.

There is no conflict between the fundamental structure of special theory ofrelativity on the one hand, and the aberration of light phenomena. First,from what is said in Chapter 1 it should be clear that special relativitydoes not require the formulated above principle of irrelevancy of velocity.Principle of special relativity applies to physical laws, not to physical facts.Not only in Einstein writings, but in every textbook on special theory ofrelativity can be found the formulation of principle of special relativity asfollows: the laws of physics are the same in all inertial frames (3).

The special principle of relativity says that the same laws should hold inall inertial frames. We interpret this to mean that the laws can be expressedby equations that have the same form in all inertial frames. According tospecial principle of relativity all inertial frames are equivalent with regardto physical laws, not with regard to physical facts. By a physical fact in thiscontext we mean properties of existing objects - in our case of interest theproperties of incoherent source. By mean a physical law we mean for exam-ple electrodynamics equations. Principle of special relativity is irrelevancyof velocity with regard to physical laws, not with regard to anything.


1. In the general case, the problem to assigning Lorentz coordinates in an in-ertial frame is complicated. Let us consider a source, arbitrary accelerating inthe inertial frame, and let us analyze its evolution with a Lorentz coordinatesystem. The permanent rest frame of the source is obviously non inertial. To

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get around that difficulty, one introduces an infinite sequence of comovingcoordinate systems. At each instant, the comoving coordinate system is aLorentz coordinate system centered on the source and moving with it. As thesource velocity changes to its new value at an infinitesimally latter instant, anew Lorentz coordinate system centered on the source and moving with it atthe new velocity is used to observe the source. We should make one furtherremark about this covariant algorithm. An opinion is sometimes expressedthat described above algorithm includes a hidden postulate. It seems neces-sary a dynamical assumption to justify attributing to an accelerated clock thesame rate as a clock in inertial motion in relation to which it is momentaryat rest. This is, in view of some authors, an extra condition that a clock mustsatisfy. It is assumed that the effect of the motion on the clock depends onlyon it instantaneous speed, not its acceleration. This condition often refereedto as a ”clock hypothesis”. To quote Brown [17]: ”If the accelerating forcesare small in relation to the internal restorative forces of the clock, then theclock’s proper time will be proportional to the Minkowski distance alongits world line. ... This condition is often referred to as the clock hypothesis,and its justification, as we have seen, rests on accelerative forces being smallin the appropriate sense.” We state that the clock hypothesis does not havestatus of independent hypothesis is not needed as an independent postulatein the theory of relativity. As discussed above, the transformations within asingle inertial frame are simply another parametrization of the observationsof the inertial observer. According to such transformations, the observer(and his clocks) motion relative to the fixed stars does not changes. In otherwords, when we discuss only the observations of an inertial observer theproblem of accelerated clock does not exist at all.

2. The ignorance of the distinction between the accelerated and the iner-tial observers is the source of much confusion in the literature. To quotee.g. French [4]: ”The apparent frequency of emitted radiation depends onthe relative motion of source and observer.” Similar statements can also befound in recently published textbooks. To quote Rafelski [5]: ”While thereis no reciprocity in time dilation, in Chap. 13 we will find reciprocity of theDoppler wavelength (or frequency) shift.” On the contrary, the measure-ments of the resonance absorption of gamma rays in a Mossbouer rotatingdisk experiment [41] demonstrates that the reciprocity can be an incorrectconcept in the Doppler shift problem.

3. By ”the same” is means form-invariance: the mathematical relations be-tween the physical quantities remain identical after a coordinate transforma-tion has been performed on these quantities. In special theory of relativitythe (form-invariant) transformations between inertial frames are Lorentz-transformations [42].

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11 Relativistic Particle Dynamics

In previous chapters we considered the kinematics of the theory of rela-tivity, which concerns the study of the four vectors of positions, velocityand acceleration. Kinematics studies trajectories as geometrical objects, in-dependently of their causes. This means that it is not possible to predict thetrajectory of a particle evolving under a given dynamical field using just akinematic treatment. In dynamics we consider the effect of interaction onmotion.

11.1 Manifestly Covariant Particle Dynamics

Dynamics equations can be expressed as tensor equations in Minkowskispace-time. When coordinates are chosen, one may work with compo-nents, instead of geometric objects. Relying on the geometric structure ofMinkowski space-time, one can define the class of inertial frames and canadopt a Lorentz frame with orthonormal basis vectors for any given inertialframe. In any Lorentz coordinate system the law of motion becomes

md2xµdτ2 = eFµν

dxνdτ

, (12)

where here the particle’s mass and charge are denoted by m and e respec-tively. The electromagnetic field is described by a second-rank, antisymmet-ric tensor with components Fµν. The coordinate-independent proper timeτ is a parameter describing the evolution of physical system under therelativistic laws of motion, Eq. (12).

The covariant equation of motion for a relativistic charged particle under theaction of the four-force Kµ = eFµνdxν/dτ in the Lorentz lab frame, Eq.(12),is a relativistic ”generalization” of the Newton’s second law. The three-dimensional Newton second law md~v/dt = ~f can always be used in theinstantaneous Lorentz comoving frame. Relativistic ”generalization” meansthat the previous three independent equations expressing Newton secondlaw are be embedded into the four-dimensional Minkowski space-time (1).

The immediate generalization of md~v/dt = ~f to an arbitrary Lorentz frameis Eq.(12), as can be checked by reducing to the rest frame. In Lorentzcoordinates there is a kinematics constraint uµuµ = c2 for the four-velocityuµ = dxµ/dτ. Because of this constraint, the four-dimensional dynamicslaw, Eq.(12), actually includes only three independent equations of motion.Using explicit expression for Lorentz force we find that the four equations

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Eq.(12) automatically imply the constraint uµuµ = c2 as it must be. To provethis, we calculate the scalar product between both sides of the equation ofmotion and uµ. Using the fact that Fµν is antisymmetric (i.e. Fµν = −Fνµ), wefind uµduµ/dτ = eFµνuµuν = 0. Thus, for the quantity Y = (u2

− c2) we finddY/dτ = 0.

11.2 Conventional Particle Tracking. Hidden Absolute Time Coordinatization

Having written down the motion equation in a 4-vector form, Eq.(12), anddetermined the components of the 4-force, we satisfied the principle ofrelativity for one thing, and, for another, we obtained the four componentsof the equation of particle motion. This is covariant relativistic generalizationof the three dimensional Newton’s equation of motion which is based onparticle proper time as the evolution parameter.

We next wish to describe a particle motion in the Lorentz lab frame using thelab time t as evolution parameter. First, we examine the reasoning presentedin textbooks. Let us determine the first three spatial components of the 4-force. We consider for this the spatial part of the dynamics equation, Eq.(12):~Q = (dt/dτ)d(mγ~v)/dt = γd(mγ~v)/dt. The prefactor γ arises from the changeof the evolution variable from the proper time τ, which is natural since ~Qis the space part of a four-vector, to the lab frame time t, which is neededto introduce the usual force three-vector ~f : ~Q = γ ~f . Written explicitly, therelativistic form of the three-force is

ddt

(m~v

√1 − v2/c2

)= e

(~E +

~vc× ~B

). (13)

The time component is

ddt

(mc2

√1 − v2/c2

)= e~E · ~v . (14)

The evolution of the particle is subject to these four equations, but also tothe constraint

E2/c2− |~p|2 = mc2 . (15)

According to the non-covariant (3+1) approach we seek for the initial value

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solution to these equations. Using explicit expression for Lorentz force wefind that the three equations Eq.(13) automatically imply the constraintEq.(15), once this is satisfied initially at t = 0. In the (3+1) approach, the fourequations of motion ”split up” into (3+1) equations and we have no mixtureof space and time parts of the dynamics equation Eq.(12). This approach torelativistic particle dynamics relies on the use of three independent equa-tions of motion Eq.(13) for three independent coordinates and velocities,”independent” meaning that equation Eq.(14) (and constraint Eq.(15)) areautomatically satisfied.

One could expect that the particle’s trajectory in the lab frame, followingfrom the previous reasoning ~x(t), should be identified with ~xcov(t). However,paradoxical result are obtained by doing so. In particular, the trajectory ~x(t)does not include relativistic kinematics effects. In the non-covariant (3+1)approach, the solution of the dynamics problem in the lab frame makes noreference to Lorentz transformations. This means that, for instance, withinthe lab frame the motion of particles in constant magnetic field looks pre-cisely the same as predicted by Newtonian kinematics: relativistic effectsdo not have a place in this description. In conventional particle tracking aparticle trajectory ~x(t) can be seen from the lab frame as the result of suc-cessive Galileo boosts that track the motion of the accelerated (in a constantmagnetic field) particle. The usual Galileo rule for addition of velocities isused to determine the Galileo boosts tracking a particular particle, instantafter instant, along its motion along the curved trajectory.

The old kinematics is especially surprising, because we are based on theuse of the covariant approach. Where does it comes from? The previouscommonly accepted derivation of the equations for the particle motion inthe three dimensional space from the covariant equation Eq.(12) includesone delicate point. In Eq.(13) and Eq.(14) the restriction ~p = m~v/

√1 − v2/c2

has already been imposed. One might well wonder why, because in theaccepted covariant approach, the solution of the dynamics problem for themomentum in the lab frame makes no reference to the three-dimensionalvelocity. In fact, equation Eq.(12) tells us that the force is the rate of changeof the momentum ~p, but does not tell us how momentum varies with speed.The four-velocity cannot be decomposed into u = (cγ, ~vγ) when we dealwith a particle accelerating along a curved trajectory in the Lorentz labframe.

Actually, the decomposition u = (cγ, ~vγ) comes from the relation uµ =dxµ/dτ = γdxµ/dt = (cγ, ~vγ). In other words, the presentation of the timecomponent as the relation dτ = dt/γ between the τ and lab time t is basedon the hidden assumption that the type of clock synchronization, whichprovides the time coordinate t in the lab frame, is based on the use of theabsolute time convention.

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It is important to stress in this point that the situation when only one clockin the comoving frame is involved in dynamics cannot be realized. TheNewton law can be written down in the proper frame only when a space-time coordinate system (x′, y′, z′, τ) has been specified. The type of clocksynchronization which provides the time coordinate in the Newton equationhas never been discussed in textbooks. It is clear that without an answerto the question about the method of distant clock synchronization used,not only the concept of acceleration, but also the dynamics law has nophysical meaning. A proper frame approach to relativistic particle dynamicsis forcefully based on a definite (Einstein) coordinatization assumption.After this, the dynamics theory states that the equation of motion in theproper frame is md2~x′/dτ2 = ~f ′. It should be stressed that in the case ofvelocity increment ∆~v′ = ∆~x′/∆τ we also deal with the distant events. Itcan be said with some abuse of language that 3-velocity vector is always a”spatially extended” object. Let us return to our consideration of the relationdτ = dt/γ between the τ and lab time t. The calculation carried out in thecase of an spatially extended object shows that the temporal coincidence oftwo distant events has absolute character: ∆τ = 0 implies ∆t = 0.

It should be note that usually the notation ”τ” arose from the proper time.Now let us recall the standard concept of the object’s proper time. Let apoint-like object move uniformly and rectilinearly relative to the lab frameK. The proper frame K′ can be fixed to the moving object. The object is atrest in this frame, so that events happening with this object are registered byone clock. This clock counts the proper time at the point where the object islocated. As we already know, the proper time on moving object flows slowerthan the time t. This phenomena was called time dilation. The proper timecan also be introduced for a point particle moving with acceleration. Thisstandard interpretation of the proper time τhas evidently nothing to do withdynamics and one may wonder where this contradiction existing betweenthe name and the content of time in dynamic law come from.

In order to avoid being to abstract for to long we have given one example.Let us discuss the case in which the velocity increment is not in the directionof the uniform motion. For example, there may be a particle which is justmoving ”upward” with velocity increment ∆y′/∆τ and the lab system ismoving ”horizontally”. Now, according to textbooks, we simply using rela-tion dτ = dt/γ with the standard result ∆vy = ∆v′y/γ. However, paradoxicalresult are obtained by doing so. In particular, the trajectory of the particledoes not include mixture of positions and time. One of the important con-clusions of the discussion presented above in the Chapter 7 is that there isno absolute notion of simultaneity in the theory of special relativity. There-fore there is no notion of an instantaneous 3-space. A coordinate axes ofreference frame K′ rotating about the axes of a Lorentz lab frame. The stan-dard presentation of the velocity increment transformation as the relation

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∆vy = ∆v′y/γ is based on the hidden assumption that y′-axis and y-axis areparallel. On the other hand Lorentz transformation makes axes (y′ and y)oblique-angled due to Wigner rotation. A difference between non-covariantand covariant point particle trajectories follows from this immediately (Wewill discuss the transformation of velocities in greater detail in Section 11.5).

11.3 Incorrect Expansion of the Relation dτ = dt/γ to an Arbitrary Motion

Authors of textbooks are dramatically mistaken in their belief about theusual momentum-velocity relation. From the theory of relativity follows thatthe equation ~pcov = m~vcov/

√1 − v2

cov/c2 does not hold for a curved trajectoryin the Lorentz lab frame. Many experts who learned the theory of relativityusing textbooks will find this statement disturbing at first sight.

How can such an unusual momentum-velocity relation come about? Weknow that the components of momentum four-vector pµ = (E/c, ~p) behaveunder transformations from one Lorentz frame to another, exactly in thesame manner as the component of the four-vector event x = (x0, ~x). Surprisescan surely be expected when we return from the four-vectors language tothe three-dimensional velocity vector ~v, which can be represented in termsof the components of four-vector as ~v = d~x/dx0. In contrast with the pseudo-Euclidean four-velocity space, the relativistic three-velocity space is a three-dimensional space with constant negative curvature, i.e. three-dimensionalspace with Lobachevsky geometry.

It is well known that for rectilinear accelerated motion the usual momentum-velocity relation holds. In fact, for the rectilinear motion the combinationof the usual momentum-velocity relation and the covariant three-velocitytransformation (according to Einstein’s law of velocity addition) is consis-tent with the covariant three-momentum transformation and both (non-covariant and covariant) approaches produce the same trajectory.

We can see why by examine the transformation of the three velocity in thetheory of relativity. For a rectilinear motion, this transformation is performedas v = (v′ + V)/(1 + v′V/c2). The relativistic factor 1/

√1 − v2/c2 is given by:

1/√

1 − v2/c2 = (1 + v′V/c2)/(√

1 − v′2/c2√

1 − V2/c2). The new momentumis then simply mv times the above expression. But we want to express thenew momentum in terms of the primed momentum and energy, and wenote that p = (p′ + E′V/c2)/

√1 − V2/c2. Thus, for a rectilinear motion, the

combination of Einstein addition law for parallel velocities and the usualmomentum-velocity relation is consistent with the covariant momentumtransformation. The dynamical evolution of the particle according to thenon-covariant particle tracking may be considered as the result of successive

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infinitesimal Lorentz transformations. The corrected Newton law is validfor each step. In other words, the two observers with rectilinear trajectorieshave common 3-space. This result was incorrectly extended to an arbitrarytrajectory.

We already know that the collinear Lorentz boosts commute. This meansthat the resultant of successive collinear Lorentz boosts is independent ofthe transformation order. On the contrary, Lorentz boosts in different direc-tions do not commute. A comparison with the three-dimensional Euclideanspace might help here. Spatial rotations do not commute either. However,also for spatial rotations there is a case where the result of two successivetransformations is independent of their order: that is, when we deal withrotation around the same axis.

As well-known, the composition of non-collinear boosts is equivalent to aboost followed by a spatial rotation. This rotation is relativistic effect thatdoes not have a non-covariant analogue. One of the consequences of non-commutativity of non-collinear Lorentz boosts is the unusual momentum-velocity relation ~pcov , m~vcov/

√1 − v2

cov/c2, which also does not have anynon-covariant analogue.

The theory of relativity shows that the unusual momentum-velocity relationdiscussed above is related with the acceleration along curved trajectories.In this case there is no notion of a common ordinary space and there is adifference between covariant and non-covariant particle trajectories. Onlythe solution of the dynamics equations in covariant form gives the correctcoupling between the usual Maxwell’s equations and particle trajectories inthe lab frame. A closer analysis of the concept of velocity, i.e. a discussion ofthe methods by which a time coordinate can actually be assigned in the labframe, opens up the possibility of a description of such physical phenomenaas radiation from a relativistic electron accelerating along a curved trajectoryin accordance with the theory of relativity.

11.4 The Usual Integration of the 4D Covariant Equation of Motion

Attempts to solve the dynamics equation Eq.(12) in manifestly covariantform can be found in the literature. The trajectory which is found does notinclude relativistic kinematics effects. Therefore, it cannot be identified with~xcov(t) even if, at first glance, it appears to be derived following covariantprescription.

First, we analyze textbooks presentation of the integration of the covariantequation of motion. Consider, for example, the motion of a particle in a given

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electromagnetic field. The simplest case, of great practical importance, isthat of an uniform electromagnetic field meaning that Fµν is constant on thewhole space-time region of interest. In particular we consider the motionof a particle in a constant homogeneous magnetic field, specified by tensorcomponents Fµν = B(eµ2 eν3 − eν2eµ3 ) where eµ2 and eµ3 are orthonormal space likebasis vectors e2

2 = e23 = −1, e2 · e3 = 0. In the lab frame of reference where

eµ0 is taken as the time axis, and eµ2 and eµ3 are space vectors the field isindeed purely magnetic, of magnitude B and parallel to the e1 axis. Let usset the initial four-velocity uµ(0) = γceµ0 +γveµ2 , where v is the initial particle’svelocity relative to the lab observer along the axis e2 at the instant τ = 0,and γ = 1/

√1 − v2/c2. The components of the equation of motion are then

du(0)/dτ = 0, du(1)/dτ = 0, du(2)/dτ = −eBu(3)/(mc), du(3)/dτ = eBu(2)/(mc). Weseek for the initial value solution to these equations as done in the existingliterature (2). A distinctive feature of the initial value problem in relativisticmechanics, is that the dynamics is always constrained. In fact, the evolutionof the particle is subject to mduµ/dτ = eFµνuν, but also to the constraintu2 = c2. However, such a condition can be weakened requiring its validityat certain values of τ only, let us say initially, at τ = 0. Therefore, if Y(τ)vanishes initially, i.e. Y(0) = 0, then Y(τ) = 0 at any τ. In other words, thedifferential Lorentz-force equation implies the constraint u2 = c2 once thisis satisfied initially. Integrating with respect to the proper time we haveuµ(τ) = γeµ0 + γv[eµ2 cos(ωτ) + eµ3 sin(ωτ)] where ω = eB/(mc). We see that γ isconstant with time, meaning that the energy of a charged particle moving ina constant magnetic field is constant. After two successive integrations wehave Xµ(τ) = Xµ(0) +γcτeµ0 + R[eµ2 sin(ωτ)− eµ3 cos(ωτ)] where R = γv/ω. Thisenables us to find the time dependence [0,X(2)(t),X(3)(t)] of the particle’sposition since t/γ = τ. From this solution of the equation of motion weconclude that the motion of a charged particle in a constant magnetic fieldis a uniform circular motion.

One could expect that the particle’s trajectory in the lab frame, followingfrom the previous reasoning [0,X(2)(t),X(3)(t)], should be identified with~xcov(t). However, paradoxical result are obtained by doing so. In particular,the trajectory [0,X(2)(t),X(3)(t)] does not include relativistic kinematics ef-fects. We found that the usual integration of the four-dimensional covariantequation of motion Eq.(12) gives particle trajectory which looks preciselythe same as in Newton dynamics and kinematics. The trajectory of the elec-tron does not include relativistic effects and the Galilean vectorial law ofaddition of velocities is actually used.

The old kinematics is especially surprising, because we are based on the useof the covariant approach. So we must have made a mistake. Notice that weare using textbooks to solve covariant equation of motion. We did not makea computational mistake in our integrations, but rather a conceptual one.We must say immediately that there is no objection to the first integration

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of Eq.(12) from initial conditions over proper time τ. With this, we findthe four-momentum. The momentum has exact objective meaning i.e. itis convention-invariant. What must be recognized is that the concept ofvelocity is only introduced in the second integration step. However, inaccepted covariant approach, the solution of the dynamics problem forthe momentum in the lab frame makes no reference to three-dimensionalvelocity. In fact, the initial condition which we used is uµ(0) = γceµ0 + γveµ2and includes γc and γv, which are actually notations for the time and spaceparts of the initial four-momentum. The three-dimensional trajectory andrespectively velocity, which are convention-dependent, are only found afterthe second integration step. Then, where does the old kinematics comesfrom? The second integration was performed using the relation dτ = dt/γ.It is only after we have made those replacement for dτ that we obtain theusual formula for conventional (non-covariant) trajectory for an electron ina constant magnetic field.

We should then expect to get results similar to those obtained in the case ofthe (3+1) non-covariant particle tracking. In fact, based on the structure ofthe four components of the equation of motion Eq.(12), we can arrive to an-other mathematically identical formulation of the dynamical problem. Thefact that the evolution of the particle in the lab frame is subject to a constrainthas already been mentioned. This means that the mathematical form of thedynamics law includes only three independent equations of motion. It iseasy to see from the initial set of four equations, du(0)/dτ = 0, du(1)/dτ = 0,du(2)/dτ = −eBu(3)/(mc), du(3)/dτ = eBu(2)/(mc), that the presentation of thetime component simply as the relation dτ = dt/γ between proper time andcoordinate time is just a simple parametrization that yields the correctedNewton’s equation Eq.(13) as another equivalent form of these four equa-tions in terms of absolute time t instead of proper time of the particle. Thisapproach to integrating dynamics equations from the initial conditions relieson the use of three independent spatial coordinates and velocities withoutconstraint and is intimately connected with old kinematics. The presenta-tion of the time component simply as the relation dτ = dt/γ between propertime and coordinate time is based on the hidden assumption that the typeof clock synchronization, which provides the time coordinate t in the labframe, is based on the use of the absolute time convention.

11.5 Covariant Particle Tracking

In the non-covariant (3+1) approach, the solution of the dynamics prob-lem in the lab frame makes no reference to Lorentz transformations. Thismeans that, for instance, within the lab frame the motion of particles inconstant magnetic field looks precisely the same as predicted by Newtonian

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kinematics: relativistic effects do not have a place in this description. Inconventional particle tracking a particle trajectory ~x(t) can be seen from thelab frame as the result of successive Galileo boosts that track the motion ofthe accelerated (in a constant magnetic field) particle. The usual Galileo rulefor addition of velocities is used to determine the Galileo boosts tracking aparticular particle, instant after instant, along its motion along the curvedtrajectory.

In order to obtain relativistic kinematics effects, and in contrast to conven-tional particle tracking, one actually needs to solve the dynamics equationin manifestly covariant form by using the coordinate-independent propertime τ to parameterize the particle world-line in space-time. Relying onthe geometric structure of Minkowski space-time, one defines the class ofinertial frames and adopts a Lorentz frame with orthonormal basis vec-tors. Within the chosen Lorentz frame, Einstein’s synchronization of distantclocks and Cartesian space coordinates are enforced. In the Lorentz labframe (i.e. the lab frame with Lorentz coordinate system) one thus has acoordinate representation of a particle world-line as (t(τ), x1(τ), x2(τ), x3(τ)).These four quantities basically are, at any τ, components of a four-vectordescribing an event in space-time. Therefore, if one chooses the lab time t asa parameter for the trajectory curve, after inverting the relation t = t(τ), oneobtains that the space position vector of a particle in the Lorentz lab framehas the functional form ~xcov(t). The trajectory ~xcov(t) is viewed from the labframe as the result of successive Lorentz transformations that depend onthe proper time. In this case relativistic kinematics effects arise.

Let us try out our algorithm for reconstructing ~xcov(t) on some example, tosee how it works. We will find formulas relating the velocity of a particlein one reference system to its velocity in a second reference system. First,we examine the reasoning presented in textbooks. Let us suppose that theK′ system moves relative to the K system with velocity V along the x axis.Let ~v = d~x/dt, be the vector of the particle velocity in the K system and~v′ = d~x′/dt′ the velocity vector of the same particle in the K′ system. FromLorentz transformation we have dx = γ(dx′ + Vdt′), dy = dy′, dz = dz′, dt =

γ(dt′ + Vdx′/c2), where γ = 1/√

1 − V2/c2. Dividing the first three equationsby the fourth we find vx = (v′x + V)/(1 + Vv′x/c2), vy = v′y/[γ(1 + Vv′x/c2)],vz = v′z/[γ(1 + Vv′x/c2)]. These formulae determine the transformation ofvelocities. According to textbooks, they describe the law of composition ofvelocities in the theory of relativity.

Nevertheless, there is argument against this commonly accepted derivationof the composition of velocities. The standard presentation of the velocitytransformation is based on the hidden assumption that (x′, y′, z′) axes and(x, y, z) axes are parallel. In other words, it is assumed that K and K′ observershave common 3-space. This is misconception. One of the consequences of

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non-commutativity of non-collinear Lorentz boosts is not the existence of acommon ordinary space. Suppose that in the Lorentz frame K′ the travelerwas observing particle motion along the y′ axis. In other words measuredvelocity of the particle v′y , and yet the frame is moving relative to the frameK along x axis. Then to an observer at rest in the Lorentz frame K, theframe K′ with the moving particle seems to rotate about the frame K. Letus consider the simple case when the velocity V approaches to that of lightand the velocity v′y is small. This means that we take limit γ 1 v′y/c. Weconsider the small parameter v′y/c and use the second order approximation.For ultrarelativistic limit γ 1 and the axes of K′ frame are then rotatedwith respect to the K frame axes by angle θw = v′y/(γc). We want to look nowat the consequences of this rotation. First we notice that there is a projectionof the velocity v′y on the x-axis, so the velocity component along the x-axis inthe frame K is going to be smaller than V. Our problem now is to work outthe decrement of the horizontal velocity in the K frame. How we can do that?First, the relativistic correction to the horizontal velocity appears only in theorder [v′y/(γc)]2. The trajectory of the particle is viewed from the K frame asthe result of successive infinitesimal Lorentz transformations. Integratingwith respect to infinitesimal value dv′y, we find that the horizontal velocityis vx = V − (v′y)2/(2γ2c). In the ultrarelativistic approximation we find thesimple result vy = v′y/γ, vx = V − v2

y/(2c), so that Lorentz boost with nonrelativistic velocity v′y simply leads to a rotation of particle velocity V of theangle vy/c = vy/V. In contrast, according to textbooks, it follows that thetotal particle speed in the Lorentz K frame increase from V to V + v2

y/(2c).Our result is at odds with the prediction from textbooks. The commonlyaccepted derivation of the composition of velocities based on the use therelation dt′ = dt/γ and does not account for the mixture of (transverse)positions and time.

It is hoped that presented example will draw wide attention to the centralrole that the Wigner rotation plays in the transformation of non-collinearvelocities. To emphasize the physical concepts involved, rather than mathe-matical formalism, we consider not the most general case of the (corrected)velocity addition theorem (3). However, the special case considered alreadyinvolves all necessary ingredients.

11.6 Convention-Invariant Particle Tracking

So far we have considered the motion of a particle in three-dimensionalspace using the vector-valued function ~x(t). We have a prescribed curve(path) along which the particle moves. The motion along the path is de-scribed by l(t), where l is a certain parameter (in our case of interest thelength of the arc). Note the difference between the notions of path and

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trajectory. The trajectory of a particle conveys more information about itsmotion because every position is described additionally by the correspond-ing time instant. The path is rather a purely geometrical notion. Completepaths or their parts may consist of, e. g., line segments, arcs, circles, helicalcurves. If we take the origin of the (Cartesian) coordinate system and weconnect the point to the point laying on the path and describing the motionof the particle, then the creating vector will be a position vector ~x(l). Thederivative of a vector is the vector tangent to the curve described by theradius vector ~x(l). The sense of the d~x(l)/dl is determined by the sense of thecurve arc l.

We already know from our discussion in Introduction that the path ~x(l) hasexact objective meaning i.e. it is convention-invariant. The components ofthe momentum four vector mu = (E/c, ~p) have also exact objective meaning.In contrast to this, and consistently with the conventionality intrinsic inthe velocity, the trajectory ~x(t) of the particle in the lab frame is conventiondependent and has no exact objective meaning.

We want now to describe how to determine the position vector ~x(l)cov incovariant particle tracking. We consider the motion in a uniform magneticfield with zero electric field. Using the Eq.(12) we obtain

d~pdτ

=e

mc~p × ~B,

dEdτ

= 0 . (16)

From dE/dτ = 0 and from the constraintE2/c2−|~p|2 = mc2 we have dp/dτ = 0,

where p = |~p| = m|d~xcov|/dτ. The unit vector ~p/p can be described by theequation ~p/p = d~xcov/|d~xcov| = d~xcov/dl, where |d~xcov| = dl is the differential ofthe path length. From the foregoing consideration follows that

d2~xcov

dl2 =d~xcov

dl×

e~Bpc

. (17)

These three equations corresponds exactly to the equations for the com-ponents of the position vector that can be found using the non-covariantparticle tracking approach, and ~x(l)cov is exactly equal to ~x(l) as it must be.The point is that both approaches describe correctly the same physical re-ality and since the curvature radius of the path in the magnetic field, andconsequently the three-momentum, has obviously an objective meaning (i.e.is convention-invariant), both approaches yield the same physical results.

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11.7 The Nature of Mass

This section concentrates on the nature of mass. The concept of mass isone of the most fundamental notions in physics, comparable in importanceonly to concept of space and time. With advent of the special theory ofrelativity, physicists and philosophers focused their attention on the conceptof relativistic mass. The remarkable progress in experimental and theoreticalphysics made during the past few decades has considerably deepened ourknowledge concerning the nature of mass. We conclude this section with abrief discussion of the physical meaning of the Minkowski geometry.

11.7.1 Phenomenology and Relativistic Extensions

In order fully to understand the meaning of the embedding of the Newton’sdynamics law in the Minkowski space-time, one must keep in mind that,above, we characterized Newton’s equation in the Lorentz comoving frameas a phenomenological law. The microscopic interpretation of the inertialmass of a particle is not given. In other words, it is generally acceptedthat Newton’s second law is a phenomenological law and the rest mass isintroduced in an ad hoc manner. The system of coordinates in which theequations of Newton’s mechanics are valid can be defined as Lorentz restframe. The relativistic generalization of the Newton’s second law to anyLorentz frame permits us to make correct predictions.

We are in the position to formulate the following general statement: anyphenomenological law, which is valid in the Lorentz rest frame, can beembedded in the four dimensional space-time only by using Lorentz co-ordinatization (i.e. Einstein synchronization convention). Suppose we donot know why a muon disintegrates, but we know the law of decay inthe Lorentz rest frame. This law would then be a phenomenological law.The relativistic generalization of this law to any Lorentz frame allows us tomake a prediction on the average distance traveled by the muon. In partic-ular, when a Lorentz transformation of the decay law is tried, one obtainsthe prediction that after the travel distance γvτ0, the population in the labframe would be reduced to 1/2 of the origin population. We may interpretthis result by saying that, in the lab frame, the characteristic lifetime ofa particle has increased from τ0 to γτ0. In contrast, in the non covariant(3+1) space and time approach there is no time dilation effect, since forGalilean transformations the time scales do not change. Therefore, in the(3+1) non covariant approach, there is no kinematics correction factor γ tothe travel distance of relativistically moving muons. The two approachesgive, in fact, a different result for the travel-distance, which must be, how-ever, convention-invariant. This glaring conflict between results of covariant

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and non covariant approaches can be explained as follows: it is a dynamicalline of arguments that explains this paradoxical situation with the relativis-tic γ factor. In fact, there is a machinery behind the muon disintegration.Its origin is explained in the framework of the Lorentz-covariant quantumfield theory. In the microscopic approach to muon disintegration, Einsteinand absolute time synchronization conventions give the same result forsuch convention-invariant observables like the average travel distance, andit does not matter which transformation (Galilean or Lorentz) is used.

11.7.2 The Relativistic Mass

In the non covariant (3+1) space and time approach, there is no time dilationnor length contraction, because for Galilean transformations time and spa-tial coordinates scales do not change. Moreover, it can easily be verified thatNewton’s second law keeps its form under Galilean transformations. There-fore, in the (3+1) non covariant approach, there is no kinematics correctionfactor γ to the mass in Newton’s second law. However, in contrast to kine-matics effects like time dilation and length contraction, the correction factorγ to the mass in the Newton’s second law has direct objective meaning. Infact, if we assign space-time coordinates to the lab frame using the absolutetime convention, the equation of motion is still given by Newton’s secondlaw corrected for the relativistic dependence of momentum on velocity eventhough, as just stated, it has no kinematical origin. Understanding this resultof the theory of relativity is similar to understanding previously discussedresults: at first we use Lorentz coordinates and later the (3+1) non covariantapproach in terms of a microscopic interpretation that must be consistentwith the principle of relativity.

It is well-known from classical electrodynamics that the electromagneticfield of an electron carries a momentum proportional to its velocity forv c, while for an arbitrary velocity v, the momentum is altered by therelativistic γ factor in the case when the absolute time convention is used.Many attempts have been made to explain the electron mass as fully orig-inating from electromagnetic fields. However, these attempts have failed.In fact, it is impossible to have a stationary non-neutral charge distributionheld together by purely electromagnetic forces. In other words, mass andmomentum of an electron cannot be completely electromagnetic in originand in order to grant stability there is a necessity for compensating elec-tromagnetic forces with non electromagnetic fields. From this viewpoint,Newton’s second law is an empirical phenomenological law where the rel-ativistic correction factor γ to the mass is introduced in an ad hoc manner.

From a microscopic viewpoint, today accepted explanation of how struc-tureless particles like leptons and quarks acquire mass is based on the

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coupling to the Higgs field, the Higgs boson having been recently experi-mentally observed at the LHC. This mechanism can be invoked to explainNewton’s second law from a microscopic viewpoint even for structurelessparticles like electrons. However, at larger scales, an interesting and intuitiveconcept of the origin of physical inertia is illustrated, without recurring tothe Higgs field, by results of Quantum Chromodynamics (QCD) for protonsand neutrons, which are not elementary and are composed of quarks andgluon fields. If an initial, unperturbed nuclear configuration is disturbed,the gluon field generates forces that tend to restore this unperturbed config-uration. It is the distortion of the nuclear field that gives rise to the force inopposition to the one producing it, in analogy to the electromagnetic case.But in contrast to the electromagnetic model of an electron, the QCD modelof a nucleon is stable, and other compensation fields are not needed. Now,the gluon field mass can be computed from the total energy (or momen-tum) stored in the field, and it turns out that the QCD version in whichquark masses are taken as zero provides a remarkably good approxima-tion to reality. Since this version of QCD is a theory whose basic buildingblocks have zero mass, the most of the mass of ordinary matter (more than90 percent) arises from pure field energy. In other words, the mass of a nu-cleon can be explained almost entirely from a microscopic viewpoint, whichautomatically provides a microscopic explanation of Newton’s second lawof motion. In order to predict, on dynamical grounds, the inertial mass ofa relativistically moving nucleon one does not need to have access to thedetailed dynamics of strong interactions. It is enough to assume Lorentzcovariance (i.e. Lorentz form-invariance of field equations) of the completeQCD dynamics involved in nucleon mass calculations.

The previous discussion, results in a most general statement: it is enough toassume Lorentz covariance of the quantum field theory involved in micro-particle (elementary or not elementary) mass calculations in order to obtainthe same result for the relativistic mass correction from the two synchro-nization conventions discussed here, and it does not matter which transfor-mation (Galilean or Lorentz) is used.

11.7.3 What does Space-Time Geometry Explain?

It is important to stress at this point that the dynamical line of argumentdiscussed here explains what the Minkowski geometry physically means.The pseudo-Euclidean geometric structure of space-time is only an interpre-tation of the behavior of the dynamical matter fields in the view of differentobservers, which is an observable, empirical fact. It should be clear that therelativistic properties of the dynamical matter fields are fundamental, whilethe geometric structure is not. Dynamics, based on the field equations, isactually hidden in the language of kinematics. The Lorentz covariance of the

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equations that govern the fundamental interactions of nature is an empiricalfact, while the postulation of the pseudo-Euclidean geometry of space-timeis a mathematical interpretation of it that yields the laws of relativistic kine-matics: at a fundamental level this postulate is, however, based on the wayfields behave dynamically.


1. Let us try to get a better understanding of the geometric restatement ofNewton’s second law. To derive the covariant form of relativistic dynamics,we should embed the three-dimensional vector relation md~v/dt = ~f intothe four-dimensional geometry of Minkowski space [43]. The idea of em-bedding is based on the principle of relativity i.e. on the fact that the usualNewton’s second law can always be used in any Lorentz frame where theparticle, whose motion we want to describe, is at rest. In other words, ifan instantaneously comoving Lorentz frame is given at some instant, onecan precisely predict the evolution of the particle in this frame during aninfinitesimal time interval. In geometric language, the Newton law is stricton a hyperplane perpendicular to the world line. However, the hyperplanetilts together with its normal uµ as one moves along the world line. Forthe embedding we need an operator P⊥ that continually projects vectors ofMinkowski space on hyperplanes perpendicular to world line. The desiredoperator is (P⊥)µν = ηµν − uµuν/u2 [43]. In the instantaneously comovingframe one can unambiguously construct a four-force Kµ = [0, ~f ]. Then, inan arbitrary Lorentz frame, the components Kµ can be found through theappropriate Lorentz transformation. In the rest frame obviously uµKµ = 0. Itfollows that, since uµKµ is an invariant, the four-force Kµ is perpendicular tothe four-velocity uµ in any Lorentz frame. The desired embedding of New-ton’s second law in hyperplanes perpendicular to the world line is found byimposing (P⊥)µν(mduν/dτ − Kν) = 0. This is a tensor equation in Minkowskispace-time that relates geometric objects and does not need coordinates tobe expressed. The evolution of a particle can be described in terms of worldline σ(τ), and the 4-velocity by u = dσ/dτ, having a meaning independentlyof any coordinate system. Similarly in geometric language, the electromag-netic field is described by the second-rank, antisymmetric tensor F, whichalso requires no coordinates for its definition. This tensor produces a 4-forceon any charged particle given by P⊥ · (mdu/dτ − eF · u) = 0 [43]. This is thebasic dynamics law for relativistic charged particles expressed in terms ofgeometric objects and automatically included the principle of relativity. Thepresence of the projector operator P⊥ suggests that we have only three inde-pendent equations. In the case of Maxwell’s equation we are able to rewritethe equations in the relativistic form without any change in the meaning at

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all, just with a change notations. It is important to noticed that the situationwith dynamics equations is more complicated. In order fully to understandthe meaning of the embedding of the dynamics law in the hyperplanesperpendicular to the world line, one must keep in mind that, above, wecharacterized the Newton’s equation in the Lorentz comoving frame as aphenomenological law. The system of coordinates in which the equationsof Newton’s mechanics are valid can be defined as Lorentz rest frame. Therelativistic generalization of the Newton’s second law to any Lorentz framepermits us to make correct predictions. The projector operator guaranteesthat this coordinate system restriction will be satisfied.

2. In general, the covariant equation of motion can be solved only by numer-ical methods; however, it is always attractive to find instances where exactsolutions can be obtained. The simplest case of great practical importance isthat the motion of a particle in a constant homogeneous magnetic field. Fromthe solution of covariant equation of motion authors of textbooks [5,43,44]conclude that the covariant motion of a charge particle in a constant mag-netic field is a uniform circular motion. The trajectory of the particle does notinclude relativistic kinematics effects and the Galilean vectorial law of addi-tion of velocities is actually used. Actually, the old kinematics comes fromthe relation dτ = dt/γ. This relation between proper time and coordinatetime is based on the hidden assumption that the type of clock synchroniza-tion, which provides the time coordinate in the lab frame, is based on theuse of the absolute time convention. It is only after the authors of textbookshave make those replacement for dτ that authors obtain the usual formulafor non-covariant trajectory.

3. The non-commutativity of the relativistic composition of non-collinearvelocities was first recognized by Mocanu [45] and latter resolved by Ungar[46], by showing role played by the Wigner rotation in the transformation.

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12 Relativity and Electrodynamics

The differential form of Maxwell’s equations describing electromagneticphenomena in the Lorentz lab frame is given by Eq.(4). Now let us usethese equations to discuss the phenomena called radiation. To evaluateradiation fields arising from an external sources in Eq. (4), we need toknow the velocity ~v and the position ~x as a function of the lab frame timet. As discussed above, it is generally accepted that one should solve theusual Maxwell’s equations in the lab frame with current and charge densitycreated by particles moving along non-covariant trajectory like ~x(t). Thetrajectory ~x(t), which follow from the solution of the corrected Newton’ssecond law Eq. (3) under the absolute time convention, does not include,however, relativistic kinematics effects.

We argue that this algorithm for solving usual Maxwell’s equations in the labframe, which is considered in all standard treatments as relativistically cor-rect, is at odds with the principle of relativity. However, the usual Maxwell’sequations in the lab frame, Eq. (4), are compatible only with covariant tra-jectories calculated by using Lorentz coordinates, therefore including suchrelativistic features as the relativity of simultaneity and the Wigner rotation.

We now consider the particle motion in a giving magnetic field. In theirusual form, Maxwell’s equations are valid only in Lorentz reference frames.According to correct coupling of fields and particles, from the Eq. 5 followsthat

ρ(~x, t) = eδ(~x − ~xcov(t)) ,~j(~x, t) = e~vcov(t)δ(~x − ~xcov(t)) , (18)

where ~vcov = d~xcov/dt. The covariant trajectory of a particle is viewed fromthe lab frame as the result of successive Lorentz transformations.

We find in this book that the Wigner rotation turns out to play an importantrole as the regulator of the velocity addition law. As we have already pointed,the four-velocity cannot be decomposed into u = (cγ, ~vγ) when we deal witha particle accelerating along a curved trajectory in the Lorentz lab frame (1).The presentation of the time component simply as the relation dτ = dt′/γbetween proper time and coordinate time is based on the hidden assumptionthat the type of clock synchronization, which provides the time coordinatet in the lab frame, is based on the use of the absolute time convention.

One of the consequences of non-commutativity of non-collinear Lorentzboosts is a difference between covariant and non covariant particle trajecto-

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ries in a giving magnetic field. One can see that this essential point has neverreceived attention by the physical community. As a result, a correction ofthe conventional radiation theory is required.

12.1 Why did the Error in Radiation Theory Remain so Long Undetected?

The difference between covariant and non-covariant particle trajectorieswas never understood. So, physicists did not appreciate that there wasa contribution to the radiation from relativistic kinematics effects. At thispoint, a reasonable question arises: why the error in radiation theory shouldhave so long remained undetected?

In order to answer this question, we shall discuss the subject more mathe-matically. For an arbitrary parameter v/c covariant calculations of the radi-ation process is very difficult. There are, however, circumstances in whichcalculations can be greatly simplified. As example of such circumstance is anon-relativistic radiation setup. The non-relativistic asymptote provides theessential simplicity of the covariant calculation. The reason is that the non-relativistic assumption implies the dipole approximation which is of greatpractical significance. In accounting only for the dipole part of the radiationwe neglect all information about the electron trajectory That means that thedipole radiation does not show any sensitivity to the difference betweencovariant and non-covariant particle trajectories.

We want now to solve electrodynamics equations mathematically in a gen-eral way and consider the radiation associated with the succeeding termsin (multi-pole) expansion of the field in powers of the ratio v/c. Radiationtheory is naturally developed in the space-frequency domain, as one is usu-ally interested in radiation properties at a given position in space and at acertain frequency. In this book we define the relation between temporal andfrequency domain via the following definition of Fourier transform pair:

f (ω) =

∞∫−∞

dt f (t) exp(iωt)↔ f (t) =1

2π

∞∫−∞

dω f (ω) exp(−iωt) . (19)

Suppose we are interested in the radiation generated by an electron andobserved far away from it. In this case it is possible to find a relatively simpleexpression for the electric field (see Appendix I). We indicate the electronvelocity in units of c with ~β, the electron trajectory in three dimensionswith ~r(t) and the observation position with ~r0. Finally, we introduce the unit

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vector

~n =~r0 − ~r(t)|~r0 − ~r(t)|

(20)

pointing from the retarded position of the electron to the observer. In thefar zone, by definition, the unit vector ~n is nearly constant in time. If theposition of the observer is far away enough from the charge, one can makethe expansion

∣∣∣~r0 − ~r(t)∣∣∣ = r0 − ~n · ~r(t) . (21)

We then obtain the following approximate expression for the the radiationfield in the space-frequency domain:

~E(~r0, ω) =iωecr0

exp[−

iωc~n · ~r0

] ∞∫−∞

dt ~n ×[~n × ~β(t)

]exp

[iω

(t +

~n · ~r(t)c

)](22)

where ω is the frequency, (−e) is the negative electron charge and we makeuse of Gaussian units. A different constant of proportionality in Eq.(22) andin the well known textbooks is to be ascribed to the use of different unitsand definition of the Fourier transform.

First we will limit our consideration to the case of sources moving in a non-relativistic fashion. According to the principle of relativity, usual Maxwell’sequations can always be used in any Lorentz frame where sources are atrest. The same considerations apply where sources are moving in non-relativistic manner. In particular, when oscillating, charge particles emitradiation, and in the non-relativistic case, when the velocities of oscillatingcharges v c, dipole radiation will be generated and described with thehelp of the Maxwell’s equations in their usual form, Eq. (4).

Let’s examine in a more detail how the dipole radiation term comes about.The time ~r(t) · (~n/c) in the integrands of the expression for the radiation fieldamplitude, Eq. (22), can be neglected in the cases where the trajectory of thecharge changes little during this time. It is easy to find the conditions forsatisfying this requirement. Let us denote by a the order of magnitude ofthe dimensions of the system. Then the time ~r(t) · (~n/c) ∼ a/c. In order toensure that the distribution of the charges in the system does not undergo asignificant change during this time, it is necessary that a λ. This conditioncan be written in the in still another form v c, where v is of the order ofmagnitude of the velocities of the charges.

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We consider the radiation associated with the first order term in the expan-sion of the Eq. (22) in power of ~r(t) · (~n/c). In doing so, we neglected allinformation about the electron trajectory ~r(t). In this dipole approximationthe electron orbit scale is always much smaller than the radiation wave-length and Eq. (22) gives fields very much like the instantaneous theory.So we are satisfied using the non-covariant approach when considering thedipole radiation theory.

But that is only the first and most practically important term. The otherterms tell us that there are higher order corrections to the dipole radiationapproximation. The calculation of this correction requires detailed informa-tion about the electron trajectory. Obviously, in order to calculate the cor-rection to the dipole radiation, we will have to use the covariant trajectoryand not be satisfied with the non-covariant approach. However, correctionto the multipole radiation is expected to be small. For example, covariant(trajectory) correction for quadrupole radiation is treated as a correction ofhigher (than quadrupole) order.

12.2 An Illustrative Example

In the next chapter we present a critical reexamination of existing syn-chrotron radiation theory. But before the discussion of this topic it wouldbe well to illustrate error in standard coupling fields and particles in ac-celerator and plasma physics by considering the relatively simple example,wherein the essential physical features are not obscured by unnecessarymathematical difficulties. This illustrative example is mainly addressed toreaders with limiting knowledge of accelerator and synchrotron radiationphysics. Fortunately, the error in standard coupling fields and particles canbe explained in a very simple way.

12.2.1 The Velocity of an Electron Accelerated by the Kicker Field

An electron kicker setup is a practical case of study for illustrating thedifference between covariant and non-covariant trajectories. Let us considerthe simple case when an ultrarelativistic electron moving with the velocityv along z-axis in the lab frame is kicked by a weak dipole magnetic fielddirected along x-axis. We assume for simplicity that the kick angle is smallcompare with 1/γ, where γ = 1/

√1 − v2/c2 is the relativistic factor. This

means that we take the limit γ 1 γvy/v. Let us start with non-covariantparticle tracking calculations. The trajectory of the electron, which followsfrom the solution of the corrected Newton’s second law under the absolutetime convention, does not include relativistic effects. Therefore, as usual

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Fig. 57. A setup for illustration the difference between covariant and non-covarianttrajectories. The motion of a relativistic electron accelerated by a kicker field. It isassumed for simplicity thatγ 1 γvy/v. According to the non-covariant particletracking, the magnetic field B~ex is only capable altering the direction of motion, butnot the speed of the electron.

for Newtonian kinematics, Galilean vectorial law of addition of velocitiesis actually used. Non-covariant particle dynamics shows that the electrondirection changes after the kick, while the speed remains unvaried (Fig.57). According to non-covariant particle tracking, the magnetic field B~ex isonly capable of altering the direction of motion, but not the speed of theelectron. After the kick, the beam velocity components are (0, vy, vz), where

vz =√

v2 − v2y. Taking the ultrarelativistic limit v ' c and using the second

order approximation we get vz = v[1 − v2y/(2v2)] = v[1 − v2

y/(2c2)].

In contrast, covariant particle tracking, which is based on the use of Lorentzcoordinates, yields different results for the velocity of the electron. Let usconsider a composition of Lorentz transformations that track the motion ofthe relativistic electron accelerated by the kicker field. Let the S be the labframe of reference and S′ a comoving frame with velocity ~v relative to S.Upstream of the kicker, the particle is at rest in the frame S′. In order tohave this, we impose that S′ is connected to S by the Lorentz boost L(~v),with ~v parallel to the z axis, which transforms a given four vector event Xin a space-time into X′ = L(~v)X. Let us analyze the particle evolution withinS′ frame. Our particle is at rest and the kicker is running towards it withvelocity −~v. The moving magnetic field of the kicker produces an electric

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field orthogonal to it. When the kicker interacts with the particle in S′ wethus deal with an electron moving in the combination of perpendicularelectric and magnetic fields.

We consider the small expansion parameter γvy/c 1, neglecting terms oforder (γvy/c)3, but not of order (γvx/c)2. In other words, we use the second-order kick angle approximation. It is easy to see that the acceleration in thecrossed fields yields a particle velocity v′y = γvy parallel to the y-axis andv′z = −v(γvy/c)2/2 parallel to the z-axis. If we neglect terms in (γvy/c)3, therelativistic correction in the composition of velocities does not appear in thisapproximation.

Let S” be a frame fixed with respect to the particle downstream the kicker. Asis known, the non collinear Lorentz boosts does not commute. In our secondorder approximation we can neglect the difference between the γvy/c andγzvy/c, where γz = 1/

√1 − v2

z/c2. Here vz = v(1 − θ2k/2) and θk = vy/v = vy/c

in our (ultrarelativistic) case of interest. Therefore we can use a sequenceof two commuting non-collinear Lorentz boosts linking X′ in S′ to X′′ inS′′ as X” = L(~e′yv′y)L(~e′zv′z)X′ = L(~e′zv′z)L(~e′yv′y)X′ in order to discuss thebeam motion in the frame S′ after the kick. Here ~e′y and ~e′z are unit vectorsdirected, respectively, along the x′ and z′ axis. Note that as observed byan observer on S′, the axes of the frame S′′ are parallel to those of S′.The relation X” = L(~e′yv′y)L(~e′zv′z)L(~ezv)X presents a step-by-step changefrom S to S′ and then to S”. For the simple case of parallel velocities, theaddition law is L(~e′zv′z)L(~ezv) = L(~ezvz). The resulting boost composition canbe represented as X” = L(~e′yv′y)L(~ezvz)X. We have already discussed thelaw of composition of velocities in Section 11.5. It was shown that velocitiesaddition in the Lorentz coordinatization is regulated by the Wigner rotation.In the ultrarelativistic approximation one finds the simple result vtot = vz, sothat a Lorentz boost with non-relativistic velocity vy leads to a rotation ofthe particle velocity vz of the angle vy/c (Fig. 58).

On the contrary, textbooks state that the magnetic field, in the Lorentzcoordinatization, is only capable altering the direction of motion, but not thespeed of the electron. It becomes clear that the contradiction is related to theWigner rotation. In fact, as we already discussed in this chapter, the relationdt′ = dt/γ cannot be used when we deal with Lorentz coordinatization.

It should be note, however, that there is another satisfactory way of thecovariant electron tracking. Let us now return to our consideration of themotion of a relativistic electron accelerated by the kicker field and let usdiscuss the observation of an inertial lab observer without passing fromone reference frame to another. In the Chapter 3 we already discussed howone can transform the absolute time coordinatization to Lorentz coordina-

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tization. The overall combination of Galileo transformation and variablechanges actually yields the Lorentz transformation in the case of absolutetime coordinatization in the lab frame. Let us analyze the resynchronizationprocess of the lab distant clocks during the acceleration of the electron. Thiswill allow us to demonstrate a direct relation between the decrease of theelectron speed after the kick in Lorentz coordinates and the time dilationphenomenon. As we already remarked, the Lorentz coordinate system isonly a mental construction: manipulations with non existing clocks are onlyneeded for the application of the usual Maxwell’s equations for synchrotronradiation calculations.

Suppose that upstream the kicker we pick a Lorentz coordinates in the labframe. Then, an instant after entering the magnetic field, the electron ve-locity changes of the infinitesimal value d~v along the y-axis. At this firststep, Eq.(13) allows us to express the differential d~v through the differentialdt in the Lorentz coordinate system assigned upstream the kicker. If clocksynchronization is fixed, this is equivalent to the application of the absolutetime convention. In order to keep Lorentz coordinates in the lab frame, asdiscussed before, we need to perform a clock resynchronization by intro-ducing an infinitesimal time shift. The simplest case is when the kick angleθk is very small, and we evaluate transformations, working only up to theorder (θkγ)2. The restriction to this order provides an essential simplicity ofcalculations in our case of interest for two reasons. First, relativistic correc-tion to compositions of non-collinear velocity increments does not appear inthis expansion order, but only in the order (γθk)3. Second, the time dilationappears in the highest order we use. Thus, Eq.(13) allows us to express thesmall velocity change ∆~v after the kick in the initial Lorentz coordinates sys-tem, and to perform clock resynchronization only downstream the kicker.Therefore, after the kick we can consider the composition of two Lorentzboosts along the perpendicular x and z directions. The first boost imparts thevelocity vθk~ey to the electron along the y-axis and the second boost impartsthe additional velocity −(vθ2

k/2)~ez along the z axis, while the restriction tosecond order assures that the boosts commute.

In order to keep a Lorentz coordinates system in the lab frame after the kick,that is equivalent to describe the kicker influence on the electron trajectoryas Lorentz transformation, we need to perform a clock resynchronization byintroducing a time shift and change the scale of time, that is the rhythm ofall clocks, from t to γyt, with γy ' 1+θ2

k/2. It is immediately understood thatthe speed of electron downstream the kicker is no longer independent ofthe electron motion in the magnetic field (Fig. 58). No relativistic correctionto the velocity component along the y-axis appears in the second order, buta correction of the longitudinal velocity component, changing vz to vz/γy

with vz = v(1 − θ2k/2) and vz/γy = v(1 − θ2

k). It follows that the total electronspeed in the lab frame, after clock resynchronization downstream the kicker,

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Fig. 58. The motion of a relativistic electron accelerated by the kicker field. In orderto describe the kicker influence on the electron trajectory as Lorentz transforma-tion, one needs to perform a clock resynchronization by introducing a time shiftand change the scale of time, that is the rhythm of all clocks, from t to γyt, withγy = 1 + θ2

k/2. It follows that the total electron speed in the Lorentz lab framedownstream the kicker decreases from v to v(1 − θ2

k/2).

decreases from v to v(1 − θ2k/2). The time dilation does not come into the

calculation of the velocity increment, but appears in the correction of theinitial (relativistic) velocity ~v = v~ez.

Note that we discuss particle tracking in the limit of a small kick angleγvy/c 1. However, even in this simple case and for a single electron weare able to demonstrate the difference between non-covariant and covariantparticle trajectories. The electron speed decreases from v to v(1−θ2

k/2). Thisresult is at odds with the prediction from non-covariant particle tracking.

12.2.2 Discussion

We now want to point out that when we accelerate the electron in the labframe by a kicker, the information about this acceleration is included intothe covariant trajectory. The result of covariant approach clearly depends onthe absolute value of the kick angle. The argument concerning the relativitymotion in our case of interest cannot be applied. Not all is relative in specialrelativity, because any change of velocity has an absolute meaning. The

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”absolute” acceleration means acceleration with respect to the fixed stars.

There are several points to be made about the above results. One interestingquestion is, where physically is the information about the electron accel-eration? We answer this question in great detail in the Section 13.5. It is adynamical line of arguments that explains this paradoxical situation withthe trajectory of the accelerated electron. Dynamics, based on the field equa-tions, is actually hidden in the language of relativistic kinematics. Withoutproof, we may state the results. The distribution of the electromagnetic fieldsfrom a fast moving charge is described by Ginzburg-Frank formula (seeAppendix IV). We remark first that for a rapidly moving electron we havenearly equal transverse and mutually perpendicular electric and magneticfields. These are indistinguishable from the fields of a beam of radiation.From microscopic viewpoint, when the electron passes through a kicker, itsfields are perturbed, and now include information about acceleration.

Let us quickly look at some other cases of interest. Above we consider thesimple case when kick angle is small. When the electron passes throughthe weak kicker there is no synchrotron radiation (to be more precise, inthis case radiation is indistinguishable from the self-electromagnetic fieldsof the electron). We expect that an electron that passes through the weakkicker is still ”field-dressed”. Suppose the ultrarelativistic electron in the labframe is kicked by dipole magnet field, but the kick angle is large comparewith 1/γ. In other words, we now consider an electron moving along astandard synchrotron radiation setup. At the exit of strong kicker we havea ”naked” (or ”field-free”) electron i.e. an electron that is not accompaniedby virtual radiation fields. There is a process of formation of the ”fielddressed” electron (i.e. formation of the fields from a fast moving charge)within the formation length from the very beginning of the straight sectiondownstream of the strong kicker. The information about acceleration isincluded now into the synchrotron radiation. But the acceleration history(together with the self fields of the ultrarelativistic electron) is washed outduring the kicking process, and now electron fields not include informationabout acceleration. That is to say, the information about acceleration is notincluded into the covariant trajectory downstream the strong kicker (wewill discuss this subject further in the Section 13.5).

12.2.3 The Motion of an Electron Accelerated by a Kicker in a Bending Magnet

In our relativistic but non-covariant study of electron motion in a givenmagnetic field, the electron has the same velocity and consequently thesame relativistic factor γ upstream and downstream of the kicker. Supposewe now put the electron through a bending magnet (i.e. a uniform mag-netic field directed along the y-axis ), Fig. 59. The motion in the bending

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Fig. 59. Geometry for radiation production from a bending magnet. The motion ofa relativistic electron accelerated by a kicker field. According to the non-covariantparticle tracking , the magnetic field B~ex of the kicker is only capable altering thedirection of motion, but not the speed of the electron.

magnet we obtained is practically the same as in the case of non-relativisticdynamics, the only difference being the appearance of the relativistic factorγ in the determination of cyclotron frequency ωc = eB/(mγ). The curvatureradius R of the trajectory is derived from the relation v⊥/R = ωc, wherev⊥ = v(1 − θ2

k/2) is the component of the velocity normal to the field of thebending magnet ~B = B~ey. According to non-covariant particle tracking, afterthe kick, the correction to the radius R is only of order θ2

k .

One could naively expect that according to covariant particle tracking, sincethe total speed of electron in the lab frame downstream of the kicker de-creases from v to v(1−θ2

k/2), this would also lead to a consequent decrease ofthe three-momentum |~p| from mγv to mγv(1−γ2θ2

k/2) in our approximation.However, such a momentum change would mean a correction to the radiusR of order γ2θ2

k so that there is a glaring conflict with the calculation of theradius according to non covariant tracking. Since the curvature radius ofthe trajectory in the bending magnet has obviously an objective meaning,i.e. it is convention-invariant, this situation seems paradoxical. The para-dox is solved taking into account the fact that in Lorentz coordinates thethree-vector of momentum ~p is transformed, under Lorentz boosts, as thespace part of the four vector pµ. Let us consider a composition of Lorentz

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boosts that track the motion of the relativistic electron accelerated by thekicker field. Under this composition of boosts the longitudinal momentumcomponent remains unchanged (with accuracy θ2

k).

Let us verify that this assertion is correct. We have pµ = [E/c, ~p]. We considerthe Lorentz frame S′ fixed with respect to the electron upstream the kicker,and in the special case when electron is at rest p′µ = [mc,~0]. We turn focus onwhat happens in S′. Acceleration in the crossed kicker fields gives rise to anelectron velocity v′y = γvy parallel to the y-axis and v′z = −v(γvy)2/2 parallelto the z-axis. Downstream of the kicker the transformed four-momentum isp′µ = [mc+mv′2y /(2c), 0,mv′y,mv′z], where we evaluate the transformation onlyup to the order (γvy/c)2, as done above. We note that, due to the transverseboost, there is a contribution to the time-like part of the four-momentumvector i.e. to the energy of the electron. In fact, the energy increases frommc2 to mc2 + m(γvy)2/2. We remind that S′ is connected to the lab frame S bya Lorentz boost. Now, with a boost to a frame moving at velocity ~v = −v~ez,the transformation of the longitudinal momentum component, normal tothe magnetic field of the bend, is pz = γ(p′z + vp′0/c) = γmv. Therefore we cansee that the momentum component along the z-axis remains unchangedin our approximation of the Lorentz transformation. We also have, fromthe transformation properties of four-vectors, that the time component p0 =γ(p′0 + vp′z) = γmc .

Let us now return to our consideration on the covariant electron trajec-tory calculation in the Lorentz lab frame when a constant magnetic fieldis applied. We analyzed a very simple (but very practical) kicker setupand we noticed that, in fact, the three-momentum is not changed; so wehave already verified that this transformation is the same as the non co-variant transformation for the three-momentum, i.e. ~pcov = ~p. We also foundthat there is a difference between covariant and non covariant output ve-locities, vcov < v. In these transformations we therefore demonstrated that~pcov , m~vcov/

√1 − v2

cov/c2 for curved trajectory in ultrarelativistic asymptotic.It is interesting to discuss what it means that there are two different (covari-ant and non covariant) approaches that produce the same particle three-momentum. The point is that both approaches describe correctly the samephysical reality and the curvature radius of the trajectory in the magneticfield (and consequently the three-momentum) has obviously an objectivemeaning, i.e. is convention-invariant. In contrast to this, the velocity of theparticle has objective meaning only up to a certain accuracy, because thefiniteness of velocity of light takes place.

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12.2.4 Red Shift of the Critical Frequency of the Synchrotron Radiation

Next we discuss the interesting problem of emission of synchrotron radia-tion in a bending magnet with and without kick. Let us consider the setuppictured in Fig. 59. Suppose that an ultrarelativistic electron moving alongthe z-axis in the lab frame is kicked by a weak dipole field directed along thex-axis before entering a uniform magnetic field directed along the y-axis, i.e.a bending magnet. An accelerated electron traveling on a curved trajectoryemits radiation. When moving at relativistic speed, this radiation is emittedas a narrow cone tangent to the path of the electron. Moreover, the radiationamplitude becomes very large in this direction. This phenomenon is knownas Doppler boosting. Synchrotron radiation is generated when a relativisticelectron is accelerated in a bending magnet. Without going into the detailsof computation, it is possible to present intuitive arguments explaining whythe characteristics of the spectrum of synchrotron radiation only depend, inthe ultrarelativistic limit, on the difference between electron and light speed.

We turn now to the radiation emitted by an ultrarelativistic electron in abending magnet. Let us discuss the case when the source is heading towardsthe observer (Fig. 60). An electromagnetic source propagates through thesystem as a function of time following a certain trajectory ~x(t′). However,an electromagnetic signal emitted at time t′ at a given position ~x(t′) arrivesat the observer position at a different time t, due to the finite speed of light.As a result, an observer sees the motion of the electromagnetic source as afunction of t. The prime used here to indicate the retarded times should notbe confused with the primes referring to a Lorentz transformed frame in theproceeding sections.

Let the coordinates of the electron be (x, y, z), with z measured along thedirection of observation. We shall still assume that the detector is veryfar from source. At a given moment in time, say the moment t′, the threecomponents of the position are x(t′), y(t′), and z(t′). The distance R is nearlyequal to R(t′) = R0 − z(t′). If the time of observation is called t then the timet′ is not the time t, but is delayed by the total distance that the light hasto go, divided by the speed of light. If disregard the uninteresting constantdelay, which just means change the origin of t by a constant, then it saysthat ct = ct′ − z(t′). Now we need to find x as a function of t, not t′, andwe can do this in the following way. Using the fact that c − v c weobtain the well-known relation dt/dt′ = (c − v cosθ)/c ' (1 − v/c + θ2/2) '(1/2)(1/γ2 +θ2), where θ is the observation angle (Fig. 60). The observer seesa time compressed motion of the source, which go from point A to point Bin an apparent time corresponding to an apparent distance 2Rθdt/dt′. Let usassume (this assumption will be justified in a moment)θ2 > 1/γ2. In this caseone has 2Rθdt/dt′ ' Rθ3. Obviously one can distinguish between radiationemitted at point A and radiation emitted at point B only when compressed

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Fig. 60. Geometry for synchrotron radiation from a bending magnet. Radiationfrom an electron passing through the setup is observed through a spectral filter bya fixed observer positioned on the tangent to the bend at point P, as shown in Fig.(a). Electromagnetic source propagates through the system, as a function of time,as shown in Fig. (b). However, electromagnetic signal emitted at time t′ at a givenposition x(t′) arrives at observer position at a different time t, due to finite speed oflight. As a result, the observer in Fig. (a) sees the electromagnetic source motion asa function of t. The apparent motion is a hypocycloid, and not the real motion x(t′).The observer sees a time-compressed motion of the sources, which go from pointA to point B in an apparent time corresponding to an apparent distance 2Rθdt/dt′.

distance Rθ3 o, i.e. for θ (o/R)1/3. This means that, as concerns the

radiative process, we cannot distinguish between point A and point B onthe bend such that Rθ < (R2o)1/3. It does not make sense at all to talk aboutthe position where electromagnetic signals are emitted within L f = (R2o)1/3

(here we assuming that the bend is longer than L f ). This characteristic lengthis called the formation length for the bend. The formation length can also beconsidered as a longitudinal size of the single electron source (in the space-frequency domain). Note that a single electron always produces diffraction-limited radiation. The limiting condition of spatially coherent radiation isa space-angle product θrd ' o, where d being the transverse size and θr

the divergence of the source. Since d ' L fθr it follows that the divergenceangle θr is strictly related to L f and o: θr '

√o/L f . One may check that

using L f = (R2o)1/3, one obtains θr ' (o/R)1/3 as it must be. In particular,at θr ' 1/γ one obtains the characteristic wavelength ocr ' R/γ3 as is wellknown for bending magnet radiation (Fig. 61).

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Fig. 61. Formation length for bend. Formation length L f can also be considered asa longitudinal size of a single-electron source. It does not make sense at all to talkabout the position where electromagnetic signals are emitted within L f .

It is clear from the above that, according to conventional synchrotron ra-diation theory, if we consider radiation, the introduction of the kick onlyamounts to a rigid rotation of the angular distribution along the new direc-tion of the electron motion. This is plausible, if one keeps in mind that afterthe kick the electron has the same velocity and emits radiation in the kickeddirection owing to the Doppler effect.

According to the correct coupling of fields and particles, there is a remark-able prediction of synchrotron radiation theory concerning the setup de-scribed above. Namely, there is a red shift of the critical frequency of thesynchrotron radiation in the kicked direction. To show this, let us considerthe covariant treatment, which makes explicit use of Lorentz transforma-tions. When the kick is introduced, covariant particle tracking predicts anon-zero red shift of the critical frequency, which arises because in Lorentzcoordinates the electron velocity decreases from v to v − vθ2

k/2, while thevelocity of light is unvaried and equal to the electrodynamics constant c.The red shift in the critical frequency can be expressed by the formula∆ωcr/ωcr ' −γ2v2

y/c2 = −γ2θ2k . We now see a second order correction θ2

k thatis, however, multiplied by a large factor γ2.

It should be note, however, that there is another satisfactory way of ex-plaining the red shift. We can reinterpret this result with the help of a non-

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covariant treatment, which deals with non- covariant particle trajectories,and with Galilean transformations of the electromagnetic field equations.According to non-covariant particle tracking the electron velocity is unvar-ied. However, Maxwell’s equations do not remain invariant with respectto Galilean transformation, and the velocity of light has increased fromc, without kick, to c(1 + θ2

k/2) with kick. The reason for the velocity oflight being different from the electrodynamics constant c is due to the factthat, according to the absolute time convention, the clocks after the kickare not resynchronized. The ratio of the electron velocity to that of light isconvention independent i.e. it does not depend on the distant clock syn-chronization or on the rhythm of the clocks. Our calculations show thatcovariant and non-covariant treatments (at the correct coupling fields andparticles) give the same result for the red shift prediction, which is obviouslyconvention-invariant and depends only on the (dimensionless) parametervcoord/ccoord = v(1 − θ2

k/2)/c = v/[c(1 + θ2k/2)], where vcoord is the coordinate

electron velocity, and ccoord is the coordinate velocity of light.

In order to confirm the predictions of our synchrotron radiation theory,we propose an experimental test at third generation synchrotron radiationsources. Synchrotron radiation from bending magnets is emitted withina wide range of frequencies. The possibility of using narrow bandwidthsources in an experimental study on the red shift in synchrotron radiationspectrum looks more attractive. This allows one to increase the sensitivity ofthe output intensity on the red shift, and to relax the requirement on beamkicker strength and photon beam line aperture. Undulators, as sources ofquasi-monochromatic synchrotron radiation, produce light in a sufficientlynarrow bandwidth for our purposes. They cause the electron beam to fol-low a periodic undulating trajectory with the consequence that interferenceeffects occur. Undulators have typically many periods. The interference ofradiation produced in different periods results in a bandwidth that scalesas the inverse number of periods. Therefore, the use of insertion devicesinstalled at third generation synchrotron radiation facilities would allow usto realize a straightforward increase in the sensitivity to the red shift at arelatively small kick angle, θk < 1/γ. In the next chapter we discuss thisexperimental test in more detail.

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1. Let us present a typical textbook statement [47] concerning the projectionof an arbitrary world line onto the Lorentz lab frame basis: ”A chargedpoint particle moving along the world line x(τ), τ being proper time, withinthe framework of Special Relativity has the velocity u(τ) = dx(τ)/dτ =(γc, γ~v). The four-velocity is normalized such that its invariant squarednorm equals c2, u2 = c2γ2(1−β2) = c2. While x(τ) and u(τ) are coordinate-freedefinitions the decomposition u = (cγ, ~vγ) presupposes the choice of a frameof reference K. The particle, which is assumed to curry the charge e, createsthe current density j(x) = ec

∫dτu(y)δ4(y − x(τ)). This is a Lorentz vector.

[...] Furthermore, in any frame of reference K, one recovers the expectedexpressions for the charge and current densities by integrating over τ bymeans of relation dτ = dt′/γ between proper time and coordinate timeand using the formula δ(y0 − x0(τ)) = δ(ct − ct′) = δ(t − t′)/c, j0(t, y) =ceδ(3)(y − x(t)) ≡ cρ(t, y), ji(t, y) = evi(t)δ(3)(y − x(t)), i = 1, 2, 3.” One cansee that the integration is performed using the relation dτ = dt′/γ. As wealready discussed, this restriction cannot be imposed when we deal with aparticle accelerating along a curved trajectory in the Lorentz lab frame.

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13 Synchrotron Radiation


Accelerator physics was always thought in terms of the old (Newtonian)kinematics that is not compatible with Maxwell’s equations. We would likenow to use our ideas about dynamics and electrodynamics to consider insome greater detail the question: ”Why did the error in the synchrotronradiation theory remain so long undetected?”

The phenomena of electromagnetic radiation which we want to study arerelatively complicated. For an arbitrary setup covariant calculations of theradiation process is very difficult. There are, however, circumstances inwhich calculations can be greatly simplified. As example of such circum-stance is a synchrotron radiation setup. Similar to the non-relativistic asymp-tote, the ultrarelativistic asymptote also provides the essential simplicity ofthe covariant calculation. The reason is that the ultrarelativistic assump-tion implies the paraxial approximation. Since the formation length of theradiation is much longer than the wavelength, the radiation is emitted atsmall angles of order 1/γ or even smaller, and we can therefore enforce thesmall angle approximation. We assume that the transverse velocity is smallcompared to the velocity of light. In other words, we use a second orderrelativistic approximation for the transverse motion. Instead of small (total)velocity parameter (v/c) in the non-relativistic case, we use a small trans-verse velocity parameter (v⊥/c). The next step is to analyze the longitudinalmotion, following the same method. We should remark that the analysis ofthe longitudinal motion in a synchrotron radiation setup is very simple. Ifwe evaluate the transformations up to second order (v⊥/c)2, the relativisticcorrection in the longitudinal motion does not appear in this approximation.

According to covariant approach, the various relativistic kinematics effectsconcerning to the synchrotron radiation setup, turn up in successive ordersof approximation.

In the first order (v⊥/c). - relativity of simultaneity. Wigner rotation, whichin the ultrarelativistic approximation appears in the first order already, andresults directly from the relativity of simultaneity.

In the second order (v⊥/c)2. - time dilation. Relativistic correction in law ofcomposition of velocities, which already appears in the second order, andresults directly from the time dilation.

The first order kinematics term (v⊥/c) plays an essential role only in thedescription of the coherent radiation from a modulated electron beam. In

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a storage ring the distribution of the longitudinal position of the electronsin a bunch is essentially uncorrelated. In this case, the radiated fields dueto different electrons are also uncorrelated and the average power radiatedis a simple sum of the radiated power from individual electrons; that is wesum intensities, not fields. In the case of incoherent synchrotron radiation,a motion of the single ultrarelativistic electron in a constant magnetic field,according to the theory of relativity, influences the kinematics terms of thesecond order (v⊥/c)2 only.

13.2 Paraxial Approximation for the Radiation Field

The general method to derive the frequency spectrum is to transform theelectric field from the time domain to the frequency domain by the use ofFourier transform. First, let us rewrite Eq. (22) as follows

~E(~r0, ω) =−iωecr0

exp[−

iωc~n · ~r0

] ∞∫−∞

dt [~β(t) − ~n] exp[iω

(t +

~n · ~r(t)c

)](23)

Eq. (22) and Eq. (23) are equivalent but include different integrands. Thisis no mistake, as different integrands can lead to the same integral (seeAppendix I).

We call z0 the observation distance along the optical axis of the system,while (x0, y0) fixes the transverse position of the observer. Using the complexnotation, in this and in the following sections we assume, in agreement withEq. (19), that the temporal dependence of fields with a certain frequency isof the form:

~E ∼ ~E(z0, x0, y0, ω) exp(−iωt) . (24)

With this choice for the temporal dependence we can describe a plane wavetraveling along the positive z-axis with

~E = ~Ea exp( iω

cz0 − iωt

). (25)

In the following we will always assume that the ultra-relativistic approxi-mation is satisfied, which is the case for SR setups. As a consequence, theparaxial approximation applies too. The paraxial approximation implies a

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slowly varying envelope of the field with respect to the wavelength. It istherefore convenient to introduce the slowly varying envelope of the trans-verse field components as

~E(z0, x0, y0, ω) = ~E(z0, x0, y0, ω) exp (−iωz0/c) . (26)

We will now replace all vectors by their components to obtain directional

dependency of the synchrotron radiation. The emission angle θ =√θ2

x + θ2y

is taking with respect to the z-axis. Here θx = x0/z0 is the observation angleprojected onto the x − z plane, θy = y0/z0 is the observation angle projectedonto the y−z plane. The components of the unit vector~n can be approximatedby nz = 1−θ2

x/2−θ2y/2, nx = θx, ny = θy, so~n·(~r0−~r) = (z0−z)(1−θ2

x/2−θ2y/2)−

xθx − yθy. We consider the motion in a static magnetic field. According tothe conventional particle tracking the magnitude of the velocity is constantand is equal v = ds/dt = const., where s(z) is the longitudinal coordinatealong the path. The transverse components of the envelope of the field inEq. (23) in the far zone and in paraxial approximation finally becomes

~E(z0,~r0, ω) =−iωec2z0

∞∫−∞

dz′exp [iΦT][(

vx(z′)c− θx

)~ex +

(vy(z′)

c− θy

)~ey

](27)

where the total phase ΦT is

ΦT = ω

[s(z′)

v−

z′

c

]+ω2c

[z0(θ2

x + θ2y) − 2θxx(z′) − 2θyy(z′) + z′(θ2

x + θ2y)]. (28)

Here vx(z′) and vy(z′) are the horizontal and the vertical components ofthe transverse velocity of the electron, x(z′) and y(z′) specify the transverseposition of the electron as a function of the longitudinal position, ~ex and ~ey

are unit vectors along the transverse coordinate axis.

13.3 Undulator Radiation

Traditionally, all courses in synchrotron radiation theory have begun in thesame way, retracing the path followed in the historical development of thesubject. One first learns about synchrotron radiation from bending magnet.We will begin in this chapter by dealing with the ”advanced” subject of the

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synchrotron radiation theory. For undulator setup covariant calculations ofthe radiation process is quite simple.

To generate specific synchrotron radiation characteristics, radiation is oftenproduced from special insertion devices called undulators. The resonanceapproximation, that can always be applied in the case of undulator radiationsetups, yields simplifications of the theory. This approximation does notreplace the paraxial one, but it is used together with it. It takes advantageof another parameter that is usually large, i.e. number of undulator periodsNw 1. The frequency emitted by a particle going through an undulatorcan be obtained by considering the interference between the parts of theradiation created at successive periods. The frequency of this field is Dopplershifted. The shortest wavelength is observed on undulator axis. In resonanceapproximation, all undulator radiation at shortest wavelength is emittedwithin an angle much smaller than 1/γ (see Fig. 62). This automaticallyselects observation angles of interest. In fact, if we consider observationangles outside the diffraction angle, we obtain zero intensity with accuracy1/Nw.

We now have to ask: Why did the error in insertion device theory remainso long undetected? We answer this question in great detail bellow. In dis-cussing this we will deal only with the radiation into the central cone (see Fig.62). In doing so, we neglected all information about the electron trajectory.In this approximation the electron oscillation amplitude is always (indepen-dently of how large the undulator strength parameter) much smaller thanthe radiation diffraction size at the undulator exit, because Nw 1. Thus,the undulator radiation theory gives fields very much like the (dipole-like)instantaneous theory. So, we are satisfied using the conventional (i.e. non-covariant) approach for describing the undulator radiation into the centralcone, that is the practical situation of interest.

However, there is one situation where the conventional theory fails. Thecovariant approach predicts a non-zero red shift of the resonance frequency,which arises when there are perturbations (kicks with respect to the lon-gitudinal axis) of the electron motion. Experimental results confirm ourcorrection for spontaneous undulator emission.

13.3.1 Conventional Theory

Eq. (27) can be used to characterize the far field from an electron movingon any path. In this section we present a simple derivation of the frequencyrepresentation of the radiated field produced by an electron in the planarundulator. The magnetic field on the undulator axis has the form

207

~B(z) = ~eyBw cos(kwz) , (29)

Here kw = 2π/λw, and λw is the undulator period. The Lorentz force is usedto derive the equation of motion of the electron in the presence of a magneticfield. Integration of this equation gives

vx(z) = −cθs sin(kwz) = −cθs

2i[exp(ikwz) − exp(−ikwz)

]. (30)

Here θs = K/γ, where K is the deflection parameter defined as

K =eλwBw

2πmc2 , (31)

m being the electron mass at rest and Bw being the maximal magnetic fieldof the undulator on axis.

In this case the electron path is given by

x(z) = rw cos(kwz) , (32)

where rw = θs/kw is the oscillation amplitude.

We write the undulator length as L = Nwλw, where Nw is the number ofundulator periods. With the help of Eq. (27) we obtain an expression, validin the far zone:

~E =iωec2z0

L/2∫−L/2

dz′exp [iΦT] exp[iωθ2z0

2c

] [Kγ

sin (kwz′)~ex + ~θ

].

(33)

Here

ΦT =

(ω

2cγ2z

+ωθ2

2c

)z′ −

Kθx

γω

kwccos(kwz′) −

K2

8γ2

ωkwc

sin(2kwz′) ,

(34)

where the average longitudinal Lorentz factor γz is defined as

γz =γ

√1 + K2/2

. (35)

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Fig. 62. Geometry for radiation production from an undulator

The choice of the integration limits in Eq. (33) implies that the referencesystem has its origin in the center of the undulator.

Usually, it does not make sense to calculate the intensity distribution fromEq. (33) alone, without extra-terms (both interfering and not) from the otherparts of the electron path. This means that one should have complete in-formation about the electron path and calculate extra-terms to be added toEq. (33) in order to have the total field from a given setup. Yet, we can findparticular situations for which the contribution from Eq. (33) is dominantwith respect to others. In this case Eq. (33), alone, has independent physicalmeaning.

One of these situations is when the resonance approximation is valid. Thisapproximation does not replace the paraxial one, based on γ2

1, but it isused together with it. It takes advantage of another parameter that is usuallylarge, i.e. the number of undulator periods Nw 1. In this case, the integralin dz′ in Eq. (33) exhibits simplifications, independently of the frequency ofinterest due to the long integration range with respect to the scale of theundulator period.

A well known expression for the angular distribution of the first harmonicfield in the far zone (see Appendix II for a detailed derivation) can be ob-

209

tained from Eq. (33). Such expression is axis-symmetric, and can, therefore,be presented as a function of a single observation angleθ, whereθ2 = θ2

x+θ2y.

One obtains the following distribution for the slowly varying envelope ofthe electric field:

~E = −Kωe

2c2z0γAJJ exp

[iωθ2z0

2c

] L/2∫−L/2

dz′ exp[i(C +

ωθ2

2c

)z′]~ex ,

(36)

Here ω = ωr + ∆ω, C = kw∆ω/ωr and

ωr = 2kwcγ2z , (37)

is the fundamental resonance frequency. Finally AJJ is defined as

AJJ = J0

(K2

4 + 2K2

)− J1

(K2

4 + 2K2

), (38)

Jn being the n-th order Bessel function of the first kind. The integration overlongitudinal coordinate can be carried out leading to the well-known finalresult:

~E(z0, ~θ) = −KωeL2c2z0γ

AJJ exp[iωθ2z0

2c

]sinc

[L2

(C +

ωθ2

2c

)]~ex ,

(39)

where sinc(·) ≡ sin(·)/(·). Therefore, the field is horizontally polarized andazimuthal symmetric. Eq. (39) describes a field with spherical wavefrontcentered in the middle of the undulator.

13.3.2 Why did the Error in Insertion Device Theory Remain so Long Undetected?

We have seen that in all generality the expression for the undulator field inthe far zone and in the ultrarelativistic (i.e. paraxial) approximation can bewritten as Eq. (109). Within the resonance approximation (Nw 1) for thefrequencies around the first harmonic it can be simplified to the well-knownexpression Eq. (39) where the field is horizontally polarized and azimuthalsymmetric. The divergence of this radiation is much smaller compared tothe angle 1/γz. The mathematical reason stems from the fact that the factor

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sin(·)/(·) represents the well-known resonance character of the undulatorradiation. If we are interested in the angular width of the peak around theobservation angle θ = 0, we can introduce an angular displacement ∆θ.Taking the first zero of the sin(·)/(·) function at C = 0 we will be able todetermine the natural angular width of the radiation for the first harmonicθc. The cone with aperture θc is usually called central cone. It can be foundthat θ2

c = 1/(2Nwγ2z) 1/γ2

z .

Now we would like to understand what is the characteristic transverse sizeof the field distribution at the exist of the undulator. The radiation frommagnetic poles always interferes coherently at zero angle with respect toundulator axis. This interference is constructive within an angle of about√

c/(ωLw). We can estimate the interference size at the undulator exit asabout

√cLw/ω. On the other hand, the electron oscillating amplitude is given

by rw = cθs/kw = cK/(γkw). It follows that r2w/(cLw/ω) = K2ω/(Lwk2

wγ2) =

K2/[πNw(1 + K2/2)] 1, where we use the fact that γ2 = (1 + K2/2)γ2z . This

inequality holds independently of the value of K, because Nw 1. Thus,the electron oscillating amplitude is always much smaller than the radiationdiffraction size at the undulator exit.

We consider the radiation associated with the first order term in the expan-sion of the Eq. (112) in power of small parameter [Kθxω/(γkwc)] (see Eq.(116)). But in doing so we miss all information about transverse electrontrajectory in the phase factor Eq. (34) since the term Kθxω cos(kwz′)/(γkwc)is neglected. In this approximation the electron orbit scale is always muchsmaller than the radiation diffraction size and Eq. (39) gives fields very muchin agreement with the dipole radiation theory. So we are satisfied using thenon covariant approach when considering the transverse electron motion.

There are several points to be made about the above result. We have justexplained that in accounting only for the radiation in the central cone, wemiss all information about the transverse electron motion. To be completewe must add an analysis of the accelerated motion along the z-direction (i.e.along the undulator axis). We assume that the transverse velocity v⊥(z) issmall compared to the velocity of light c. We consider the small expansionparameter v⊥/c, neglecting terms of order (v⊥/c)3, but not of order (v⊥/c)2. Inother words we use a second order relativistic approximation for transversemotion. We should remark that the analysis of the longitudinal motion in theultrarelativistic approximation is much simpler than in the case of transversemotion. It is easy to see that the acceleration in the constant magnetic fieldyields an transverse electron velocity v⊥ and ∆vz = −v(v⊥/c)2/2 parallel tothe z-axis.

If we evaluate the transformations up to the second order (v⊥/c)2, the rela-tivistic correction in the longitudinal motion does not appear. So one should

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not be surprised to find that, in this approximation, there is no influence ofthe difference between the non-covariant and covariant constrained electrontrajectories on the undulator radiation in the central cone.

13.3.3 Influence of the Kick According to Conventional Theory

Eq. (39) can be generalized to the case of a particle with a given offset~l anddeflection angle ~η with respect to the longitudinal axis, assuming that themagnetic field in the undulator is independent of the transverse coordinateof the particle (see Appendix III). Although this can be done using Eq.(122) directly, it is sometimes possible to save time by getting the answerwith some trick. For example, in the undulator case one takes advantageof the following geometrical considerations, which are in agreement withrigorous mathematical derivation. First, we consider the effect of an offset~l on the transverse plane, with respect to the longitudinal axis z. Since themagnetic field experienced by the particle does not change, the far-zone fieldis simply shifted by a quantity ~l. Eq. (39), can be immediately generalizedby systematic substitution of the transverse coordinate of observation, ~r0

with ~r0 −~l. This means that ~θ = ~r0/z0 must be substituted by ~θ −~l/z0, thus

yielding

E(z0,~l, ~θ

)=−

KωeL2c2z0γ

AJJ exp

iωz0

2c

∣∣∣∣∣∣~θ − ~lz0

∣∣∣∣∣∣2 sinc

ωL

∣∣∣∣~θ − (~l/z0

)∣∣∣∣24c

. (40)

Let us now discuss the effect of a deflection angle ~η. Since the magnetic fieldexperienced by the electron is assumed to be independent of its transversecoordinate, the path followed is still sinusoidal, but the effective undulatorperiod is now given by λw/ cos(η) ' (1 +η2/2)λw. This induces a relative redshift in the resonant wavelength ∆λ/λ ∼ η2/2. In practical cases of interestwe may estimate η ∼ 1/γ. Then, ∆λ/λ ∼ 1/γ2 should be compared withthe relative bandwidth of the resonance, that is ∆λ/λ ∼ 1/Nw, Nw being thenumber of undulator periods. For example, if γ > 103, the red shift due tothe deflection angle can be neglected in all situations of practical relevance.As a result, the introduction of a deflection angle only amounts to a rigidrotation of the entire system. Performing such rotation we should accountfor the fact that the phase factor in Eq. (40) is indicative of a sphericalwavefront propagating outwards from position z = 0 and remains thusinvariant under rotations. The argument in the sinc(·) function in Eq. (40),instead, is modified because the rotation maps the point (z0, 0, 0) into thepoint (z0,−ηxz0,−ηyz0). As a result, after rotation, Eq. (40) transforms to

212

E(z0, ~η,~l, ~θ

)= −

KωeLAJJ

2c2z0γexp

iωz0

2c

∣∣∣∣∣∣~θ − ~lz0

∣∣∣∣∣∣2 sinc

ωL

∣∣∣∣~θ − (~l/z0

)− ~η

∣∣∣∣24c

(41)

Finally, in the far-zone case, we can always work in the limit for l/z0 1, thatallows one to neglect the term ~l/z0 in the argument of the sinc(·) function,as well as the quadratic term in ωl2/(2cz0) in the phase. Thus Eq. (41) can befurther simplified, giving the generalization of Eq. (39) in its final form:

E(z0, ~η,~l, ~θ

)=−

KωeLAJJ

2c2z0γexp

[iωc

(z0θ2

2− ~θ ·~l

)]sinc

ωL

∣∣∣∣~θ − ~η∣∣∣∣24c

. (42)

It is clear from the above that, according to conventional synchrotron radi-ation theory, if we consider radiation from one electron at detuning C fromresonance, the introduction of a kick only amounts to a rigid rotation of theangular distribution along the new direction of the electron motion. This isplausible, if one keeps in mind that after the kick the electron has the samevelocity and emits radiation in the kicked direction owing to the Dopplereffect. After such rotation, Eq. (39) transforms into Eq. (42)

13.3.4 Influence of the Kick According to Correct Coupling of Fields and Particles

According to the correct coupling of fields and particles, there is a remark-able prediction of undulator radiation theory concerning to the undulatorradiation from the single electron with and without kick. Namely, when akick is introduced, there is a red shift in the resonance wavelength of the un-dulator radiation in the velocity direction. To show this, let us consider thecovariant treatment, which makes explicit use of Lorentz transformations.

When the kick is introduced, covariant particle tracking predicts a non-zero red shift of the resonance frequency, which arises because in Lorentzcoordinates the electron velocity decreases from v to v−vθ2

k/2 after the kick,while the velocity of light is unvaried and equal to the electrodynamicsconstant c.

Now the formula Eq. (124) is not quite right, because we should have usednot the velocity of electron v but v − vθ2

k/2. The shift in the total phaseΦT under the integral Eq. (122) can be expressed by the formula ∆ΦT =ωθ2

kz′/(2c), where we account to that v ' c in ultrarelativistic approximation.

213

Suppose that without kick the electron moves along the constrained trajec-tory parallel to the undulator axis. The field which produces this electron inthe far zone is given by Eq. (39). Referring back to the Eq. (42), we see thatthe conventional undulator radiation theory gives the following expressionfor radiation field after the kick

~E = −KωeL2c2z0γ

AJJ exp[iωθ2z0

2c

]sinc

L2

C +ω

∣∣∣∣~θ − ~θk

∣∣∣∣22c

~ex .

(43)

The covariant equations say that, when the kick is introduced, the radiationfield in question is given by the formula


AJJ exp[iωθ2z0

2c

]sinc

L2

C +ωθ2

k

2c+ω

∣∣∣∣~θ − ~θk

∣∣∣∣22c

~ex ,

(44)

This formula has nearly, but not quite the same form as Eq. (43), the dif-ference consisting in the term ωθ2

k/(2c) in the argument of sinc function.Attention must be called to the difference in resonance frequency betweenthe undulator radiation setup with and without the kick. Remembering thedefinition of the detuning parameter C = kw∆ω/ωr, we can write the redshift in resonance frequency as ∆ω/ωr = −ωrθ2

k/(2kwc). With this we alsopointed out that the red shift can be written as ∆ω/ωr = −γ2θ2

k/(1 + K2/2).We now see a second order correction θ2

k that is, however, multiplied by thelarge factor γ2.

We are now ready to investigate, more generally, what form the field ex-pression takes under the introduction of a kick. Suppose that, without kick,the electron moves along the trajectory with angle ~η with respect to theundulator axis. The field produced by this electron is given by Eq. (42). Welet ~θk be the kick angle of the electron with respect to its initial motion. Theconventional approach gives the following expression for the field after thekick


AJJ exp[iωθ2z0

2c

]sinc

L2

C +ω

∣∣∣∣~θ − ~η − ~θk

∣∣∣∣22c

~ex .

214

(45)

In contrast, the covariant approach gives


AJJ exp[iωθ2z0

2c

]sinc

L2

C +ωθ2

k

2c+ω

∣∣∣∣~θ − ~η − ~θk

∣∣∣∣22c

~ex ,

(46)

Now this all leads to an interesting situation. According to the conventionaltheory, the resonance wavelength depends only on the observation anglewith respect to the electron velocity direction. Equation (45) says that for anykick angle ~θk and for any angle ~η between the undulator axis and the initialelectron velocity direction, the radiation along the velocity direction has nored shift. We would like to emphasize a very important difference betweenconventional and covariant theory. The result of the covariant approachEq. (46) clearly depends on the absolute value of the kick angle θk and theradiation along the velocity direction has the red shift only when the kickangle has nonzero value.

We must conclude that when we accelerate the electron in the lab frameupstream the undulator, the information about this acceleration is includedinto the covariant trajectory.

13.3.5 Experimental Test of SR Theory in 3rd Generation Light Source

One way to demonstrate incompatibility between the standard approachto relativistic electrodynamics, which deals with the usual Maxwell’s equa-tions, and particle trajectories calculated by using non-covariant particletracking, is to make a direct laboratory test of synchrotron radiation theory.In other words, we are stating here that, despite the many measurementsdone during decades, synchrotron radiation theory is not an experimentallywell-confirmed theory.

Let us analyze the potential for exploiting synchrotron radiation sources inorder to confirm the predictions of corrected synchrotron radiation theory.The emittance of the electron beam in new generation synchrotron radia-tion sources is small enough, so that one can neglect finite electron beamsize and angular divergence in the soft X-ray wavelength range, and suchsynchrotron radiation source can be examined under the approximation ofa filament electron beam. This allows us to take advantage of analyticalpresentations for single electron synchrotron radiation fields.

215

Fig. 63. Basic setup for the proposed critical experimental test of synchrotron radi-ation theory with third generation light source. Top: the case for an electron beamwithout kick. Bottom: the case for the electron beam kicked by an angle η. In bothcases, the X-ray pulse is filtered by a monochromator and the total energy recordedby a detector as a function of the undulator detuning.

The basic setup for a test experiment is sketched in Fig 63. The soft X-ray un-dulator beam line should be tuned to a minimum photon energy (typicallythis limit is related with the so called ”water window” wavelength range).The radiation pulse goes through a monochromator filter F and its energyis subsequently measured by the detector. No precise monochromatizationof the undulator radiation is required in this case: a monochromator linewidth ∆ω/ω ' 10−3 is sufficient. In order for proposed test experiment tobe curried out, it is necessary to control the beam kicking e.g. by correctormagnet. In the case of no kick the maximum pulse energy registered bythe detector will coincide with the monochromator line tuned to resonance.When the kick is introduced the conventional synchrotron radiation theorystill predicts a zero red shift in the resonance wavelength. In contrast to this,one of the immediate consequences of the corrected theory is the occurrenceof a non-zero red shift.

The proposed experimental procedure is relatively simple, because is basedon relative measurements in the (electron beam) velocity direction with andwithout transverse kick. Such a measurement is critical, in the sense thatthe prediction of conventional theory is the absence of red shift and (to ourknowledge) has never been performed at synchrotron radiation facilities.

216

However, XFEL based experiment confirm our correction for spontaneousundulator emission (1).

13.4 Synchrotron Radiation from Bending Magnets

Consider a single relativistic electron moving on a circular orbit. The ob-server in the standard treatment is assumed to be located in a vertical planetangent to the circular trajectory at the origin, at an angle θ above the levelof the orbit. In other words, in this geometry the z axis is not fixed, butdepends on the observer’s position. Note that the geometry of the electronmotion has a cylindrical symmetry, with the vertical axis going through thecenter of the circular orbit. Because of this symmetry, in order to calculatespectral and angular photon distributions, it is not necessary to consider anobserver at a more general location.

There are a number of remarkable effects which are a consequence of thecylindrical symmetry and the paraxial approximation. In discussing this wewill demonstrate that there is no needs to use covariant particle tracking forderivation of the bending magnet radiation. However, there is one situa-tion where the conventional theory fails. The covariant approach predicts anon-zero red shift of the critical frequency, which arises when there are per-turbations of the electron motion along the magnetic field (i.e with respectto nominal orbit).

It would be well to begin with bird’s view of the main results. We want tosolve the electrodynamics problem based on Maxwell’s equations in theirusual form. In this case we should analyze the particle evolution within theframework of special relativity, where the problem of assigning Lorentz co-ordinates to the lab frame in the case of accelerating motion is complicated.The only possibility to introduce Lorentz coordinates in this situation con-sists in introducing individual coordinate systems (i.e. individual rule-clockstructure) for each point of the path.

In order to keep Lorentz coordinates in the lab frame, as discussed be-fore, we need only to perform a clock resynchronization by introducing atime shift and change the scale of time, that is the rhythm of clocks. Be-cause of cylindrical symmetry, the observer is assumed to be located inthe vertical plane tangent to the circular trajectory at the origin. Becausethe formation length is very short, it is necessary to know the velocity andposition over only a small arc of the trajectory. In the ultrarelativistic (parax-ial) approximation, the observer sees a uniform acceleration of the electrona = v2/R = c2/R in the transverse direction. We can, then, write velocity andoffset of the electron as follows v⊥ = at, r⊥ = at2/2. Relativity of simultaneity

217

gives time shift ∆t = v⊥r⊥/c2 = v2⊥

t/(2c2). But a correction of the rhythmis t

√1 − v2

⊥/c2 ' t − v2

⊥t/(2c2). We have a beautiful cancellation. This is a

coincidence. It is because we have deal with uniform acceleration in thetransverse direction using a second order (paraxial) approximation whenan electron is moving along an arc of a circle.

13.4.1 Conventional Theory

Consider a single relativistic electron moving on a circular orbit and anobserver. It is worth to underline the difference between the more generalgeometry which we use and the geometry used in most synchrotron radi-ation textbooks for the treatment of bending magnet radiation. In standardtreatments of bending magnet radiation, the horizontal observation angleθx is always equal to zero. The reason for this is that most textbooks focuson the calculation of the intensity radiated by a single electron in the farzone, which involves the square modulus of the field amplitude, but do notanalyze, for instance, situations like source imaging.

We can use Eq. (27) to calculate the far zone field of radiation from a rela-tivistic electron moving along an arc of a circle. Assuming a geometry witha fixed z we can write the transverse position of the electron as a functionof the curvilinear abscissa s as

~r(s) = −R (1 − cos(s/R)) ~ex (47)

and

z(s) = R sin(s/R) (48)

where R is the bending radius.

Since the integral in Eq. (27) is performed along z we should invert z(s) inEq. (48) and find the explicit dependence s(z):

s(z) = R arcsin(z/R) ' z +z3

6R2 (49)

so that

~r(z) = −z2

2R~ex , (50)

where the expansion in Eq. (49) and Eq. (50) is justified, once again, in theframework of the paraxial approximation.

218

With Eq. (27) we obtain the radiation field amplitude in the far zone:

~E =iωec2z0

∞∫−∞

dz′eiΦT

(z′ + Rθx

R~ex + θy~ey

)(51)

where

ΦT = ω

θ2x + θ2

y

2cz0

+

12γ2c

+θ2

x + θ2y

2c

z′

+(θx

2Rc

)z′2 +

( 16R2c

)z′3

]. (52)

One can easily reorganize the terms in Eq. (52) to obtain

ΦT = ω

θ2x + θ2

y

2cz0

− Rθx

2c

(1γ2 +

θ2x

3+ θ2

y

)+

(1γ2 + θ2

y

)(z′ + Rθx)

2c+

(z′ + Rθx)3

6R2c

]. (53)

With redefinition of z′ as z′ + Rθx under integral we obtain the final result:

~E =iωec2z0

eiΦseiΦ0

∞∫−∞

dz′(z′

R~ex + θy~ey

)× exp

iω

[z′

2γ2c

(1 + γ2θ2

y

)+

z′3

6R2c

], (54)

where

Φs =ωz0

2c

(θ2

x + θ2y

)(55)

and

Φ0 = −ωRθx

2c

(1γ2 +

θ2x

3+ θ2

y

). (56)

In standard treatments of bending magnet radiation, the phase term exp(iΦ0)is absent. In fact, the horizontal observation angle θx is always equal to zero.

219

13.4.2 Why did the Error in Synchrotron Radiation Remain so Long Undetected?

Our case of interest is an ultrarelativistic electron accelerating in a circle.As already remarked, in conventional (non-covariant) particle tracking thedescription of the dynamical evolution in the lab frame is based on theuse of the absolute time convention. In this case simultaneity is absolute,and we only need one set of synchronized clocks in the lab frame, to beused for the description of the accelerated motion. However, the use ofthe absolute time convention automatically implies the use of much morecomplicated field equations, and these equations are different for each valueof the particle velocity i.e. for each point along its path. This is the reason toprefer the covariant approach within the framework of both dynamics andelectrodynamics.

We start by considering an electron moving along a circular trajectory thatlies in the (x, z)-plane and tangent to the z axis. Because of cylindrical sym-metry, in order to calculate spectral and angular photon distributions, it isnot necessary to consider an observer at general location. The observer isassumed to be located in the vertical plane tangent to the circular trajectoryat the origin. In ultrarelativistic (paraxial) approximation we evaluate trans-formations working only up to the order of v2

x/c2. The restriction to this orderprovides an essential simplicity of calculations. We can interpret manipula-tion with rule-clock structure in the lab frame simply as a change of vari-ables according to the transformation Eq. (7): xd = γxx, td = (t/γx +γxxvx/c2).We are dealing with a second order approximation and γx = 1 + v2

x/(2c2).The overall combination of Galilean transformation and variable changesactually yields to the transverse Lorentz transformation. Since the Galileantransformation, completed by the introduction of the new variables, is math-ematically equivalent to a Lorentz transformation, it obviously follows thattransforming to new variables leads to the usual Maxwell’s equations.

In order to keep Lorentz coordinates in the lab frame, as discussed before,we need only to perform a clock resynchronization by introducing the timeshift ∆t = td − t = −[v2

x/(2c2)]t + xvx/c2. The relativistic correction to the par-ticle’s offset ”x” does not appear in this expansion order, but only in orderof v3

x/c3 and xd = x in our case of interest. Although we have only shownthat time shift in one rather special case, the result is right for any offset and(transverse) velocity direction: ∆t = td − t = −[|~v⊥|2/(2c2)]z′/c + ~r⊥ · ~v⊥/c2.To finish our analysis we need only find a relativistic correction to the lon-gitudinal motion. We remark again that if we evaluate the transformationsup to the second order (v⊥/c)2, the relativistic correction in the longitudinalmotion does not appear in this approximation. We have demonstrated thecovariant method that can be used for any trajectory - a general way offunding what happens directly in space-frequency domain and in paraxialapproximation.

220

Let us now see how to apply this covariant method to a special situation.Let’s use our knowledge of the relativistically correct method for calculatingsynchrotron radiation emission to find the photon angular-spectral densitydistributions from a bending magnet. In the ultrarelativistic approximation,we have a uniform acceleration of the electron a = v2/R = c2/R in thetransverse direction. We can, then, write velocity and offset of the electronas follows vx = at = az′/v = az′/c, x = at2/2 = az′2/(2c2). We have now allquantities we wanted. Let us put them all together in relativistic time shift:∆t = td − t = −a2z′3/(2c5) + a2z′3/(2c5) = 0. There is no difference! We do notneed to use covariant particle tracking for derivation of the bending magnetradiation. Usually, such a beautiful cancellation is found to stem from adeep underlying principle. Nevertheless, in this case there does not appearto be any such profound implication. This is a coincidence.

This cancellation is not surprising, if one analyzes the general expressionfor the radiation field from bending magnet in the far zone Eq.(54). In ourprevious discussion of undulator radiation, we learned that the relativisticcorrection appears only when the transverse electron trajectory is includedin the total phase ΦT under the integral Eq.(27). Referring back to Eq.(28) forthe phase factor ΦT , we see that the term which depends on the transverseposition of the electron can be written as exp i(ω/c)[θxx(z′) + θyy(z′)]. Weconclude that the observation angle in the total phase factor under theintegral must be related with the contribution of the transverse electrontrajectory. Now look at Eq.(54). This equation includes only the observationangleθy in the phase factor under the integral. This means that the transverseconstraint motion of the electron in the bending magnet does not affectsynchrotron radiation. So we are justified using a non-covariant approachfor considering the constrained electron motion along the nominal orbit in(x, z)-plane.

We point out that the cancellation in relativistic time shift and the indepen-dence of the Fraunhofer propagator (to be more precise, in space-frequencydomain we are dealing with a paraxial approximation of Green’s function ofnonhomogeneous Helmholtz equation) on the observation angle θx in thefar zone can be regarded as the two sides of the same coin: they are manifes-tation of the cylindrical symmetry when an electron is moving along an arcof a circle. Because of cylindrical symmetry, in order to calculate spectral andangular photon distributions in the far zone, it is not necessary to consideran observer at a general location. The observer is assumed to be located inthe vertical plane tangent to the circular trajectory at the origin. In this caseobservation angle θx = 0 and the observation angle θy is above the levelof the orbit. In other words, in this very special geometry the z-axis is notfixed, but depends on the observer position. However, this way of proceed-ing can hardly help to obtain radiation fields in the near zone. Indeed, inthe near zone we are dealing with the Fresnel propagator, which obviously

221

depends on the constrained motion of the electron. We use far-zone argu-ments only to show that there is no influence of the difference between thenon-covariant and covariant trajectories on the synchrotron radiation frombending magnets. The cancellation in the relativistic time shift leads to thesame outcome in the near zone as it must be.

13.4.3 Influence of the Kick According to Conventional Theory

Up to this point we considered an electron moving along a circular trajectorythat lies in the (x, z)-plane and tangent to the z axis. The phase differencein the fields will be determined by the position of the observer and by theelectron trajectory. Let us now discuss the bending magnet radiation froma single electron with arbitrary angular deflection and offset with respect tothe nominal orbit.

Approximation for the electron path (see Eq. (120), Eq. (121) in the AppendixIII) can be used to characterize the field from an electron moving on anytrajectory. Using Eq. (49) and Eq. (50) an approximated expression for s(z)can be found:

s(z) = z +z3

6R2 +z2ηx

2R+

zη2x

2+

zη2y

2(57)

so that

~v(z) =(−

vzR

+ vηx

)~ex +

(vηy

)~ey (58)

and

~r(z) =

(−

z2

2R+ ηxz + lx

)~ex +

(ηyz + ly

)~ey . (59)

It is evident that the offsets lx and ly are always subtracted from x0 andy0 respectively: a shift in the particle trajectory on the vertical plane isequivalent to a shift of the observer in the opposite direction. With this inmind we introduce angles θx = θx − lx/z0 and θy = θy − ly/z0 to obtain

~E =iωec2z0

∞∫−∞

dz′eiΦT

(z′ + R(θx − ηx)

R~ex + (θy − ηy)~ey

)(60)

and

222

ΦT = ω

θ2x + θ2

y

2cz0

+ω2c

(1γ2 +

(θx − ηx

)2+

(θy − ηy

)2)

z′

+

(ω(θx − ηx)

2Rc

)z′2 +

(ω

6R2c

)z′3 . (61)

One can easily reorganize the terms in Eq. (61) to obtain

ΦT = ω

θ2x + θ2

y

2cz0

− ωR(θx − ηx)2c

×

(1γ2 + (θy − ηy)2 +

(θx − ηx)2

3

)+

(1γ2 + (θy − ηy)2

)ω

(z′ + R(θx − ηx)

)2c

+ω

(z′ + R(θx − ηx)

)3

6R2c. (62)

Redefinition of z′ as z′ + R(θx − ηx) gives the result

~E =iωec2z0

eiΦseiΦ0

∞∫−∞

dz′(z′

R~ex + (θy − ηy)~ey

)× exp

iω

[z′

2γ2c

(1 + γ2(θy − ηy)2

)+

z′3

6R2c

], (63)

where

Φs =ωz0

2c

(θ2

x + θ2y

)(64)

and

Φ0 = −ωR(θx − ηx)

2c

(1γ2 + (θy − ηy)2 +

(θx − ηx)2

3

). (65)

In the far zone we can neglect terms in lx/z0 and ly/z0, which leads to

~E =iωec2z0

eiΦseiΦ0

∞∫−∞

dz′(z′

R~ex +

(θy − ηy

)~ey

)× exp

iω

[z′

2γ2c

(1 + γ2

(θy − ηy

)2)

+z′3

6R2c

], (66)

223

where

Φs =ωz0

2c

(θ2

x + θ2y

)(67)

and

Φo ' −ωR(θx − ηx)

2c

(1γ2 + (θy − ηy)2 +

(θx − ηx)2

3

)−ωc

(lxθx + lyθy) . (68)

It is clear from the above that the field distribution in the far zone dependsonly on the observation angle with respect to the electron velocity direction.

According to the conventional (incorrect) coupling of fields and particles,there is a prediction of radiation theory concerning to the bending magnetradiation from a single electron with and without kick. Namely, when a kickis introduced, there is a rigid rotation of the angular distribution in the farzone.

13.4.4 Influence of the Kick According to Correct Coupling of Fields and Particles

Let us discuss the covariant treatment, which makes explicit use of Lorentztransformations. Consider the bending magnet radiation from a single elec-tron with a kick with respect to the nominal orbit in (x, z)-plane. In thiscase, we additionally have a translation along the y-axis with constantvelocity vy = vθk. We can, then, write the offset of the electron as fol-lows y = θkz′. Let’s put velocity and offset in the relativistic time shift:∆t = td − t = −θ2

kz′/(2c) + θ2kz′/c = θ2

kz′/(2c). So, the shift in the totalphase under the integral along the path can be expressed by the formula∆ΦT = ωθ2

kz′/(2c). The result agrees with our red shift calculation in theundulator case when the kick is introduced, as it must be.

We would like to make a historical note. The difference between covariantand non-covariant particle trajectories was never understood. So, accel-erator physicists did not appreciate that there was a contribution to thesynchrotron radiation from relativistic kinematics effects. The question nowarises how can storage rings actually operate. The point is that this exam-ple deals with a situation where electron beam kinetics is determined bythe emission of synchrotron radiation from bending magnets. However, be-cause of the cylindrical symmetry, covariant and non-covariant solutionsfor the electron motion along an arc of a circle yield similar properties ofsynchrotron radiation except the following modification. The covariant ap-proach predicts a non-zero red shift of the critical frequency, which ariseswhen there are perturbations of the electron motion in the vertical direction.

224

But synchrotron radiation from bending magnets is emitted within a widerange of frequencies, and the output intensity is not sensitive on the redshift.

13.5 How to Solve Problems Involving Many Trajectory Kicks

We shall now discuss the situation where there are n arbitrary spaced kick-ers, all different from one another in terms of the rotation angle introduced.Let us consider how we may apply covariant particle tracking in this cir-cumstance, and try to understand what is happening when we have forexample an undulator downstream of the kicker setup. Formally, if onewants to calculate the radiation from the undulator one should take intoaccount all kicks in the electron trajectory, from the generation of the elec-tron. However, this situation is not surprising, if one analyzes the generalexpression for the radiation field from a single electron Eq.(22). In fact, weshould note that, in general, one needs to know the entire history of theelectron from t′ = −∞ to t′ = ∞ since the integration in Eq.(22) is performed(at least formally) between these limits. However, this statement should beinterpreted physically, depending on the situation under study: integrationshould in fact be performed from and up to times when the electron doesnot contribute to the field anymore.

We should pointed out that it is the electrodynamics theory, which ultimatelydecides what part of the particle trajectory is important for calculating undu-lator radiation and what part can be neglected. The most important, generalstatement concerning the relevant part of the particle trajectory, is that itmust be calculated according to the covariant method (if one wants to usethe usual Maxwell’s equations).

Let us consider the ultrarelativistic assumption 1/γ2 1, which is verified

for synchrotron radiation setups. In general, the introduction of a smallparameter in any theory brings simplifications. The ultrarelativistic approx-imation implies a paraxial approximation and Eq.(22) can be simplified toEq.(27). Suppose that we take a situation in which the rotation angle of thefirst bending magnet upstream of the undulator is much larger than 1/γ. Inother words, we now consider an electron moving along a standard syn-chrotron radiation setup. The electron enters the setup via a bending magnet,passes through a straight section, an undulator, and another straight sec-tion. Finally, it leaves the setup via another bend. Note that, although theintegration in Eq.(27) is performed from −∞ to ∞, the only (edge) part ofthe trajectory into the bending magnets contributing to the integral is oforder of the radiation formation length L f . Mathematically, it is reflectedin the fact that ΦT(z′) in Eq.(27) exhibits more and more rapid oscillations

225

as z′ becomes larger than the formation length. At the critical wavelengththe formation length is simply of order of R/γ, R being the radius of thebend. That simply corresponds to an orbiting angular interval ∆θ ' 1/γ.Typically, the critical wavelength of the radiation from a bending magnet insynchrotron radiation source is about 0.1 nm and the formation length inthis case is only few millimeters.

Note that for ultrarelativistic systems in general, the formation length isalways much longer than the radiation wavelength. This counterintuitiveresult follows from the fact that for ultrarelativistic systems one cannotlocalize sources of radiation within a macroscopic part of the trajectory.The formation length can be considered as the longitudinal size of a singleelectron source. It does not make sense at all to talk about the position whereelectromagnetic signals are emitted within the formation length. This meansthat, as concerns the radiative process in the bending magnet, we cannotdistinguish between radiation emitted at point A and radiation emitted atpoint B when the distance between these two points is shorter than theformation length L f . Let us now consider the case of a straight section oflength L inserted between the bending magnet and the undulator. One canstill use the same reasoning considered for the bend to define a region ofthe trajectory where it does not make sense to distinguish between differentpoints. As in bending magnet case, the observer sees a time compressedmotion of the source and in the case of straight motion the apparent timecorresponds to an apparent distance oγ2. At the critical wavelength thebending magnet formation length L f ' R/γ is simply order of the straightline formation length oγ2.

Intuitively, bending magnets act like switchers for the ultrarelativistic elec-tron trajectory. We consider the case when switchers are presented in theform of bending magnets, but other setups can be considered where switch-ers have different physical realizations. The only feature that these differentrealizations must have in common, by definition of switcher, is that theswitching process must depends exponentially on the distance from thebeginning of the process. Then, a characteristic length ds can be associatedto any switcher. Consider, for example, a plasma accelerator where an elec-tron is accelerated with high-gradient fields. In this case it is the acceleratoritself that switches on the relativistic electron trajectory, since accelerationin the GeV range takes place within a few millimeters only. In the (soft)X-ray range the acceleration distance da is shorter than the formation lengthoγ2 for the following straight section. In this particular case length da playsthe role of the characteristic length of the switcher ds, which switch on theultrarelativistic electron trajectory.

Let us now return to our consideration of the standard synchrotron radiationsetup and let us analyze the radiation process in an insertion device (un-

226

Fig. 64. Standard undulator radiation setup. When the electron passes througha bending magnet there is the synchrotron radiation, washing out the (Frank-Ginzburg) fields of the fast moving charge. At the exit of the bending magnetwe have ”naked” electron. There is a process of formation of the field-dressedelectron within the formation length from the very beginning of the straight sectiondownstream the bending magnet. The field-dressed ultrarelativistic electron has avisible transverse size of order a few microns for third generation synchrotronradiation sources.

dulator). We have actually the ”creation” of the relativistic electron withina distance of order oγ2 from the very beginning of the straight section up-stream the undulator. It is assumed that the length of the straight sectionL is much longer than the formation length oγ2 that is clearly always thecase in the X-ray range. When the switching distance ds . oγ2

L, thenature of the switcher is not important for describing the radiation from theundulator installed within the straight section (Fig. 64).

Downstream of the switcher we have a uniformly moving electron. Thefields associated to an electron with a constant velocity exhibit an interest-ing behavior when the speed of the charge approaches that of light. Namely,in the space-frequency domain there is an equivalence of the fields of a rel-ativistic electron and those of a beam of electromagnetic radiation. In fact,for a rapidly moving electron we have nearly equal transverse and mutu-ally perpendicular electric and magnetic fields. These are indistinguishablefrom the fields of a beam of radiation. This virtual radiation beam has a

227

macroscopic transverse size of order oγ (see Appendix IV). At the exit ofthe switcher we have a ”naked” (or ”field-free”) electron i.e. an electron thatis not accompanied by virtual radiation fields. There is a process of forma-tion of the ”field-dressed” electron (i.e. the formation of the fields from a fastmoving charge) within the distance of order oγ2 from the very beginning ofthe straight section downstream of the switcher.

The electron trajectory being divided into two essentially different parts:before and after the switcher. If we accelerate the electron in the lab frameupstream of the switcher, the information about this acceleration is includedinto the first part of the covariant trajectory. But this acceleration prehistory(together with the fields of the ultrarelativistic electron) is washed out dur-ing the switching process and at the entrance of the straight section we havea ”naked” electron.

We start with the description of the field formation process along the straightsection downstream of the switcher, based on the covariant approach. Firstof all we have to synchronize distant clocks within the lab frame. The syn-chronization procedure that follows is the usual Einstein synchronizationprocedure. It is assumed that in the Lorentz lab frame the electron proceedsfollowing a rectilinear trajectory with velocity v. This assumption is used asinitial condition. Then we can analyze situation downstream the switcherby using the usual Maxwell’s equations.

When one analyzes the process of ”field-dressed” electron formation fromthe viewpoint of the non covariant approach, one assumes the same initialconditions (rectilinear trajectory with velocity v) for the electron motion.Then one solves the electrodynamics problem of fields formation by us-ing the usual Maxwell’s equations. We already mentioned that the type ofclock synchronization which results in time coordinate t in an electron tra-jectory ~x(t) is never discussed in accelerator physics. However, we knowthat the usual Maxwell’s equations are only valid in the Lorentz frame.The non covariant approach is obviously based on a definite synchroniza-tion assumption, but this is actually a hidden assumption. In our case ofinterest, within the lab frame the Lorentz coordinates are then automati-cally enforced. So one should not be surprised to find that in this simplecase of rectilinear motion there is no difference between covariant and noncovariant calculations of the initial conditions at the undulator entrance.

Because of the characteristics of undulator radiation, in order to calculatethe radiation field within the central cone, we simply use the instantaneous(i.e. dipole-like) theory of radiation. So we are satisfied using a non covari-ant approach for considering the constrained motion along the undulator.We conclude that it does not matter which approach is used to describethe standard synchrotron radiation setup. The two approaches, treated ac-

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cording to Einstein’s or absolute time synchronization conventions give thesame result for the radiation within the central cone.

Let us now see what happens with a weak dipole magnet (a kicker), which isinstalled in the straight section upstream of the undulator and is character-ized by a small kick angle (γθk)2

1. What do we expect for the undulatorradiation? At first glance the situation is similar to the switcher setup andthe electron trajectory is again divided into two parts: before and after thekicker. The most important difference, however, is that electrodynamicsnow dictates that both trajectories are important for the calculation of theundulator radiation. When the electron passes through the kicker there isno synchrotron radiation (to be more precise, in this case radiation is indis-tinguishable from the self-electromagnetic fields of the electron), washingout the virtual radiation fields like in the switcher case. We expect that anelectron that passes through a kicker is still ”field-dressed”, but we havean electron whose fields has been perturbed, and now include informationabout the acceleration with respect to an inertial frame.

According to the conventional theory, as usual for Newtonian kinematics,the Galilean vectorial law of addition of velocities is actually used. Non-covariant particle dynamics shows that the direction of the electron trajec-tory changes after the kick, while its speed remains unvaried. In contrast,covariant particle tracking, which is based on the use of Lorentz coordinates,yields different results for the trajectory of the electron. The electron speeddecreases from v to v(1−θ2

k/2). This result is at odds with the prediction fromnon-covariant particle tracking, because the corrected relativistic additionlaw for non-parallel velocities is used to calculate the electron trajectory.

According to the conventional algorithm for solving electrodynamics fieldequations, which deals with the usual Maxwell’s equations, and particletrajectories calculated by using non-covariant particle tracking, the undu-lator radiation along the velocity direction has no red shift of resonancefrequency for any kick angle θk. According to the correct coupling of fieldsand particles, there is a remarkable prediction of synchrotron radiation the-ory concerning the setup described above. Namely, there is a red shift of theresonance frequency of the undulator radiation in the kicked direction. Thered shift can be expressed by the formula ∆ωr/ωr = −γ2θ2

k/(1 + K2/2).

13.6 Synchrotron Radiation in the Case of Particle Motion on a Helix

The presence of red shift in bending magnet radiation automatically impliesthe same problem for conventional cyclotron radiation theory. In the ultra-relativistic limit, there are well-known analytical formulas that describe the

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spectral and angular distribution of cyclotron radiation emitted by an elec-tron moving in a constant magnetic field having a non-relativistic velocitycomponent parallel to the field, and an ultrarelativistic velocity componentperpendicular to it. According to the conventional approach, exactly as forthe bending magnet case, the angular-spectral distribution of radiation is afunction of the total velocity of the particle due, again, to the Doppler effect.In contrast, the covariant approach predicts a non-zero red shift of the criticalfrequency, which arises when there are perturbations of the electron motionin the magnetic field direction. It should be note that cyclotron-synchrotronradiation emission is one of the most important processes in plasma physicsand astrophysics and our corrections are important for a much wider partof physics than that of synchrotron sources.

13.6.1 Existing Theory

Let us discuss in some detail the relativistic cyclotron radiation. Here weshall only give some final results and discuss their relation with the conven-tional synchrotron radiation theory from bending magnet. In the case of anuniform translation motion with non-relativistic velocity along the magneticfield direction (Fig. 65), a widely accepted (in astrophysics) expression forthe angular and spectral distributions of radiation from an ultra-relativisticelectron on a helical orbit is given by (2)

~E(χ, α) ∼~ex

(ξ2 + ψ2)K2/3

ω2ωc

(1 +

ψ2

ξ2

)3/2−i~ey

(ξ2 + ψ2)1/2ψK1/3

ω2ωc

(1 +

ψ2

ξ2

)3/2 , (69)

where K1/3 and K2/3 are the modified Bessel functions, ξ = 1/γ, ψ = χ − α,(χ is the angle between ~v and ~B and α that between ~n and ~B); the angle ψ isclearly the angular distance between the direction of the electron velocity ~vand the direction of observation ~n. Here the ωc is defined by 3eBγ2/(2mc).

Actually we have already discussed radiation from an ultrarelativistic elec-tron on a helical orbit in the previous section. Equation Eq. (66) is the resultwe worked out above for the bending magnet radiation from a single elec-tron with angular deflection with respect to nominal orbit. Eq. ( 69) doesnot look the same as Eq. (66). It will, however, if we now define the smalldeflection angle ηy = π/2 − χ and the observation angle θy = π/2 − α (theobserver is also assumed to be located in the vertical plane tangent to thetrajectory i.e. θx, ηx = 0). The integrals in Eq. (66) can be expressed in termsof the modified Bessel functions:

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Fig. 65. Geometry for radiation production from helical motion.

∞∫0

x sin[(3/2)α(x + x3/3)]dx = (1/√

3)K2/3(α) ,

∞∫0

cos[(3/2)α(x + x3/3)]dx = (1/√

3)K1/3(α) . (70)

Then, making the necessary variable changes, the formula reduces to Eq.(69).

13.6.2 Methodology of Solving Problems Involving Boosts

The calculation leading to Eq. ( 69) is rather elaborate. It is therefore de-sirable to have an independent derivation. The simplest way of analyz-ing the radiation for an ultrarelativistic helical motion makes use of thetheory of relativity and involves practically no calculations. The way forcomputing the radiation in the case of uniform translation is simple. Onedescribes a complicated situation by finding a coordinate system wherethe analysis is already done (radiation in the case of circular motion) andtransforms back to the old coordinate system. The reference system S′ inwhich the electron moves in circular motion can be transformed to a ref-erence system S in which the electron proceeds following a helical trajec-tory. Eq. ( 69) holds, indeed, in the frame S for a particle whose velocityis (vx, vy, vz) = (v0 sinχ sinφ, v0 cosχ, v0 sinχ cosφ). The Lorentz transfor-mation, which leads to the value vy = v0 cosχ for the y-component ofthe velocity yields (vx, vy, vz) = (v′ sinφ′/γy, vy, v′ cosφ′/γy), where γy =

1/√

1 − v2y/c2, v′ is the velocity of the electron in the frame S′ and the phase

angle φ′ = φ is invariant. This means that, in order to end up in S with

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a transverse (to the magnetic field direction) velocity v⊥ = v0 sinχ, onemust start in S′ with v′ = γyv0 sinχ. In the ultrarelativistic approximationγ2⊥

= 1/(1 − v2⊥/c2) 1, and one finds the simple result v0 = v′, so that

a Lorentz boost with non-relativistic velocity vy leads to a rotation of theparticle velocity ~v0 of the angle η = π/2 − χ ' vy/c 1 (if angle η is smalland v0 ' c, we would write γy sinχ ' 1). If one transforms the radiationfield for a particle in a circular motion in the system S′, one obtains the resultthat the effect of a boost amounts to a rigid rotation of the angular-spectraldistribution of the radiation emitted by the electron moving with velocityv0 on a circle that is, once more, Eq. ( 69) (3) . The notation ”⊥” used here toindicate the velocity in the transverse to the magnetic field direction shouldnot be confused with the ”⊥” referring to an acceleration of the electron inthe transverse direction in the proceeding sections.

It comes out quite naturally that the covariant way of analyzing the radia-tion for helical motion considered above is based on the Lorentz transfor-mation. In other words, within the lab frame the Lorentz coordinates areautomatically enforced. It assumed that in the Lorentz lab frame the electronproceeds following a helical trajectory with velocity v0. This is employedas initial condition. In the ultrarelativistic approximation a Lorentz boostalong the field direction with non relativistic velocity vy leads to the circularmotion of the electron with the same velocity v0. Thus the single boost alongthe field direction will leave the radiation properties unchanged.

Let us discuss the synchrotron radiation from a single electron with a kickwith respect to the nominal orbit in (x, z)-plane. In this case, the accelerationyields a particle velocity increment vy = vθk parallel to the y-axis and ∆vz =−vθ2

k/2 parallel to the z-axis. The restriction to the second order providesan essential simplicity of the calculations. We can use a sequence of twocommuting non-collinear Lorentz boosts in order to discuss the particlemotion downstream the kicker.

When the kick is introduced, the covariant particle tracking predicts achange of the initial condition at the entrance of the synchrotron radia-tion setup. If we will keep the Lorentz coordinate system in the lab framedownstream of the kicker, we will find that the covariant velocity on thehelical orbit after the kick decreases from v to v − vθ2

k/2 and the covariantway of analyzing the radiation for a helical motion with covariant veloc-ity v0 = v − vθ2

k/2 considered above will leads to a red shift in the criticalwavelength.

It should be note that the single passive Lorentz boost to the referencesystem S′ discussed above is only a mathematical trick. A good way tothink of this transformation is to regard it as a result of transformationto new variables. According to this kinematic transformation, the electron

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(and the observer) motion relative to the fixed stars does not changes. Anabsolute (i.e. physically real) acceleration is always a dynamical process.We must conclude that when we accelerate with respect to the fixed starsan electron in the lab inertial frame upstream the uniform magnetic field,this acceleration is absolute and the information about this acceleration isincluded into the covariant trajectory.

13.6.3 On the Advanced ”Paradox” Related to the Coupling Fields and Particles

We now want to point out that there are two different sets of initial conditionsresulting in the same uniform translation along the magnetic field directionin the Lorentz lab frame. We start by considering an electron moving alonga circular trajectory that lies in the (x, z)-plane. We then rotate the magneticfield vector ~B in the (y, z)-plane by an angle θ0, assuming that rotation angleis small (θ0γ)2

1. We consider a situation in which the electron is inuniform motion with velocity vθ0 along the magnetic field direction. It isclear that if we consider the radiation from an electron moving on a circularorbit, the introduction of the magnetic field vector rotation will leave theradiation properties unchanged. This is plausible, if one keeps in mind thatafter rotating the bending magnet, the electron has the same velocity andemits radiation in the velocity direction owing to the Doppler effect. Afterthe rotation, correction to the curvature radius R is only of order θ2

0 and canbe neglected.

Now we consider another situation. Let us see what happens with a kicker,which is installed in the straight section upstream of the bending magnet andis characterized by a kick angle θk = θ0. When the kick in the y directionis introduced, there is a red shift of the critical wavelength which arisebecause, according to corrected addition velocities law, the electron velocitydecreases from v to v − vθ2

0/2 after the kick. The red shift of the criticalfrequency ωc can be expressed by the formula ∆ωc/ωc = −(3/2)γ2θ2

0. We seea second order correction θ2

0 that is, however, multiplied by a large factor γ2.The result of the covariant approach clearly depends on the absolute valueof the kick angle θ0 and the radiation along the velocity direction has a redshift only when the kick angle has nonzero value. The implicit ”absolute”acceleration means acceleration relative to the fixed stars.

The difference between these two situations, ending with a final uniformtranslation along the direction of the magnetic field is very interesting. Itcomes about as the result of the difference between two Lorentz coordinatesystems in the lab frame. By trying to accelerate the electron upstream thebending magnet we have changed Lorentz coordinates for that particularsource. We know that in order to keep a Lorentz coordinates system in thelab frame after the kick we need to perform a clock resynchronization. So we

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Fig. 66. Two sets of initial conditions resulting in the same uniform motion alongthe magnetic field direction in the case of absolute time coordinatization in the labframe. The magnitude of the electron velocity and the orientation of the velocityvector with respect to the magnetic field vector are identical in both setups.

should expect the electron velocity to be changed. The difference betweenthe two setups is understandable. When we do not perturb the electronmotion (relative to the fixed stars) upstream of the bending magnet, noclock resynchronization takes place, while when we do perturb the motion,clock resynchronization is introduced.

We would now like to describe an apparent paradox. Because of our usageof Galilean transformations within special theory of relativity we have someapparent paradoxes. An analysis of paradoxes leads to a better understand-ing of the four-dimensional geometrical significance of the concepts of spaceand time.

The argument that the difference between these two situations, ending witha final uniform translation along the magnetic fields direction, is paradoxicalcan be summarized in the following way: in the case of absolute time coor-dinatization in the lab frame, the initial conditions at the bending magnetentrance are apparently identical. In fact, the magnitude of the electron ve-locity and the orientation of the velocity vector with respect to the magneticfield vector are identical in both setups. We must conclude that when weaccelerate the electron in the lab frame upstream the bending magnet, theinformation about this acceleration is not included into the non-covariant

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trajectory. Where is the information about the electron acceleration recordedin the case of absolute time coordinatization? Since an electron is a struc-tureless particle, the situation seems indeed paradoxical.

The above statement includes one delicate point. It is not true that an ultra-relativistic electron is a ”structureless particle”. Electrodynamics deals withobservable quantities. Let us consider the measurement of the red shift inthe bending magnet radiation from our kicked electron. We can measurethe accurate value of the red shift using a spectrometer in the lab frame, andthis leads to a description of the setup in the space-frequency domain.

Suppose we have a uniformly moving electron. The fields associated toan electron with constant velocity exhibit an interesting behavior when thespeed of the charge approaches that of light. Namely, in the space-frequencydomain there is an equivalence of the fields of a relativistic electron and thoseof a beam of electromagnetic radiation. In fact, for a rapidly moving electronwe have nearly equal transverse and mutually perpendicular electric andmagnetic fields. These are indistinguishable from the fields of a beam of radi-ation. This virtual radiation beam has a macroscopic transverse size of orderoγ. An ultrarelativistic electron at synchrotron radiation facilities, emittingat nanometer-wavelengths (we work in the space-frequency domain) hasindeed a macroscopic transverse size of order of 1 µm. The field distributionof the virtual radiation beam is described by the Ginzburg-Frank formula(see Appendix IV).

When the electron passes through a kicker, its fields are perturbed, and nowinclude information about the acceleration. According to the old kinemat-ics, the orientation of the virtual radiation phase front is unvaried. How-ever, Maxwell’s equations do not remain invariant with respect to Galileantransformations and, as discuss throughout this book, the choice of the oldkinematics implies using anisotropic field equations. As a result, the phasefront remains plane but the direction of propagation is not perpendicu-lar to the phase front. In other words, the radiation beam motion and theradiation phase front normal have different directions. Then, having cho-sen the absolute time synchronization, electrodynamics predicts that thevirtual radiation beam propagates in the kicked direction with the phasefront tilt θk. This is the key to the ”paradox” discussed here. The informa-tion about the electron acceleration is recorded in the perturbation of theself-electromagnetic fields of the electron. Mathematically information isrecorded in the phase front tilt of the virtual radiation beam.

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1. It should be note that results of the beam splitting experiment at LCLSconfirm our correction for spontaneous undulator emission [53]. It appar-ently demonstrated that after a modulated electron beam is kicked on a largeangle compared to the divergence of the XFEL radiation, the modulationwavefront is readjusted along the new direction of the motion of the kickedbeam. Therefore, coherent radiation from the undulator placed after thekicker is emitted along the kicked direction practically without suppression(see the Chapter 13 for more detail). In the framework of the conventionaltheory, there is a second outstanding puzzle concerning the beam splittingexperiment at the LCLS. In accordance with conventional undulator radia-tion theory, if the modulated electron beam is at perfect resonance withoutkick, then after the kick the same modulated beam must be at perfect reso-nance in the velocity direction. However, experimental results clearly showthat when the kick is introduced there is a red shift in the resonance wave-length. The maximum power of the coherent radiation is reached when theundulator is detuned to be resonant to the lower longitudinal velocity afterthe kick [53]. It should be remarked that any linear superposition of a givenradiation field from single electrons conserves single-particle characteris-tics like parametric dependence on undulator parameters and polarization.Consider a modulated electron beam kicked by a weak dipole field beforeentering a downstream undulator. Radiation fields generated by this beamcan be seen as a linear superposition of fields from individual electrons. Nowexperimental results clearly show that there is a red shift in the resonancewavelength for coherent undulator radiation when the kick is introduced.It follows that the undulator radiation from the single electron has red shiftwhen the kick is introduced as well. This argument suggests that results ofthe beam splitting experiment in reference [53] confirm our correction forspontaneous undulator emission.

2. A widely accepted expression for the angular and spectral distributions ofradiation from an ultra-relativistic electron on a helical orbit were calculatedin [48,49]. At present, relativistic cyclotron radiation results are textbookexamples (see e.g. [50]) and do not require a detail description.

3. The covariant way of analyzing the radiation for helical motion wasconsidered in [51]. It is generally believed that ~x(t) = ~x(t)cov and this isthe reason why in the [51] there is no distinction between the two (noncovariant and covariant) approaches to describe the electron motion on ahelix downstream of the kicker setup.

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14 Relativity and X-Ray Free Electron Lasers


In the previous chapter we attempted to answer the question of why theerror in radiation theory should have so long remained undetected. Ac-cording to covariant approach, the various relativistic kinematics effectsconcerning to the synchrotron radiation setup, turn up in successive ordersof approximation. Instead of small (total) velocity parameter (v/c) in thenon-relativistic case, we use a small transverse velocity parameter (v⊥/c).In our previous discussion of bending magnet radiation, we learned that amotion of the single ultrarelativistic electron in a constant magnetic field,according to the theory of relativity, influences the kinematics terms of thesecond order (v⊥/c)2 only. It is demonstrated that due to a combination ofthe ultrarelativistic (i.e. paraxial) approximation and a very special symme-try of the conventional synchrotron radiation setup there is a cancellation ofthe second order relativistic kinematics effects except the non-zero red shiftof the critical frequency, which arises when there are perturbations of theelectron motion in the (bending) magnetic field direction. But synchrotronradiation from a bending magnets is emitted within a wide range of frequen-cies, and the output intencity is not sensitive on the red shift. That meansthat the spontaneous synchrotron radiation does not show sensitivity to thedifference between covariant and non-covariant particle trajectories.

But in the 21st century with the operation XFELs this situation changes.An XFEL is an example where the first order kinematics term (v⊥/c) playsan essential role in the description of the XFEL radiation and, in this case,covariant coupling of fields and particles predicts an effect in completecontrast to the conventional treatment. In this chapter we present a criticalreexamination of existing XFEL theory. The main emphasize of this chapteris on coherent undulator radiation from the modulated electron beam. Thischapter mainly addressed to readers with limiting knowledge of acceleratorand XFEL physics.

The usual XFEL theory based on the use of old Newtonian kinematics forparticle dynamics and the Einstein’s kinematics for the electrodynamics. Infact, the usual theoretical treatment of relativistic particle dynamics involvesonly a corrected Newton’s second law and is based on the use Galileanedition of velocities. For rectilinear motion of the modulated electron beam,non-covariant and covariant approaches produce the same trajectories, andMaxwell’s equations are compatible with the result of conventional particletracking. However, one of the consequences of the relativity of simultaneity(i.e. mixture of positions and time) is a difference between covariant and

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Fig. 67. A well-known result of conventional (non-covariant) particle tracking. Amicro-bunching electron beam passing through a weak dipole magnet (kicker) andundergoes a kick of an angle θk. The propagation axis of the electron beam isdeflected, while the wavefront orientation is preserved.

non-covariant kinematics of a modulated electron beam in a given magneticfield. The theory of relativity shows that discussed above difference relatedwith the acceleration along curved trajectories.

There are several cases where the first order relativistic effect can occur inXFELs, mainly through the introduction of an trajectory kick (1). The mostelementary of the effect that represents a crucial test of the correct couplingfields and particles is a problem involves the production of coherent un-dulator radiation by modulated ultrarelativistic electron beam kicked by aweak dipole field before entering a downstream undulator.

It would be well to begin with bird’s view of some of the main results. Letus now move on to consider the predictions of the existing XFEL theoryin the case of non-collinear electron beam motion. As well-known result ofconventional particle tracking states that after an electron beam is kicked bya weak dipole magnet there is a change in the trajectory of the electron beam,while the orientation of the modulated wavefront remains as before (Fig.67). In other words, the kick results in a difference between the directionsof the electron motion and the normal to the modulation wavefront (i.e. ina wavefront tilt). In existing XFEL theory the wavefront tilt is consideredas real. According to conventional treatment, a transverse kick does not

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Fig. 68. A result of covariant particle tracking. In the ultrarelativistic limit, the ori-entation of the modulation wavefront, i.e. the orientation of plane of simultaneity,is always perpendicular to the electron beam velocity when the evolution of themodulation electron beam is treated using Lorentz coordinates. The theory of rel-ativity dictates that a modulated electron beam in the ultrarelativistic limit has thesame kinematics, in Lorentz coordinates, as laser beam. According to Maxwell’sequations, the wavefront of a laser beam is always orthogonal to the propagationdirection.

change the orientation of a modulation wavefront, and hence suppressesthe radiation emitted in the direction of the electron motion (2).

The covariant approach within the framework of both mechanics and elec-trodynamics predicts an effect in complete contrast to the conventionaltreatment. Namely, in the ultrarelativistic limit, the wavefront of modula-tion, that is a plane of simultaneity, is always perpendicular to the electronbeam velocity (Fig. 68). As a result, the Maxwell’s equations predict strongemission of coherent undulator radiation from the modulated electron beamin the kicked direction. Experiments show that this prediction is, in fact, true(3). The results of XFEL experiments demonstrated that even the direction ofemission of coherent undulator radiation is beyond the predictive power ofthe conventional theory (4).

It is worth remarking that the absent of a dynamical explanation for themodulation wavefront readjusting in the Lorentz coordinatization has dis-turbed some XFEL experts. We only wish to emphasize that a good wayto think of the modulation wavefront readjusting is to regard it as a result

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of transformation to a new time variable in the framework of the Galilean(”single frame”) electrodynamics.

14.2 Modulation Wavefront Orientation

Let us suppose that a modulated electron beam moves along the z-axis of aCartesian (x, y, z) system in the lab frame. As an example, suppose that themodulation wavefront is perpendicular to the velocity vz. How to measurethis orientation? A moving electron bunch changes its position with time.The natural way to do this is to answer the question: when does each electroncross the y-axis of the reference system? If we have adopted a method fortiming distant events (i.e. a synchronization convention), we can also spec-ify a method for measuring the orientation of the modulation wavefront:if electrons located at the position with maximum density cross the y-axissimultaneously at certain position z, then the modulation wavefront is per-pendicular to z-axis. In other words, the modulation wavefront is definedas a plane of simultaneous events (the events being the arrival of particleslocated at maximum density): in short, a ”plane of simultaneity”.

Let us formulate the initial conditions in the lab frame in terms of orientationof the modulation wavefront and beam velocity. Suppose that vz is thevelocity of the comoving frame R(τ) with respect to the lab frame K(τ) alongthe common z-axis in positive direction. In the lab frame we select a specialtype of coordinate system, a Lorentz coordinate system to be precise. Withina Lorentz frame (i.e. inertial frame with Lorentz coordinates), Einstein’ssynchronization of distant clocks and Cartesian space coordinates (x, y, z)are enforced. In order to have this, we impose that R is connected to K bythe Lorentz boost L(~vz), with ~vz, which transforms a given four vector eventX in space-time into XR = L(~vz)X.

We now consider the acceleration of the beam in the lab frame up to velocityvy along the y-axis. The question arises how to assign synchronization inthe lab frame after the beam acceleration. Before acceleration we picked aLorentz coordinate system. Then, after the acceleration, the beam velocitychanges of an small value vy along the y-axis. Without changing synchro-nization in the lab frame after the particle acceleration we have a complicatedsituation as concerns electrodynamics of moving charges. As a result of suchboost, the transformation of time and spatial coordinates has the form of aGalilean transformation. In order to keep a Lorentz coordinate system in thelab frame after acceleration, one needs to perform a clock resynchronizationby introducing the time shift t→ t+yvy/c2. This form of time transformationis justified by the fact that we are dealing with a first order approximation.Therefore, vy/c is so small that v2

y/c2 can be neglected and one arrives at the

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coordinate transformation y → y + vyt, t → t + yvy/c2. The Lorentz trans-formation just described differs from a Galilean transformation just by theinclusion of the relativity of simultaneity, which is only relativistic effect thatappearing in the first order in vy/c. The relation XR = L(~vz)L(~vy)X presents astep-by-step change from the lab reference frame K(τ+ dτ) to K(τ) and thento the proper reference frame R. The shift in the time when electrons locatedat the position with maximum density cross the y-axis of the lab frame∆t = yvy/c2 has the effect of a rotation the modulation wavefront on theangle vz∆t/y = vzvy/c2 in the first order approximation. In ultrarelativisticlimits, vz ' c, and the modulation wavefront rotates exactly as the velocityvector ~v.

What does this wavefront readjustment mean in terms of measurements? Inthe absolute time coordinatization the simultaneity of a pair of events hasabsolute character. The absolute character of the temporal coincidence oftwo events is a consequence of the absolute time synchronization conven-tion. According to this old kinematics, the modulation wavefront remainsunvaried. However, according to the covariant approach we establish a cri-terion for the simultaneity of events, which is based on the invariance of thespeed of light. It is immediately understood that, as a result of the motionof electrons along the y axis (i.e. along the plane of simultaneity before theboost) with the velocity vy, the simultaneity of different events is no longerabsolute, i.e. independent of the kick angle θ = vy/c. This reasoning is inanalogy with Einstein’s train-embankment thought experiment.

The wavefront orientation has no exact objective meaning, because the rela-tivity of simultaneity takes place. The statement that the wavefront orienta-tion has objective meaning to within a certain accuracy can be visualized bythe picture of wavefront in the proper orientation with approximate angleextension (blurring) given by ∆θ ' vz(vy/c2). This relation specifies the lim-its within which the non relativistic theory can be applied. In fact, it followsthat for a very non relativistic electron beam for which v2

z/c2 is very small,the angle ”blurring” becomes very small too. In this case angle of wavefronttilt θ = vy/vz is practically sharp ∆θ/θ ' v2

z/c2 1. This is a limiting case

of non-relativistic kinematics. The angle ”blurring” is a peculiarity of rela-tivistic beam motion. In the ultrarelativistic limit when vz ' c, the wavefronttilt has no exact objective meaning at all since, due to the finiteness of thespeed of light, we cannot specify any experimental method by which thistilt could be ascertained.

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14.3 XFEL Radiation Setup

The most elementary of the effect that represents a crucial test of the cor-rect coupling fields and particles is a problem involves the production ofcoherent undulator radiation by modulated ultrarelativistic electron beamkicked by a weak dipole field before entering a downstream undulator. Wewant to study the process of emission of coherent undulator radiation fromsuch setup.

The key element of a XFEL source is the udulator, which forces the electronsto move along curved periodical trajectories. There are two popular undu-lator configurations: helical and planar. To understand the basic principlesof undulator source operation, let us consider the helical undulator. Themagnetic field on the axis of the helical undulator is given by

~Bw = ~exBw cos(kwz) − ~eyBw sin(kwz) , (71)

where kw = 2π/λw is the undulator wavenumber and ~ex,y are unit vectorsdirected along the x and y axes. We neglected the transverse variation of themagnetic field. It is necessary to mention that in XFEL engineering we dealwith a very high quality of the undulator systems, which have a sufficientlywide good-field-region, so that our studies, which refer to a simple modelof undulator field nevertheless yields a correct quantitative description inlarge variety of practical problems. The Lorentz force ~F = −e~v× ~Bw/c is usedto derive the equation of motion of electrons with charge −e and mass m inthe presence of magnetic field

mγdvx

dt=

ec

vzBy = −ec

vzBw sin(kwz) ,

mγdvy

dt= −

ec

vzBx = −ec

vzBw cos(kwz) . (72)

Introducing v = vx + ivy, dz = vzdt we obtain

mγdvdz

= −iec

(Bx + iBy) = −iec

Bw exp(−ikwz) . (73)

Integration of the latter equation gives

242

vc

= θw exp(−kwz) , (74)

where θw = K/γ and K = eBw/(kwmc2) is the undulator parameter. The ex-plicit expression for the electron velocity in the field of the helical undulatorhas the form

~v = cθw[~ex cos(kwz) − ~ey sin(kwz)] , (75)

This means that the reference electron in the undulator moves along theconstrained helical trajectory parallel to the z axis. As a rule, the electronrotation angle θw is small and the longitudinal electron velocity vz is closeto the velocity of light, vz =

√v2 − v2

⊥' v(1 − θ2

w/2) ' c.

Let us consider a modulated ultrarelativistic electron beam moving alone thez axis in the field of the helical undulator. In the present study we introducethe following assumptions. First, without kick the electrons move alongconstrained helical trajectories in parallel with the z axis. Second, electronbeam density at the undulator entrance is simply

n = n0(~r⊥)[1 + a cosω(z/vz − t)] , (76)

where a = const. In other words we consider the case in which there are novariation in amplitude and phase of the density modulation in the trans-verse plane. Under these assumptions the transverse current density maybe written in the form

~j⊥ = −e~v⊥(z)n0(~r⊥)[1 + a cosω(z/vz − t)] . (77)

Even through the measured quantities are real, it is generally more conve-nient to use complex representation, starting with real ~j⊥, one defines thecomplex transverse current density:

jx + i jy = −ecθwn0(~r⊥) exp(−ikwz)[1 + a cosω(z/vz − t)] . (78)

The transverse current density has an angular frequency ω and two wavestraveling in the same direction with variations exp i(ωz/vz − kwz − ωt) andexp−i(ωz/vz + kwz − ωt) will add to give a total current proportional toexp(−ikwz)[1 + a cosω(z/vz− t)]. The factor exp i(ωz/vz− kwz−ωt) indicates a

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fast wave, while the factor exp−i(ωz/vz+kwz−ωt) indicates a slow wave. Theuse of the word ”fast” (”slow”) here implies a wave with a phase velocityfaster (slower) than the beam velocity.

Having defined the sources, we now should consider the electrodynamicsproblem. Maxwell equations can be manipulated mathematically in manyways in order to yield derived equations more suitable for certain appli-cations. For example, from Maxwell equations Eq.(4) we can obtain anequation which depends only on the electric field vector ~E (in Gaussianunits):

c2~∇ × (~∇ × ~E) = −∂2~E/∂t2− 4π∂~j/∂t . (79)

With the help of the identity

~∇ × (~∇ × ~E) = ~∇(~∇ · ~E) − ∇2~E (80)

and Poisson equation

~∇ · ~E = 4πρ (81)

we obtain the inhomogeneous wave equation for ~E

c2∇

2~E − ∂2~E/∂t2 = 4πc2~∇ρ + 4π∂~j/∂t . (82)

Once the charge and current densities ρ and ~j are specified as a function oftime and position, this equation allows one to calculate the electric field ~E ateach point of space and time. Thus, this nonhomogeneous wave equationis the complete and correct formula for radiation. However we want toapply it to still simpler circumstance in which second term (or, the currentterm) in the right-hand side provides the main contribution to the valueof the radiation field. It is relevant to remember that our case of interestis the coherent undulator radiation and the divergence of this radiation ismuch smaller compared to the angle 1/γ. It can be shown that when thiscondition is fulfilled the gradient term, 4πc2~∇ρ, in the right-hand side of thenonhomoheneous wave equation can be neglected. Thus we consider thewave equation

c2∇

2~E − ∂2~E/∂t2 = 4π∂~j⊥/∂t . (83)

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We wish to examine the case when the phase velocity of the current wave isclose to the velocity of light. This requirement may be met under resonanceconditionω/c = ω/vz−kw. This is the condition for synchronism between thetransverse electromagnetic wave and the fast transverse current wave withthe propagation constant ω/vz − kw. With the current wave traveling withthe same phase speed as electromagnetic wave, we have the possibility ofobtaining a spatial resonance between electromagnetic wave and electrons.If this the case, a cumulative interaction between modulated electron beamand transverse electromagnetic wave in empty space takes place. We aretherefore justified in considering the contributions of all the waves exceptthe synchronous one to be negligible as long as the undulator is made of alarge number of periods.

Here follows an explanation of the resonance condition which is elemen-tary in the sense that we can see what is happening physically. The fieldof electromagnetic wave has only transverse components, so the energy ex-change between the electron and electromagnetic wave is due to transversecomponent of the electron velocity. For effective energy exchange betweenthe electron and the wave, the scalar product −e~v⊥ · ~E should be kept nearlyconstant along the whole undulator length. We see that required synchro-nism kw + ω/c − ω/vz = 0 takes place when the wave advances the electronbeam by the wavelength at one undulator period λw/vz = λ/(c − vz), whereλ = 2π/ω is the radiation wavelength. This tells us that the angle betweenthe transverse velocity of the particle ~v⊥ and the vector of the electric field~E remains nearly constant. Since vz ' c this resonance condition may bewritten as λ ' λw/(2γ2

z) = λw(1 + K2)/(2γ2).

We will use an adiabatic approximation that can be taken advantage of, inall practical situations involving XFELs, where the XFEL modulation wave-length is much shorter than the electron bunch length σb, i.e. σbω/c 1.Since we are interested in coherent emission around the modulation wave-length the theory of coherent undulator radiation is naturally developed inthe space-frequency domain. In fact, in this case one is usually interestedinto radiation properties at fixed modulation frequency.

We first apply a temporal Fourier transformation to the inhomogeneouswave equation to obtain the inhomogeneous Helmholtz equation

c2∇

2~E + ω2~E = −4πiω~j⊥, (84)

where ~j⊥

(~r, ω) is the Fourier transform of the current density ~j⊥(~r, t). Thesolution can be represented as a weighted superposition of solutions cor-responding to a unit point source located at ~r′. The Green function for the

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inhomogeneous Helmholtz equation is given by (for unbounded space andoutgoing waves)

4πG(~r, ~r′, ω) =1

|~r − ~r′|exp

[iωc|~r − ~r′|

], (85)

with |~r − ~r′| =√

(x′ − x)2 + (y′ − y)2 + (z′ − z)2. With the help of this Greenfunction we can write a formal solution for the field equation as:

~E =

∫d~r′ G(~r, ~r′)

[−4πi

ωc2~j⊥]. (86)

This is just a mathematical description of the concept of Huygens’ secondarysources and waves, and is of course well-known, but we still recalled how itfollows directly from the Maxwell’s equations. We may consider the ampli-tude of the beam radiated by plane of oscillating electrons as a whole to bethe resultant of radiated spherical waves. This is because Maxwell’s theoryhas no intrinsic anisotropy. The electrons lying on the plane of simultaneitygives rise to spherical radiated wavelets, and these combine according toHuygens’ principle to form what is effectively a radiated wave. If the planeof simultaneity is the xy-plane (i.e. beam modulation wavefront is perpen-dicular to the z- axis), then the Huygens’ construction shows that planewavefronts will be emitted along the z-axis.

In summary: according to Maxwell’s electrodynamics, coherent radiationis always emitted in the direction normal to the modulation wavefront.We already stressed that Maxwell’s equations are valid only in a Lorentzreference frame, i.e. when an inertial frame where the Einstein synchroniza-tion procedure is used to assign values to the time coordinates. Einstein’stime order should be applied and kept in consistent way in both dynamicsand electrodynamics. Our previous description implies quite naturally thatMaxwell’s equations in the lab frame are compatible only with covarianttrajectories ~xcov(t), calculated by using Lorentz coordinates and, therefore,including relativistic kinematics effects.

Let us go back to the modulated electron beam, kicked transversely withrespect to the direction of motion, that was discussed before. Conventionalparticle tracking shows that while the electron beam direction changes afterthe kick, the orientation of the modulation wavefront stays unvaried. Inother words, the electron motion and the wavefront normal have differentdirections. Therefore, according to conventional coupling of fields and par-ticles that we deem incorrect, the coherent undulator radiation in the kicked

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direction produced in a downstream undulator is expected to be dramati-cally suppressed as soon as the kick angle is larger than the divergence ofthe output coherent radiation.

In order to estimate the loss in radiation efficiency in the kicked directionaccording to the conventional coupling of fields and particles, we make theassumption that the spatial profile of the modulation is close to that of theelectron beam and has a Gaussian shape with standard deviation σ. A mod-ulated electron beam in an undulator can be considered as a sequence ofperiodically spaced oscillators. The radiation produced by these oscillatorsalways interferes coherently at zero angle with respect to the undulator axis.When all the oscillators are in phase there is, therefore, strong emission inthe direction θ = 0. If we have a triangle with a small altitude r ' θz andlong base z, than the diagonal s is longer than the base. The difference is∆ = s− z ' zθ2/2. When ∆ is equal to one wavelength, we get a minimum inthe emission. This is because in this case the contributions of various oscilla-tors are uniformly distributed in phase from 0 to 2π. In the limit for a smallsize of the electron beam, σ→ 0, the interference will be constructive withinan angle of about ∆θ w

√c/(ωLw) = 1/(

√4πNwγz) 1/γ, where Lw = λwNw

is the undulator length. In the limit for a large size of the electron beam, theangle of coherence is about ∆θ w c/(ωσ) instead. The boundary betweenthese two asymptotes is for sizes of about σdif w

√cLw/ω. The parameter

ωσ2/(cLw) can be referred to as the electron beam Fresnel number. It is worthnoting that, for XFELs, the transverse size of electron beam σ is typicallymuch larger than σdif (i.e electron beam Fresnel number is large). Thus, wecan conclude that the angular distribution of the radiation power in thefar zone has a Gaussian shape with standard deviation σθ w c/(

√2ωσ).

However, still according to the conventional treatment, after the electronbeam is kicked we have the already-mentioned discrepancy between di-rection of the electron motion and wavefront normal. Then, the radiationintensity along the new direction of the electron beam can be approximatedas I w I0 exp[−θ2

k/(2σ2θ)], where I0 is the on-axis intensity without kick and

θk is the kick angle. The exponential suppression factor is due to the tiltof the modulation wavefront with respect to the direction of motion of theelectrons.

We presented a study of very idealized situation for illustrating the differ-ence between conventional and covariant coupling of fields and particles.We solved the dynamics problem of the motion of a relativistic electrons inthe prescribed force field of weak kicker magnet by working only up to theorder of γθk. This approximation is of particular theoretical interest becauseit is relatively simple and at the same time forms the basis for understandingrelativistic kinematic effects such as relativity of simultaneity.

Let us discuss the region of validity of our small kick angle approxima-

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tion θkγ 1. Since in XFELs the Fresnel number is rather large, we canalways consider a kick angle which is relatively large compared to thedivergence of the output coherent radiation, and, at the same time, it is rel-atively small compared to the angle 1/γ. In fact, from ωσ2/(cLw) 1, withsome rearranging, we obtain σ2

θ ' c2/(ω2σ2) c/(ωLw). Then we recall that√c/(ωLw) = 1/(

√4πNwγz) 1/γ. Therefore, the first order approximation

used to investigate the kicker setup in this chapter is of practical interest inXFEL engineering.

It is one of the aims of this chapter is to demonstrate the kind of experimen-tal predictions we are expecting from our corrected radiation theory. Weworked out a very simple case in order to illustrate all the essential physi-cal principles very clearly. Surprisingly, the first order approximation usedto investigate the kicker setup in this section has also important practicalapplications.

Above we have shown that our correct coupling of fields and particles pre-dicts an effect in complete contrast to the conventional treatment. Namely,in the ultrarelativistic limit, the plane of simultaneity, that is wavefront ori-entation of the modulation, is always perpendicular to the electron beamvelocity. As a result, we predict strong emission of coherent undulator radi-ation from the modulated electron beam in the kicked direction.

XFEL experts actually witnessed an apparent wavefront readjusting due tothe relativistic kinematics effect, but they never drew this conclusion. In thisbook, we are actually first in considering the idea that results of the conven-tional theory of radiation by relativistically moving charges are not consis-tent with the principle of relativity. In previous literature, identification ofthe trajectories in the source part of the usual Maxwell’s equations with thetrajectories calculated by conventional particle tracking in the (”single”) labframe has always been considered obvious.

14.4 Modulation Wavefront Tilt in Maxwell’s Theory. Logical Inconsistency

In existing literature theoretical analysis is presented, of an XFEL drivenby an electron beam with wavefront tilt, and this analysis is based on theexploitation of usual Maxwell’s equations and standard simulation codes.Using only a kicker setup (i.e. without undulator radiation setup) we candemonstrate that the coupling fields and particles in the conventional XFELtheory is intrinsically incorrect.

The existing XFEL theory based on the use of the absolute time convention(i.e. old kinematics) for particle dynamics. Here we will give a simple proof

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Fig. 69. Basic setup for the experimental test of XFEL theory. The correct couplingfields and particles predicts an effect in complete contrast to the conventionaltreatment. According to the covariant approach, in the ultrarelativistic limit, thewavefront of modulation is always perpendicular to the electron beam velocity.As a result, the Maxwell’s equations predict strong emission of coherent undulatorradiation from the modulated electron beam in the kicked direction. Experimentsshow that this prediction is, in fact, true.

of the conflict between conventional particle tracking and Maxwell’s elec-trodynamics. The purpose is to show how one can demonstrate in a simpleway that the conventional XFEL theory is absolutely incapable of correctlydescribing the distribution of the electromagnetic fields from a fast movingmodulated electron beam downstream the kicker.

Under the Maxwell’s electrodynamics, the fields of a modulated electronbeam moving with a constant velocity exhibit an interesting behavior whenthe velocity of charges approaches that of light: namely, in the space-timedomain they resemble more and more close of a laser beam (see AppendixIV). In fact, for a rapidly moving modulated electron beam we have nearlyequal transverse and mutually perpendicular electric and magnetic fields:in the limit v → c they become indistinguishable from the fields of a laserbeam, and according to Maxwell’s equations, the wavefront of the laserbeam is always perpendicular to the propagation direction. This is indeedthe case for virtual laser-like radiation beam in the region upstream thekicker.

Let us now consider the effect of the kick on the electron modulation wave-

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front. If we rely on the conventional particle tracking, the kick results in adifference between the directions of electron motion and the normal to themodulation wavefront, i.e. in a tilt of the modulation wavefront.

This is already a conflict result, because we now conclude that, accordingto the conventional ”single frame” approach, the direction of propagationafter the kick is not perpendicular to the radiation beam wavefront. In otherwords, the radiation beam motion and the radiation wavefront normal havedifferent directions. The virtual radiation beam (which is indistinguishablefrom a real radiation beam in the ultrarelativistic asymptote) propagates inthe kicked direction with a wavefront tilt. This is what we would get for thecase when our analysis is based on the conventional particle tracking, andis obviously absurd from the viewpoint of Maxwell’s electrodynamics.

In existing literature theoretical analysis is presented, of an XFEL driven byan electron beam with wavefront tilt, and this analysis is based on the ex-ploitation of usual Maxwell’s equations and standard simulation codes. Us-ing only a kicker setup (i.e. without undulator radiation setup) we demon-strated that the coupling fields and particles in the conventional XFEL theoryis intrinsically incorrect.

The difficulty above is a part of the continual problem of XFEL physics,which started with coherent undulator radiation from an ultrarelativisticmodulated electron beam in the kicked direction, and now has been focusedon the wavefront tilt of the self-electromagnetic fields of the modulatedelectron beam.

In existing XFEL theory the wavefront tilt is considered as real. However,there is a common mistake made in accelerator physics connected withthe wavefront tilt. In the ultrarelativistic domain the wavefront tilt has noexact objective meaning. The angle of wavefront tilt depends on the choiceof a procedure for clock synchronization in the lab frame, as a result ofwhich it can be given any preassigned values within the interval (0, θk). Forinstance, in the ultrarelativistic domain, the orientation of the modulationwavefront is always perpendicular to the electron beam velocity (i.e.θtilt = 0)when the evolution of the modulated electron beam is treated using Lorentzcoordinates. No physical effects may depends on an arbitrary constant oran arbitrary function (5).

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1. An angular kick is often an essential part of many XFEL related diag-nostic or experimental procedures. The standard gain length measurementprocedure in XFELs employs such kicks. Other applications include ”beam-splitting” schemes where different polarization components are separatedby means of an angular kick to the modulated electron beam [52,53].

2. In a typical configuration for an XFEL, the orbit of a modulated electronbeam is controlled to avoid large excursions from the undulator axis. Allexisting XFEL codes are based on a model in which the modulated electronbeam moves only along the undulator axis. However, random errors in thefocusing system can cause angular trajectory errors (or ”kicks”). The discrep-ancy between directions of the electron motion and wavefront normal afterthe kick have been discussed in the literature. One particular consequencethat received attention following the [54] is the effect of the trajectory er-ror (single kick error) on the XFEL amplification process. It was pointed outthat coherent radiation is emitted towards the wavefront normal of the beammodulation. Thus, according to conventional coupling of fields and particles(which we claimed incorrect), the discrepancy between the two directionsdecreases the radiation efficiency [54]. Analysis of the trajectory errors onthe XFEL amplification process showed that any XFEL undulator magneticfield must satisfy stringent requirements. However, semi-analytical stud-ies of this critical aspect in the design of a XFEL sources are based on anincorrect coupling of fields and particles. The pleasant surprise is that thetolerances predicted are more stringent than they need be according to thecorrected XFEL theory. This can be considered as one of the reason for theexceptional progress in XFEL developments over last decades.

3. The fact that our theory predicts reality in a satisfactory way is well-illustrated by comparing the prediction we just made with the results of anexperiment involving ”X-ray beam splitting” of a circularly-polarized XFELpulse from the linearly-polarized XFEL background pulse, a technique usedin order to maximize the degree of circular polarization. The XFEL exper-iments apparently demonstrated that after a modulated electron beam iskicked on a large angle compared to the divergence of the XFEL radiation,the modulation wavefront is readjusted along the new direction of motion ofthe kicked beam. This is the only way to justify coherent radiation emissionfrom the short undulator placed after the kicker and along the kicked direc-tion, see Fig. 14 in [55]. These results came unexpectedly, but from a practicalstandpoint, the ”apparent wavefront readjusting” immediately led to the re-alization that the unwanted, linearly-polarized radiation background couldbe fully eliminated without extra-hardware.

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4. In existing literature a theoretical analysis of XFELs driven by an elec-tron beam with wavefront tilt was presented in [56,57], based on the usethe usual Maxwell’s equations. In fact, the Maxwell solver was used as apart of the standard simulation code. We state that this approach is funda-mentally incorrect. In ultrarelativistic asymptote a modulation wavefronttilt is absurd with the viewpoint of Maxwell’s electrodynamics. In the caseof an XFEL we deal with an ultrarelativistic electron beam and within theLorentz lab frame (i.e. within the validity of the Maxwell’s equations) thetilted modulation wavefront is at odds with the principle of relativity.

5. In existing XFEL theory the wavefront tilt is considered as real. Let usconsider one example. One finds some papers (see e.g. [54]) which saythat a wavefront tilt leads to significant degradation of the electron beammodulation in XFELs. First, suppose that modulation wavefront is perpen-dicular to the beam velocity v. The effect of betatron oscillations, whichcan influence the operation of the XFEL, has its origin in an additional lon-gitudinal velocity spread. Particles with equal energies, but with differentbetatron angles, have different longitudinal velocities. In other words, ontop of the longitudinal velocity spread due to the energy spread, there isan additional source of velocity spread. To estimate the importance of thelast effect, we should calculate the dispersion of the longitudinal velocitiesdue to both effects. The deviation of the longitudinal velocity from nominalvalue is ∆vz = v∆γ/γ3

− v∆θ2/2. The finite angular spread of the electronbeam results in a difference in time when each electron arrives at the samelongitudinal position, and this spoils the phase coherence. This is so callednormal debunching effect. From the viewpoint of the existing XFEL theory,the time difference is enhanced by the kick angle θk. In this case, accordingto conventional (non-covariant) particle tracking, the angle of wavefront tiltis θtilt = θk. It is a widespread belief that the wavefront tilt has physicalmeaning, and that the deviation of the longitudinal velocity component(i.e. velocity component which is perpendicular to the modulation wave-front within the framework of Galilean kinematics) is now given by theexpression ∆vz = −v|∆~θ + ~θk|

2/2. If such picture is correct, the crossed termv~θk ·∆~θ leads to a significant degradation of the modulation amplitude. Thismechanism is called smearing of modulation and should be distinguishedfrom the normal debunching [54]. Many experts would like to think thatany debunching process obviously has objective meaning. The theory ofrelativity says, however, that normal debunching has objective meaning,but smearing effects not exist at all. The explanation of the new debunchingmechanism clearly demonstrates the essential dependence of the smear-ing effect on the choice of the coordinate system in the four-dimensionalspace, which from the physical point of view is meaningless. Now let usunderstand physically why the new debunching mechanism does not ex-ist in framework of Galilean kinematics. In this old kinematics the crossed

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term v~θk · ∆~θ leads to a degradation of modulation amplitude in the for-ward direction. Our Galilean transformed electrodynamics says, however,that by making a measurement on the coherent radiation, one can observeonly radiation in the kicked direction. But the crossed term is absent in theexpression for the deviation of the velocity component along the kickeddirection. It comes out quite naturally that the smearing effect is not a realphenomenon.

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15 Kinematics of the Wigner Rotation


The subject of this chapter is the Wigner rotation (1). We have already dis-cussed the Wigner rotation in Chapter 7. Our earlier discussion is reallyabout aberration of light and particles. The Wigner rotation is also relatesthe angular rotation velocity of the spin of an elementary particle follow-ing a curvilinear orbit. The Wigner rotation is a relativistic kinematic effect,which consists in that a coordinate axes of a reference frame, moves alonga curvilinear trajectory rotating about the axes of a Lorentz lab frame. Thepoint is that two observers with different trajectories have different 3-spaces.In other words, in the space measurement of one observer there is mixingof space and time, as seen by the other. When we studied aberration of lightand particles, we begun by writing down an expression for Wigner rota-tion angle. To deduce this expression is the main purpose of this chapter.We have tried to keep the mathematical complexity of the discussion to aminimum.

15.2 The Commutativity of Collinear Lorentz Boosts

Lorentz transformations are essential to the further mathematical develop-ment of the Wigner rotation theory, so the next two subsections detail theusual applications together with some physical discussion.

Let us now consider a relativistic particle, accelerating in the lab frame,and let us analyze its evolution within Lorentz coordinate systems. Thepermanent rest frame of the particle is obviously not inertial and any trans-formation of observations in the lab frame, back to the rest frame, cannotbe made by means of Lorentz transformations. To get around that difficultyone introduces an infinite sequence of comoving frames. At each instant,the rest frame is a Lorentz frame centered on the particle and moving withit. As the particle velocity changes to its new value at an infinitesimally laterinstant, a new Lorentz frame centered on the particle and moving with it atthe new velocity is used to observing the particle.

Let us denote the three inertial frames by K,R(τ),R(τ+dτ). The lab frame is K,R(τ) is the rest frame with velocity ~v = ~v(τ) relative to K, and R(τ+ dτ) is therest frame at the next instant of proper time τ + dτ, which moves relative toR(τ) with infinitesimal velocity d~v′. All inertial reference frames are assumedto be Lorentz reference frames. In order to have this, we impose that R(τ)is connected to K by the Lorentz boost L(~v), with ~v, which transforms a

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given four vector event X in a space-time into XR = L(~v)X. The relationXR = L(d~v′)L(~v)X presents a step-by-step change from K to R(τ) and then toR(τ + dτ).

There is another composition of reference-frame transformations which de-scribes the same particle evolution in the Minkowski space-time. Let K(τ)be an inertial frame with velocity d~v relative to the lab frame K(τ + dτ). Weimpose that K(τ) is connected to K(τ + dτ) by the Lorentz boost L(d~v). TheLorentz rest frame R is supposed to move relative to the Lorentz frame K(τ)with velocity ~v. The relation XR = L(~v)L(d~v)X presents a step-by-step changefrom K(τ + dτ) to K(τ) and then to the rest frame R.

Let us examine the transformation of the three-velocity in the theory ofrelativity. For a rectilinear motion along the z axis it is performed in ac-cordance with the following equation: vz(τ + dτ) = (dvz + vz)/(1 + vzdvz/c2).Like it happens with the composition of Galilean boosts, collinear Lorentzboosts commute: L(dvz)L(vz) = L(vz)L(dvz). This means that the resultant ofsuccessive collinear Lorentz boosts is independent of which transformationapplies first.

15.3 The Noncommutativity of Two Lorentz Boosts in Nonparallel Directions

In contrast with the case of Lorentz boosts in collinear directions, Lorentzboosts in different directions do not commute. While the successive applica-tion of two Galilean boosts is Galilean boost and the successive application oftwo rotations is a rotation, the successive application of two non-collinearLorentz boosts is not a Lorentz boost. The composition of non-collinearboosts will results to be equivalent to a boost, followed by spatial rotation,the Wigner rotation.

Let us compare the succession K → R(τ) → R(τ + dτ) with the successionK(τ + dτ) → K(τ) → R in the case when the acceleration in the rest frameis perpendicular to the line of flight of the lab frame in the rest frame. Theframe R(τ+dτ) is supposed to move relative to R(τ) with velocity d~v′x. Now,since we can write the result in terms of succession L(~vz)L(d~vx) as well as interms of succession L(d~v′x)L(~vz), there is a need to clarify a number of ques-tions associated with these compositions of Lorentz frames. We can easilyunderstand that the operational interpretation of the succession L(~vz)L(d~vx)is particular simple, involving physical operation used in the measurementof the particle’s velocity increment d~vx in the lab frame. We should be able tounderstand the operational interpretation of the succession L(d~v′x)L(~vz). Webegin by making an important point: the laws of physics in any one inertialreference frame should be able to account for all physical phenomena, in-

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cluding the observations made by moving observers. The lab observer seesthe time dilation in the Lorentz frame which moves with respect to the labframe with velocity ~vz: dt/γ = dτ. What velocity increment d~vR is measuredby moving observer? As viewed from the lab frame the moving observermeasures the increment d~vR = γd~vx.

Because of time dilation in the moving frame, the velocity increment inthe lab frame dvx corresponds to a velocity γdvx and −γdvx in the framesR(τ) and R(τ + dτ) respectively. The resulting boost compositions can berepresented as XR = L(d~v′x)L(~vz)X = L(~vz)L(d~vx)X. In other words, Lorentzboosts in different direction do not commute: L(~vz)L(d~vx) , L(d~vx)L(~vz).

15.4 Wigner Rotation

15.4.1 Expression for the Wigner Rotation in the Case of an Arbitrary Velocity

A large number of incorrect expressions for the Wigner rotation can befound in the literature. Therefore, there is a need to consider the Wignerrotation in detail. Rather than working out all the transformation matricesfor four-vector components, we would like to show a geometrical approachthat is very useful for this problem.

Consider the succession of inertial frame systems K→ R(τ)→ R(τ+ dτ). Asviewed from the lab frame the observer in the proper frame measures thevelocity−vz and the velocity increment d~vR(τ+dτ) = −γd~vx. The correspondingrotation of the velocity direction in the proper frame R(τ + dτ) is γdvx/vz

(Fig. 70). In the lab frame the velocity rotation angle would be dvx/vz =dθ. Both rotations are in the same direction. Using the line motion as areference line, the lab observer can then calculate the difference betweenthese velocity rotation angles to find the rotation angle of the lab frameaxes in the proper frame. This difference γdvx/vz − dvx/vz = (γ − 1) dθ isthe Wigner rotation angle in the proper frame. In vector form this is seen tobe d~ΦR = (γ − 1)~v × d~v/v2. Transformed back to the lab frame this becomed~Φ = (1−1/γ)~v×d~v/v2 so that this is the Wigner rotation angle of the properframe axes in the lab frame (2). We note that owing to the relativistic effect oftime dilation in the reference frame that moves to the lab frame, the Wignerrotation angle in the proper frame is always γ time higher than in the labframe (3).

We derived the exact relation d~Φ = (1−1/γ)~v×d~v/v2 using only rudimentaryknowledge of special relativity. In textbooks on the theory of relativity, thespatial rotation associated with the composition of two Lorentz boosts innon-parallel directions is often introduced using the algebraic approach.

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Fig. 70. The interpretation of the Wigner rotation about the proper frame axes. Labframe view of the observation of the proper observer. The lab observer is able toaccount for the observation the velocity rotation angle in the proper frame (γdvx/vz)and the observation of the rotation angle made in the lab frame (dvx/vz). In 3D spacethe proper frame moves with respect to the lab frame along the line motion. Usingthe line motion as reference line, the lab observer sees that the observer in theproper frame measures the rotating angle of the lab frame axes with respect toproper frame axes δθw = δθ(γ − 1).

This is one of the reason why authors of textbooks obtained an incorrectexpression for the Wigner rotation. They describe the rotation of a movingobject without geometrical meaning of such rotation and encounter seriousdifficulties in the interpretation of the applied calculations and of the results.

15.5 How to Measure a Wigner rotation?

The Wigner rotation is a relativistic kinematic effect, which consists in thata coordinate axes of a reference frame, moves along a curvalinear trajectoryrotating about the axes of a Lorentz lab frame. In the Lorentz lab frame,for infinitely small transformations (due to acceleration) we obtained theformula

d~Φ = (1 − 1/γ)~v × d~v/v2 =

(1 −

1γ

)~δθ , (87)

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Fig. 71. The interpretation of the Wigner rotation about the lab frame. A rod directedalong x-axis in the comoving frame. After the first boost the motion takes place alongthe z-axis. The lab frame (xr, zr) coordinate system rotated with respect to the initiallab frame (x, z) coordinate system on the angle δθ. The projection of the moving rodon the zr-axis is simply Lδθ. After the second boost at velocity dvx along the x-axisof the lab frame this projection will be contracted down to Lδθ/γ. It is assumedthat infinitesimal angle δθ = dvx/vz. According to the contracted projection thecomoving frame is rotated with respect to the initial lab frame axes by the Wignerangle equal to δθ(1 − 1/γ).

where d~v is the vector of small velocity change due to acceleration, Φ is theWigner rotation angle of the spatial coordinate axes of the proper systemrelative to Lorentz lab frame, and θ is the orbital angle. But how to measurethis orientation? A moving coordinate system changes its position in time.The question arises whether it is possible to give an experimental inter-pretation of the rotation of a moving coordinate system. We illustrate theproblem of how to represent orientation of the moving coordinate systemwith a simple example.

The execution of successive transformations from K(τ+dτ) to K(τ) at velocityd~vx and from from K(τ) to R at velocity ~vz equivalent to the compositionof boost L(~v + d~v) and rotation. The Wigner rotation Eq.(87) is performedadditionally to the Lorentz boost at velocity ~vz + d~vx. The interpretation ofthis rotation about the lab frame of reference is closely associated with thelength contraction.

Suppose that the lab observer after the Lorentz boost at velocity vz rotates

258

the coordinate system on the angle ~δθ = ~vz × d~vx/v2z and now xr locates

orthogonally to the vector ~vz + d~vx. Similarly, the axis zr is parallel to thevector ~vz + d~vx. Consider a rod directed along x-axis in the comoving frame.The motion takes place in the plane (x, z) and the rod located perpendicularlyto the velocity ~vz. After the rotation of the lab frame axes, the projection ofthe rod on the zr axes will be simply lδθ, where l is the rod length in the Rframe and also in the lab frame after the first boost along the z-axis. Afterthe second Lorentz boost at velocity d~vx this projection will be contracteddown to Lδθ/γ (Fig. 71). Let the observer in the lab frame fix the position ofthe axes of the comoving frame. In ultrarelativistic limit γ → ∞ these axeswill be parallel to the rotated lab frame axes (xr, zr). In fact, projection of therod on the zr axis will be zero. In the case of an arbitrary velocity, axes of thecomoving frame turn out to not only parallel to the rotated lab frame axes(xr, zr). According to contracted projection, the angle will be− ~δθ/γ. And onecan verify directly that the axes of the comoving frame are actually rotatedwith respect to the initial lab frame axes (x, z) by the angle equal to δθ−δθ/γ,which is just the Wigner rotation angle in accordance with equation Eq.(87).

Here we only wished to show how naturally Lorentz transformations leadto the Wigner rotation phenomenon. We have come to the conclusion thatwhat are usually considered advanced parts of the theory of relativity are, infact, quite simple. Indeed, we demonstrated that the Wigner rotation resultsdirectly from the length contraction. Why our derivation of the expressionfor the Wigner rotation is so simple? The reason is that we employed anew method that is very useful in this kind of problem. What we did wasto analyze the physical operations used in the measurement of Wignerrotation. In fact, the operations for performing measurements on a movingobject are not the same as those for measuring an object at rest, and anabsolute significance has been attributed to the concept of simultaneity.

15.6 Wigner Rotation and the Reciprocity

There are several points to be made about the Wigner rotation theory. Wewish to remark that the expression for the Wigner rotation angle in theproper frame (as viewed from the Lorentz lab frame) can be presented inthe form d~ΦR = (1 − 1/γ)~vR × d~vR/v2

R. Thus, the Wigner rotation angle inthe proper frame is expressed in terms of the lab frame velocity ~vR andits increment d~vR in the proper frame. We have just demonstrated that thereciprocity is the correct concept in the case of infinitesimal Wigner rotation,as it must be. It should be note that presented reciprocal formulas can beconsidered only as a first order approximation in dvR/vR. In the theory of(infinitesimal) Wigner rotation we must also consider the relativistic pa-rameter vR/c = v/c. If we cannot neglect v2

R/c2 = v2/c2 compared with 1, the

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special relativity says that particle (or spin) kinematics cannot be consideredas a topic in the classical theory. On the other hand, the theory of specialrelativity and experiments demonstrate that the reciprocity can be incorrectconcept in the clock problem. According to asymmetry between the inertialand accelerated frames, the accelerated observer cannot observe the effectof slowness of the clock and red shifting of observer’s source located in thelab frame.

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1. As known, a composition of noncollinear Lorentz boosts does not resultsin a different boost but in a Lorentz transformation involving a boost and aspatial rotation, the latter being known as Wigner rotation [58,59]. Wignerrotation is sometimes called Thomas rotation (see e.g. [13,60]).

2. The correct expression for the Thomas (Wigner) rotation was first ob-tained by V. Ritus [61]. In deriving expressions for the Thomas (Wigner)rotation, the majority of authors (see e.g. [60]) were supposedly guidedby the incorrect expression for Thomas (Wigner) rotation from Moeller’smonograph [13]. The expression obtained by Moeller is given by ~δΦ =

(1 − γ)~v × d~v/v2 = (1 − γ) ~δθ (and subsequently expression for Thomas pre-cession ΩT = (1 − γ)ω0). It should be note that, in his monograph, Moellerstated several times that this expression valid in the lab Lorentz frame.Clearly, this expression and correct result Eq. (87) differ both in sign andin magnitude. An analysis of the reason why Moeller obtained an incorrectexpression for the Thomas (Wigner) rotation in the lab frame is the focus ofRitus paper [62]. As shown in [62], the Moeller’s mistake is not computa-tional, but conceptual in nature. In review [63] it is shown that the correctresult was obtained in the works of several authors, which were publishedmore than half century ago but remained unnoticed against the backgroundof numerous incorrect works. The authors of some papers believe that theincorrect result for Wigner rotation in the lab frame presented in textbooksd~Φ = −(γ−1)~v×d~v/v2 is only incorrectly interpreted with the understandingthat it should be reinterpreted as a Wigner rotation of the lab frame in theproper frame. We note that such reinterpreted expression for Wigner rota-tion in the proper frame d~Φ = −(γ − 1)~v × d~v/v2

→ d~ΦR = −(γ − 1)~v × d~v/v2

is also incorrect in sign.

3. We note that in 1986, M. Stranberg obtained an expression for the Thomas(Wigner) rotation correct both in the lab inertial frame and the referenceframe comoving with a spinning particle [64]. It is noteworthy that [64] isone of the few papers that explicitly states that the angle of the Thomas(Wigner) rotation in the comoving reference frame is γ times higher than inthe lab frame.

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16 Relativistic Spin Dynamics


In 1959, a paper by Bargmann, Michel, and Telegdi was published, whichdealt with the motion of elementary charged spinning particles with ananomalous magnetic moment in electromagnetic field (1). The extremelyprecise measurements of the magnetic-moment anomaly of the electronmade on highly relativistic electrons are based on the BMT equation. Theanomalous magnetic moment can be calculated by use of quantum electro-dynamics. The theoretical result agrees with experiments to within a veryhigh accuracy. This can be regarded as a direct test of BMT equation.

The existing textbooks suggest that the experimental test of the BMT equa-tion is a direct test of what we consider the incorrect expression for Wignerrotation in the Lorentz lab frame. We claim that the inclusion of this incorrectexpression as an integral part of the BMT equation in most texts is based onan incorrect physical argument. In this chapter we will investigate in detailthe reason why this is the case (2).

16.2 Magnetic Dipole at Rest in an Electromagnetic Field

Let us consider at first the spin precession for a non relativistic charge par-ticle. The proportionality of magnetic moment ~µ and angular momentum~s has been confirmed in many ”gyromagnetic” experiments on many dif-ferent systems. The constant of proportionality is one of the parameterscharactering a particular system. It is normally specified by giving the gyro-magnetic ratio or g factor, defined by ~µ = ge~s/(2mc). This formula says thatthe magnetic moment is parallel to the angular momentum and can haveany magnitude. For an electron g is very nearly 2.

Suppose that a particle is at rest in an external magnetic field ~BR. The equa-tion of motion for the angular momentum in its rest frame is d~s/dτ = ~µ×~BR =

eg~s× ~BR/(2mc) = ~ωs ×~s. In other words, the spin precesses around the direc-tion of magnetic field with the frequencyωs = −eg~BR/(2mc). In the same nonrelativistic limit the velocity processes around the direction of ~BR with thefrequency ωp = −(e/mc)~BR: d~v/dτ = (e/mc)~v × ~BR. Thus, for g = 2 spin andvelocity precess with the same frequency, so that the angle between them isconserved.

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16.3 Derivation of the Covariant (BMT) Equation of Motion of Spin

Spin dynamics equations can be expressed as tensor equations in Minkowskispace-time. We shall limit ourselves to the case of a particle with a mag-netic moment ~µ in a microscopically homogeneous electromagnetic field.Evidently the torque affects only the spin and the force affects only the mo-mentum. It follows that the motion of the system as a whole in any frame isdetermined entirely by its charge, independent of magnetic dipole moment.This part of the motion has been treated in the Chapter 11. We need nowonly consider the spin motion.

In seeking the equation for the spin motion, we shall be guided by the knowndynamics in the rest frame and the known relativistic transformation laws.We emphasize that spin is defined in a particular frame (the rest frame).Therefore, to form expressions with known transformation behavior, weneed to introduce a four-quantity related to the spin. A convenient choice isa four- (pseudo)-vector S defined by the requirement that in the rest frameits space-like components are the spin components, while the time-likecomponent is zero. We shall call S four-spin; when normalized by dividingby its invariant length, it will be called polarization four-vector. It is space-like, and therefore in no frame does it space-like part vanish.

Let the spin of the particle be represented in the rest frame by ~s. The four-vector Sα is by definition required to be purely spatial at time τ in an in-stantaneous Lorentz rest frame R(τ) of the particle and to coincide at thistime with the spin ~s(τ) of the particle; that is SαR(τ) = (0, ~SR(τ)) = (0,~s(τ)). Ata later instant τ + ∆τ in an instantaneous inertial rest frame R(τ + ∆τ), wehave similarly SαR(τ + ∆τ) = (0, ~SR(τ + ∆τ)) = (0,~s(τ + ∆τ)).

The BMT equation is manifestly covariant equation of motion for a four-vector spin Sα in an electromagnetic field Fαβ:

dSα

dτ=

ge2mc

[FαβSβ +

1c2 uα

(SλFλµuµ

)]−

1c2 uα

(Sλ

duλ

dτ

), (88)

where uµ = dxµ/dτ is the four-dimensional particle velocity vector. WithEq.(12), one has (1)

dSα

dτ=

emc

[g2

FαβSβ +g − 22c2 uα

(SλFλµuµ

)], (89)

The BMT equation is valid for any given inertial frame, and consistently

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describes, together with the covariant-force law, the motion of a chargedparticle with spin and magnetic moment. If Fµν , 0, even with g = 0, wesee that dSµ/dτ , 0. Thus, a spinning charged particle will precess in anelectromagnetic field even if it has no magnetic moment. This precession ispure relativistic effect.

The covariant equation of spin motion for a relativistic particle under theaction of the four-force Qµ = eFµνuν in the Lorentz lab frame, Eq.(88), is arelativistic ”generalization” of the equation of motion for a particle angularmomentum in its rest frame. Relativistic ”generalization” means that thethree independent equations expressing the Larmor spin precession are beembedded into the four-dimensional Minkowski space-time. The idea ofembedding is based on the principle of relativity i.e. on the fact that theclassical equatuion of motion for particle angular momentum d~s/dτ = eg~s×~BR/(2mc) can always be used in any Lorentz frame where the particle, whosemotion we want to describe, is at rest. In other words, if an instantaneouslycomoving Lorentz frame is given at some instant, one can precisely predictthe evolution of the particle spin in this frame during an infinitesimal timeinterval.

In Lorentz coordinates there is a kinematics constraint Sµuµ = 0, whichis orthogonality condition of four-spin and four-velocity. Because of thisconstraint, the four-dimensional dynamics law, Eq.(88), actually includesonly three independent equations of motion. Using the explicit expressionfor Lorentz force we find that the four equations Eq.(88) automatically implythe constraint Sµuµ = 0 as it must be. To prove this we may point out that onehas in every Lorentz frame S0 = ~S · ~v. While S0 vanishes in the rest frame,dS0/dτ need not. In fact d(Sµuµ)/dτ = 0 implies dS0/dτ = ~S · d~v/dτ. Theimmediate generalization of d~s/dτ = eg~s × ~BR/(2mc) and dS0/dτ = ~S · d~v/dτto arbitrary Lorentz frames is Eq.(88) as can be checked by reducing to therest frame. A methodological analogy with the relativistic generalization ofthe Newton’s second law emerges.

In order to fully understand the meaning of embedding of the spin dynamicslaw in the Minkowski space-time, one must keep in mind that, above, wecharacterized the spin dynamics equation in the Lorentz comoving frameas a phenomenological law. The microscopic interpretation of the magneticmoment of a particle is not given. In other words, it is generally acceptedthat the spin dynamics law is a phenomenological law and the magneticmoment is introduced in an ad hoc manner. The system of coordinates inwhich the classical equations of motion for particle angular momentum arevalid can be defined as Lorentz rest frame. The relativistic generalization ofthe three-dimensional equation d~s/dτ = eg~s×~BR/(2mc) to any Lorentz framepermits us to make correct predictions.

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16.4 Change Spin Variables

The equation Eq.(89) is more complex than one might think. In fact, it iscomposed by a set of coupled differential equations. To find solution directlyfrom the system seems quite difficult, even for a very symmetric, uniformmagnetic field setup.

In order to apply Eq.(89) to specific problems it is convenient to introduce athree vector ~s by the equation ~s = ~S + S0~pc/(E + mc2). With the help of thisrelation one can work out the equation of motion for ~s . In the importantcase of a uniform magnetic field with no electric field in the lab frame onehas, after a somewhat lengthly calculations:

d~sdτ

= −e

2m

[(g − 2 +

2γ

)γ~B −

(g − 2)γγ + 1

~vc

(~vc· γ~B

)]× ~s , (90)

What must be recognized is that in the accepted covariant approach (in-deed, Eq.(89) is obviously manifestly covariant), the solution of the dy-namics problem for the spin in the lab frame makes no reference to thethree-dimensional velocity. In fact, the Eq.(90) includes relativistic factor γand vector ~v/c, which are actually notations: γ = E/(mc2), ~v/c = ~pc/E. Allquantities E, ~p, ~B are measured in the lab frame and have exact objectivemeaning i.e. they are convention-independent. The evolution parameter τis also measured in the lab frame and has exact objective meaning . Forinstance, it is not hard to demonstrate that dτ = mdl/|~p|, where dl is thedifferential of the path length.

Spin vector~s is not part of a four-vector, and depends on both ~S and S0. Whilenot being a four-vector, it is effectively a three-dimensional object (havingzero time component in the inertial frame in question) and the spatial partof this object undergoes pure rotation with constant rate for the example ofmotion along a circle in special relativity. If we perform an arbitrary velocitymapping, ~s will have to be recomputed from the transformed values Sµ andpµ. However, this new ~s will satisfy an equation of the form Eq.(90), with ~Bcomputed from the transformed Fµν.

Let us restrict our treatment of spinning particle dynamics to purely trans-verse magnetic fields. This means that the magnetic field vector ~B is orientednormal to the particle line motion. If the field is transverse, then equationEq.(90) is reduced to

265

d~sdτ

= ~Ω × ~s = −e

2mc

[(g − 2 +

2γ

)γ~B

]× ~s , (91)

Now we have an equation in the most convenient form to be solved. Supposewe let the charged spinning particle in the lab frame through a bendingmagnet with the length dl. We know that dθ = −eBdl/(|~p|c) is the orbitalangle of the particle in the lab frame. Note that dτ = mdl/|~p|. Then, Eq.(91)tells us that we may write the spin rotation angle with respect to the labframe axes Ωdτ as Ωdτ = [(g/2 − 1)γdθ + dθ].

This tell us that in the lab frame the spin of a particle ~s changes the angle φwith its line motion. The rate of change of the angle φwith the orbital angleis (g/2 − 1)γ, so we can write dφ = (g/2 − 1)γdθ.

We would like to discuss the following question: Is the vector ~s merely adevice which is useful in making calculations - or is it a real quantity ( i.e. aquantity which has direct physical meaning)? Knowing that there is a simplealgebraic relation between ~s and the standard spin vector, the spin vector ~scan be used as an intermediate step to easily find the standard spin vectorSµ. There is, however, also a direct physical meaning to the spin vector ~s .The spin vector ~s directly gives the spin as perceived in a comoving system.

16.5 An Alternative Approach to the BMT Theory

16.5.1 BMT Equation Transformed to the Rest Frame

When Bargman, Michel and Telegdi first discovered the correct laws of spindynamics, they wrote a manifestly covariant equation in Minkowski space-time, Eq.(88), which describes the motion of the four spin Sµ. The derivationof this equation was very similar to the four-tensor equations that werealready known to relativistic particle dynamics. How to solve this four-tensor equation is an interesting question. In relativistic spin dynamics it isdone in one particular way, which is very convenient. In order to apply four-tensor equation Eq.(88) to specific problems it is convenient to transformthis equation to the rest frame as of that instant. Should one be surprisedthat the starting point of Bargman, Michel and Telegdi was the particle restframe and the classical equation of motion for particle angular momentum,which they generalized to the Lorentz lab frame and then transformed backto the rest frame?

We want to emphasize that the equation Eq.(91) for the proper spin ~s andthe BMT equation Eq.(88) for the four spin Sµ are completely equivalent,they both determine the behaviour of the spin from the point of view of

266

the lab frame. With Eq.(91) have what we need to know - the evolution ofthe proper spin vector ~s with respect to the lab frame axes. Starting fromthe classical equation d~s/dτ = eg~s × ~BR/(2mc), which describes the Larmorprecession with respect to the proper frame axes, we have derived theequation Eq.(90), which describes the spin motion with respect to the labframe axes in the proper frame and reduced to Eq.(91) in the case of purelytransverse magnetic fields. That means that we know the orientation of theproper spin with respect to the lab coordinate system which is moving withvelocity −~v and acceleration −γd~v/dτ in the proper frame.

Above we described the BMT equation, Eq.(91), in the standard manner. Ituses a spin quantity defined in the proper frame but observed with respectto the lab frame axes. Let’s look at what the equation Eq.(91) says in a littlemore detail. It will be more convenient if we rewrite this equation as

d~s = ~Ωdτ × ~s = −egγ~Bdτ/(2mc) × ~s + e(γ − 1)~Bdτ/(mc) × ~s . (92)

16.5.2 Relativistic Kinematic Addition to the Larmor Rotation

Now let’s see how we can write the right-hand side of Eq.(92) . The firstterm is that we would expect for the spin rotation due to a torque withrespect to the proper frame axes d~φL = −egγ~Bdτ/(2mc) = (g/2)γd~θ. Hered~θ = −eBdl/(|~p|c) is the angle of the velocity rotation in the lab frame. Ithas also been made evident by our analysis in the previous Chapter 15 thatangle of rotation d~φW = −e(γ − 1)~Bdτ/(mc) = (γ − 1)d~θ corresponds to theWigner rotation of the lab frame axes with respect to the proper frame axes.With this definitions, we have

d~s = ~Ωdτ × ~s = d~φL × ~s − d~φW × ~s , (93)

which begins to look interesting.

16.5.3 Wigner Rotation in the Proper Frame. First Practical Application

Now we introduce our new approach to the BMT theory, finding anotherway in which our complicated problem can be solved. We know that d~φL

and d~φW are the rotations with respect to the proper frame axes. Actuallywe only need to find the spin motion with respect to the lab frame axes.Now we must be careful about signs of rotations.

267

There is a good mnemonic rule to learn the signs of different rotations. Therule says, first, that the direction of the velocity rotation in the proper frameis the same as the direction of the velocity rotation in the lab frame. Second,the direction of the lab frame rotation in the proper frame is the same asthe direction of the velocity rotation in the proper frame. Third, the signof the spin rotation due to a torque at g > 0 (it is handy to remember thatfor an electron g is positive and very nearly 2) means that the direction ofthe rotation in the proper frame is the same as the direction of the velocityrotation in the proper frame.

We now ask about the proper spin rotation with respect to to the lab frameaxes. This is easy to find. The relative rotation angle is d~φL − d~φW. So webegin to understand the basic machinery behind spin dynamics. We see whythe Wigner rotation of the lab frame axes in the proper frame must be takeninto account if we need to know the proper spin dynamics with respect tothe lab frame axes.

Why the new derivation of the BMT equation is so simple? The reason isthat the splitting of the particle spin motion with respect to the lab frameaxes into the dynamic (Larmor) and kinematic (Wigner) parts can only berealized in the proper frame. In the proper frame, we do not need to knowany more about a relativistic ”generalization” of the (phenomenological)classical equation of motion for the particle angular momentum. In thiscase, it is possible to separate the spin dynamics problem into the trivialdynamic problem and into the kinematic problem of Wigner rotation of thelab frame in the proper frame.

16.6 Spin Tracking

Having written down the spin motion equation in a 4-vector form, Eq.(89),and determined the components of the 4-force, we satisfied the principle ofrelativity for one thing, and, for another, we obtained the four componentsof the equation for the spin motion. This is a covariant relativistic generaliza-tion of the usual three dimensional equation of magnetic moment motion,which is based on the particle proper time as the evolution parameter. Wenext wish to describe the spin motion with respect to the Lorentz lab frameusing the lab time t as the evolution parameter.

16.6.1 Conventional spin tracking. Hidden absolute time coordinatization

When going from the proper time τ to the lab time t, the frequency of spinprecession with respect to the lab frame can be obtained using the well-

268

known formula dτ = dt/γ. We then find

d~sdt

= ~$ × ~s = −e

2mc

[(g − 2 +

2γ

)~B]× ~s . (94)

The frequency of spin precession can be written in the form

$ = ω0[1 + γ(g/2 − 1)] , (95)

where ω0 is the particle revolution frequency. Now the time-like part of thefour-velocity is decomposed to cγ = c/

√1 − v2/c2 and the trajectory does

not include relativistic kinematics effects. In particular, the Galilean vectoriallaw of addition of velocities is actually used. So we must have made a jumpto the absolute time coordinatization.

The previous commonly accepted derivation of the equations for the spinprecession in the lab frame from the covariant equation Eq.(88) has the samedelicate point as the derivation of the equation of particle motion from thecovariant equation Eq.(12). The four-velocity cannot be decomposed into u =(cγ, ~vγ) when we deal with a particle accelerating along a curved trajectoryin the Lorentz lab frame. One of the consequences of non-commutativity ofnon-collinear Lorentz boosts is the unusual momentum-velocity relation. Inthis case there is a difference between covariant and non-covariant particletrajectories.

The old kinematics comes from the relation dτ = dt/γ. The presentation ofthe time component simply as the relation dτ = dt/γ between proper timeand coordinate time is based on the hidden assumption that the type ofclock synchronization that provides the time coordinate t in the lab frame isbased on the use of the absolute time convention.

16.6.2 Convention-Invariant Spin Tracking

In the Chapter 11 we saw that the particle path ~x(l) has an exact objectivemeaning i.e. it is convention-invariant. The spin orientation ~s at each pointof the particle path ~x(l) has also exact objective meaning. In contrast to this,and consistently with the conventionality of the three-velocity, the function~s(t) describing the spinning particle in the lab frame has no exact objectivemeaning.

We now want to describe how to determine the spin orientation along thepath ~s(l) in covariant spin tracking. Using the covariant equation Eq.(88) we

269

obtain Eq.(91). If we use the relation dτ = mdl/|~p| our convention-invariantequation of spin motion reads

d~sdl

= −eE

m|~p|c3

[(g2− 1 +

mc2

E

)~B]× ~s =

[(g2− 1

)E

mc2 + 1] d~θ

dl× ~s , (96)

which is based on the path length l as the evolution parameter. These threeequations corresponds exactly to the equations for components of the properspin vector that can be found from the non-covariant spin tracking equationEq.(94). We want to emphasize that there are two different (covariant andnon covariant) approaches that produce the same spin orientation~s(l) alongthe path. The point is that both approaches describe correctly the samephysical reality and the orientation of the proper spin ~s at any point ofparticle path in the magnetic field has obviously an objective meaning, i.e.is convention-invariant.

Now we take an example, so it can be seen that we do not need to askquestions about the function ~s(t) of a spinning particle experimentally. Justthink of experiments related with accelerator physics. Suppose we wantto calibrate the beam energy in a storage ring based on measurement ofspin precession frequency of polarized electrons. To measure the precessionfrequency $, a method of beam resonance depolarization by an oscillatingelectromagnetic field can be used (2). There are many forms of depolarizers,but we will mention just one, which especially simple. It is a depolarizerwhose operation depends on the radio-frequency longitudinal magneticfield which is produced by a current-curring loop around a ceramic sectionof the vacuum chamber.

Suppose the observer in the lab frame performs the beam energy measure-ment. We should examine what parts of the measured data depends on thechoice of synchronization convention and what parts do not. Clearly, physi-cally meaningful results must be convention-invariant. One might think thatthis is a typical time-depending measurement of function ~s(t). However, westate that the precession frequency $ has no intrinsic meaning - its meaningis only being assigned by a convention. It is not possible to determine theprecession frequency$ uniquely, because there is always some arbitrarinessin the ~s(t). For instance, it is always possible to make an arbitrary changein the rhythm of the clocks (i.e. scale of the time). But our problem is todetermine the energy for an electron beam. So one needs to measure alsothe revolution frequency ω0 by using the same space-time grid. What thisall means physically is very interesting. The ratio $/ω0 is convention in-dependent i.e it does not depend on the distant clocks synchronization oron the rhythm of the clocks. It means, for example, that if we observe the

270

dimensionless frequency$/ω0, we can find out the value of the convention-invariant beam energy E. The (g/2 − 1) factor can be calculated by use ofquantum electrodynamics.

Let us now return to our examination of the measured data in experimentsrelated with the calibration of the beam energy in a storage ring. The spin ~sof a particle makes the angle φ with it velocity. From Eq.(96) we have beenable to write the angle φ in therm of orbital angle θ(l) in a form φ = φ(θ).We thus use the orbital angle θ as evolution parameter. Suppose that thedepolarizer is placed at an azimuth θ0. During a period of velocity rotation,the spin will rotate through an angle of ∆φ = φ(θ0 +2π)−φ(θ0). The point isthat depolarizer measurements are made to determine the observable ∆φ.Let us see how equation Eq.(96) gives the observable ∆φ. It can be writtenin integral form ∆φ =

∫dθ[(g/2− 1)E/(mc2)] = 2π[(g/2− 1)E/(mc2)]. We can

already conclude something from these results. The convention-invariantobservation ∆φ is actually a geometric parameter. It comes quite naturallythat in experiments related with spin dynamics in a storage ring we do notneed to ask question about the function ~s(t) experimentally.

16.7 Spin Rotation in the Limit g→ 0 as Dynamics Effect

16.7.1 Spin Tracking at g→ 0

It is generally accepted that spin dynamics law is a phenomenological lawand that the magnetic moment is introduced in an ad hoc manner. Let usconsider the special case with g→ 0. The BMT equation for a particle withsmall g factor is

dSα

dτ= −

1c2 uα

(Sλ

duλ

dτ

)= −

emc3 uα

(SλFλµuµ

). (97)

It is often more convenient to write this equation as the equation of motionfor ~s. If the field is transverse, then the equation Eq.(97) is reduced to

d~sdτ

=[(E

mc2 − 1) e

mc~B]× ~s , (98)

Note that the equation Eq.(98) for the proper spin ~s and the BMT equationEq.(97) for four spin Sµ are completely equivalent. Eq.(98) is the result oftransformation to new spin variables.

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Conventional spin tracking treats the space-time continuum in a non rela-tivistic format, as a (3+1) manifold. In the conventional spin tracking, weassign absolute time coordinate and we have no mixture of positions andtime. This approach to relativistic spin dynamics relies on the use of threeequations for the spin motion

d~sdt

=

[(1 −

1γ

)e

mc~B]× ~s = −

[(γ − 1

)~ω0

]× ~s , (99)

which are based on the use of the absolute time t as the evolution parameter.Here, ~ω0 = −e~B/(mcγ) is the particle angular frequency in the lab frame. Nowthe time-like part of the four-velocity is decomposed to cγ = c/

√1 − v2/c2.

This decomposition is a manifestation of the absolute time convention.

There are two different (covariant and non covariant) approaches that pro-duce the same spin orientation ~s(l) along the path. Using the Eq.(98) orEq.(99) we obtain

d~sdl

= −[E

mc2 − 1] d~θ

dl× ~s , (100)

Both approaches describe correctly the same physical reality, and the ro-tation of the proper spin ~s with respect to the lab frame axes at g → 0 isconvention-invariant.

The relativistic kinematic effects such as Wigner rotation, Lorentz-Fitzgeraldcontraction, time dilation and relativistic corrections to the law of composi-tion of velocities are coordinate (i.e. convention-dependent) effects and haveno exact objective meaning. In the case of the Lorentz coordinatization, onewill experience e.g. the Wigner rotation phenomenon. In contrast to this, inthe case of absolute time coordinatization there are no relativistic kinematicseffects, and no Wigner rotation will be found (3).

However, the spin orientation at each point of the particle path has exactobjective meaning. In fact, Eq.(100) is convention-invariant i.e includes onlyquantities which have exact objective meaning. Understanding this resultof the theory of relativity is similar to understanding the previously dis-cussed result for relativistic mass correction. We find that the evolution ofa particle along its path is still given by the corrected Newton’s second laweven though the relativistic mass correction has no kinematical origin. Amethodological analogy with the spin dynamics equation Eq.(100) emergeby itself. The spin rotation in the lab frame at g → 0 is relativistic effect (as

272

the relativistic mass correction) but it has no kinematical origin.

16.8 Incorrect Expression for Wigner Rotation. Myth About Experimental Test

16.8.1 Terminology. Thomas Precession: Correct and Incorrect Solutions

The expression for the Wigner rotation in the lab frame obtained by au-thors of textbooks is given by ~δΦ = (1 − γ)~v × d~v/v2 = (1 − γ) ~δθ, whichoften presented as ~ωTh = d~Φ/dt = (1 − γ)[~v × d~v/dt]/v2. In other words, theproper frame coordinate performs a precession relative to the lab framewith the velocity of precession ~ωTh, where d~v/dt is the acceleration of thespinning particle in the lab frame. This precession phenomenon is calledThomas precession. From the viewpoint of the generally accepted terminol-ogy, Thomas precession is actually a manifestation of the Wigner (Thomas)rotation. According to expression for Thomas precession in the lab framepresented in textbooks, the comoving frame precesses in the opposite direc-tion with respect to the direction of the angular velocity of the precession~ω0 = ~v×d~v/dt/v2 andωTh → −∞ in the limit γ −→ ∞. The theory of relativityshows us that the textbook expression for the Thomas precession in the labframe and correct result ~ωTh = (1 − 1/γ)[~v × d~v/dt]/v2 actually differ both insign and magnitude.

16.8.2 Incorrect Interpretation of the Correct BMT Result

The existence of the usual incorrect expression for the Thomas precessionin the lab frame has led to incorrect interpretation of the BMT result and, inparticular, of the spin dynamics equation Eq.(99). Using the incorrect resultfor the Thomas precession, the BMT result for a small g factor, Eq.(99), isusually presented as

d~sdt

= −[(γ − 1

)~ω0

]× ~s = ~ωTh × ~s , (101)

Frequently, the first stumbling blocks in the process of understanding andaccepting the correct Wigner (Thomas) rotation theory is a widespread beliefthat the experimental test of the BMT equation is a direct test of the incorrectexpression for Thomas precession. There are many physicists who havealready received knowledge about the Thomas precession from well-knowntextbooks and who would say, ”The extremely precise measurements ofthe magnetic-moment anomaly of the electron made on highly relativisticelectrons are based on the BMT equation, of which the Thomas precession

273

is an integral part, and can be taken as experimental confirmations of thestandard expression for the Thomas precession.” This misconception aboutexperimental test of the incorrect expression for the Thomas precession inthe lab frame is common and pernicious.

The interpretation of Eq.(99) as the Thomas precession Eq.(101) is presentedin textbooks as alternative approach to the already developed BMT theory(3). Authors of textbooks got the correct BMT result by using the incorrectexpression for the Thomas precession and an incorrect physical argument.This wrong argument is an assumption about the splitting of spin precessionin the lab frame into dynamics (Larmor) and kinematics (Thomas) parts. Inthis chapter we demonstrated that this splitting can not be obtained in thelab frame. It is possible to perform this splitting only in the Lorentz properframe where the spinning particle is at rest and the Lorentz lab referenceframe moves with velocity −~v and acceleration −γd~v/dτ with respect to theproper frame axes.

This splitting can not be realized in the lab frame for the following reason:the starting point of the BMT theory is the phenomenological dynamics lawd~s/dτ = eg~s × ~BR/(2mc), which is the equation of motion for the angularmomentum in its rest frame (i.e. with respect to the rest frame axes). It isphenomenological because the microscopic interpretation of the (anoma-lous) magnetic moment of a particle is not given. The BMT equation is arelativistic generalization of the phenomenological dynamics law. It is validfor any given Lorentz frame, for example for the Lorentz lab frame. In thelab frame, the BMT equation is a phenomenological dynamics equation forthe spin motion with respect to the lab frame axis even at g→ 0 (4).

It is generally believed that if a particle with spin has no magnetic moment(g → 0), the spin rotation in the lab frame is purely kinematic in nature (4).It is simply not true in physics. The argument for the dynamics origin ofthe spin rotation in the lab frame at g → 0 is that the relativistic kinematiceffects are coordinate (i.e. convention-dependent) effects and have no exactobjective meaning. In the case of Lorentz coordinatization, one will expe-rience e.g. the Thomas precession phenomenon. In contrast to this, in thecase of absolute time coordinatization there are no relativistic kinematicseffects, and therefore no Thomas precession will be found. However, thespin orientation at g→ 0 with respect to the lab frame axes at each point ofthe particle path in the lab frame has exact objective meaning (i.e. it has nokinematical origin).

With the wording ”spin orientation with respect to the lab frame axes inthe lab frame” we mean that the lab observer can directly measure the spinorientation with respect to the lab frame axes using a polarimeter. In thelab frame, the polarimeter is at rest and the field theory involved in the

274

polarimeter operation description is isotropic. We do not need to know anymore about the polarimeter operation. It is in this sense, we can discuss inthe lab frame about the spin orientation with respect to the lab frame axes asphysical reality. In contrast, the spin orientation with respect to the lab frameaxes in the proper frame ( and also the spin orientation with respect to theproper frame axes in the lab frame) has no direct physical meaning. In fact,if we use the Lorentz coordinatization in the proper frame there is a Thomasprecession of the lab frame axes in the proper frame. In the case of absolutetime coordinatization in the proper frame, there is no kinematic Thomasprecession of the lab frame axes with respect to the proper frame axes. Thetwo coordinatizations give, in fact, a different result for spin rotation withrespect to the lab frame axes.

At this point a reasonable question arises: why in the lab frame the spinorientation with respect to the lab frame axes has physical meaning, butthe same orientation in the proper frame does not? The answer is that inthe lab frame an observer who performs spin orientation measurement isat rest with respect to the lab reference frame. This situation is symmetricalwith respect to a change of the reference frames. In fact, the spin orientationmeasurement with respect to the proper frame axes in the proper frame hasexact objective meaning (i.e. it has dynamical origin) and we observe thesame result no matter how the lab frame rotates with respect to the properframe.

The argument that the result of spin orientation measurements with respectto the lab frame axes in the proper frame is paradoxical runs somethinglike this: the laws of physics in any one reference frame should be ableto account for all physical phenomena, including the observations made bymoving observers. Suppose that an observer in the lab frame performs a spinrotation measurement. Viewed from the proper frame, the two proper framecoordinatizations give a different result for the spin rotation with respect tothe lab frame axes in the lab frame which must be convention-invariant.

Nature doesn’t see a paradox, however, because the proper observer seesthat the lab polarimeter is moving on an accelerated motion, and the lab ob-server, moving with the polarimeter, performs the spin direction measure-ment. In order to predict the result of the moving polarimeter measurementone does not need to have access to the detailed dynamics of the particleinto the polarimeter. It is enough to assume the Lorentz covariance of thefield theory involved in the description of the polarimeter operation.

In the Lorentz proper frame the field theory involved in the description ofthe (lab) polarimeter operation is isotropic. Clearly, in the case of Lorentzcoordinatization we can discuss in the proper frame about the spin orienta-tion with respect to the lab frame axes as a prediction of the measurement

275

made by the lab observer.

Now the question is, what is the prediction of the proper observer in the caseof absolute time coordinatization? How shall we describe the polarimeteroperation after the Galilean transformations? After the Galilean transfor-mations we obtain the complicated (anisotropic ) field equations. The newterms that have to be put into the field equations due to the use of Galileantransformations lead to the same prediction as concerns experimental re-sults: the spin of the particle is rotated with respect to the lab frame axesaccording to the Lorentz coordinatization prediction. Let us examine in alittle more detail how this spin rotation comes about. As usual, in the case ofabsolute time coordinatization, we are going to make a mathematical trickfor solving the field equations with anisotropic terms. In order to eliminatethese terms we make a change of the variables. Using new variables weobtain the phenomenon of spin rotation with respect to the lab frame axesin the lab frame.

16.9 Reasons that Underlie the Derivation of the Incorrect Expressions

A large number of incorrect expressions for the Wigner (Thomas) rotationcan be found in the literature. We believe there is a good reason from themethodological point of view to analyze the complicated situation relat-ing to this relativistic kinematic effect. The main conclusion drawn fromthe literature is as follows. The situation relating to the use of the Wigner(Thomas) rotation theory in the relativistic spin dynamics is extremely com-plex. We have already seen in the discussion of this chapter that the startingpoint of Bargman, Michel and Telegdi was the particle rest frame and theclassical equation of motion for particle angular momentum, which theygeneralized to the Lorentz lab frame (Eq.(88)) and then transformed back tothe rest frame.

This brings up an interesting question: Why? The explanation is deep downin relativistic quantum mechanics. The approach in which we deal withthe proper spin is much preferred in the experimental practice due to clearphysical meaning of the vector ~s. Unlike momentum, which has definitecomponents in each reference frame, angular momentum is defined only inone particular reference frame. It does not transform. Any statement aboutit refers to the rest frame as of that instant. If we say that in the lab framethe spin of a particle makes the angle φ with its velocity, we mean that inthe particle’s rest frame the spin makes this angle with the line motion ofthe lab frame.

In the Chapter 7 we demonstrated that the analysis of physical phenomena

276

in non-inertial frames of reference can be described in an inertial framewithin standard (Einstein) special relativity taking advantage of well knownrelativistic kinematic effects. For example, when viewed from the inertialframe, the aberration of light effect in the rotating frame is easy to calculatein the framework of the special theory of relativity taking advantage of theWigner rotation theory. In contrast, in order to compute the aberration oflight effect in non-inertial systems we used a metric tensor.

The difference between these two (aberration of light and spin dynamics)applications of the Wigner rotation theory is very interesting. In the caseof spin dynamics one has no occasion to make measurements with non-inertial devices. Common textbooks presentations of the spin dynamicsuse the approach which deals only with observations of the inertial (lab)observer. The conventional method used to explain the spin dynamics in thelab frame is very unusual. First, we already pointed that the laws of physicsin the lab frame are able to account for all physical phenomena, includingthe observation made by non-inertial observer. In particular, it is possibleto determine the result of spin orientation measurements with respect tothe lab frame axes in the proper frame, which is moving with velocity −vand acceleration −γd~v/dτ in the proper frame. In this case, when viewedfrom the lab frame, the interpretation of the spin rotation experiments inthe lab frame is simple and may be derived from the prediction of theproper observer as concerns the measurement made by the lab observer.Therefore, the spin rotation measurement by a polarimeter in the lab frameis interpreted with viewpoint of the proper observer as viewing this of thelab observer. This complexity is one of the reason why authors of textbooksobtained an incorrect expression for the Wigner rotation.

277


1. The motion of the classical spin in an external electromagnetic field ispresented by the Bargmann-Michel-Telegdi (BMT) equation [65]. The BMTequation is manifestly covariant and can be used in any inertial frame. It isthe law of motion of the four-spin for a particle in a uniform electric andmagnetic fields.

2. A method for measuring the particle energy in an electron-positron stor-age ring by means of resonance depolarization by a radio-frequency longi-tudinal magnetic field is described in [66].

3. The results in the Bargmann-Michel-Telegdi paper [65] were obtainedby the method of semi-classical approximation of the Dirac equation. TheWigner rotation was not considered in [65] at all, because the Dirac equationallow obtaining the solution for the total particle’s spin motion without anexplicit splitting it into the Larmor and Wigner parts.

4. It is generally believed that ”If the particle with spin has no magneticmoment (g = 0), the spin precession is purely kinematic in nature ... Forexample, the 235

92 U nucleus has a rather small g factor (g = −0.26), which issmaller than that of the normal gyro-magnetic ratio by a factor 8. For suchobject, the kinematic effect dominates over the dynamics one.”[67]. Thisstatement presented in most published papers and textbooks is misleading.The reason is that a splitting of particle spin motion into the dynamic andkinematic (Wigner) parts cannot be performed in the Lorentz lab frame.In the Lorentz lab frame, Eq.(97), is a relativistic ”generalization” of theequation of motion for a particle angular momentum in its rest frame. Inother words, Eq.(97) is a dynamics equation. The relativistic kinematicseffects, such as Wigner rotation, are coordinate effects and have no exactobjective meaning. However, the spin orientation with respect to the labframe axes at each point of the particle path in the lab frame has exactobjective meaning.

278

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281

Appendix I. Radiation by Moving Charges

We start with the solution of Maxwell’s equation in the space-time domain,the well-known Lienard-Wiechert expression, and we subsequently applya Fourier transformation. The Lienard-Wiechert expression for the electricfield of a point charge (−e) reads (see, e.g. [60]):

~E(~ro, t) =−e~n − ~β

γ2(1 − ~n · ~β)3|~ro − ~r′|2−

ec~n × [(~n − ~β) × ~β]

(1 − ~n · ~β)3|~ro − ~r′|. (102)

R = |~ro − ~r′(t′)| denotes the displacement vector from the retarded positionof the charge to the point where the fields are calculated. Moreover, ~β =

~v/c, ~β = ~v/c, while ~v and ~v denote the retarded velocity and accelerationof the electron. Finally, the observation time t is linked with the retardedtime t′ by the retardation condition R = c(t − t′). As is well-known, Eq.(102) serves as a basis for the decomposition of the electric field into asum of two quantities. The first term on the right-hand side of Eq. (102)is independent of acceleration, while the second term linearly depends onit. For this reason, the first term is called ”velocity field”, and the second”acceleration field” [60]. The velocity field differs from the acceleration fieldin several respects, one of which is the behavior in the limit for a very largedistance from the electron. There one finds that the velocity field decreaseslike R−2, while the acceleration field only decreases as R−1. Let us apply aFourier transformation:

~E(~ro, ω) =−e

∞∫−∞

dt′~n − ~β

γ2(1 − ~n · ~β)2|~ro − ~r′|2exp

[iω

(t′ +|~ro − ~r′(t′)|

c

)]

−ec

∞∫−∞

dt′~n × [(~n − ~β) × ~β]

(1 − ~n · ~β)2|~ro − ~r′|exp

[iω

(t′ +|~ro − ~r′(t′)|

c

)]. (103)

As in Eq. (102) one may formally recognize a velocity and an accelerationterm in Eq. (103) as well. Since Eq. (103) follows directly from Eq. (102), thatis valid in the time domain, the magnitude of the velocity and accelerationparts in Eq. (103), that include terms in 1/R2 and 1/R respectively, do notdepend on the wavelengthλ. It is instructive to take advantage of integrationby parts. With the help of

1c

ddt′|~ro − ~r′(t′)| = −~n · ~β and

d~ndt′

=c

|~ro − ~r′(t′)|

[−~β + ~n

(~n · ~β

)], (104)

Eq. (103) can be written as

282

~E (~ro, ω) = −e

∞∫−∞

dt′~n

|~ro − ~r′(t′)|2exp

[iω

(t′ +|~ro − ~r′(t′)|

c

)]

+ec

∞∫−∞

dt′d

dt′

~β − ~n

(1 − ~n · ~β)|~ro − ~r′(t′)|

exp[iω

(t′ +|~ro − ~r′(t′)|

c

)].

(105)

Eq. (105) may now be integrated by parts. When edge terms can be droppedone obtains [68]

~E(~ro, ω) =−iωec

∞∫−∞

dt′ ~β − ~n

|~ro − ~r′(t′)|−

icω

~n

|~ro − ~r′(t′)|2

× exp

iω

(t′ +|~ro − ~r′(t′)|

c

). (106)

The only assumption made going from Eq. (103) to Eq. (106) is that edgeterms in the integration by parts can be dropped. This assumption canbe justified by means of physical arguments in the most general situationaccounting for the fact that the integral in dt′ has to be performed over theentire history of the particle and that at t′ = −∞ and t′ = +∞ the electrondoes not contribute to the field anymore. Let us give a concrete example foran ultra-relativistic electron. Imagine that bending magnets are placed at thebeginning and at the end of a given setup, such that they deflect the electrontrajectory of an angle much larger than the maximal observation angle ofinterest for radiation from a bending magnet. This means that the magnetswould be longer than the formation length associated with the bends, i.e.Lfb ∼ (cρ2/ω)1/3, where ρ is the bending radius. In this way, intuitively, themagnets act like switches: the first magnet switches the radiation on, thesecond switches it off. Then, what precedes the upstream bend and whatfollows the downstream bend does not contribute to the field detected atthe screen position. With these caveat Eq. (106) is completely equivalent toEq. (103).

The derivation of Eq. (106) is particularly instructive because shows thateach term in Eq. (106) is due to a combination of velocity and accelerationterms in Eq. (103). In other words the terms in 1/R and in 1/R2 in Eq. (106)appear as a combination of the terms in 1/R (acceleration term) and 1/R2

(velocity term) in Eq. (103). As a result, one can say that there exist contri-butions to the radiation from the velocity part in Eq. (103). The presentationin Eq. (106) is more interesting with respect to that in Eq. (103) (althoughequivalent to it) because the magnitude of the 1/R2-term in Eq. (106) can

283

directly be compared with the magnitude of the 1/R-term inside the integralsign.

The bottom line is that physical sense can be ascribed only to the integral inEq. (103) or Eq. (106). The integrand is, in fact, an artificial construction. Inthis regard, it is interesting to note that the integration by parts giving Eq.(106) is not unique. First, we find that [68]

~n × [(~n − ~β) × ~β]

|~ro − ~r′|(1 − ~n · ~β)2=

1

|~ro − ~r′|

ddt′

~n × (~n × ~β)

(1 − ~n · ~β)

−

~n(~n · ~β) + ~n(~n · ~β) − ~n(~n · ~β)2− ~β(~n · ~β)

|~ro − ~r′|(1 − ~n · ~β)2

. (107)

Note that Eq. (107) accounts for the fact that ~n = (~ro − ~r′(t′))/|~ro − ~r′(t′)| is nota constant in time. Using Eq. (107) in the integration by parts, we obtain

~E(~ro, ω) =−iωec

∞∫−∞

dt′−~n × (~n × ~β)

|~ro − ~r′(t′)|+

icω

~β − ~n − 2~n(~n · ~β)

|~ro − ~r′(t′)|2

× exp

iω

(t′ +|~ro − ~r′(t′)|

c

). (108)

Similarly as before, the edge terms have been dropped. Eq. (103), Eq. (106)and Eq. (108) are equivalent but include different integrands. This is nomistake, as different integrands can lead to the same integral.

If the position of the observer is far away enough from the charge, one canmake the expansion Eq. (21). Using Eq. (108), we obtain Eq. (22).

284

Appendix II. Undulator Radiation in Resonance Approximation. Far Zone

Calculations pertaining undulator radiation are well established see e.g.[69]. In all generality, the field in Eq. (33) can be written as

~E = exp[iωθ2z0

2c

]iωec2z0

×

L/2∫−L/2

dz′

K2iγ

[exp (2ikwz′) − 1

]~ex + ~θ exp (ikwz′)

× exp[i(C +

ωθ2

2c

)z′ −

Kθx

γω

kwccos(kwz′) −

K2

8γ2

ωkwc

sin(2kwz′)].

(109)

Here ω = ωr + ∆ω, C = kw∆ω/ωr and

ωr = 2kwcγ2z , (110)

is the fundamental resonance frequency.

Using the Anger-Jacobi expansion:

exp[ia sin(ψ)

]=

∞∑p=−∞

Jp(a) exp[ipψ

], (111)

where Jp(·) indicates the Bessel function of the first kind of order p, to writethe integral in Eq. (109) in a different way:

~E = exp[iωθ2z0

2c

]iωec2z0

∞∑m,n=−∞

Jm(u)Jn(v) exp[ iπn

2

]

×

L/2∫−L/2

dz′ exp[i(C +

ωθ2

2c

)z′]

×

K

2iγ[exp (2ikwz′) − 1

]~ex + ~θ exp (ikwz′)

exp [i(n + 2m)kwz′] ,

(112)

285

where 1

u = −K2ω

8γ2kwcand v = −

Kθxωγkwc

. (113)

Up to now we just re-wrote Eq. (33) in a different way. Eq. (33) and Eq. (112)are equivalent. Of course, definition of C is suited to investigate frequenciesaround the fundamental harmonic but no approximation is taken besidesthe paraxial approximation.

Whenever

C +ωθ2

2c kw , (114)

the first phase term in z′ under the integral sign in Eq. (112) is varyingslowly on the scale of the undulator period λw. As a result, simplificationsarise when Nw 1, because fast oscillating terms in powers of exp[ikwz′]effectively average to zero. When these simplifications are taken, resonanceapproximation is applied, in the sense that one exploits the large parameterNw 1. This is possible under condition (114). Note that (114) restricts therange of frequencies for positive values of C independently of the obser-vation angle θ, but for any value C < 0 (i.e. for wavelengths longer thanor = c/ωr) there is always some range of θ such that Eq. (114) can be applied.Altogether, application of the resonance approximation is possible for fre-quencies around ωr and lower than ωr. Once any frequency is fixed, (114)poses constraints on the observation region where the resonance approxima-tion applies. Similar reasonings can be done for frequencies around higherharmonics with a more convenient definition of the detuning parameter C.

Within the resonance approximation we further select frequencies such that

|∆ω|ωr 1 , i.e. |C| kw . (115)

Note that this condition on frequencies automatically selects observationangles of interest θ2

1/γ2z . In fact, if one considers observation angles

outside the range θ2 1/γ2

z , condition (114) is not fulfilled, and the inte-grand in Eq. (112) exhibits fast oscillations on the integration scale L. As a

result, one obtains zero transverse field, ~E = 0, with accuracy 1/Nw. Underthe constraint imposed by (115), independently of the value of K and for

1 Here the parameter v should not be confused with the velocity.

286

observation angles of interest θ2 1/γ2

z , we have

|v| =K|θx|

γω

kwc=

(1 +

∆ωωr

) 2√

2K√

2 + K2γz|θx| . γz|θx| 1 . (116)

This means that, independently of K, |v| 1 and we may expand Jn(v) inEq. (112) according to Jn(v) ' [2−n/Γ(1 + n)] vn, Γ(·) being the Euler gammafunction

Γ(z) =

∞∫0

dt tz−1 exp[−t] . (117)

Similar reasonings can be done for frequencies around higher harmonicswith a different definition of the detuning parameter C. However, aroundodd harmonics, the before-mentioned expansion, together with the appli-cation of the resonance approximation for Nw 1 (fast oscillating terms inpowers of exp[ikwz′] effectively average to zero), yields extra-simplifications.

Here we are dealing specifically with the first harmonic. Therefore, theseextra-simplifications apply. We neglect both the term in cos(kwz′) in thephase of Eq. (109) and the term in ~θ in Eq. (109). First, non-negligible termsin the expansion of Jn(v) are those for small values of n, since Jn(v) ∼ vn, with|v| 1. The value n = 0 gives a non-negligible contribution J0(v) ∼ 1. Then,since the integration in dz′ is performed over a large number of undulatorperiods Nw 1, all terms of the expansion in Eq. (112) but those for m = −1and m = 0 average to zero due to resonance approximation. Note thatsurviving contributions are proportional to K/γ, and can be traced back tothe term in~ex only, while the term in ~θ in Eq. (112) averages to zero for n = 0.Values n = ±1 already give negligible contributions. In fact, J±1(v) ∼ v.Then, the term in ~ex in Eq. (112) is v times the term with n = 0 and isimmediately negligible, regardless of the values of m. The term in ~θ wouldsurvive averaging when n = 1, m = −1 and when n = −1, m = 0. However,it scales as ~θv. Now, using condition (115) we see that, for observation anglesof interest θ2

1/γ2z , |~θ| |v| ∼ (

√2 K /

√

2 + K2 ) γzθ2 K/γ. Therefore, the

term in ~θ is negligible with respect to the term in ~ex for n = 0, that scales asK/γ. All terms corresponding to larger values of |n| are negligible.

Summing up, all terms of the expansion in Eq. (111) but those for n = 0 andm = −1 or m = 0 give negligible contribution. After definition of

AJJ = J0

(ωK2

8kwcγ2

)− J1

(ωK2

8kwcγ2

), (118)

287

that can be calculated at ω = ωr since |C| kw, we have

~E = −Kωe

2c2z0γAJJ exp

[iωθ2z0

2c

] L/2∫−L/2

dz′ exp[i(C +

ωθ2

2c

)z′]~ex .

(119)

288

Appendix III. Approximation for the Electron Path

Let us now discuss the case of the radiation from a single electron withan arbitrary angular deflection ~η and an arbitrary offset ~l with respect toa reference orbit defined as the path through the origin of the coordinatesystem, that is x(0) = y(0) = 0.

If the magnetic field in the setup does not depend on the transverse coor-dinates, i.e. B = B(z), an initial offset x(0) = lx, y(0) = ly shifts the path ofan electron of ~l. Similarly, an angular deflection ~η = (ηx, ηy) at z = 0 tiltsthe path without modifying it. Cases when the magnetic field of SR sourcesinclude focusing elements (or the natural focusing of insertion devices) areout of the scope of this paper. Assuming further that ηx 1 and ηy 1,which is typically justified for ultrarelativistic electron beams, one obtainsthe following approximation for the electron path:

x(z) = xr(z) + ηxz + lx , y(z) = yr(z) + ηyz + ly ,(120)

where the subscript ‘r’ refers to the reference path. This gives a parametricdescription of the path of a single electron with offset~l and deflection ~η. Thecurvilinear abscissa on the path can then be written as

s(z) =

z∫0

dz′1 +

(dxdz′

)2

+

(dydz′

)21/2

'

z∫0

dz′1 +

12

(dxr

dz′

)2

+12

(dyr

dz′

)2

+12

(η2

x + η2y

)+ ηx

dxr

dz′+ ηy

dyr

dz′

= sr(z) +

η2z2

+ xr(z)ηx + yr(z)ηy , (121)

where we expanded the square root around unity in the first passage, wemade use of Eq. (120), and of the fact that the curvilinear abscissa along thereference path is sr(z) ' z +

∫ z

0dz′[(dxr/dz′)2/2 + (dyr/dz′)2/2].

We now substitute Eq. (120) and Eq. (121) into Eq. (27) to obtain:

~E(z0,~r0, ω) = −iωec2z0

∞∫−∞

dz′exp [iΦT]

×

[(vx(z′)

c− (θx − ηx)

)~ex +

(vy(z′)

c− (θy − ηy)

)~ey

],

289

(122)

where the total phase ΦT is

ΦT = ω

[sr(z′)

v+η2z′

2v+

1v

[xr(z′)ηx + yr(z′)ηy] −z′

c

]+ω2c

[z0(θ2

x + θ2y) − 2θxxr(z′) − 2θxηxz′ − 2θxlx

−2θyy(z′) − 2θyηyz′ − 2θyly + z′(θ2x + θ2

y)], (123)

which can be rearranged as

ΦT ' ω

[sr(z′)

v−

z′

c

]−ωc

(θxlx + θyly)

+ω2c

[z0(θ2

x + θ2y) − 2(θx − ηx)xr(z′)

−2(θy − ηy)yr(z′) + z′((θx − ηx)2 + (θy − ηy)2

)].

(124)

290

Appendix IV. Self-Electromagnetic Fields of the Modulated Electron Beam

In general, the electrodynamical theory is based on the exploitation, for theultra-relativistic particles, of the small parameter 1/γ2. By this, Maxwell’sequations are reduced to much simpler equations with the help of paraxialapproximation.

Whatever the method used to present results, one needs to solve Maxwell’sequations in unbounded space. We introduce a cartesian coordinate system,where a point in space is identified by a longitudinal coordinate z andtransverse position ~r⊥. Accounting for electromagnetic sources, i.e. in aregion of space where current and charge densities are present, the followingequation for the field in the space-frequency domain holds in all generality:

c2∇

2~E + ω2~E = 4πc2~∇ρ − 4πiω~j , (125)

where ρ(z,~r⊥, ω) and ~j(z,~r⊥, ω) are the Fourier transforms of the chargedensity ρ(z,~r⊥, t) and of the current density ~j(z,~r⊥, t). Eq. (125) is the well-known Helmholtz equation. Here ~E indicates the Fourier transform of theelectric field in the space-time domain.

Eq. (125) can be solved with the help of an appropriate Green’s functionG(z − z′,~r⊥ − ~r′⊥) yielding

~E(z,~r⊥, ω) =−4π

∞∫−∞

dz′∫

d~r′⊥( iω

c2~j − ∇′ρ

)G(z − z′,~r⊥ − ~r′⊥) , (126)

the integration in d~r′⊥ being performed over the entire transverse plane. Anexplicit expression for the Green’s function to be used in Eq. (126) is givenby

G(z − z′,~r⊥ − ~r′⊥) =−

exp

i(ω/c)[∣∣∣~r⊥ − ~r′⊥∣∣∣2 + (z − z′)2

]1/2

4π[∣∣∣~r⊥ − ~r′⊥∣∣∣2 + (z − z′)2

]1/2 , (127)

that automatically includes the proper boundary conditions at infinity.

The transverse field ~E⊥ can be treated in terms of paraxial Maxwell’s equa-tions in the space-frequency domain (see e.g. [68]). From the paraxial ap-

proximation follows that the electric field envelope ~E⊥ = ~E⊥ exp [−iωz/c]

291

does not vary much along z on the scale of the reduced wavelength o =λ/(2π). As a result, the following field equation holds:

D

[~E⊥(z,~r⊥, ω)

]= ~g(z,~r⊥, ω) , (128)

where the differential operatorD is defined by

D ≡

(∇⊥

2 +2iω

c∂∂z

), (129)

∇⊥2 being the Laplacian operator over transverse cartesian coordinates. Eq.

(128) is Maxwell’s equation in paraxial approximation.

Eq. (125), which is an elliptic partial differential equation, has thus beentransformed into Eq. (128), that is of parabolic type. Note that the appli-cability of the paraxial approximation depends on the ultra-relativistic as-sumption γ2

1 but not on the choice of the z axis. If, for a certain choiceof the longitudinal z direction, part of the trajectory is such that γ2

z ∼ 1,the formation length is very short (L f ∼ o), and the radiated field is prac-tically zero. As a result, Eq. (125) can always be applied, i.e. the paraxialapproximation can always be applied, whenever γ2

1.

Complementarily, it should also be remarked here that the status of theparaxial equation Eq. (128) in Synchrotron Radiation theory is differentfrom that of the paraxial equation in Physical Optics. In the latter case,the paraxial approximation is satisfied only by small observation angles.For example, one may think of a setup where a thermal source is studiedby an observer positioned at a long distance from the source and behind alimiting aperture. Only if a small-angle acceptance is considered the paraxialapproximation can be applied. On the contrary, due to the ultra-relativisticnature of the emitting electrons, contributions to the SR field from parts ofthe trajectory with formation length L f o (the only non-negligible) arehighly collimated. As a result, the paraxial equation can be applied at anyangle of interest, because it practically returns zero field at angles where itshould not be applied.

The source-term vector ~g(z,~r⊥) is specified by the trajectory of the sourceelectrons, and can be written in terms of the Fourier transform of the trans-verse current density, ~j

⊥(z,~r⊥, ω), and of the charge density, ρ(z,~r⊥, ω), as

~g = −4π exp[−

iωzc

] ( iωc2~j⊥− ~∇⊥ρ

). (130)

292

~j⊥

and ρ are regarded as given data. We will treat ~j⊥

and ρ as macroscopicquantities, without investigating individual electron contributions. In thetime domain, we may write the charge density ρ(~r, t) and the current density~j(~r, t) as

ρ(~r, t) =1vρ⊥(~r⊥) f

(t −

zv

)(131)

and

~j(~r, t) =1v~vρ⊥(~r⊥) f

(t −

zv

),

(132)

where v denote the velocity of the electron. The quantity ρ⊥ has the meaningof transverse electron beam distribution, while f is the longitudinal chargedensity distribution.

In the space-frequency domain, Eq. (131) and Eq. (132) transform to:

ρ(~r⊥, z, ω) = ρ⊥(~r⊥

)f (ω) exp [iωz)/v] , (133)

and

~j(~r⊥, z, ω) = ~vρ⊥(~r⊥

)f (ω) exp [iωz/v] . (134)

It should be remarked that ρ and ~j = ρ~v satisfy the continuity equation. Inother words, one can find ~∇ · ~j = iωρ.

We find an exact solution of Eq. (128) without any other assumption aboutthe parameters of the problem. A Green’s function for Eq. (128), namely thesolution corresponding to the unit point source can be written as (see e.g.[68]):

G(z − z′; ~r⊥ − ~r′⊥) =−1

4π(z − z′)exp

iω| ~r⊥ − ~r′⊥ |

2

2c(z − z′)

, (135)

assuming z − z′ > 0. When z − z′ < 0 the paraxial approximation does nothold, and the paraxial wave equation Eq. (128) should be substituted, in thespace-frequency domain, by a more general Helmholtz equation. Yet, the

293

radiation formation length for z−z′ < 0 is very short with respect to the casez− z′ > 0, i.e. we can neglect contributions from sources located at z− z′ < 0.

Since it is assumed that electrons are moving along the z-axis, we have~j⊥

= 0. Thus, after integration by parts, we obtain the solution

~E⊥(z,~r⊥) =−iωc

f (ω)

z∫0

dz′∫

d~r′⊥ exp

iω[| ~r⊥ − ~r′⊥ |2

2c(z − z′)

]+ i

ωz′

2cγ2

×1

z − z′ρ⊥

(~r′⊥

) (~r⊥ − ~r′⊥z − z′

).

(136)

Eq. (136) describes the field at any position z.

First, we make a change in the integration variable from z′ to ξ ≡ z − z′. Inthe limit for z −→ ∞, corresponding to the condition z γ2o, we can writefor the transverse field

~E⊥(z,~r⊥) =−iω f (ω)

c

∫d~r′⊥ρ⊥

(~r′⊥

)exp

[iωz2cγ2

] icω

~r⊥ − ~r′⊥| ~r⊥ − ~r′⊥ |

·d

d[| ~r⊥ − ~r′⊥ |

]×

∞∫0

dξξ

exp[+iω| ~r⊥ − ~r′⊥ |2

2cξ−

iωξ2cγ2

] (137)

We now use the fact that, for any real number α > 0:

∞∫0

dξ exp [i (−ξ + α/ξ)] /ξ = 2K0

(2√α), (138)

where K0 is the zero order modified Bessel function of the second kind.Using Eq. (138) we can write Eq. (137) as

~E⊥(z,~r⊥) =iω f (ω)

cexp

[iωz2cγ2

] ∫d~r′⊥ρ⊥

(~r′⊥

)×

[icω

~r⊥ − ~r′⊥| ~r⊥ − ~r′⊥ |

]2γzo

K1

(| ~r⊥ − ~r′⊥ |

γo

) , (139)

294

where K1(·) is the modified Bessel function of the first order.

Let us assume a Gaussian transverse charge density distribution of theelectron bunch with rms size σ i.e. ρ⊥ = (2πσ2)−1 exp[−r2

⊥/(2σ2)]. Within the

deep asymptotic region when the transverse size of the modulated electronbeam σ oγ the Ginzburg-Frank formula can be applied [70]

~E⊥(z,~r⊥) = −2ωec2γ

exp[

iωz2cγ2

]~r⊥r⊥

K1

(ωr⊥cγ

). (140)

Analysis of Eq.(140) shows a typical scale related to the transverse fielddistribution of order oγ in dimensional units. Here λ is the modulationwavelength. In this asymptotic region radiation can be considered as virtualradiation from a filament electron beam (with no transverse dimensions).

However, in XFEL practice we only deal with the deep asymptotic re-gion where σ oγ. Then, it can be seen that the field distribution inthe space-time domain is essentially a convolution in the space domainbetween the transverse charge distribution of the electron beam and thefield spread function described by the Ginzburg-Frank formula. Assuminga Gaussian (azimuthally-symmetric) transverse density distribution of theelectron beam we obtain the radially polarized virtual radiation beam.

295

evgeny saldin - arxiv.org

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