iAPPLICATIONS OF

CLASSICAL PHYSICS

Roger D. Blandford and Kip S. Thorne

version 1200.1.K.pdf, January 28, 2013

Preface

Please send comments, suggestions, and errata via email to kip@caltech.edu, or on paper to

Kip Thorne, 350-17 Caltech, Pasadena CA 91125

This book is an introduction to the fundamentals and 21st-century applications of all themajor branches of classical physics except classical mechanics, electromagnetic theory, andelementary thermodynamics (which we assume the reader has already learned elsewhere).

Classical physics and this book deal with physical phenomena on macroscopic scales:scales where the particulate natures of matter and radiation are secondary to the behaviorof particles in bulk; scales where particles statistical as opposed to individual properties areimportant, and where matters inherent graininess can be smoothed over. In this book, weshall take a journey through spacetime and phase space, through statistical and continuummechanics (including solids, fluids, and plasmas), and through optics and relativity, bothspecial and general. In our journey, we shall seek to comprehend the fundamental lawsof classical physics in their own terms, and also in relation to quantum physics. Usingcarefully chosen examples, we shall show how the classical laws are applied to important,contemporary, 21st-century problems and to everyday phenomena, and we shall uncoversome deep connections among the various fundamental laws, and connections among thepractical techniques that are used in different subfields of physics.

Many of the most important recent developments in physicsand more generally inscience and engineeringinvolve classical subjects such as optics, fluids, plasmas, randomprocesses, and curved spacetime. Unfortunately, many physicists today have little under-standing of these subjects and their applications. Our goal, in writing this book, is to rectifythat. More specifically:

We believe that every masters-level or PhD physicist should be familiar with the basicconcepts of all the major branches of classical physics, and should have had someexperience in applying them to real-world phenomena; this book is designed to facilitatethat.

A large fraction of physics, astronomy and engineering graduate students in the UnitedStates and around the world use classical physics extensively in their research, and evenmore of them go on to careers in which classical physics is an essential component; thisbook is designed to facilitate that research and those careers.

Many professional physicists and engineers discover, in mid-career, that they need anunderstanding of areas of classical physics that they had not previously mastered. This

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book is designed to help them fill in the gaps, and to see the relationship of topics theystudy to already familiar topics.

In pursuit of these goals, we seek, in this book, to give the reader a clear understandingof the basic concepts and principles of classical physics. We present these principles inthe language of modern physics (not nineteenth century applied mathematics), and presentthem for physicists as distinct from mathematicians or engineers though we hope thatmathematicians and engineers will also find our presentation useful. As far as possible, weemphasize theory that involves general principles which extend well beyond the particularsubjects we study.

In this book, we also seek to teach the reader how to apply classical physics ideas. We doso by presenting contemporary applications from a variety of fields, such as

fundamental physics, experimental physics and applied physics,

astrophysics and cosmology,

geophysics, oceanography and meteorology,

biophysics and chemical physics,

engineering, optical science & technology, radio science & technology, and informationscience & technology.

Why is the range of applications so wide? Because we believe that physicists should haveat their disposal enough understanding of general principles to attack problems that arise inunfamiliar environments. In the modern era, a large fraction of physics students will go onto careers away from the core of fundamental physics. For such students, a broad exposureto non-core applications will be of great value. For those who wind up in the core, such anexposure is of value culturally, and also because ideas from other fields often turn out tohave impact back in the core of physics. Our examples will illustrate how basic concepts andproblem solving techniques are freely interchanged between disciplines.

Classical physics is defined as the physics where Plancks constant can be approximatedas zero. To a large extent, it is the body of physics for which the fundamental equationswere established prior to the development of quantum mechanics in the 1920s. Does thisimply that it should be studied in isolation from quantum mechanics? Our answer is, mostemphatically, No!. The reasons are simple:

First, quantum mechanics has primacy over classical physics: classical physics is anapproximation, often excellent, sometimes poor, to quantum mechanics. Second, in recentdecades many concepts and mathematical techniques developed for quantum mechanics havebeen imported into classical physics and used to enlarge our classical understanding andenhance our computational capability. An example that we shall discuss occurs in plasmaphysics, where nonlinearly interacting waves are treated as quanta (plasmons), despite thefact that they are solutions of classical field equations. Third, ideas developed initially forclassical problems are frequently adapted for application to avowedly quantum mechanicalsubjects; examples (not discussed in this book) are found in supersymmetric string theoryand in the liquid drop model of the atomic nucleus. Because of these intimate connections

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between quantum and classical physics, quantum physics will appear frequently in this book,in many ways.

The amount and variety of material covered in this book may seem overwhelming. If so,please keep in mind the key goals of the book: to teach the fundamental concepts, whichare not so extensive that they should overwhelm, and to illustrate those concepts. Our goalis not to provide a mastery of the many illustrative applications contained in the book, butrather to convey the spirit of how to apply the basic concepts of classical physics. To helpstudents and readers who feel overwhelmed, we have labeled as Track Two sections thatcan easily be skipped on a first reading, or skipped entirely but are sufficiently interestingthat many readers may choose to browse or study them. Track-Two sections are labeled

by the symbol T2 . To keep Track One manageable for a one-year course, the Track-Oneportion of each chapter is no longer than 40 pages (including many pages of exercises) andoften somewhat shorter.

This book will also seem much more manageable and less overwhelming when one realizesthat the same concepts and problem solving techniques appear over and over again, in avariety of different subjects and applications. These unifying concepts and techniques arelisted in outline form in Appendix B, along with the specific applications and section numbers

in this book, where they arise. The reader may also find Appendix A useful. It contains anoutline of the entire book based on concepts an outline complementary to the Table ofContents.

This book is divided into seven parts; see the Table of Contents:

I. Foundations which introduces a powerful geometric point of view on the laws ofphysics (a viewpoint that we shall use throughout this book), and brings readers up tospeed on some concepts and mathematical tools that we shall need. Many readers willalready have mastered most or all of the material in Part I, and may find that theycan understand most of the rest of the book without adopting our avowedly geometricviewpoint. Nevertheless, we encourage such readers to browse Part I, at least briefly,before moving onward, so as to become familiar with our viewpoint. It does have greatpower.

Part I is split into two chapters: Chap. 1 on Newtonian Physics; Chap. 2 on SpecialRelativity. Since the vast majority of Parts IIVI is Newtonian, readers may chooseto skip Chap. 2 and the occasional special relativity sections of subsequent chapters,until they are ready to launch into Part VII, General Relativity. Accordingly Chap. 2is labeled Track Two, though it becomes Track One when readers embark on Part VII.

II. Statistical physics including kinetic theory, statistical mechanics, statistical ther-modynamcs, and the theory of random processes. These subjects underly some por-tions of the rest of the book, especially plasma physics and fluid mechanics. Amongthe applications we study are the statistical-theory computation of macroscopic prop-erties of matter (equations of state, thermal and electric conductivity, viscosity, ...);phase transitions (boiling and condensation, melting and freezing, ...); the Ising modeland renormalization group; chemical and nuclear reactions, e.g. in nuclear reactors;Bose-Einstein condensates; Olbers Paradox in cosmology; the Greenhouse effect and

vits influence on the earths climate; noise and signal processing, the relationship be-tween information and entropy; entropy in the expanding universe; and the entropy ofblack holes.

III. Optics by which we mean classical waves of all sorts: light waves, radio waves,sound waves, water waves, waves in plasmas, and gravitational waves. The major con-cepts we develop for dealing with all these waves include geometric optics, diffraction,interference, and nonlinear wave-wave mixing. Some of the applications we will meetare gravitational lenses, caustics and catastrophes, Berrys phase, phase-contrast mi-croscopy, Fourier-transform spectroscopy, radio-telescope interferometry, gravitational-wave interferometers, holography, frequency doubling and phase conjugation in non-linear crystals, squeezed light, and how information is encoded on BDs, DVDs andCDs.

IV. Elasticity elastic deformations, both static and dynamic, of solids. Here some of ourapplications are bifurcations of equilibria and bifurcation-triggered instabilities, stress-polishing of mirrors, mountain folding, buckling, seismology and seismic tomography,and elasticity of DNA molecules.

V. Fluid Dynamics with the fluids including, for example, air, water, blood, andinterplanetary and interstellar gas. Among the fluid concepts we study are vorticity,turbulence, boundary layers, subsonic and supersonic flows, convection, sound waves,shock waves and magnetohydrodynamics. Among our applications are the flow ofblood through constricted vessels, the dynamics of a high-speed spinning baseball,how living things propel themselves, convection in stars, helioseismology, supernovae,nuclear explosions, sedimentation and nuclear winter, the excitation of ocean wavesby wind, salt fingers in the ocean, tornados and water spouts, the Sargasso Sea andthe Gulf Stream in the Atlantic Ocean, nonlinear waves in fluids (solitons and theirinteractions), stellerators, tokamaks, and controlled thermonuclear fusion.

VI. Plasma Physics with the plasmas including those in earth-bound laboratories andtechnological devices, the earths ionosphere, stellar interiors and coronae, and inter-planetary and interstellar space. In addition to magnetohydrodynamics (treated inPart V), we develop three other physical and mathematical descriptions of plasmas:kinetic theory, two-fluid formalism, and quasi-linear theory which we express in thequantum language of weakly coupled plasmons and particles. Among our plasma appli-cations are: some of the many types of waves (plasmons) that a plasma can supportboth linear waves and nonlinear (soliton) waves; the influence of the earths ionosphereon radio-wave propagation; the wide range of plasma instabilities that have plagued thedevelopment of controlled thermonuclear fusion; and wave-particle (plasmon-electronand plasmon-ion) interactions, including the two-stream instability for fast coronal elec-trons in the solar wind, isotropization of cosmic rays via scattering by magnetosonicwaves, and Landau damping of electrostatic waves.

VII. General Relativity the physics of curved spacetime, including the laws by whichmass-energy and momentum curve spacetime, and by which that curvature influences

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the motion of matter and inflluences the classical laws of physics (e.g., the laws of fluidmechanics, electromagntic fields, and optics). Here our applications include, amongothers, gravitational experiments on earth and in our solar system; relativistic starsand black holes, both spinning (Kerr) and nonspinning (Schwarzschild); the extrac-tion of spin energy from black holes; interactions of black holes with surrounding andinfalling matter; gravitational waves and their generation and detection; and the large-scale structure and evolution of the universe (cosmology), including the big bang, theinflationary era, and the modern era. Throughout, we emphasize the physical contentof general relativity and the connection of the theory to experiment and observation.

This books seven Parts are semi-independent of each other. It should be possible to readand teach the parts independently, if one is willing to dip into earlier parts occasionally, asneeded, to pick up an occasional concept, tool or result. We have tried to provide enoughcross references to make this possible.

Track One of the book has been designed for a full-year course at the first-year graduatelevel; and that is how we have used it, covering Part I in the first week, and then on averageone chapter per week thereafter. (Many fourth-year undergraduates have taken our coursesuccessfully, but not easily.)

Exercises are a major component of this book. There are five types of exercises:

1. Practice. Exercises that give practice at mathematical manipulations (e.g., of tensors).

2. Derivation. Exercises that fill in details of arguments or derivations which are skippedover in the text.

3. Example. Exercises that lead the reader step by step through the details of someimportant extension or application of the material in the text.

4. Problem. Exercises with few if any hints, in which the task of figuring out how to setthe calculation up and get started on it often is as difficult as doing the calculationitself.

5. Challenge. An especially difficult exercise whose solution may require that one readother books or articles as a foundation for getting started.

We urge readers to try working many of the exercises, and read and think about all ofthe Example exercises. The Examples should be regarded as continuations of the text; theycontain many of the most illuminating applications. We label with double stars, **, Exampleexercises that are especially important.

A few words on units: In this text we will be dealing with practical matters and willfrequently need to have a quantitative understanding of the magnitudes of various physicalquantities. This requires us to adopt a particular unit system. Students we teach are aboutequally divided in preferring cgs/Gaussian units or MKS/SI units. Both of these systemsprovide a complete and internally consistent set for all of physics and it is an often-debatedissue as to which is the more convenient or aesthetically appealing. We will not enter thisdebate! Ones choice of units should not matter and a mature physicist should be able to

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change from one system to another with only a little thought. However, when learning newconcepts, having to figure out where the 4pis go is a genuine impediment to progress. Oursolution to this problem is as follows: We shall use the units that seem most natural forthe topic at hand or those which, we judge, constitute the majority usage for the subculturethat the topic represents. We shall not pedantically convert cm to m or vice versa atevery juncture; we trust that the reader can easily make whatever translation is necessary.However, where the equations are actually different, for example in electromagnetic theory,we shall sometimes provide, in brackets or footnotes, the equivalent equations in the otherunit system and enough information for the reader to proceed in his or her preferred scheme.As an aid, we also give some unit-conversion information in Appendix C, and values ofphysical constants in Appendix D.

We have written this book in connection with a full-year course that we and others havetaught at Caltech nearly every year since the early 1980s. We conceived that course and thisbook in response to a general concern at Caltech that our PhD physics students were beingtrained too narrowly, without exposure to the basic concepts of classical physics beyondelectricity and magnetism, classical mechanics, and elementary thermodynamics. Coursesbased on parts of this book, in its preliminary form, have been taught by various physicists,not only at Caltech but also at a few other institutions in recent years, and since moving toStanford in 2003, Blandford has taught from it there. Many students who took our Caltechcourse, based on early versions of our book, have told us with enthusiasm how valuable itwas in their later careers. Some were even enthusiastic during the course.

Many generations of students and many colleagues have helped us hone the books presen-tation and its exercises through comments and criticisms, sometimes caustic, usually helpful;we thank them. Most especially:

For helpful advice about presentations and/or exercises in the book, and/or materialthat went into the book, we thank Professors Richard Blade, Yanbei Chen, Michael Cross,Steven Frautschi, Peter Goldreich, Steve Koonin, Sterl Phinney, David Politzer, and DavidStevenson at Caltech (all of whom taught portions of our Caltech course at one time oranother), and XXXXX [ROGER: WHO ELSE SHOULD WE BE LISTING?]

Over the years, we have received extremely valuable advice about this book from theteaching assistants in our course: XXXXXXX[KIP IS ASSEMBLING A LIST]XXXXXXXXWe are very indebted to them.

We hope that this book will trigger a significant broadening of the training of physicsgraduate students elsewhere in the world, as it has done at Caltech, and will be of wide useto mature physicists as well.

Roger D. Blandford and Kip S. ThorneStanford University and Caltech, December 2012

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CONTENTS

[For an alternative overview of this book, See Appendix A. Concept-Based Outline (doesnot exist yet)]

Preface

I. FOUNDATIONS1. Newtonian Physics: Geometric Viewpoint

1.1 Introduction

1.2 Foundational Concepts

1.3 Tensor Algebra Without a Coordinate System

Box: [T2] Vectors and tensors in quantum theory

1.4 Particle Kinetics and Lorentz Force in Geometric Language

1.5 Component Representation of Tensor Algebra

1.6 Orthogonal Transformations of Bases

1.7 Directional Derivatives, Gradients, Levi-Civita Tensor, Cross Product and Curl

**Examples: Rotation in x, y Plane; Vector identities for cross product and curl

1.8 Volumes, Integration, and Conservation Laws

1.9 The Stress Tensor and Conservation of Momentum

**Examples: Equations of motion for a perfect fluid; Electromagnetic stress tensor

1.10 Geometrized Units and Relativistic Particles for Newtonian Readers

2. [T2] Special Relativity: Geometric Viewpoint

2.1 Overview

2.2 Foundational Concepts

Boxes: Measuring the speed of light without light; Propagation speeds of other waves;

Proof of invariance of the interval for timelike separations

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2.3 Tensor Algebra Without a Coordinate System

2.4 Particle Kinetics and Lorentz Force Without a Reference Frame

**Examples: Frame-independent expressions for energy, momentum and velocity; 3-

metric as a projection tensor; Doppler shift derived without Lorentz transformations

2.5 Component Representation of Tensor Algebra

2.6 Particle Kinetics in Index Notation and in a Lorentz Frame

2.7 Lorentz Transformations

**Exercise: General boosts and rotations

2.8 Spacetime Diagrams for Boosts

2.9 Time Travel

2.10 Directional Derivatives, Gradients, Levi-Civita Tensor

2.11 Nature of Electric and Magnetic Fields; Maxwells Equations

2.12 Volumes, Integration and Conservation Laws

2.13 The Stress-Energy Tensor and Conservation of 4-Momentum

**Example: Stress-energy tensor and energy-momentum conservation for a perfect fluid

and for the electromagnetic field; Inertial mass per unit volume;

II. STATISTICAL PHYSICS

3. Kinetic Theory

3.1 Overview

3.2 Phase Space and Distribution Function: number density in phase space; distri-bution function for particles in a plasma; distribution function for photons; meanoccupation number

**Examples: [T2] Distribution function for particles with a range of rest masses;

Regimes of particulate and wave-like behaviorX-rays from Cygnus X-1 and gravi-

tational waves from a supernova

3.3 Thermal Equilibrium Distribution Functions

**Examples: Maxwell velocity distribution; [T2] Observations of cosmic microwave ra-

diation from earth

3.4 Macroscopic Properties of Matter as Integrals Over Momentum Space: Newto-nian particle density, flux and stress tensor; relativistic number-flux 4-vector andstress-energy tensor

x3.5 Isotropic Distribution Functions and Equations of State: density, pressure, en-ergy density, equation of state for nonrelativistic hydrogen gas, for relativisticdegenerate hydrogren gas, and for radiation

3.6 Evolution of the Distribution Function: Liouvilles Theorem, the CollisionlessBoltzmann Equation, and the Boltzmann Transport Equation

Box: [T2] Sophisticated derivation of relativistic collisionless Boltzmann equation

**Examples: Solar heating of the earththe Greenhouse effect; Olbers paradox and

solar furnace

3.7 Transport Coefficients: diffusive heat conduction inside a star, analyzed in orderof magnitude and via the Boltzmann transport equation

Box: Two lengthscale expansions

**Examples: Solution of diffusion equation in an infinte, homogeneous medium; Dif-

fusion equation for temperature; Viscosity of a monatomic gas; Neutron diffusion in a

nuclear reactor

4. Statistical Mechanics

4.1 Overview

4.2 Systems, Ensembles, and Distribution Functions

Box: [T2] Density operator and quantum statistical mechanics

4.3 Liouvilles Theorem and the Evolution of the Distribution Function

4.4 Statistical Equilibrium: canonical ensemble and distribution; general ensemble;Gibbs ensemble; grand canonical ensemble; Bose-Einstein and Fermi-Dirac dis-tributions; equipartition theorem

4.5 The Microcanonical Ensemble

4.6 The Ergodic Hypothesis

4.7 Entropy and the Evolution into Statistical Equilibrium: the second law of ther-modynamics; what causes entropy to increase?

Box: [T2] Entropy increase due to discarding quantum correlations

**Exercises: Entropy of a thermalized mode of a field; Entropy of mixing, indistin-

guishability of atoms, and the Gibbs paradox

4.8 Entropy Per Particle

**Exercise: Primordial element formation

4.9 Bose-Einstein Condensate

**Exercise: Onset of Bose-Einstein condensation; Discontinuous change of specific heat

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4.10 [T2] Statistical Mechanics in the Presence of Gravity: Galaxies, Black Holes, theUniverse, and Structure Formation in the Early Universe

4.11 [T2] Entropy and Information: information gained in measurements; informa-tion in communication theory; examples of information content; some propertiesof information; capacity of communication channels; erasing information fromcomputer memories

5. Statistical Thermodynamics

5.1 Overview

5.2 Microcanonical Ensemble and the Energy Representation of Thermodynamics

5.3 Grand Canonical Ensemble and the Grand Potential Representation of Thermo-dynamics: computation of thermodynamic properties; van der Waals gas

Box: [T2] Derivation of van der Waals grand potential

5.4 Canonical Ensemble and the Physical Free-Energy Representation of Thermody-namic; ideal gas with internal degrees of freedom

5.5 The Gibbs Representation of Thermodynamics; Phase Transitions and ChemicalReactions

**Exercises: Electron-positron equilibrium at low temperatures; Saha equation for ion-

ization equilibrium

5.6 Fluctuations away from Satistical Equilibrium

5.7 Van der Waals Gas: Volume Fluctuations and Gas-To-Liquid Phase Transition

**Exercise: Out-of-equilibrium Gibbs potential for water; surface tension and nucleation

5.8 [T2] Magnetic materials: Paramagnetism, Ising Model for Ferromagnetism,Renormalization Group, and Monte Carlo Methods

6. Random Processes

6.1 Overview

6.2 Fundamental Concepts: random variables and processes, probability distribu-tions, ergodic hypothesis

6.3 Markov Processes and Gaussian Processes; central limit theorem; random walk;Doobs theorem

**Exercises: Diffusion of a particle; Random walks

6.4 Correlation Functions and Spectral Densities; the Wiener-Khintchine theorem;light spectra; noise in a gravitational-wave detector

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**Exercise: Cosmological density fluctuations

6.5 [T2] Two-Dimensional Random Processes

6.6 Noise and its Types of Spectra; Noise in atomic Clocks; Information missing fromthe spectral density

6.7 Filtering Random Processes; Brownian motion and random walks; Extracting aweak Signal from noise; Band-pass filter; Signal to noise ratio; Shot noise

**Exercises: Wieners optimal filter; Allan variance for clock noise

6.8 Fluctuation-Dissipation Theorem; Langevin equation; Johnson noise in a resistor;Relaxation time for Brownian motion; [T2] Generalized fluctuation dissipationtheorem

**Exercises: Detectability of a sinusoidal force acting on an oscillator with thermal

noise; [T2] Standard quantum limit for minimum noise in a linear measuring device,

and how to evade it

6.9 Fokker-Planck Equation; Optical molasses (doppler cooling of atoms)

**Exercise: [T2] Solution of Fokker-Planck equation for thermal noise in an oscillator

III. OPTICS

7. Geometrical Optics

7.1 Overview

7.2 Waves in a Homogeneous Medium: monochromatic plane waves; dispersion rela-tion; wave packets; group and phase velocities

Applications to: (i) EM waves in isotropic, dielectric medium, (ii) sound waves in a

solid or fluid, (iii) waves on the surface of a deep body of water, (iv) flexural waves on

a stiff beam or rod, (v) Alfven waves in a magnetized plasma

**Exercise: Gaussian wave packet and its dispersion

7.3 Waves in an Inhomogeneous, Time-Varying Medium: The Eikonal approximation;geometric optics; relation to quantum theory; relation to wavepackets; breakdownof geometric optics; Fermats principle

Box: Bookkeeping parameter in two-lengthscale expansions

**Exercises for dispersionless waves: Amplitude propagation; energy density, energy

flux, and adiabatic invariant

**Exercise: Geometric optics for Schrodinger equation; Hamilton-Jacobi theory

Applications: sound waves in wind; spherical scalar waves; flexural waves; Alfven waves;

light through a lens; self-focusing optical fibers

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7.4 Paraxial Optics

Applications: lenses and mirrors, telescope, microscope, optical cavity, converging mag-

netic lens for charged particle beam

7.5 Catastrophe Optics: Multiple Images; Formation of Caustics and their Properties

Applications: Imperfect lens, sunlights caustic patterns on the bottom of a swimming

pool and through a water glass

**Exercises: Catastrophe theory the five elementary catastrophes applied to optical

caustics and to the van der Waals phase change

7.6 [T2] Gravitational Lenses: Refractive index model; Multiple Images and Caustics

[T2] Applications: Lensing by a point mass; lensing of a quasar by a galaxy

7.7 Polarization: its geometric-optics propagation

[T2] Application: The Geometric Phase

8. Diffraction

8.1 Overview

8.2 Helmholtz-Kirchhoff Integral: diffraction by an aperture; spreading of the wave-front

8.3 Fraunhofer Diffraction: diffraction grating; Airy pattern; Hubble space telescope;Babinets principle

Other Applications: Pointilist paintings; light scattering by large opaque particle; mea-

suring thickness of human hair via diffraction

8.4 Fresnel Diffraction: Fresnel integrals and Cornu spiral; lunar occultation of aradio source; circular apertures, Fresnel zones and zone plates

Other Applications: seeing (stellar scintillation) in the atmosphere; multiconjugate

adaptive optics; spy satellites

8.5 Paraxial Fourier Optics: coherent illumination; point spread functions; Abbetheory of image formation by a thin lens; spatial filtering in the focal plane of alens; Gaussian beams; gravitational-wave interferometers

Other applications of paraxial Fourier optics high-pass filter to clean a laser beam;

low-pass filter to enhance sharp features in an image; notch filter to remove pixellations

from an image; phase contrast microscopy; spatial pattern recognition; convolution;

Gaussian beams in an optical fiber; scattered-light noise in LIGO

**Example: Transmission electron microscope

8.6 Diffraction at Caustics; scaling laws

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9. Interference

9.1 Overview

9.2 Coherence: Youngs slits; extended source; van Cittert-Zernike theorem; spatiallateral coherence; 2-dimensional coherence; Michelson stellar interferometer andatmospheric seeing; temporal coherence; Michelson interferometer; Fourier trans-form spectroscopy; degree of coherence

**Example: Complex random processes and van Cittert-Zernike theorem

9.3 Radio Telescopes: two-element interferometer; multiple element interferometer;closure phase; angular resolution

9.4 Etalons and Fabry-Perot Interferometers: multiple-beam interferometry; modesof a Fabry-Perot cavity

Fabry-Perot applications: spectrometer, laser, mode-cleaning cavity, beam-shaping cav-

ity, PDH laser stabilization, optical frequency comb

**Examples: transmission and reflection coefficients; reciprocity relations; antireflection

coating; Sagnac interferometer

9.5 [T2] Laser Interferometer Gravitational Wave Detectors

9.6 Power Correlations and Photon Statistics: Hanbury Brown & Twiss IntensityInterferometer

10. Nonlinear Optics

10.1 Overview

10.2 Lasers: Basic Principles; Types of Lasers their performances, and applications;details of Ti:Sapp mode-locked laser

10.3 Holography: recording a hologram and reconstructing a 3D image from it

Other types of holography and applications: phase holography, volume holography, re-

flection holography, white-light holography, computational holograms, full-color holog-

raphy, holographic interferometry, holographic lenses

**Problem: CDs, DVDs and Blu Ray disks

10.4 Phase-Conjugate Optics

10.5 Maxwells Equations in a Nonlinear Medium: nonlinear dielectric susceptibilities;electro-optic effects

Box: properties of some anisotropic, nonlinear crystals

10.6 Three-Wave Mixing in Anisotropic, Nonlinear Crystals: resonance conditions;evolution equations in a medium that is isotropic at linear order; three-wavemixing in a birefringent crystal: phase matching and evolution equations

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10.7 Applications of Three-Wave Mixing: Frequency doubling; optical parametric am-plification; squeezed light

**Example: frequency doubling in a green laser pointer; qqueezing by children who

swing; squeezed states of light

10.8 Four-wave mixing in isotropic media: third-order susceptibilities and fieldstrengths; phase conjugation via four-wave mixing in Cs2; optical Kerr effectand four-wave mixing in an optical fiber.

Table: materials used in four-wave mixing; **Problems: spontaneous oscillation in four-

wave mixing; squeezed light produced by phase conjugation

IV. ELASTICITY

11. Elastostatics

11.1 Overview

11.2 Displacement and Strain; expansion, rotation, and shear

11.3 Stress and Elastic Moduli: stress tensor; elastostatic stress balance; energy ofdeformation; molecular origin of elastic stress

11.4 Youngs Modulus and Poisson Ratio for an Isotropic Material

11.5 [T2] Cylindrical and Spherical Coordinates: connection coefficients and compo-nents of strain

11.6 [T2] Solving the 3-Dimensional Elastostatic Equations in Cylindrical Coordinates:simple methodspipe fracture and torsion pendulum; separation of variables andGreens functionsthermoelastic noise in a LIGO mirror

11.7 Reducing the Elastostatic Equations to One Dimension for a Bent Beam: can-tilever bridges; elastica

11.8 Bifurcation of Equilibria; Buckling and Mountain Folding

11.9 [T2] Reducing the Elastostatic Equations to Two Dimensions for a Deformed ThisPlate: stress-polishing a telescope mirror

12. Elastodynamics

12.1 Overview

12.2 Conservation Laws

12.3 Basic Equations of Elastodynamics: equation of motion; elastodynamic waves;longitudinal sound waves; transverse shear waves; energy of elastodynamic waves

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12.4 Waves in Rods, Strings and Beams: compression waves; torsion waves; waves onstrings; flexural waves on a beam; bifurcation of equilibria and buckling (oncemore)

12.5 Body and Surface Waves Seismology: body waves; edge waves; Greens func-tion for a homogeneous half space; free oscillations of solid bodies; seismic to-mography

12.6 The Relationship of Classical Waves to Quantum Mechanical Excitations

V. FLUID DYNAMICS

13. Foundations of Fluid Dynamics

13.1 Overview

13.2 The Macroscopic Nature of a Fluid: Density, Pressure, Flow Velocity; Fluids vs.gases

13.3 Hydrostatics: Archimedes law; stars and planets; rotating fluids

13.4 Conservation Laws

13.5 The Dynamics of an Ideal Fluid: mass conservation; momentum conservation;Euler equation; Bernoulli theorem; conservation of energy; Joule-Kelvin cooling

13.6 Incompressible Flows

13.7 Viscous Flows with Heat Conduction: decomposition of the velocity gradient intoexpansion, vorticity, and shear; Navier-Stokes equation; energy conservation andentropy production; molecular origin of viscosity; Reynolds number; pipe flow

13.8 [T2] Relativistic Dynamics of an Ideal Fluid: stress-energy tensor and equationsof relativistic fluid mechanics; relativistic Bernoulli equation and ultrarelativisticastrophysical jets; nonrelativistic limt

14. Vorticity

14.1 Overview

14.2 Vorticity and Circulation: vorticity transport; vortex lines; tornados; Kelvinstheorem; diffusion of vortex lines; sources of vorticity

14.3 Low Reynolds Number Flow Stokes flow: sedimentation; nuclear winter

14.4 High Reynolds Number Flow Laminar Boundary Layers: similarity solution;vorticity profile; separation

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14.5 Nearly Rigidly Rotating Flow Earths Atmosphere and Oceans: fluid dynam-ics in a rotating reference frame; geostrophic flows; Taylor-Proudman theorem;Ekman boundary layers

14.6 [T2] Instabilities of Shear Flows Billow Clouds, Turbulence in the Stratosphere:discontinuous flow, Kelvin-Helmholtz instability; discontinuous flow with gravity;smoothly stratified flows, Rayleigh and Richardson criteria for instability

15. Turbulence

15.1 Overview

15.2 The Transition to Turbulence Flow past a Cylinder

15.3 Empirical Description of Turbulence: the role of vorticity in turbulence

15.4 Semi-Quantitative Analysis of Turbulence: weak turbulence; turbulent viscos-ity; turbulent wakes and jets; entrainment and the Coanda effect; Kolmogorovspectrum

15.5 Turbulent Boundary Layers: profile of a turbulent boundary layer; instability ofa laminar boundary layer; the flight of a ball

15.6 The Route to Turbulence Onset of Chaos: Couette flow; Feigenbaum sequenceand onset of turbulence in convection

16. Waves

16.1 Overview

16.2 Gravity Waves on Surface of a Fluid: deep water waves; shallow water waves;surface tension; capillary waves; tsunamis; helioseismology

16.3 Nonlinear Shallow Water Waves and Solitons: Korteweg-deVries equation; phys-ical effects in the kdV equation; single soliton solution; two soliton solution;solitons in contemporary physics

16.4 Rossby Waves in a Rotating Fluid

16.5 Sound Waves: wave energy; sound generation; [T2] radiation reaction, runawaysolutions and matched asymptotic expansions

17. Compressible and Supersonic Flow

17.1 Overview

17.2 Equations of Compressible Flow

17.3 Stationary, Irrotational Flow: quasi-one-dimensional flow; setting up a stationarytransonic flow; rocket engines

17.4 One Dimensional, Time-Dependent Flow: Riemann invariants; shock tube

xviii

17.5 Shock Fronts: shock jump conditions; Rankine-Hugoniot relations; internal struc-ture of a shock; jump conditions in perfect gas with constant ; Mach cone

17.6 Self-Similar Solutions Sedov-Taylor Blast Wave: atomic bomb; supernovae

18. Convection

18.1 Overview

18.2 [T2] Diffusive Heat Conduction: cooling a nuclear reactor; thermal boundarylayers

18.3 [T2] Boussinesq Approximation

18.4 [T2] Rayleigh-Benard Convection: mantle convection and continental drift

18.5 Convection in Stars

18.6 [T2] Double Diffusion: salt fingers

19. Magnetohydrodynamics

19.1 Overview

19.2 Basic Equations of MHD: Maxwells equations in MHD approximation; momen-tum and energy conservation; boundary conditions; magnetic field and vorticity

19.3 Magnetostatic Equilibria: controlled thermonuclear fusion; Z pinch; pinch; toka-mak

19.4 Hydromagnetic Flows: electromagnetic brake; MHD power generator; flow meter;electromagnetic pump; Hartmann flow

19.5 Stability of Hydromagnetic Equilibria: linear perturbation theory; Z pinch sausage and kink instabilities; energy principle

19.6 Dynamos and Magnetic Field Line Reconnection: Cowlings theorem; kinematicdynamos; magnetic reconnection

19.7 Magnetosonic Waves and the Scattering of Cosmic Rays

VI. PLASMA PHYSICS

20. The Particle Kinetics of Plasmas

20.1 Overview

20.2 Examples of Plasmas and their Density-Temperature Regimes: ionization bound-ary; degeneracy boundary; relativistic boundary; pair production boundary; ex-amples of natural and man-made plasmas

xix

20.3 Collective Effects in Plasmas: Debye shielding; collective behavior; plasma oscil-lations and plasma frequency

20.4 Coulomb Collisions: collision frequency; Coulomb logarithm; thermal equilibra-tion times

20.5 Transport Coefficients: anomalous resistivity and anomalous equilibration

20.6 Magnetic field: Cyclotron frequency and Larmor radius; validity of the fluidapproximation; conductivity tensor

20.7 Adiabatic invariants: homogeneous time-independent electric and magnetic fields;inhomogeneous time-independent magnetic field; a slowly time-varying magneticfield

21. Waves in Cold Plasmas: Two-Fluid Formalism

21.1 Overview

21.2 Dielectric Tensor, Wave Equation, and General Dispersion Relation

21.3 Two-Fluid Formalism

21.4 Wave Modes in an Unmagnetized Plasma: dielectric tensor and dispersion rela-tion for a cold plasma; electromagnetic plasma waves; Langmuir waves and ionacoustic waves in a warm plasma; cutoffs and resonances

21.5 Wave Modes in a Cold, Magnetized Plasma: dielectric tensor and dispersionrelation; parallel propagation; perpendicular propagation

21.6 Propagation of Radio Waves in the Ionosphere

21.7 CMA Diagram for Wave Modes in Cold, Magnetized Plasma

21.8 Two-Stream Instability

22. Kinetic Theory of Warm Plasmas

22.1 Overview

22.2 Basic Concepts of Kinetic Theory and its Relationship to Two-Fluid Theory:distribution function and Vlasov equation; Jeans theorem

22.3 Electrostatic Waves in an Unmagnetized Plasma and Landau Damping: formaldispersion relation; two-stream instability; the Landau contour; dispersion re-lation for weakly damped or growing waves; Langmuir waves and their Landaudamping; ion acoustic waves and conditions for their Landau damping to be weak

22.4 Stability of Electromagnetic Waves in an Unmagnetized Plasma

22.5 Particle Trapping

xx

22.6 [T2] N-Particle Distribution Function: BBKGY hierarchy, two-point correlationfunction, Coulomb correction to plasma pressure

23. Nonlinear Dynamics of Plasmas

23.1 Overview

23.2 Quasi-Linear Theory in Classical Language: classical derivation of the theory;summary of the theory; conservation laws; generalization to three dimensions

23.3 Quasilinear Theory in Quantum Mechanical Language: plasmon occupation num-ber ; evolution of plasmons via interaction with electrons; evolution electronsvia interaction with plasmons; emission of plasmons by particles in presence ofa magnetic field; relationship between classical and quantum formalisms; three-wave moxing

23.4 Quasilinear Evolution of Unstable Distribution Function The Bump in Tail:instability of streaming cosmic rays

23.5 Parametric Instabilities

23.6 Solitons and Collisionless Shock Waves

VII. GENERAL RELATIVITY

24. From Special to General Relativity

24.1 Overview

24.2 Special Relativity Once Again: geometric, frame-independent formulation; iner-tial frames and components of vectors, tensors and physical laws; light speed, theinterval, and spacetime diagrams

24.3 Differential Geometry in General Bases and in Curved Manifolds: nonorthonor-mal bases; vectors as differential operators; tangent space; commutators; differ-entiation of vectors and tensors; connection coefficients; integration

24.4 Stress-Energy Tensor Revisited

24.5 Proper Reference Frame of an Accelerated Observer: relation to inertial co-ordinates; metric in proper reference frame; transport law for rotating vec-tors; geodesic equation for freely falling particle; uniformly accelerated observer;Rindler coordinates for Minkowski spacetime

25. Fundamental Concepts of General Relativity

25.1 Overview

xxi

25.2 Local Lorentz Frames, the Principle of Relativity, and Einsteins EquivalencePrinciple

25.3 The Spacetime Metric, and Gravity as a Curvature of Spacetime

25.4 Free-fall Motion and Geodesics of Spacetime

25.5 Relative Acceleration, Tidal Gravity, and Spacetime Curvature: Newtonian de-scription of tidal gravity; relativistic description; comparison of descriptions

25.6 Properties of the Riemann curvature tensor

25.7 Curvature Coupling Delicacies in the Equivalence Principle, and some Non-gravitational Laws of Physics in Curved Spacetime

25.8 The Einstein Field Equation

25.9 Weak Gravitational Fields: Newtonian limit of general relativity; linearized the-ory; gravitational field outside a stationary, linearized source; conservation lawsfor mass, momentum and angular momentum; tidal and frame-drag fields

26. Relativistic Stars and Black Holes

26.1 Overview

26.2 Schwarzschilds Spacetime Geometry

26.3 Static Stars: Birkhoffs theorem; stellar interior; local energy and momentumconservation; Einstein field equation; stellar models and their properties; embed-ding diagrams

26.4 Gravitational Implosion of a Star to Form a Black Hole: tidal forces at thegravitational radius; stellar implosion in Eddington-Finkelstein coordinates; tidalforces at r = 0 the central singularity; Schwarschild black hole

26.5 Spinning Black Holes: the Kerr metric for a spinning black hole; dragging ofinertial frames; light-cone structure and the horizon; evolution of black holes rotational energy and its extraction; [T2] tendex and vortex lines

26.6 The Many-Fingered Nature of Time

27. Gravitational Waves and Experimental Tests of General Relativity

27.1 Overview

27.2 Experimental Tests of General Relativity: equivalence principle, gravitationalredshift, and global positioning system; perihelion advance of Mercury; gravita-tional deflection of light, Fermats principle and gravitational lenses; Shapiro timedelay; frame dragging and Gravity Probe B; binary pulsar

xxii

27.3 Gravitational Waves Propagating Through Flat Spacetime: weak plane waves inlinearized theory; measuring a gravitational wave by its tidal forces; tendex andvortex lines for a gravitational wave; gravitons and their spin and rest mass

27.4 Gravitational Waves Propagating Through Curved Spacetime: gravitationalwave equation in curved spacetime; geometric-optics propagation of gravitationalwaves; energy and momentum in gravitational waves

27.5 The Generation of Gravitational Waves: multipole-moment expansion; quadru-pole moment formalism; quadrupolar wave strength, energy, angular momen-tum and radiation reaction; gravitational waves from a binary star system; [T2]gravitational waves from binaries made of black holes and/or neutron stars numerical relativity

27.6 The Detection of Gravitational Waves: frequency bands and detection techniques;gravitational-wave interfereomters: overview and elementary treatment; [T2] in-terferometer analyzed in TT gauge; [T2] interferometer analyzed in proper refer-ence frame of beam splitter; [T2] realistic interferometers

28. Cosmology

28.1 Overview

28.2 Homogeneity and Isotropy of the Universe Robertson-Walker line element

28.3 The Stress-energy Tensor and the Einstein Field Equation

28.4 Evolution of the Universe: constituents of the universe cold matter, radiation,and dark energy; the vacuum stress-energy tensor; evolution of the densities;evolution in time and redshift; physical processes in the expanding universe

28.5 Observational Cosmology: parameters characterizing the universe; local Lorentzframe of homogenous observers near Earth; Hubble expansion rate; primordialnucleosynthesis; density of cold dark matter; radiation temperature and density;anisotropy of the CMB: measurements of the Doppler peaks; age of the universe constraint on the dark energy; magnitude-redshift relation for type Ia supernovae confirmation that the universe is accelerating

28.6 The Big-Bang Singularity, Quantum Gravity and the Intial Conditions of theUniverse

xxiii

28.7 Inflationary Cosmology: amplification of primordial gravitational waves by infla-tion; search for primordial gravitational waves by their influence on the CMB;probing the inflationary expansion rate

APPENDICES

Appendix A: Concept-Based Outline of this Book

Appendix B: Unifying Concepts

Appendix C: Units

Appendix D: Values of Physical Constants

Contents

I FOUNDATIONS ii

1 Newtonian Physics: Geometric Viewpoint 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.1 The Geometric Viewpoint on the Laws of Physics . . . . . . . . . . . 21.1.2 Purposes of this Chapter . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.3 Overview of This Chapter . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Foundational Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Tensor Algebra Without a Coordinate System . . . . . . . . . . . . . . . . . 51.4 Particle Kinetics and Lorentz Force in Geometric Language . . . . . . . . . . 81.5 Component Representation of Tensor Algebra . . . . . . . . . . . . . . . . . 9

1.5.1 Slot-Naming Index Notation . . . . . . . . . . . . . . . . . . . . . . . 101.5.2 Particle Kinetics in Index Notation . . . . . . . . . . . . . . . . . . . 12

1.6 Orthogonal Transformations of Bases . . . . . . . . . . . . . . . . . . . . . . 131.7 Directional Derivatives, Gradients, Levi-Civita Tensor, Cross Product and Curl 151.8 Volumes, Integration and Integral Conservation Laws . . . . . . . . . . . . . 191.9 The Stress Tensor and Conservation of Momentum . . . . . . . . . . . . . . 221.10 Geometrized Units and Relativistic Particles for Newtonian Readers . . . . . 26

i

Part I

FOUNDATIONS

ii

iii

In this book, a central theme will be a Geometric Principle: The laws of physics mustall be expressible as geometric (coordinate-independent and reference-frame-independent) re-

lationships between geometric objects (scalars, vectors, tensors, ...), which represent physical

entitities.

There are three different conceptual frameworks for the classical laws of physics, andcorrespondingly three different geometric arenas for the laws; see Fig. 1. General Relativityis the most accurate classical framework; it formulates the laws as geometric relationshipsbetween geometric objects in the arena of curved 4-dimensional spacetime. Special Relativityis the limit of general relativity in the complete absence of gravity; its arena is flat, 4-dimensional Minkowski spacetime1. Newtonian Physics is the limit of general relativity when(i) gravity is weak but not necessarily absent, (ii) relative speeds of particles and materials aresmall compared to the speed of light c, and (iii) all stresses (pressures) are small compared tothe total density of mass-energy; its arena is flat, 3-dimensional Euclidean space with timeseparated off and made universal (by contrast with relativitys reference-frame-dependenttime).

In Parts IIVI of this book (statistical physics, optics, elasticity theory, fluid mechanics,plasma physics) we shall confine ourselves to the Newtonian formulations of the laws (plusspecial relativistic formulations in portions of Track 2), and accordingly our arena will beflat Euclidean space (plus flat Minkowski spacetime in portions of Track 2). In Part VII, weshall extend many of the laws we have studied into the domain of strong gravity (generalrelativity), i.e., the arena of curved spacetime.

In Parts II and III (statistical physics and optics), in addition to confining ourselves toflat space (plus flat spacetime in Track 2), we shall avoid any sophisticated use of curvilinearcoordinates. Correspondingly, when using coordinates in nontrivial ways, we shall confineourselves to Cartesian coordinates in Euclidean space (and Lorentz coordinates in Minkowskispacetime).

Part I of this book contains just two chapters. Chapter 1 is an introduction to our ge-ometric viewpoint on Newtonian physics, and to all the geometric mathematical tools that

1so-called because it was Hermann Minkowski (1908) who identified the special relativistic invariantinterval as defining a metric in spacetime, and who elucidated the resulting geometry of flat spacetime.

Special Relativity Classical Physics in the absence of gravity

Arena: Flat, Minkowski spacetime

vanishinggravity

General Relativity The most accurate framework for Classical Physics

Arena: Curved spacetime

weak gravitysmall speedssmall stresses

Newtonian Physics Approximation to relativistic physics Arena: Flat, Euclidean 3-space, plus universal time

low speedssmall stresses

add weak gravity

Fig. 1: The three frameworks and arenas for the classical laws of physics, and their relationship toeach other.

iv

we shall need in Parts part:StatisticalPhysics and III for Newtonian Physics in its arena, 3-

dimensional Euclidean space. Chapter 2 introduces our geometric viewpoint on Special Rela-

tivistic Physics, and extends our geometric tools into special relativitys arena, flat Minkowski

spacetime. Readers whose focus is Newtonian Physics will have no need for Chap. 2; and ifthey are already familiar with the material in Chap. 1 but not from our geometric viewpoint,they can successfully study Parts IIVI without reading Chap. 1. However, in doing so, theywill miss some deep insights; so we recommend they at least browse Chap. 1 to get somesense of our viewpoint, then return to Chap. 1 occasionally, as needed, when encounteringan unfamiliar geometric argument.

In Parts IV, V, and VI, when studying elasticity theory, fluid mechanics, and plasmaphysics, we will use curvilinear coordinates in nontrivial ways. As a foundation for this,at the beginning of Part IV we will extend our flat-space geometric tools to curvilinearcoordinate systems (e.g. cylindrical and spherical coordinates). Finally, at the beginning ofPart VII, we shall extend our geometric tools to the arena of curved spacetime.

Throughout this book, we shall pay close attention to the relationship between classicalphysics and quantum physics. Indeed, we shall often find it powerful to use quantum me-chanical language or formalism when discussing and analyzing classical phenomena. Thisquantum power in classical domains arises from the fact that quantum physics is primaryand classical physics is secondary. Classical physics arises from quantum physics, not con-versely. The relationship between quantum frameworks and arenas for the laws of physics,and classical frameworks, is sketched in Fig. 2.

Quantum Gravity(string theory?)

General RelativityQuantum Field Theory in Curved Spacetime

Quantum Field Theory in Flat Spacetime

NonrelativisticQuantum Mechanics

Special Relativity

Newtonian Physics

classicalgravity

nogravity

low speeds,particles notcreated or destroyed

classical limit

classical limit

classical limit

nogravity

low speeds,small stresses,

add weak gravity

classical limit

Fig. 2: The relationship of the three frameworks for classical physics (on right) to four frameworksfor quantum physics (on left). Each arrow indicates an approximation. All other frameworks areapproximations to the ultimate laws of quantum gravity (whatever they may be perhaps a variantof string theory).

Chapter 1

Newtonian Physics: Geometric

Viewpoint

Version 1201.1.K, 7 September 2012

Please send comments, suggestions, and errata via email to kip@caltech.edu, or on paper to

Kip Thorne, 350-17 Caltech, Pasadena CA 91125

Box 1.1

Readers Guide

This chapter is a foundation for almost all of this book. Many readers already know the material in this chapter, but from a viewpointdifferent from our geometric one. Such readers will be able to understand almostall of Parts IIVI of this book without learning our viewpoint. Nevertheless, thatgeometric viewpoint has such power that we encourage them to learn it, e.g., bybrowsing this chapter and focusing especially on Secs. 1.1.1, 1.2, 1.3, 1.5, 1.7, and1.8.

The stress tensor, introduced and discussed in Sec. 1.9 will play an important rolein Kinetic Theory (Chap. 3), and a crucial role in Elasticity Theory (Part IV),Fluid Mechanics (Part V) and Plasma Physics (Part VI).

The integral and differential conservation laws derived and discussed in Secs. 1.8and 1.9 will play major roles throughout this book.

The Box labeled T2 is advanced material (Track 2) that can be skipped in atime-limited course or on a first reading of this book.

1

21.1 Introduction

1.1.1 The Geometric Viewpoint on the Laws of Physics

In this book, we shall adopt a different viewpoint on the laws of physics than that in mostelementary and intermediate texts. In most textbooks, physical laws are expressed in termsof quantities (locations in space, momenta of particles, etc.) that are measured in somecoordinate system. For example, Newtonian vectorial quantities are expressed as tripletsof numbers, e.g., p = (px, py, pz) = (1, 9,4), representing the components of a particlesmomentum on the axes of a Cartesian coordinate system; and tensors are expressed as arraysof numbers, e.g.

I =

Ixx Ixy IxzIyx Iyy Iyz

Izx Izy Izz

(1.1)

for the moment of inertia tensor.By contrast, in this book, we shall express all physical quantities and laws in a geometric

form, i.e. a form that is independent of any coordinate system or basis vectors.For example, a particles velocity v and the electric and magnetic fields E and B that itencounters will be vectors described as arrows that live in the 3-dimensional, flat Euclideanspace of everyday experience. They require no coordinate system or basis vectors for theirexistence or descriptionthough often coordinates will be useful.

We shall insist that the Newtonian laws of physics all obey a Geometric Principle: theyare all geometric relationships between geometric objects (primarily scalars, vectors andtensors), expressible without the aid of any coordinates or bases. An example is the Lorentzforce law mdv/dt = q(E+ vB) a (coordinate-free) relationship between the geometric(coordinate-independent) vectors v, E and B and the particles scalar mass m and chargeq. As another example, a bodys moment of inertia tensor I can be viewed as a vector-valued linear function of vectors (a coordinate-independent, basis-independent geometricobject). Insert into the tensor I the bodys angular velocity vector and you will get outthe bodys angular momentum vector: J = I(). No coordinates or basis vectors are neededfor this law of physics, nor is any description of I as a matrix-like entity with componentsIij . Components are secondary; they only exist after one has chosen a set of basis vectors.Components (we claim) are an impediment to a clear and deep understanding of the laws ofphysics. The coordinate-free, component-free description is deeper, andonce one becomesaccustomed to itmuch more clear and understandable.

By adopting this geometric viewpoint, we shall gain great conceptual power, and oftenalso computational power. For example, when we ignore experiment and simply ask whatforms the laws of physics can possibly take (what forms are allowed by the requirement thatthe laws be geometric), we shall find that there is remarkably little freedom. Coordinateindependence and basis independence strongly constrain the laws of physics.1

1Examples are the equation of elastodynamics (12.4b) and the Navier-Stokes equation of fluid mechanics(13.69), which are both dictated by momentum conservation plus the form of the stress tensor [Eqs. (11.18)and (13.68)] forms that are dictated by the irreducible tensorial parts (Box 11.2) of the strain and rateof strain.

3This power, together with the elegance of the geometric formulation, suggests that insome deep sense, Natures physical laws are geometric and have nothing whatsoever to dowith coordinates or components or vector bases.

1.1.2 Purposes of this Chapter

The principal purpose of this foundational chapter is to teach the reader this geometric view-

point.

The mathematical foundation for our geometric viewpoint is differential geometry (alsocalled tensor analysis by physicists). Differential geometry can be thought of as an exten-sion of the vector analysis with which all readers should be familiar. A second purpose ofthis chapter is to develop key parts of differential geometry in a simple form well adapted to

Newtonian classical physics.

1.1.3 Overview of This Chapter

In this chapter, we lay the geometric foundations for the Newtonian laws of physics inflat Euclidean space. We begin in Sec. 1.2 by introducing some foundational geometricconcepts: points, scalars, vectors, inner products of vectors, distance between points. Thenin Sec. 1.3, we introduce the concept of a tensor as a linear function of vectors, and wedevelop a number of geometric tools: the tools of coordinate-free tensor algebra. In Sec. 1.4,we illustrate our tensor-algebra tools by using them to describewithout any coordinatesystemthe kinematics of a charged point particle that moves through Euclidean space,driven by electric and magnetic forces.

In Sec. 1.5, we introduce, for the first time, Cartesian coordinate systems and their basisvectors, and also the components of vectors and tensors on those basis vectors; and weexplore how to express geometric relationships in the language of components. In Sec. 1.6,we deduce how the components of vectors and tensors transform, when one rotates onesCartesian coordinate axes. (These are the transformation laws that most physics textbooksuse to define vectors and tensors.)

In Sec. 1.7, we introduce directional derivatives and gradients of vectors and tensors,thereby moving from tensor algebra to true differential geometry (in Euclidean space). Wealso introduce the Levi-Civita tensor and use it to define curls and cross products, and welearn how to use index gymnastics to derive, quickly, formulae for multiple cross products.In Sec. 1.8, we use the Levi-Civita tensor to define vectorial areas and scalar volumes,and integration over surfaces. These concepts then enable us to formulate, in geometric,coordinate-free ways, integral and differential conservation laws. In Sec. 1.9 we discuss,in particular, the law of momentum conservation, formulating it in a geometric way withthe aid of a geometric object called the stress tensor. As important examples, we use thisgeometric conservation law to derive and discuss the equations of Newtonian fluid dynamics,and the interaction between a charged medium and an electromagnetic field. We concludein Sec. 1.10 with some concepts from special relativity that we shall need in our discussionsof Newtonian physics.

41.2 Foundational Concepts

The arena for the Newtonian laws is a spacetime composed of the familiar 3-dimensionalEuclidean space of everyday experience (which we shall call 3-space), and a universal timet. We shall denote points (locations) in 3-space by capital script letters such as P and Q.These points and the 3-space in which they live require no coordinates for their definition.

A scalar is a single number. We are interested in scalars that represent physical quantities,e.g., temperature T . These are generally real numbers, and when they are functions oflocation P in space, e.g. T (P), we call them scalar fields.

A vector in Euclidean 3-space can be thought of as a straight arrow that reaches fromone point, P, to another, Q (e.g., the arrow x of Fig. 1.1a). Equivalently, x can bethought of as a direction at P and a number, the vectors length. Sometimes we shall selectone point O in 3-space as an origin and identify all other points, say Q and P, by theirvectorial separations xQ and xP from that origin.

The Euclidean distance between two points P and Q in 3-space can be measuredwith a ruler and so, of course, requires no coordinate system for its definition. (If one doeshave a Cartesian coordinate system, then can be computed by the Pythagorean formula,a precursor to the invariant interval of flat spacetime, Sec. 2.2.2.) This distance is alsothe length |x| of the vector x that reaches from P to Q, and the square of that length isdenoted

|x|2 (x)2 ()2 . (1.2)Of particular importance is the case when P and Q are neighboring points and x is a

differential (infinitesimal) quantity dx. By traveling along a sequence of such dxs, layingthem down tail-at-tip, one after another, we can map out a curve to which these dxs aretangent (Fig. 1.1b). The curve is P(), with a parameter along the curve; and theinfinitesimal vectors that map it out are dx = (dP/d)d.

The product of a scalar with a vector is still a vector; so if we take the change of locationdx of a particular element of a fluid during a (universal) time interval dt, and multiply it by1/dt, we obtain a new vector, the fluid elements velocity v = dx/dt, at the fluid elementslocation P. Performing this operation at every point P in the fluid defines the velocity fieldv(P). Similarly, the sum (or difference) of two vectors is also a vector and so taking thedifference of two velocity measurements at times separated by dt and multiplying by 1/dtgenerates the acceleration a = dv/dt. Multiplying by the fluid elements (scalar) mass m

P

Q

P

Q

x

x

x OC

(a) (b)

Fig. 1.1: (a) A Euclidean 3-space diagram depicting two points P and Q, their vectorial separationsxP and xQ from the (arbitrarily chosen) origin O, and the vector x = xQ xP connecting them.(b) A curve C generated by laying out a sequence of infinitesimal vectors, tail-to-tip.

5gives the force F = ma that produced the acceleration; dividing an electrically producedforce by the fluid elements charge q gives another vector, the electric field E = F/q, andso on. We can define inner products [Eq. (1.4a) below] of pairs of vectors at a point (e.g.,force and displacement) to obtain a new scalar (e.g., work), and cross products [Eq. (1.22a)]of vectors to obtain a new vector (e.g., torque). By examining how a differentiable scalarfield changes from point to point, we can define its gradient [Eq. (1.15b)]. In this fashion,which should be familiar to the reader and will be elucidated and generalized below, we canconstruct all of the standard scalars and vectors of Newtonian physics. What is importantis that these physical quantities require no coordinate system for their definition. They aregeometric (coordinate-independent) objects residing in Euclidean 3-space at a particulartime.

It is a fundamental (though often ignored) principle of physics that the Newtonian physicallaws are all expressible as geometric relationships between these types of geometric objects,

and these relationships do not depend upon any coordinate system or orientation of axes, nor

on any reference frame (i.e., on any purported velocity of the Euclidean space in which themeasurements are made).2 We shall call this the Geometric Principle for the laws of physics,and we shall use it throughout this book. It is the Newtonian analog of Einsteins Principleof Relativity (Sec. 2.2.2 below).

1.3 Tensor Algebra Without a Coordinate System

In preparation for developing our geometric view of physical laws, we now introduce, in acoordinate-free way, some fundamental concepts of differential geometry: tensors, the innerproduct, the metric tensor, the tensor product, and contraction of tensors.

We have already defined a vectorA as a straight arrow from one point, say P, in our spaceto another, say Q. Because our space is flat, there is a unique and obvious way to transportsuch an arrow from one location to another, keeping its length and direction unchanged.3

Accordingly, we shall regard vectors as unchanged by such transport. This enables us toignore the issue of where in space a vector actually resides; it is completely determined byits direction and its length.

7.95 T

Fig. 1.2: A rank-3 tensor T.

A rank-n tensor T is, by definition, a real-valued, linear function of n vectors.4 Pictorially

2By changing the velocity of Euclidean space, one adds a constant velocity to all particles, but this leavesthe laws, e.g. Newtons F = ma, unchanged.

3This is not so in curved spaces, as we shall see in Sec. 25.7.4This is a different use of the word rank than for a matrix, whose rank is its number of linearly independent

rows or columns.

6we shall regard T as a box (Fig. 1.2) with n slots in its top, into which are inserted n vectors,and one slot in its end, which prints out a single real number: the value that the tensor Thas when evaluated as a function of the n inserted vectors. Notationally we shall denote thetensor by a bold-face sans-serif character T

T( , , , ) (1.3a) n slots in which to put the vectors.

This is a very different (and far simpler) definition of a tensor than one meets in moststandard physics textbooks [e.g., Thornton and Marion (2004), Goldstein, Poole and Safko(2002), Griffiths (1999), and Jackson (1999)]. There a tensor is an array of numbers thattransform in a particular way under rotations. We shall learn the connection between thesedefinitions in Sec. 1.6 below.

If T is a rank-3 tensor (has 3 slots) as in Fig. 1.2, then its value on the vectors A,B,Cis denoted T(A,B,C). Linearity of this function can be expressed as

T(eE+ fF,B,C) = eT(E,B,C) + fT(F,B,C) , (1.3b)

where e and f are real numbers, and similarly for the second and third slots.We have already defined the squared length (A)2 A2 of a vector A as the squared

distance between the points at its tail and its tip. The inner product A B of two vectors isdefined in terms of this squared length by

A B 14

[(A+B)2 (AB)2] . (1.4a)

In Euclidean space, this is the standard inner product, familiar from elementary geometry.One can show that the inner product (1.4a) is a real-valued linear function of each of

its vectors. Therefore, we can regard it as a tensor of rank 2. When so regarded, the innerproduct is denoted g( , ) and is called the metric tensor. In other words, the metric tensorg is that linear function of two vectors whose value is given by

g(A,B) A B . (1.4b)

Notice that, because A B = B A, the metric tensor is symmetric in its two slots; i.e., onegets the same real number independently of the order in which one inserts the two vectorsinto the slots:

g(A,B) = g(B,A) . (1.4c)

With the aid of the inner product, we can regard any vector A as a tensor of rank one:The real number that is produced when an arbitrary vector C is inserted into As slot is

A(C) A C . (1.4d)

In Newtonian physics, we rarely meet tensors of rank higher than two. However, second-rank tensors appear frequentlyoften in roles where one sticks a single vector into the second

7slot and leaves the first slot empty, thereby producing a single-slotted entity, a vector. Anexample that we met in Sec. 1.1.1 is a rigid bodys moment-of-inertia tensor I( , ), whichgives us the bodys angular momentum J( ) = I( ,) when its angular velocity isinserted into its second slot.5 Another example is the stress tensor of a solid, a fluid, aplasma, or a field (Sec. 1.9 below).

From three (or any number of) vectors A, B, C we can construct a tensor, their tensorproduct (also called outer product in contradistinction to the inner product A B), definedas follows:

ABC(E,F,G) A(E)B(F)C(G) = (A E)(B F)(C G) . (1.5a)Here the first expression is the notation for the value of the new tensor, ABC evaluatedon the three vectors E, F, G; the middle expression is the ordinary product of three realnumbers, the value of A on E, the value of B on F, and the value of C on G; and thethird expression is that same product with the three numbers rewritten as scalar products.Similar definitions can be given (and should be obvious) for the tensor product of any twoor more tensors of any rank; for example, if T has rank 2 and S has rank 3, then

T S(E,F,G,H,J) T(E,F)S(G,H,J) . (1.5b)

One last geometric (i.e. frame-independent) concept we shall need is contraction. Weshall illustrate this concept first by a simple example, then give the general definition. Fromtwo vectors A and B we can construct the tensor product A B (a second-rank tensor),and we can also construct the scalar product A B (a real number, i.e. a scalar, i.e. a rank-0tensor). The process of contraction is the construction of A B from AB

contraction(AB) A B . (1.6a)One can show fairly easily using component techniques (Sec. 1.5 below) that any second-ranktensor T can be expressed as a sum of tensor products of vectors, T = AB+CD+ . . .;and correspondingly, it is natural to define the contraction of T to be contraction(T) =A B+C D+ . . .. Note that this contraction process lowers the rank of the tensor by two,from 2 to 0. Similarly, for a tensor of rank n one can construct a tensor of rank n 2 bycontraction, but in this case one must specify which slots are to be contracted. For example,if T is a third rank tensor, expressible as T = A B C + E F G + . . ., then thecontraction of T on its first and third slots is the rank-1 tensor (vector)

1&3contraction(ABC+ E FG+ . . .) (A C)B+ (E G)F+ . . . . (1.6b)

All the concepts developed in this section (vectors, tensors, metric tensor, inner product,tensor product, and contraction of a tensor) can be carried over, with no change whatsoever,into any vector space6 that is endowed with a concept of squared length for example, tothe four-dimensional spacetime of special relativity (next chapter).

5Actually, it doesnt matter which slot since I is symmetric.6or, more precisely, any vector space over the real numbers. If the vector spaces scalars are complex

numbers, as in quantum mechanics, then slight changes are needed.

81.4 Particle Kinetics and Lorentz Force in Geometric Lan-

guage

In this section, we shall illustrate our geometric viewpoint by formulating Newtons laws ofmotion for particles.

In Newtonian physics, a classical particle moves through Euclidean 3-space as universaltime t passes. At time t it is located at some point x(t) (its position). The function x(t)represents a curve in 3-space, the particles trajectory. The particles velocity v(t) is the timederivative of its position, its momentum p(t) is the product of its mass m and velocity, itsacceleration a(t) is the time derivative of its velocity, and its energy is half its mass timesvelocity squared:

v(t) =dx

dt

, p(t) = mv(t) , a(t) =dv

dt

=d2x

dt2, E(t) =

1

2mv2 . (1.7a)

Since points in 3-space are geometric objects (defined independently of any coordinate sys-tem), so also are the trajectory x(t), the velocity, the momentum, the acceleration and theenergy. (Physically, of course, the velocity has an ambiguity; it depends on ones standardof rest.)

Newtons second law of motion states that the particles momentum can change only ifa force F acts on it, and that its change is given by

dp/dt = ma = F . (1.7b)

If the force is produced by an electric field E and magnetic field B, then this law of motionin SI units takes the familiar Lorentz-force form

dp/dt = q(E+ v B) . (1.7c)(Here we have used the vector cross product, which will not be introduced formally untilSec. 1.7 below.) The laws of motion (1.7) are geometric relationships between geometricobjects.

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EXERCISES

Exercise 1.1 Practice: Energy change for charged particle

Without introducing any coordinates or basis vectors, show that, when a particle with chargeq interacts with electric and magnetic fields, its energy changes at a rate

dE/dt = q v E . (1.8)Exercise 1.2 Practice: Particle moving in a circular orbit

Consider a particle moving in a circle with uniform speed v = |v| and uniform magnitudea = |a| of acceleration. Without introducing any coordinates or basis vectors, show thefollowing:

9(a) At any moment of time, let n = v/v be the unit vector pointing along the velocity, andlet s denote distance that the particle travels in its orbit. By drawing a picture, showthat dn/ds is a unit vector that points to the center of the particles circular orbit,divided by the radius of the orbit.

(b) Show that the vector (not unit vector) pointing from the particles location to thecenter of its orbit is (v/a)2a.

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1.5 Component Representation of Tensor Algebra

In the Euclidean 3-space of Newtonian physics, there is a unique set of orthonormal basisvectors {ex, ey, ez} {e1, e2, e3} associated with any Cartesian coordinate system {x, y, z} {x1, x2, x3} {x1, x2, x3}. [In Cartesian coordinates in Euclidean space, we will usually placeindices down, but occasionally we will place them up. It doesnt matter. By definition, inCartesian coordinates a quantity is the same whether its index is down or up.] The basisvector ej points along the xj coordinate direction, which is orthogonal to all the othercoordinate directions, and it has unit length (Fig. 1.3), so

ej ek = jk . (1.9a)

Any vector A in 3-space can be expanded in terms of this basis,

A = Ajej . (1.9b)

Here and throughout this book, we adopt the Einstein summation convention: repeatedindices (in this case j) are to be summed (in this 3-space case over j = 1, 2, 3), unlessotherwise instructed. By virtue of the orthonormality of the basis, the components Aj of Acan be computed as the scalar product

Aj = A ej . (1.9c)

[The proof of this is straightforward: A ej = (Akek) ej = Ak(ek ej) = Akkj = Aj.]

x y

z

e 1

e 3e 2

Fig. 1.3: The orthonormal basis vectors ej associated with a Euclidean coordinate system inEuclidean 3-space.

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Any tensor, say the third-rank tensor T( , , ), can be expanded in terms of tensorproducts of the basis vectors:

T = Tijkei ej ek . (1.9d)The components Tijk of T can be computed from T and the basis vectors by the generalizationof Eq. (1.9c)

Tijk = T(ei, ej, ek) . (1.9e)

[This equation can be derived using the orthonormality of the basis in the same way asEq. (1.9c) was derived.] As an important example, the components of the metric aregjk = g(ej, ek) = ej ek = jk [where the first equality is the method (1.9e) of comput-ing tensor components, the second is the definition (1.4b) of the metric, and the third is theorthonormality relation (1.9a)]:

gjk = jk . (1.9f)

The components of a tensor product, e.g. T( , , ) S( , ), are easily deduced byinserting the basis vectors into the slots [Eq. (1.9e)]; they are T(ei, ej, ek)S(el, em) = TijkSlm[cf. Eq. (1.5a)]. In words, the components of a tensor product are equal to the ordinaryarithmetic product of the components of the individual tensors.

In component notation, the inner product of two vectors and the value of a tensor whenvectors are inserted into its slots are given by

A B = AjBj , T(A,B,C) = TijkAiBjCk , (1.9g)

as one can easily show using previous equations. Finally, the contraction of a tensor [say, thefourth rank tensor R( , , , )] on two of its slots [say, the first and third] has componentsthat are easily computed from the tensors own components:

Components of [1&3contraction of R] = Rijik (1.9h)

Note that Rijik is summed on the i index, so it has only two free indices, j and k, and thusis the component of a second rank tensor, as it must be if it is to represent the contractionof a fourth-rank tensor.

1.5.1 Slot-Naming Index Notation

We now pause, in our development of the component version of tensor algebra, to introducea very important new viewpoint:

Consider the rank-2 tensor F( , ). We can define a new tensor G( , ) to be thesame as F, but with the slots interchanged; i.e., for any two vectors A and B it is truethat G(A,B) = F(B,A). We need a simple, compact way to indicate that F and G areequal except for an interchange of slots. The best way is to give the slots names, say aand b i.e., to rewrite F( , ) as F( a, b) or more conveniently as Fab; and then towrite the relationship between G and F as Gab = Fba. NO! some readers might object.This notation is indistinguishable from our notation for components on a particular basis.GOOD! a more astute reader will exclaim. The relation Gab = Fba in a particular basis is

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Box 1.2

T2 Vectors and Tensors in Quantum Theory

The laws of quantum theory, like all other laws of Nature, can be expressed as geometricrelationships between geometric objects. Most of quantum theorys geometric objects,like those of classical theory, are vectors and tensors:

The quantum state | of a physical system (e.g. a particle in a harmonic-oscillatorpotential) is a Hilbert-space vectorthe analog of a Euclidean-space vector A. Thereis an inner product, denoted |, between any two states | and |, analogous toB A; but, whereas B A is a real number, | is a complex number (and we add andsubtract quantum states with complex-number coefficients). The Hermitian operatorsthat represent observables (e.g. the Hamiltonian H for the particle in the potential) aretwo-slotted (second-rank), complex-valued functions of vectors; |H| is the complexnumber that one gets when one inserts and into the first and second slots of H . Justas, in Euclidean space, we get a new vector (first-rank tensor) T( ,A) when we insert thevector A into the second slot of T, so in quantum theory we get a new vector (physicalstate) H| (the result of letting H act on |) when we insert | into the secondslot of H . In these senses, we can regard T as a linear map of Euclidean vectors intoEuclidean vectors, and H as a linear map of states (Hilbert-space vectors) into states.

For the electron in the Hydrogen atom, we can introduce a set of orthonormal basisvectors {|1, |2, |3, ...}, e.g. the atoms energy eigenstates, with m|n = mn. But bycontrast with Newtonian physics, where we only need three basis vectors because ourEuclidean space is 3-dimensional, for the particle in a harmonic-oscillator potential weneed an infinite number of basis vectors, since the Hilbert space of all states is infinitedimensional. In the particles quantum-state basis, any observable (e.g. the particlesposition x or momentum p) has components computed by inserting the basis vectorsinto its two slots: xmn = m|x|n, and pmn = m|p|n. The observable xp (which mapsstates into states) has components in this basis xjkpkm (a matrix product); and thenoncommutation of position and momentum [x, p] = i~ (an important physical law) hascomponents xjkpkm pjkxkm = i~mn.

a true statement if and only if G = F with slots interchanged is true, so why not use thesame notation to symbolize both? This, in fact, we shall do. We shall ask our readers tolook at any index equation such as Gab = Fba like they would look at an Escher drawing:momentarily think of it as a relationship between components of tensors in a specific basis;then do a quick mind-flip and regard it quite differently, as a relationship between geometric,basis-independent tensors with the indices playing the roles of names of slots. This mind-flipapproach to tensor algebra will pay substantial dividends.

As an example of the power of this slot-naming index notation, consider the contrac-tion of the first and third slots of a third-rank tensor T. In any basis the components of1&3contraction(T) are Taba; cf. Eq. (1.9h). Correspondingly, in slot-naming index notationwe denote 1&3contraction(T) by the simple expression Taba. We can think of the first andthird slots as strangling or killing each other by the contraction, leaving free only the

12

second slot (named b) and therefore producing a rank-1 tensor (a vector).We should caution that the phrase slot-naming index notation is unconventional (as

are killing and strangling). You are unlikely to find it in any other textbooks. However, welike it. It says precisely what we want it to say.

1.5.2 Particle Kinetics in Index Notation

As an example of slot-naming index notation, we can rewrite the equations of particle kinetics(1.7) as follows:

vi =dxi

dt

, pi = mvi , ai =dvi

dt

=d2xi

dt2, E =

1

2mvjvj ,

dpi

dt

= q(Ei + ijkvjBk) . (1.10)

(In the last equation ijk is the so-called Levi-Civita tensor, which is used to produce thecross product; we shall learn about it in Sec. 1.7 below.)

Equations (1.10) can be viewed in either of two ways: (i) as the basis-independent geo-metric laws v = dx/dt, p = mv, a = dv/dt = d2x/dt2, E = 1

2mv2, and dp/dtq(E+vB)

written in slot-naming index notation; or (ii) as equations for the components of v, p, a, Eand B in some particular Cartesian coordinate system.

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EXERCISES

Exercise 1.3 Derivation: Component Manipulation Rules

Derive the component manipulation rules (1.9g) and (1.9h).

Exercise 1.4 Example and Practice: Numerics of Component Manipulations

The third rank tensor S( , , ) and vectors A and B have as their only nonzero compo-nents S123 = S231 = S312 = +1, A1 = 3, B1 = 4, B2 = 5. What are the components of thevector C = S(A,B, ), the vector D = S(A, ,B) and the tensor W = AB?[Partial solution: In component notation, Ck = SijkAiBj , where (of course) we sum overthe repeated indices i and j. This tells us that C1 = S231A2B3 because S231 is the onlycomponent of S whose last index is a 1; and this in turn implies that C1 = 0 since A2 = 0.Similarly, C2 = S312A3B1 = 0 (because A3 = 0). Finally, C3 = S123A1B2 = +1 3 5 = 15.Also, in component notation Wij = AiBj , so W11 = A1 B1 = 3 4 = 12 and W12 =A1B2 = 35 = 15. Here the is numerical multiplication, not the vector cross product.]

Exercise 1.5 Practice: Meaning of Slot-Naming Index Notation

(a) The following expressions and equations are written in slot-naming index notation;convert them to geometric, index-free notation: AiBjk, AiBji, Sijk = Skji, AiBi =AiBjgij.

13

(b) The following expressions are written in geometric, index-free notation; convert themto slot-naming index notation: T( , ,A); T( ,S(B, ), ).

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1.6 Orthogonal Transformations of Bases

Consider two different Cartesian coordinate systems {x, y, z} {x1, x2, x3}, and {x, y, z} {x1, x2, x3}. Denote by {ei} and {ep} the corresponding bases. It is possible to expand thebasis vectors of one basis in terms of those of the other. We shall denote the expansioncoefficients by the letter R and shall write

ei = epRpi , ep = eiRip . (1.11)

The quantities Rpi and Rip are not the components of a tensor; rather, they are the elementsof transformation matrices

[Rpi] =

R11 R12 R13R21 R22 R23R31 R32 R33

, [Rip] =

R11 R12 R13R21 R22 R23R31 R32 R33

. (1.12a)

(Here and throughout this book we use square brackets to denote matrices.) These twomatrices must be the inverse of each other, since one takes us from the barred basis to theunbarred, and the other in the reverse direction, from unbarred to barred:

RpiRiq = pq , RipRpj = ij . (1.12b)

The orthonormality requirement for the two bases implies that ij = ei ej = (epRpi) (eqRqj) = RpiRqj(ep eq) = RpiRqjpq = RpiRpj. This says that the transpose of [Rpi] is itsinversewhich we have already denoted by [Rip];

[Rip] Inverse ([Rpi]) = Transpose ([Rpi]) . (1.12c)This property implies that the transformation matrix is orthogonal; i.e., the transformationis a reflection or a rotation [see, e.g., Goldstein, Poole and Safko (2002)]. Thus (as should beobvious and familiar), the bases associated with any two Euclidean coordinate systems arerelated by a reflection or rotation. Note: Eq. (1.12c) does not say that [Rip] is a symmetricmatrix. In fact, most rotation matrices are not symmetric; see, e.g., Eq. (1.14) below.

The fact that a vector A is a geometric, basis-independent object implies that A =Aiei = Ai(epRpi) = (RpiAi)ep = Apep; i.e.,

Ap = RpiAi , and similarly Ai = RipAp ; (1.13a)

and correspondingly for the components of a tensor

Tpqr = RpiRqjRrkTijk , Tijk = RipRjqRkrTpqr . (1.13b)

14

It is instructive to compare the transformation law (1.13a) for the components of a vectorwith those (1.11) for the bases. To make these laws look natural, we have placed thetransformation matrix on the left in the former and on the right in the latter. In Minkowskispacetime (Chap. 2), the placement of indices, up or down, will automatically tell us theorder.

If we choose the origins of our two coordinate systems to coincide, then the vector xreaching from the common origin to some point P, whose coordinates are xj and xp, hascomponents equal to those coordinates; and as a result, the coordinates themselves obey thesame transformation law as any other vector

xp = Rpixi , xi = Ripxp . (1.13c)

The product of two rotation matrices, [RipRps] is another rotation matrix [Ris], whichtransforms the Cartesian bases es to ei. Under this product rule, the rotation matrices forma mathematical group: the rotation group, whose group representations play an importantrole in quantum theory.

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EXERCISES

Exercise 1.6 **Example and Practice: Rotation in x, y Plane7

Consider two Cartesian coordinate systems rotated with respect to each other in the x, yplane as shown in Fig. 1.4.

(a) Show that the rotation matrix that takes the barred basis vectors to the unbarred basisvectors is

[Rpi] =

cos sin 0 sin cos 0

0 0 1

,