sr lectures.dvi

7/28/2019 SR Lectures.dvi

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Special Relativity & Relativistic Electrodynamics

Sourav Sur

Dept. of Physics & Astrophysics, University of Delhi, Delhi - 110 007, India

Lecture Notes for M.Sc. First Semester, University of Delhi.

Contents

1 Early conceptions: Galilean Relativity 31.1 Galilean Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Galilean Principle of Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Formulation of Spatial Relativity 62.1 The Postulates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Lorentz Transformations for a Boost along one axis . . . . . . . . . . . . . . . . . . . . . . . . 62.3 The invariant Line element and the Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Poincare Transformations for a Boost along one axis . . . . . . . . . . . . . . . . . . . . . . . . 102.5 Lorentz Transformations for a Boost in an arbitrary direction . . . . . . . . . . . . . . . . . . 112.6 Lorentz Transformations for arbitrary Boosts and Rotations . . . . . . . . . . . . . . . . . . . 122.7 Successive Lorentz Boosts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.8 Relativistic Velocity Addition Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.9 General Rule of Transformation of Coordinate Differentials . . . . . . . . . . . . . . . . . . . . 14

3 Lightcone, World-lines, Intervals and the Proper Time 163.1 The Light-cone and the World-lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2 Time-like, Space-like and Null Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3 The Proper Time and Time Dilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4 Tensor Analysis 194.1 Tensors of various Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.1.1 Tensor of Rank 0 : Scalar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.1.2 Tensor of Rank 1 : Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.1.3 Tensors of Rank 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.2 Tensor Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.2.1 Linear Combination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.2.2 Multiplication: Direct Product, Contracted Product and Scalar Product . . . . . . . . 204.2.3 Index raising/lowering, Norm and Trace . . . . . . . . . . . . . . . . . . . . . . . . . . 214.2.4 Relation between Contravariant and Covariant components of a Tensor . . . . . . . . . 21

4.3 (Anti-)Symmetry of Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.3.1 Symmetric Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.3.2 Antisymmetric Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.3.3 Symmetrization and Antisymmetrization of Tensors . . . . . . . . . . . . . . . . . . . . 22Email: [email protected], [email protected], [email protected]

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CONTENTS

4.3.4 Salient features of Symmetric and Antisymmetric Tensors . . . . . . . . . . . . . . . . . 234.3.5 The Levi-Civita Tensor and the Generalized Kronecker Delta . . . . . . . . . . . . . . . 244.3.6 Tensor Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.4 Tensor Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.4.1 Tensor Differentiation: Grad, Div, Curl and DAlembertian operators . . . . . . . . . . 25

4.4.2 The Four Volume and Tensor Integration . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5 Special Relativistic Particle Dynamics 275.1 Four Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.2 Four Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.3 Four Momentum and Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.4 Conservation of Energy and Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.5 Four Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

6 Special Relativistic Electrodynamics 316.1 Four Current Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

6.2 Covariant Lorentz Force Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316.3 Covariant formulation of Maxwells Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 326.4 Transformation of Electromagnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.5 Electromagnetic Scalar invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.6 Potential formulation of Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

6.6.1 Electromagnetic Four Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356.6.2 Gauge Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356.6.3 Gauge conditions: Lorentz and Coulomb gauges . . . . . . . . . . . . . . . . . . . . . . 36

7 Wave Solutions in Electrodynamics the Retarded Potentials 387.1 Solutions for scalar and three-vector gauge potentials the Non-covariant approach . . . . . 38

7.1.1 Solution technique in the Lorentz gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . 387.1.2 Solution technique in the Coulomb gauge . . . . . . . . . . . . . . . . . . . . . . . . . . 397.1.3 Solutions of the Propagator equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407.1.4 General Solutions for the Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

7.2 Solutions for the four vector gauge potential the Covariant approach . . . . . . . . . . . . . 437.2.1 Potential expansion in terms of Greens function, and Solutions . . . . . . . . . . . . . 437.2.2 Covariant formulation of Retarded and Advarnced Greens functions . . . . . . . . . . 467.2.3 Retarded and Advanced Solutions for the Potential . . . . . . . . . . . . . . . . . . . . 47

8 Radiation by Moving Charges 488.1 Scalar and vector potentials due to a point charge the Lienard-Wiechart potentials . . . . . 488.2 Retarded Electric and Magnetic fields due to a moving charge . . . . . . . . . . . . . . . . . . 50

8.2.1 Time and space derivatives in the frames of the retarded source and the observer . . . 508.2.2 Electric and Magnetic fields from the Lienard-Wiechart potentials . . . . . . . . . . . . 518.2.3 Electromagnetic field tensor and components from generic four potential expression . . 538.2.4 Acceleration-dependent parts of the fields and Electromagnetic Radiation . . . . . . . . 54

8.3 Power radiated by an accelerated charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558.3.1 Non-relativistic radiant power formula the Larmor result . . . . . . . . . . . . . . . 558.3.2 Relativistic radiant power formula the Lienard result . . . . . . . . . . . . . . . . . . 56

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1 Early conceptions: Galilean Relativity

1.1 Galilean Transformations

Consider two systems of reference and . is characterized by coordinates (x,y ,z) and time t, whereas

is characterized by coordinates (x

, y

, z

) and time t

. Let the origins of the two systems coincide at t = t

= 0and is moving relative to with constant velocity v in an arbitrary direction (See Fig. 1).

x

z

x

z

v = constant

y

O

O

y

Figure 1: Galilean transformations for frame motion in an arbitrary direction, preserving the axes orientation, .

The Galilean transformations are a set of linear coordinate transformations given by

t = t

x = x vxty = y vytz = z vzt

or, in vector notation:t = t

x = x vt , (1.1)

vx, vy, vz being the components of v along x, y and z respectively.

1.2 Galilean Principle of Relativity

All laws of mechanics are invariant under the Galilean transformations.

Example: System of particles interacting via two-body central potentials

Equation of motion of the ith particle in the reference frame :

midvidt

= i

j

Vijxi xj (1.2)

mi being the mass of the ith particle, and Vij is potential through which it interacts with the j

th particle.

From Eqs. (1.1) it follows:

t = t , xi xj = xi xj , (1.3)so that

vi =dxidt

=dxidt

v = vi v = dvi

dt=

dvidt

. (1.4)

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1 EARLY CONCEPTIONS: GALILEAN RELATIVITY

Now, to work out the relations between the coordinate differentials in the two frames and , we proceed asfollows

Consider a scalar function [ = (t,x,y,z) in , and = (t, x, y, z) in ]. Then

d =

t

dt +

x

dx +

y

dy +

z

dz =

t

dt +

xdx +

ydy +

zdz . (1.5)

Or,

tdt +

xdx +

ydy +

zdz =

tdt +

x(dx vxdt) +

y (dy vydt) +

z (dz vzdt) . (1.6)

Equating separately the coefficients of dt,dx,dy and dz on both sides

x=

x,

y =

y,

z =

z, and

t=

t+ vx

x+ vy

y+ vz

z. (1.7)

Or, in vectorial form

= ,

t =

t + v . (1.8)Using Eqs. (1.3), (1.4) and the first relation of Eqs. (1.8), we can write Eq. (1.2) as

midvidt

= i

j

Vij |xi xj | (1.9)

which is the equation of motion of the ith particle in the reference frame .So, the Galilean transformation equations preserve the form of Newtonian equations of motion.

Counter-example: Homogeneous wave equation for a scalar field

Let us again consider a scalar field which satisfies the wave equation1

c22

t2 2

= 0 (1.10)

in the reference frame .Then, under the Galilean transformations, using Eqs. (1.8), we have in the reference frame

1

c2

2

t2+ 2v

t+ (v ) (v )

2

= 0 , (1.11)

which shows that the form of the wave equation is not preserved under the Galilean transformations.For sound waves, this isnt a problem, since sound waves are compressions and rarefactions in a given

medium (e.g. air). Therefore, the preferred reference frame in which Eq. (1.10) is valid, is obviously theframe in which the medium is at rest. The same can also hold for light waves, if they propagate through amedium. However, the essential difference between the sound and the light wave propagation is that: the framemotion directly influences the propagation of sound (by virtue of mechanical vibrations described by Newtonianclassical mechanics), whereas for light, the intervening medium, if exists, has no such manifestation, other thanto support the wave propagation.

Now, the light wave propagation could be shown to be due to transverse electric and magnetic fields,whose dynamics is governed by the Maxwells equations. So there seemed to be an incompatibility betweenthe Maxwells equations and Galilean relativity, in the sense that the Galilean transformations fail to preserve

the form of the Maxwells equations (or, the wave equations corresponding to every component of the electricand the magnetic field vectors) which involve a fixed parameter c, that can be interpreted as a fixed speed ofpropagation c of light signals.

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1.2 Galilean Principle of Relativity

Possible resolutions

I. Maxwells theory is incorrect and a proper theory of electromagnetism must have its equations invariantunder Galilean transformations.

II. Galilean relativity applies to the laws of Newtonian mechanics, and the Maxwells theory is correct butbizarre, in the sense that the electromagnetic wave propagation always requires a preferred frame, whichis the one where the intervening medium the so-called luminiferous ther is at rest.

III. There should exist a relativity principle, other than the Galilean one, for both the laws of mechanics andthe equations of electrodynamics.

The first alternative was hardly viable, given the immense success of the Maxwells theory of electromagnetism,otherwise.

The no-go establishment of the ther-hypothesis (following the works of Michaelson and Morley in 1886)ruled out the second alternative as well.

So, what remained is to find out a set of transformation relations so that the Maxwells equations are of the

same form, with the same speed parameter c, in all frames moving with uniform relative speeds (i.e. inertialframes).

The first step towards obtaining such transformation relations, and hence to formulate a new theory ofrelativity, was to abolish the Newtonian concept of absolute space, and to keep the time and the space inequal footing. That is to say, one should now invoke the concept of space-time (instead of treating spaceand time separately) represented by all the spatial coordinates, as well as time multiplied by a constant speed(from dimensional arguments). Such a speed parameter must be a universal constant, which is argued to bethe speed of light c in vacuum, as we see below.

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2 FORMULATION OF SPATIAL RELATIVITY

2 Formulation of Spatial Relativity

2.1 The Postulates

I. All laws of physics (except that for gravitation) are independent of uniform translational motion of thesystem, i.e., the reference frame (RF), where they operate Principle of Relativity:

Physical LawsInv.

(RF)

v=const.= (RF)

. (2.1)

Corollary: The mathematical equations expressing the laws of nature must be covariant, i.e., invariant in formunder the transformations that leave invariant the infinitesimal separation between two events in space-time.

II. The speed of light (c) in empty space is finite and independent of the motion of the source:

cInv.

(RF)

v=const.= (RF)

. (2.2)

Corollary: For physical entities in every inertial frame, there is a finite limiting speed, equal to c (as found

experimentally): vlim = c = const. . (2.3)

The most general set of coordinate transformations which preserve the infinitesimal separation between twospace-time events, side-by-side maintaining the constancy of the speed of light, are the Poincare transfor-mations. A sub-class of these are the so-called (homogeneous) Lorentz transformations which preserve theseparation between two events which are not necessarily very close to each other, i.e., the separation is not nec-essarily infinitesimal. We discuss below in detail the form and properties of both the Lorentz and the Poincaretransformation equations in certain simplified, as well as generic, scenarios.

2.2 Lorentz Transformations for a Boost along one axis

Consider two reference frames and represented by (ct,x,y,x) and (ct, x, y, z) respectively, as shown in

Fig. 2. Let us suppose that

I. The axes of the two frames are parallely oriented, i.e, ct||ct, x||x, y||y, z||z.II. Frame moves with constant velocity v relative to the frame along, say, the positive x- (x-) axis.

III. Origins of the two frames coincide at t = t = 0.

xO

y y

Ox

v = constant

zz

Figure 2: Lorentz transformations for frame motion, preserving the axes orientation, and along one axis.

Consider now an event P, whose coordinates are (ct, x, y, z) in frame and (ct,x,y,z) in frame . Asthe origin of moves relative to with speed v along positive x-direction

x = 0 =

x = vt . (2.4)

Conversely, as the origin of moves relative to with speed v along the negative x-direction

x = 0 = x = vt . (2.5)

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2.2 Lorentz Transformations for a Boost along one axis

Now, the transformation equations between the two systems must be linear, so that for an event in there isa unique associated event in , and vice versa. The most general relations satisfying Eqs. (2.4) and (2.5) are

x = (x vt) , x = x + vt [ = dimensionless constant] . (2.6)Or,

dx = (dx

vdt) , dx = dx + vdt . (2.7)

Therefore,dx

dt=

dx

dt v dt

dt

=

dx

dt v

dt

dt,

dx

dt=

dx

dt+ v

dt

dt

=

dx

dt+ v

dt

dt. (2.8)

In the limit dx/dt c and dx/dt c, Eqs. (2.8) reduce todt

dt=

c

(c v) ,dt

dt=

c

(c + v). (2.9)

Solving for , subject to the requirement that the transformation equations we seek must reduce to the Galileantransformation equations in the limit v/c 0, we get

= 1 21/2 , where = v

c

. (2.10)

One can now write down the full set of transformation equations by solving Eqs.(2.6) for x and t, and notingthat there is no frame motion in y- or z-directions in the simplified set-up we are considering presently:

(i) ct = (ct x)(ii) x = (ct + x)(iii) y = y

(iv) z = z

(2.11)

These are Lorentz transformation (LT) equations in the simplest possible scenario where there is a framemotion, or Lorentz boost, along only one spatial direction. Denoting the coordinates as:

x0 ct , x1 x , x2 y , and x3 z (2.12)we write the above equations (2.11) in matrix form as

x0

x1

x2

x3

=

0 0 0 0

0 0 1 00 0 0 1

x0

x1

x2

x3

. (2.13)

Or, in compact formx = x

, [, = 0, 1, 2, 3] , (2.14)

where x and are the elements of

x {x} =

x0

x1

x2

x3

: 1 4 Position (column) vector in (3 + 1)-dimensional spacetime , (2.15)

{} =

0 0 0 0

0 0 1 00 0 0 1

:

4 4 (homogeneous) LT matrix in a (3 + 1)-dimensionalspace-time for single Lorentz boost along positive x1-axis; denotes the columns and denotes the rows.

(2.16)

In the above Eq. (2.14), Einsteins summation convention is implied, i.e., repeated indices are summed over:

x

3=0

x [ = 0, 1, 2, 3] . (2.17)

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The inverse transformation equations are given by

x =

x , [, = 0, 1, 2, 3] , (2.18)

where x and are the elements of

x1

{x

}= (x

0, x

1, x

2, x

3) : 4

1 Position (row) vector in (3 + 1)-dimensional spacetime , (2.19)

1 = []1 : 4 4 Inverse LT matrix for a boost along negative x1 axis . (2.20)The distinction between x0 and x0, x

1 and x1, and so forth, will be explained later on, when we will make a general

study of tensors and their components. For the time being, however, lets stick to the matrix terminology and focus on

the properties of the LT matrix , where denotes columns and denotes rows.

Following are the two two important properties of the LT matrix for a single boost along one of the axes:

(i) T = = T1 = T1 = I = = , and (ii) det = 1 . (2.21)

To observe physically the meaning of a boost, let us make the following parametrization:

=v

c= tanh , =

1 21/2 = cosh , (2.22)

where is a dimensionless constant, called the Boost or Rapidity parameter. Under the above parameteri-zation, the LT equations (2.11) take the form:

x0 = x0 cosh x1 sinh ,x1 = x0 sinh + x1 cosh ,x2 = x2 , and x3 = x3 , (2.23)

i.e., the LT matrix (due to a boost along the positive x1-direction with constant speed v) takes the form:

{} =

cosh

sinh 0 0

sinh cosh 0 00 0 1 00 0 0 1

. (2.24)Under a further transformation, and reparameterization:

x0 x0 = ix0 i.e., t t = it ; xk xk = xk [k = 1, 2, 3] , = i , (2.25)the above LT matrix (2.24) reduces to

{} =

cos sin 0 0

sin cos 0 00 0 1 00 0 0 1

= {R} , (2.26)

where

R

are the elements of the matrix corresponding to the ordinary (orthonormal) rotation of the x0

x1

plane in the (3 + 1)-dimensional space-time (see Fig. 3):

x0= ix

0

x1= x

1

x1= x

1

x0= ix

0

O

Figure 3: Simple Lorentz transformation as (orthonormal) rotation of the ix0 x1 plane.

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Now, the line element ds2 being quadratic in the differentials of the coordinates (dx ; = 0, 1, 2, 3), one mayexpress it (as any other quadratic):

ds2 = g dx dx , (2.32)

where the coefficients g (, = 0, 1, 2, 3) form the elements of the so-called (covariant) Metric tensor of rank

2, or simply the metric. We will give a detailed account of the definition and properties of tensors of varioustypes later on. However, for the time being, consider g forming an object which could in general be a functionof the coordinates, and is symmetric under the interchange of its indices, i.e., g = g , so that there are atmost 4(4 + 1) /2 = 10 independent components of the in 3 + 1 dimensions:

ds2 = g00

dx02

ONE term for

= = 0.

+ 2g0i dx0 dxi

Sum of THREE terms for

= 0, = i.The factor of2 is for symmetry g0i = gi0.

+ gij dxi dxj

Sum of SIX terms for

= i, = j, due tosymmetry gij = gji .

. (2.33)

In the above (and throughout) we use the following notations:

Greek indices ( , , , . . . ): run from 0 to 3, i.e., over the entire (3 + 1)-dimensional space-time, Latin indices (i , j , k , . . . ): run from 1 to 3, i.e., over the 3-dimensional spatial hypersurface.

Special Relativity (SR) alludes to a special form the metric tensor: g , referred to as the Minkowskimetric, which is diagonal with each (diagonal) element normalized to unity in magnitude:

= diag (1, 1, 1, 1) i.e., || = . (2.34)Or, explicitly: 00 = 1 , 0i = 0 ( i = 1, 2, 3) , ij = ij ( i, j = 1, 2, 3) . (2.35)The line element in Special Relativity is thus

ds2SR

= dx dx =

dx02 dx12 dx22 dx32 . (2.36)

2.4 Poincare Transformations for a Boost along one axis

The necessary condition for the constancy of the speed of light c, in a relativity theory governed by certaintransformation equations, is that the line element ds2 must remain invariant under such transformations.Lorentz transformations (LT) preserve not only ds2, but s2 [Eq. (2.30)] as well, which is not necessarilyrequired. The invariance of s2 is due to the third assumption made in the derivation of the LT equations insubsection 2.2, viz., the frames of reference and coincide at t = t = 0. If we relax this assumption,then the most general linear transformations that leave ds2 is invariant are the Poincare transformations (PT),which in the simple case of a single Lorentz boost along (say) the x-axis, with constant speed v, are given by

the equations:(i) ct = (ct x) + at(ii) x = (ct + x) + ax(iii) y = y + ay

(iv) z = z + az

(2.37)

where (at, ax, ay, az) are constant shifts between the origins of and at t = 0, along t,x,y and z respectively.

Denotinga0 at , a1 ax , az ay , a3 az (2.38)

the above equations are expressed in compact form as

x = x + a . (2.39)

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2.5 Lorentz Transformations for a Boost in an arbitrary direction

2.5 Lorentz Transformations for a Boost in an arbitrary direction

Consider again two reference frames and characterized by coordinates

x0, x1, x2, x3

and

x0, x1, x2, x3

respectively.

I. The axes of the two frames are parallely oriented, i.e, x0

||x0, x1

||x1, x2

||x2, x3

||x3.

II. Origins of the two frames coincide at x0 = x0 = 0.

III. Frame moves with constant velocity v relative to the frame in an arbitrary direction.

x2

x1

x3

x2

x1

x3

O

O

v = const.

Figure 4: Lorentz transformation for frame motion, preserving the axes orientation, but in an arbitrary direction.

As such, the generic Lorentz boost with velocity v may be thought of as the resultant of three boosts

v1, v2, v3 along x1, x2, x3 respectively, and in order to derive this generic boost transformation matrixelements (, = 0, 1, 2, 3), we resort to the following two equations:

I. Lorentz transformation equations (in differential form):

dx = dx = 0 dx

0 + i dxi . (2.40)

II. Invariance of the metric under Lorentz transformations:

ds2 = ds2 = dxdx = dxdx =

dx

dx

[by Eqs. (2.40)]

=

=

(2.41)

Let us now suppose a particle to be at rest w.r.to the frame , i.e.,

dxi = 0 (i = 1, 2, 3) .

Then, since the origin of the frame moves relative to the frame with constant velocity v, the particle isseen to move with velocity v in the frame , i.e.,

dxi

dt= vi (i = 1, 2, 3)

From Eqs. (2.40) we get

dx

0

=

0

0 dx

0

, dx

i

=

i

0 dx

0

(i = 1, 2, 3) , (2.42)

= i0 = i 00 ,

where i =vi

c= 1

c

dxi

dt= dx

i

dx0

. (2.43)

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Moreover, by setting = = 0 in Eq. (2.41), we have

00 =

0

0 = 00

002

+ ij i

0 j

0 =

002 i0 j0 ij = 1 . (2.44)

Solving Eqs. (2.42) and (2.44) simultaneously, we obtain:

00 = , i

0 = i ,

where =

1 ij ij1/2

=

1 ||21/2

. (2.45)

The other elements of the LT matrix are not uniquely determined. However, one convenient choice is

0i = ijj , ij = ij +( 1)

||2jk

ik . (2.46)

We leave it to the reader to verify that these indeed satisfy Eq. (2.41). It is also easy to see that in the ordinarythree-vectorial notation, the above elements of imply that the LT equations are given by

x0 =

x0 x

, x = x +( 1)

||2

x

x0 , (2.47)

with =

1, 2, 3

=

v

c=

v1

c,

v2

c,

v3

c

, =

1 ||2

1/2. (2.48)

So the LT involves three boosts, along all three spatial axes

x1, x2, x3

with speeds

v1, v2, v3

respectively.

2.6 Lorentz Transformations for arbitrary Boosts and Rotations

If, in addition to the scenario in previous subsection, the axes of the two frames ,

are not taken be parallelyoriented (as in Fig. 5), then the corresponding LT matrix is in its most general form, which involves not onlythe three boosts, but three spatial rotations as well.

x2

x1

x3

x2

x1

x3

O

O

v = const.

Figure 5: Lorentz transformation for frame motion, not preserving the axes orientation, and in an arbitrary direction.

Even for the most general LT matrix, the first of the properties (2.21), viz.,

T1 = T1 = I = , (2.49)

can be shown to hold. As to the second property, taking determinant of both sides of Eq.(2.41), we have

det (det)2 = det det = 1 . (2.50)

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2.7 Successive Lorentz Boosts

However, our interest is in the so-called proper Lorenetz transformations for which

00 0 , det = 1 (2.51)and which can always be converted to the identitiy transformation by continuous variation of parameters of. In fact, the full LT could be decomposed into: an ordinary spatial rotation, followed by a boost, followedby a further ordinary rotation. The first rotation lines up one of the spatial axes, say the x1-axis (see Fig. 5),of with the velocity v of . Then a boost in this direction with speed v = |v| transforms to a frame whichis at rest relative to . Finally, another rotation lines up the coordinate frame with that of .

2.7 Successive Lorentz Boosts

Two frames of reference and , with coordinates

x0, x1, x2, x3

and

x0, x1, x2, x3

, have their axesparallely oriented and origins coincident at x0 = x0 = 0. Frame moves with constant velocity v in anarbitrary direction, relative to the frame .

x2

x3

x2

x1

x3

P

v

vv

x1

O

O

Figure 6: Successive Lorentz Boosts

Consider now a particle P which moves with constant velocity v relative to . Let v be the particles velocityrelative to , and be the particles rest frame, i.e, the coordinate system

x0, x1, x2, x3

attached to the

particle. Then

I. For the relative motion between and :

x = () x ,

=

v

c

. (2.52)

where (k) are the LT matrix elements for boosts with speeds vk = ck.

II. For the relative motion between and :

x = (

) x ,

=

v

c

. (2.53)


III. For the relative motion between and :

x = (

) x ,

=

v

c

. (2.54)


Substituting for x

from Eq. (2.52) in Eq. (2.53), and comparing with Eq. (2.54), we have

x = (

) (

) x = (

) x = (k) = (k) (k) . (2.55)

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Thus a Lorentz boost along the direction of

= v/c is the product of the Lorentz boosts along

= v/c

and = v/c. One may verify that the product of the boosts is non-commutative:

(k) (

k) = (k) (k) . (2.56)

2.8 Relativistic Velocity Addition Theorem

Refer back to the situation in the previous subsection. If the relative frame velocity v and the particles velocityv w.r.to the frame are known, then what would be the particles velocity v w.r.to the frame ?To determine this, let us re-write Eq. (2.55) as

(k) = 0(

k) 0(k) + i(

k) i(k) , (2.57)

and recall from 2.5, that the transformation matrix elements for a generic Lorentz boost, in an arbitrarydirection, are given by

00 = , i0 = i , 0i = ikk , ij = ij +(

1)

||2 jk ik , (2.58)

where =

1 ikik1/2

=

1 ||21/2

.

Using these, the (0 - 0) and (i - 0) components of the above Eq. (2.57) give

00(k) = 00(

k) 00(k) + 0i(

k) i0(k) = =

1 + ik

ik

, (2.59)

i0(k) = i0(

k) 00(k) + ij(

k) j0(k)

=

i =

i + i1

1

i

jk j k

||2i. (2.60)

One may however note that:

ill

i jk

j k

||2i

=

||2= 0 . (2.61)

Since ill is arbitrary and not necessarily zero, we have

i jk jk

||2i = 0 . (2.62)

Substituting this in Eq. (2.60), and using Eq. (2.59), we finally get

i =i + i

1 + jk j k= v = v

+ v

1 + (v v/c2) . (2.63)

This is the relativistic velocity addition formula.

2.9 General Rule of Transformation of Coordinate Differentials

Consider again a reference frame (RF) with

Coordinates: x0 ct,x1 x, x2 y, x3 z. (3 + 1)-D position vector: x = x0, xi = (ct, x).

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2.9 General Rule of Transformation of Coordinate Differentials

3-D (spatial) position vector: xi = x.

Let there be another RF with

Coordinates: x0

ct, x1

x, x2

y, x3

z.

(3 + 1)-D position vector: x = x0, xi = (ct, x). 3-D (spatial) position vector: xi = x.

Let the coordinates in be related to those in by a coordinate transformation:

x = x

x0, x1, x2, x3

, = 0, 1, 2, 3 . (2.64)

Then the necessary and sufficient condition that the four components of x form a set of independent, realfunctions ofx

0, x1, x2, x3 is that their Jacobian:

J(x x) =xx

=

x0

x0. . . x

3

x0...

...x0

x3. . . x

3

x3

= 0 . (2.65)

In such case, one can have the inverse coordinate transformations as:

x = x

x0, x1, x2, x3

, = 0, 1, 2, 3 . (2.66)

At any point in a general (curved, Riemannian) spacetime, the direction (of a four-vector) in a coordinateframe, say , is determined by the coordinate differentials dx defined in that frame. In another frame, say ,

the direction is determined by the associated coordinate differentials dx

. The relation between the coordinatedifferentials in the two frames is given by the transformation rules:

dx =x

xdx , and dx =

x

xdx . (2.67)

While carrying out such a transformation, we have

x

xx

x= , and

x

xx

x= . (2.68)

So, if one considers x/x as a matrix the transformation matrix, for transformation from the x

coordinate system to the x coordinate system then x/x is its inverse, i.e., the transformation matrix

for the inverse transformation from the x coordinate system to the x coordinate system. Consequently, theJocobians of the transformation and the inverse transformation satisfy:

J(x x)J(x x) =xx

xx

= 1 . (2.69)In Special Relativity (SR), the transformation matrices are the LT matrices, and therefore the Jacobian isidentically equal to unity:

x

xSR= , J(x x) =

x

x

SR= det () = 1 . (2.70)

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3 LIGHTCONE, WORLD-LINES, INTERVALS AND THE PROPER TIME

3 Lightcone, World-lines, Intervals and the Proper Time

3.1 The Light-cone and the World-lines

One of the intriguing concepts of relativity is that of a Light-cone, described as follows:

Consider first, for simplicity, the (1 + 1)-D spacetime diagram where one axis is of course x0 = ct andthe other axis is one of the space axes, say x1 = x, as shown in Fig. 7. The remaining axes x2 = y andx3 = z have been suppressed.

At time t = 0, a particle (or a system of particles) is supposed to be at the origin O.

Elsewhere

Future

B

O

A

x0= ct

x1= x

Elsewhere

Past

Figure 7: (1 + 1)-D version of the Light-cone that is contained in a (1 + 1)-D spacetime with the axes: x0 = ct and oneof the space axes, say x1 = x (the other axes are suppressed). Essentially, the light-cones include the regions above andbelow the x-axis and within the angles formed by the lines that make 45o with the axes.

Since, according to the Special Relativistic second law, the speed of light c in vacuum is the upper limitof the magnitude of speeds of all material particles or systems, in the (1 + 1) spacetime diagram a specialstatus is given to the lines which make 45o angles with the axes, i.e., whose equations are given by:x1 = x0 implying |dx/dt| = c. These lines are called light-lines, as they denote light signals travelingout at 45o after being emitted from the origin O.

The light-lines divide the entire (1 + 1)-D spacetime into three regions:(I) The (1+1)-D version of the upper and lower light-cones(or, rather light-triangleswhich are the regionsabove and below the x-axis and within the angles formed by the light-lines.

(II) The light-lines themselves, which make 45o angles with the axes, and which form the edges of thelight-triangles.

(III) The regions on the left and right outside the light-triangles, called elsewhere.

The (1 +1)-D light-cones (or, the light-triangles) are so named not only because their edges are formed bythe light-lines, but also due to the fact that the spacetime evolution trajectories of all material particlesor systems, which have positive definite and finite net mass squared, and whose speeds cannot exceed

the speed of light c (by the Special Relativistic postulate) are confined within them. More specifically,since all material particles have the magnitude of speed |v| = |dx/dt| < c, as time proceeds the eventsassociated with a material particle or a system would trace out a path, say OB, as shown in Fig. 7, inside

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3.2 Time-like, Space-like and Null Intervals

the upper half (1 + 1) D cone (or, triangle), called the future cone (t > 0). The locus of all pointson the curve OB precisely satisfy |x| < ct which implies |dx/dt| < c. Similarly, the history of all eventsassociated with a particle or a system, upto the chosen origin O, could be depicted by another curve, sayAO, in the spacetime diagram in Fig. 7. As the locus of all points on OA which satisfy |x| < ct, fort < 0 inside the lower half (1 + 1)

D cone (or, triangle), the latter is called the past cone.

The entire line AOB is called the particles or the systems World line. In general the world line could becurved, as in Fig. 7, implying non-uniform evolution of the particle or system in space and time. Uniformmotion with constant speed is denoted by a straight world line. If further, the straight line is parallel tothe x0 = ct axis, then that implies the particle or the system at rest.

The region outside the (1 + 1)-D light-cones (or, the light-triangles) is called elsewhere. No materialparticle or system can have an associated event taking place there.

FutureB

A

O

x0 = ct

x1 = x

Past

Figure 8: The Light-cone in a (2 + 1)-D spacetime with the axes: x0 = ct and two of the space axes, say x1 = x andx2 = y (not shown), the third space axis is suppressed. The edges of the lightcone are formed by the light-surfaces thatmake 45o with the axes.

In (2 + 1)-D, the light-triangle becomes a (2 + 1)-D light-cone, which is an ordinary three dimensionaldouble cone, as shown in Fig. 8. The top and base of the light-cone are the circles with equation(x1)2 + (x2)2 = (x0)2, or x2 + y2 = c2t2. The edges of the light-cone are now the 2-D surfaces, calledlight-surfaces, which make 45o with the axes. All the above arguments for the (1 + 1)-D line-cone holdfor (2 + 1)-D light-cone as well.

The full (3+1)-D spacetime diagram contains a (3+1)-D light cone, which is a higher double-cone whosetop and base are spheres (x1)2 + (x2)2 + (x3)2 = (x0)2, or x2 + y2 + z2 = c2t2. The edges of the light-coneare 3-D hypersurfaces, which make 45o with the axes.

3.2 Time-like, Space-like and Null Intervals

Consider now the separation sAB between two events PA(tA, xA) and PB(tB , xB) in spacetime:

s2AB = c2 (tA tB)2 |xA xB|2 . (3.1)

For any two such events, there are three possibilities:

1. s2AB > 0 : In which case, the events are said to have a time-like separation.

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3 LIGHTCONE, WORLD-LINES, INTERVALS AND THE PROPER TIME

Since s2AB is invariant under Lorentz transformation (LT), it is always possible to find a referenceframe (RF) in which xA = x

B , whence s

2AB = c

2 (tA tB)2 > 0 (guaranteed). In the new frame , the events occur at the same position, but are separated in time. For e.g., one

event may be at the origin, and the other in the past or future region of the light-cone.

Accordingly, the particles which have their event-trajectories inside the light-cone, as for instancethe line AOB in Fig. 5 or 6, are called time-like particles. All material particles moving with speedv < c are time-like.

2. s2AB < 0 : In which case, the events are said to have a space-like separation.

It is always possible to find a RF in which tA = tB, whence s2AB = |xA xB |2 < 0 (guaranteed). In the new frame , the events occur at different space points at the same instant of time. For e.g.,

one event may be at the origin, and the other in the elsewhere region (outside the light-cone).

Accordingly, the particles which have their event-trajectories outside the light-cone are called space-like particles. No material particle could be space-like, only some hypothetical particles moving with

speed v > c are space-like.

3. s2AB = 0 : In which case, the events are said to have a light-like or null separation.

The events occur on the light-cone and can be connected only by light signals. The corresponding particles are the null particles or light particles (photons).

3.3 The Proper Time and Time Dilation

Consider a particle moving with instantaneous velocity v(t) relative to some inertial frame :

In time dt, the change in the position: dx = v(t) dt. Corresponding infinitesimal invariant interval:

ds2 = c2dt2 |dx|2 = c2dt2

1 (t)2 , where (t) = v(t)

c. (3.2)

In another frame , in which the particle is supposed to instantaneously at rest:

ds2 = c2dt2 = c2d2 , where d = dt

1 (t)2 = dt

(t): Lorentz invariant . (3.3)

The time variable is called the Proper time. It is the time as measured in the rest frame of a particle,or the time measured in a frame moving synchronously with a particle.Consider now, two successive events PA and PB associated with a particle, which are separated by the propertime interval (B A) in the particles rest frame. In another frame , relative to which the particle is movingwith a velocity v and where the time coordinate is t, the events are temporally separated by:

(tB tA) =B

A

() d . (3.4)

Since =

1

2

> 1 always, the above equation implies that (tB tA) > (B A), i.e., the temporal

separation between the events is elongated in the moving frame a phenomenon called Time dilation.

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4 Tensor Analysis

Definition: Tensors (of specific rank and type) are physical entities associated with a spacetime point, whichfollow linear, homogeneous laws of transformation under the transformation of coordinates.

4.1 Tensors of various Rank

Most generally, a tensor carries a number of superscript and subscript indices, the total number of whichsignifies the so-called rank of the tensor:

T12...m12...n Rank = (m + n) . (4.1)

4.1.1 Tensor of Rank 0 : Scalar

Rank 0 tensors are scalars which remain invariant under all coordinate transformations:

t, xi

Inv. Coordinate transformations x x . (4.2)

In SR: a scalar, which is invariant under the Lorentz transformations, is often called a Lorentz scalar, e.g., theinvariant distance squared s2 = c2t2 |x|2.

4.1.2 Tensor of Rank 1 : Vector

Rank 1 tensors are vectors of two types:

I. Contravariant vector:A =

A0, Ai

=

A0, A

. (4.3)

Under a coordinate transformation, this type of vectors transform similar to the coordinate differentials:

A

=

x

x A

Similar

to dx

=

x

x dx

. (4.4)

In SR: the above transformation rule becomes

A = A . (4.5)

II. Covariant vector:B = (B0, Bj ) . (4.6)

Under a coordinate transformation, this type of vectors transform similar to the partial derivatives of thecoordinate scalars:

B =x

xB

Similarto

x=

x

x

x. (4.7)

In SR: the above transformation rule becomesB =

B . (4.8)

4.1.3 Tensors of Rank 2These are of three types:

I. Contravariant tensor: T12...m , which transforms like a direct product of m contravariant vectors:

T1...m =x1

x1. . .

xm

xmT1...m

Similarto

A1 . . . Am =x1

x1. . .

xm

xmA1 . . . Am . (4.9)


T1...m = 11 . . . m

m T1...m . (4.10)

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4 TENSOR ANALYSIS

II. Covariant tensor: T12...n, which transforms like a direct product of n covariant vectors:

T1...n =x1

x1. . .

xn

xnT1...n

Similarto

B1 . . . Bn =

x1

x1. . .

xn

xnB1 . . . Bn . (4.11)

In SR: the above transformation rule becomesT1...n =

11 . . .

nn T1...n . (4.12)

III. Mixed tensor: T12...m12...n, which transforms like a direct product of m contravariant and ncovariant vectors:

T1...m1...n =

x1

x1. . .

xm

xm

x1

x1. . .

xn

xn

T1...m1...n (4.13)

Similarto

A1 . . . Am

B1 . . . B

n

=

x1

x1. . .

xm

xm

x1

x1. . .

xn

xn

(A1 . . . Am) (B1 . . . Bn) .


T1...m1...n =

11 . . . m

m

11 . . .

nn

T1...m1...n . (4.14)

4.2 Tensor Algebra

4.2.1 Linear Combination

Only tensors of the same rank and type (contravariant, covariant or mixed) could be added, to produce anothertensor of identical rank and type. The same is true, in fact, for any linear combination of tensors of same rankand type. For e.g., given two covariant tensors of rank 2, A and B, the linear combination

T = A + B , [, : Spacetime scalars] , (4.15)also transforms like a covariant tensor of rank 2.

4.2.2 Multiplication: Direct Product, Contracted Product and Scalar Product

Unlike addition, the multiplication of tensors of any type and rank, same or different, is possible. Accordingly,one can define three kinds of tensor products:

I. Direct Product: The aggregate obtained via multiplication of any two tensors of the same or differenttype, and of ranks m and n say, provides another tensor of rank (m + n). For e.g.,

A B = Ta , A

B = T , A B = T

, . (4.16)II. Contracted Product: If two different types of tensors of ranks m and n say, have p number of matchingcontra- and covariant indices, then the direct product of the tensors reduces to another tensor of rank (m+n2p)with no matching contra- and covariant indices. For e.g.,

A B = T

, A

B

= T

, A

B

= T , . (4.17)

Contraction implies the reduction of any mixed tensor of rank n (> 1), and with p matching contra- andcovariant indices, to a new tensor of rank (n 2p) after summing over the p matched indices. For e.g.,

T = T , T

= T , T

= T

, . (4.18)III. Scalar Product: Scalar function which is the outcome of a fully contracted product of two tensors ofthe same rank and when the contra(co)variant indices of one is identical to the co(contra)variant indices of the

other. For e.g.,A B = A B , S T = S T , Q R = Q R , . (4.19)

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4.2 Tensor Algebra

4.2.3 Index raising/lowering, Norm and Trace

Raising and lowering of tensor indices are done by contracting with the metric tensor g. For e.g.,

A = g A , A = g A ;

T = gg T

, T

= g

g

T ;T ...... =

gg g . . .

T...... , . (4.20)

Accordingly, the scalar products are expressed as

A B = AB = gAB = AaB , S T = ST = gg ST = ST , . (4.21)The norm of a tensor is its scalar product with itself. For e.g.,

|A|2 = AA , |S|2 = SS , |T|2 = TT , . (4.22)The trace of a tensor of even rank ( 2) is its full contraction in the mixed form. For e.g.,

tr

S

= gS = S , tr

T

= gg T

= T , . (4.23)

For the metric tensor g itself, the norm and the trace are the same:

|g|2 tr (g) = g = = 4 in (3 + 1)-D , (4.24)where in the fourth step, we have used a unique property of the metric tensor

g g = ,

=

1 for =

0 otherwise: (3 + 1)-D Kronecker delta . (4.25)

4.2.4 Relation between Contravariant and Covariant components of a Tensor

In Special Relativity (SR), g = .

A. Components of a Vector A = A:

A0 = 0 A = 00A

0 = A0 , Ai = iA = ijA

j = ij Aj . (4.26)Therefore,

A :=

A0, Ai A0, A = A := (A0, Ai) = A0, ijAj A0, A . (4.27)

B. Components of a second rank Tensor T = T:

T00 = 00T = (00)

2 T00 = T00 ,

T0i = 0iT = 00ijT

0j = ijT0j , Ti0 = ijTj0 [Similarly] ,Tij = ijT

= ikjl Tkl = ikjl T

kl . (4.28)

Corollary: If we choose, T , then for the Minkowski metric tensor :00 =

00 = 1 , 0i = 0 = 0i , ij = ij ij = ij . (4.29)

= and have the same set of components.

For higher rank tensors, the relationship between the contra- and covariant components could be found in asimilar manner as above.

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4 TENSOR ANALYSIS

4.3 (Anti-)Symmetry of Tensors

For contravariant or covariant (but not mixed) tensors of rank 2, the symmetry or the antisymmetry underthe interchange of pair(s) of indices are defined as follows:

4.3.1 Symmetric TensorsRank 2 tensors: S = S , S = S . (4.30)Rank 3 tensors:

Partially Symmetric: S = S , or S = S , or S = S ; (4.31)

Completely Symmetric: S = S = S = S = S = S . (4.32)

Similarly, for higher rank tensors one can define partial symmetry (under the exchange of a given pair of indices)and complete symmetry (under the exchange of any pair of indices). Symmetry for covariant tensors alwaysimplies symmetry for their contravariant counterparts, and vice versa.

4.3.2 Antisymmetric Tensors

Rank 2 tensors: A = A , A = A = A = A = 0. (4.33)Rank 3 tensors:

Partially Antisymmetric: A = A , or A = A , or A = A ; (4.34)Completely Antisymmetric: A = A = A = A = A = A . (4.35)

Similarly, for higher rank tensors one can define partial antisymmetry (under the exchange of a given pair ofindices) and complete antisymmetry (under the exchange of any pair of indices). Antisymmetry for covarianttensors always implies antisymmetry for their contravariant counterparts, and vice versa.

4.3.3 Symmetrization and Antisymmetrization of Tensors

Rank 2 tensors: Any tensor T (in general asymmetric) could be split up into a symmetric part S andantisymmetric part A:

T = S + A , S = T() , A = T[] , (4.36)

whereT() =

1

2(T + T) , T[] =

1

2(T T ) , (4.37)

are called the symmetrizationand the antisymmetrizationofT. The above splitting procedure can be adoptedfor a contravariant tensors of rank 2, T, as well. But for tensors of rank > 2 such a splitting into completely

symmetric (or, symmetrized) and completely antisymmetric (or, antisymmetrized) parts is not possible.Rank 3 tensors: Symmetrization and Antisymmetrization of third rank covariant tensor T can be doneas follows:

T() =1

3!(T + T + T + T + T+ T ) ,

T[] =1

3!(T + T + T T T T ) . (4.38)

By definition, T() and T[] are completely symmetric and completely antisymmetric tensors, respectively,and one may note that T[] has non-vanishing components only for = = . However, as mentioned above,mere addition (or any linear combination) of T() and T[] does not give the original tensor T.

The procedure of symmetrization and antisymmetrization can be adopted for higher ranked tensors as well. Ingeneral, for any covariant or contravariant tensor of rank p ( 2), in a D-dimensional spacetime (i.e., 1 time +(D 1) space), the symmetrization and the antisymmetrization schemes are as follows:

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4.3 (Anti-)Symmetry of Tensors

Rank p 2 tensors: Completely symmetric and antisymmetric tensors constructed out of tensor T12...pare given respectively by:

T(12...p) =1

p!

P

TP(1)

P(2)

...P(p)

, (4.39)

T[12...p] =1

p!

P

(

1)P T

P(1)

P(2)

...P(p)

, (4.40)

where the summations are for all permutations P of the indices (12 . . . p) and the factor (1)P indicatesthe parity equal to 1 of odd permutations P, and +1 for even permutations P.

4.3.4 Salient features of Symmetric and Antisymmetric Tensors

A few features of symmetric and antisymmetric tensors, and of the symmetrization and the antisymmetrizationof tensors are in order:

1. The scalar product of a symmetric and antisymmetric tensor vanishes identically. For e.g., consider therank 2 symmetric covariant and antisymmetric contravariant tensors S and A

respectively. Then

SA = SA : Interchanging the dummy indices ,

= S

A

: Using the symmetry and antisymmetry properties of S and A

= 0 . (4.41)

2. If the tensor T1...p is completely symmetric, then T1...p = T(1...p), whereas if the tensor T1...p iscompletely antisymmetric, then T1...p = T[1...p].

3. In D-dimensions, the total number of independent components of the rank p tensor T1...p and of itssymmetrized and antisymmetrized forms are given respectively by

N= Dp

, Nsym = D + p

1

p

=

(D + p

1)!

p! (D 1)! , Nantisym = D

p

=

D!

p! (D p)! . (4.42)For e.g., if p = 2, then T1...p = T12 has N = D2 components, and the number of independentcomponents of the symmetric tensor T(12) could be computed as

Nsym[p = 2] = N(diag) + 12N(off-diag) = D +

D2 D

2=

D (D + 1)

2, (4.43)

which matches with that obtained from the formula forNsym in Eq. (4.42), viz., (D+21)!/[2!(D1)!] =D(D + 1)/2. Similarly, keeping in mind that no diagonal elements exist for the antisymmetric tensorT[12], the number of its independent components can be computed as

Nantisym[p = 2] =

1

2N

(off-diag) = D2 D

2 =D (D 1)

2, (4.44)

which again matches with that one gets using the formula for Nantisym in Eq. (4.42). One may furthernote that: since D(D + 1)/2 + D(D 1)/2 = D2, the symmetrized and antisymmetrized forms T(12)and T[12] add up to give the original tensor T12 . This is a unique feature of the rank-2 tensors in anygeneral D-dimensional spacetime.

4. The above expression for Nantisym in Eq. (4.42) is also suggestive from the point of view of its validityonly for tensors of rank p D. This is evident from the reduced expression:

Nantisym = D!p! (D p)! =

D (D 1) (D 2) . . . (D p + 1)p!

, (4.45)

which vanishes whenever p = D + 1. Therefore, for any tensor whose rank exceeds the dimensionality,

antisymmetrization identically leads to zero. For e.g., in 3-dimensions (i.e., D = 3), the maximum rank ofa completely antisymmetric tensor is 3 (i.e., the tensor is T[ijk]), and the tensor has only one non-vanishingindependent component T123. The other non-vanishing components are all equal to it magnitude (e.g.,

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4 TENSOR ANALYSIS

T231, |T213|). The reason is that complete antisymmetry implies all the indices of a given tensor must bedifferent. So when the rank equals the dimensionality the number of tensor indices equals the numberof dimensional indices, and there is only one choice for all the tensor indices to be unequal. Thus, in(3 + 1)-dimensions (i.e., D = 4), the maximum rank of a completely antisymmetric tensor is 4 (i.e., thetensor is T[]), and the only non-vanishing independent component of the tensor is T0123.

4.3.5 The Levi-Civita Tensor and the Generalized Kronecker Delta

In D-dimensions, the maximum ranked completely antisymmetric tensor is proportional to the Levi-Civitatensor defined by

T[12...D] 12...D =

1 for 12 . . . D an even permutation of 01 . . . D+1 for 12 . . . D an odd permutation of 01 . . . D

0 otherwise

, (4.46)

T[12...D] 12...D =

+1 for 12 . . . D an even permutation of 01 . . . D

1 for 12 . . . D an odd permutation of 01 . . . D

0 otherwise

. (4.47)

The following contraction identities exist between and in (3 + 1)-dimensions:

= 4! , = 3! , = 2! , = , (4.48)where

=

, =

, . . . , (4.49)are the Generalized Kronecker Delta tensors.

4.3.6 Tensor Duality

In D-dimensions the total number of independent components of a totally antisymmetric tensor of rank p isthe same as that of a totally antisymmetric tensor of rank (D p) due to the fact that

N(D, p) = D!p! (D p)! =

D!

(D p)! [D (D p)]! = N(D, D p) . (4.50)

This implies that there exists a duality between a rank p totally antisymmetric tensor and a rank (D p)totally antisymmetric tensor in D-dimensions. Such a duality relationship is given by

A12...p = A[12...p] = p! 12...pp+1...DAp+1...D , (4.51)

where Ap+1...D is called the (Hodge-)dual of A12...p . The inverse duality relation is given by

A12...p =A[12...p] =

1

p!12...pp+1...D A

p+1...D . (4.52)

Most appropriately, the definitions hold for the covariant and contravariant Levi-Civita tensor densities 12...D and12...D . Tensor densities are the most general entities which follow linear, homogeneous laws of transformation under thetransformation of coordinates, and tensors form only a sub-class of tensor densities. The difference between a tensor and tensordensity is that the latter involves in its transformation law an extra factor depending on a certain integral power W (called theweight) of the Jacobian |x/x|:

T =

xxW

x

x

x

x T .

However, since the Jacobian is identically equal to unity for the Lorentz transformations in Special Relativity, the above transfor-mation law for the tensor density T reduces to the transformation law for a tensor T

:

T

=

x

x

x

x T

.Accordingly, there is no distinction between the Levi-Civita tensor density 12...D of weight W = 1 and the tensor 12...D ,and also between the Levi-Civita tensor density 12...D of weight W = +1 and the tensor 12...D , in Special Relativity.

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4.4 Tensor Calculus

The covariant indices of A... andA... could be raised to the contravariant indices in the usual way using

the metric tensor g. For e.g.,

In 3-D:Aijk = 3! ijk

A = Aijk = 3! ijk A , Aijk = 13!

ijk A = Aijk = 13!

ijk A ,

Aij = 2! ijk Ak = Aij = 2! ijk Ak , Aij = 12! ijk Ak = Aij = 12! ijk Ak ,Ai = ijk

Ajk = Ai = ijk Ajk , Ai = ijk Ajk = Ai = ijk Ajk ,A = ijk

Aijk = ijk Aijk ,A = ijk A

ijk = ijk Aijk . (4.53)

In (3 + 1)-D:

A = 4!A = A = 4! A , A = 1

4! A = A = 1

4! A ,

A = 3!A = A = 3! A , A = 1

3! A

= A = 13!

A ,

A = 2!A =

A = 2! A ,

A =1

2! A

=

A =

1

2! A ,

A = A = A = A , A = A = A = A ,

A = A = A ,

A = A = A . (4.54)

4.4 Tensor Calculus

4.4.1 Tensor Differentiation: Grad, Div, Curl and DAlembertian operators

Differentiation of tensors in (3 + 1)-D is analogous to that in ordinary 3-D. We consider only the SpecialRelativistic scenario, where the component structure of the differential operators are as follows:

Covariant and Contravariant Differential operators:

x

=

x0

, xi (0, ) ,

x=

x0

, xi

0, . (4.55)where /xi is the ordinary three-gradient operator. Note that the above co(contra)variant operators havecomponent structures just opposite to the co(contra)variant four-vectors, viz., A =

A0, A

, A =

A0, A

.

Covariant Gradient: For a four-scalar (xk, t), the (covariant) gradient is defined by

x

=

x0,

=

1

c

t,

. (4.56)

Covariant Divergence: For a contravariant four-vector A(xk, t) (i.e., a contravariant four-tensor of rank 1),the (covariant) divergence defined by

A A

x= A

0

x0+ A

i

xi= 1

cA

0

t+ A , (4.57)

is a four-scalar (i.e., a four-tensor of rank 0). Similarly for a contravariant four tensor T of rank 2, the(covariant) divergence defined by

T T

x=

T0

x0+

Ti

xi=

0T00 + iT

i0

,

0T0j + iT

ij

, (4.58)

is a contravariant four-vector (i.e., a contravariant four-tensor of rank 1).

In general, for any contravariant four-tensor of rank m, the operation of (covariant) divergence yields acontravariant four-tensor of rank (m 1).

The contravariant divergence is similar to the covariant divergence, e.g.,A

= A , T = T , T

= T , . . . . (4.59)

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4 TENSOR ANALYSIS

Covariant Curl: For a covariant four-vector A(xk, t), the (covariant) curl is a contravariant antisymmetric

tensor of rank 2, defined by

C = C[] =1

2 (A A) = A . (4.60)

Recall this from the analogy of the ordinary three-curl of a three-vector Ak(xl) in 3-D, being a three-vector

itself with the i-th component as (j Ak kAj), i.e., A

i=

1

2ijk (j Ak kAj ) = ijk jAk . (4.61)

The DAlembertian Operator: This is the four dimensional version of the Laplacian operator, defined as

2 = 00 + ii = 1

c2d2

dt2 2 . (4.62)

4.4.2 The Four Volume and Tensor Integration

The invariant Four Volume element: The volume element in (3 + 1)-dimensions is given by

dV = d4x dt d3x . (4.63)

Under a general coordinate transformation: x x = x(x0, x1, x2, x3) [cf. Eq. (2.64)], the volume elementtransforms to:

dV = d4x =xx

d4x . (4.64)i.e., d4x is actually a scalar density (or, a tensor density of rank 0).

Now, by the transformation rule of the metric tensor g we have

g =x

xx

xg = g det

g

= xx

2 det (g) xx

2 g . (4.65)Therefore, using this relation, we find the invariant four-volume element:

g dV = g d4x g dt d3x , (4.66)

which is a scalar (or, a tensor of rank 0).

In Special Relativity, g = det () = 1, and the invariant four-volume element is simply d4x = dt d3x.

Tensor Integration: In principle, integrations of tensors over the invariant four-volume yield tensor-likequantities. However, one may define surface and volume integrals, by defining infinitesimal surface and volumeelements, in a (3 + 1)-D spacetime, in a similar way as in ordinary 3-D space. One may also have the general-izations of the three-space Divergence and Stokes theorems in (3 + 1)-D. However, these are beyond the scopeof these lectures.

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5 Special Relativistic Particle Dynamics

5.1 Four Velocity

In a covariant formalism one can construct a four vector, whose spatial components are proportional to thecomponents of the ordinary three-velocity of a particle (or a system of particles) and the temporal componentis proportional to the invariant speed of light c. The contravariant projections of this four velocity or worldvelocity vector are:

u =dx

d=

dx

dt/=

dx0

dt,

dxi

dt

= (c, v) . (5.1)

Since the proper time is invariant under a Lorentz transformation (LT), the four velocity u transforms likethe coordinate differentials dx under LT.

The norm of u is a universal constant:

uu = 2 c2 |v|2 = c2 . (5.2)

5.2 Four Acceleration

In a covariant formalism one can define the four acceleration, or the world acceleration of a particle as therate of change of the four velocity w.r.to the proper time along the particles world line. The contravariantprojections of the four acceleration are:

a =du

d=

du

dt=

c , v + v

. (5.3)

Now,

=

d

dt

1 v

v

c21/2

= 3 v

v

c2 . (5.4)

So

a = 2

2 v v

c, v + 2

v v

v

c2

. (5.5)

One may explicitly verify thataa

< 0 and au = 0 . (5.6)

5.3 Four Momentum and Energy

In a covariant formalism one can define the four momentum of a particle as the product of the particlesrest mass m0 and its four velocity. The spatial components of the four momentum correspond to those for therelativistic three momentum of the particle, and the temporal component is proportional to the total relativisticenergy. The contravariant projections of the four momentum are:

P = m0 u = m0

dx

d= (m0c, m0v) = (mc, mv) =

Ec

, ppp

, (5.7)

wherem m0 : Relativistic mass ,ppp = mv = m0v : Relativistic three-momentum ,

E = mc2 = m0c2 : Relativistic total (i.e., rest + kinetic) energy .The interpretations ofppp and Eas relativistic momentum and energy come from the following considerations:

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5 SPECIAL RELATIVISTIC PARTICLE DYNAMICS

Expanding ppp and Ein powers of (v/c), one finds that in the non-relativistic limit (v/c 0):ppp m0v : Non-relativistic three momentum ,

E m0c2 + 12

m0v2 : Rest energy + Non-relativistic kinetic energy . (5.8)

ppp and Ecan be shown to be the only functions of the velocity v, whose conservations are Lorentz invariant. The conservation of the ppp under a Lorentz transformation (LT) necessarily implies the conservation ofE.

This is shown in the following subsection.

The norm of the four momentum P is given by

PP = P

P = 00

P02

+ ijPiPj =

E2c2

|ppp|2 . (5.9)But, since

PP = m20uu

= m202

c2 v2 = m20c2 , (5.10)we have

E2

= |ppp|2

c2

+ m2

0c4

. (5.11)Therefore, for massless particles the total energy is simply E= |ppp| c.

5.4 Conservation of Energy and Momentum

Since the four momentum P = m0 dx/d transforms as a four vector under a LT (due to the fact that m0

and are invariants and dx is a four vector), in any reaction the change of the sum of the four momenta P

of all the involved particles (say n in number), also transforms as a four vector:

P = P =

n

P(n) =

n

P(n) . (5.12)

Now, the conservation of the relativistic three-momentum ppp under a LT means that:

n

ppp(n) = 0 in one inertial frame =

n

ppp(n) = 0 in all other inertial frames .i.e.,

n

Pi(n) = 0 =

n

Pi(n) = 0 = i

n

P(n)

= i0

n

P0(n) + ij

n

Pj(n)

. (5.13)

Since ij are constants (i, j = 1, 2, 3),

n Pi(n) = 0 implies that the last term on the r.h.s. of the above

equation is automatically zero. So we are left with

i 0

n

P0(n) = 0 =

n

P0(n) = 0 =1

c

n

E(n) , (5.14)

as i

0 (i = 1, 2, 3) are not necessarily zero.Thus, the conservation of the relativisitic three-momentum p, implies conservation of the total relativistic

energy E, and both in tern imply conservation of the four momentum P. In other words,

n

P(n) = 0 i.e.

n

P(n)I =

n

P(n)F follows from:

(i)

n

ppp(n) = 0 i.e.

n

ppp(n)I =

n

ppp(n)F and (ii)

n

E(n) = 0 i.e.

n

E(n)I =

n

E(n)F ,

(5.15)

where the subscript I (F) denotes initial (final) configuration. The conservation of the total relativistic energy,by definition, implies the conservation of the total relativistic mass:

n

m(n) = 0 i.e.

n

m(n)I =

n

m(n)F [m = m0] . (5.16)

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5.4 Conservation of Energy and Momentum

Illustrative Example: Compton effect

A photon hits a stationary particle of rest mass m0 as shown in Fig. 9. We need to determine the angle ofdeflection of the photon in terms of its initial and final energies (EI and EF respectively), or its initial andfinal frequencies (

Iand

Frespectively).

Recoil

EI

EF

m0

Figure 9: Illustration of the Compton effect.

Initial and final four momenta (P(I) and P(F) respectively) of the photon are given by

P(I) =

EIc

, pppI

, P(F) =

EFc

, pppF

, (5.17)

where pppI and pppF are the initial and the final (relativistic) three-momenta of the photon. As, for the photon,EI = c pppI and EF = c pppF, we have

P(I) = (|pppI| , pppI) , P(F) = (|pppF| , pppF) . (5.18)

For the massive particle, since its initial energy is equal to its rest energy m0c2, its initial four momentum (P(I)say) is given as

P(I) =

m0c , 0

. (5.19)

Let its final four momentum be P(F). Then since the norm of the four momentum is a Lorentz invariantquantity, we have

P(F)P(F) = P(I)P

(I) = m

20c

2 . (5.20)

Now, by the conservation of four momenta

P(I) + P(I) = P

(F) + P

(F) = P(F) = (m0c + |pppI| |pppF| , pppI pppF) , (5.21)

using Eqs. (5.18) and (5.19). Again, by the Eq. (5.20)

P(F)P(F) = m

20c

2 = (m0c + |pppI| |pppF|)2 + |pppI pppF|2

= 2m0c (|pppI| |pppF|) + (|pppI| |pppF|)2 = |pppI|2 + |pppF|2 + 2 pppI pppF= 1|pppI|

1|pppF|=

2

m0csin2

2, (5.22)

where is the angle between pppI and pppF.Since, EI = c pppI = h I and EF = c pppF = h F, h being the Plancks constant, we finally obtain

= 2 s in1

m0c22h

1

F

1I

1/2. (5.23)

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5 SPECIAL RELATIVISTIC PARTICLE DYNAMICS

5.5 Four Force

In a covariant formalism, the four force on a particle as the rate of change of the four momemtum w.r.to theproper time along the particles world line. The contravariant projections of the four force are:

F = dP

d= dP

dt=

Ec

, FFF

, (5.24)

whereFFF = ppp dppp

dt: Relativistic three-force . (5.25)

By defintion

F = ma = m0a = m0 du

dt, (5.26)

which can be regarded as a relativistic generalization to the Newtons second law of motion in classical me-chanics.

Now, for a particle moving with instantaneous velocity v, the time derivative of the total energy is equalto the total power dissipated:

E = FFF v = ppp v . (5.27)Therefore,

F =

Ec

, FFF

=

FFF v

c, FFF

. (5.28)

Moreover, since the components of the four-velocity are given by: u0 = c and u = v, we have

F

=

1

c

FFF u , u0

FFF

. (5.29)

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6 Special Relativistic Electrodynamics

6.1 Four Current Density

Consider a system ofn charged particles, with fixed charges q(n) at locations x(n)(t). The ordinary three-chargedensity and three-current density are defined respectively by

(x, t) =

n

q(n) 3

x x(n)(t)

jjj (x, t) =

n

q(n) 3

x x(n)(t) dx(n)(t)

dt

(6.1)

where 3

x x(n)

is the three-dimensional Dirac delta function. In analogy with the three-current densityjjj (x, t) one may define the four current densityvector J(x, t) which unites both the densities and jjj as follows:

J(x, t) = n

q(n) 3 x x(n)(t)

dx(n)(t)

dt

. (6.2)

To verify that this is indeed a four vector, we set x0(n)(t) = ct ( n), and use the property of the delta function:dx f(x) (x y) = f(y) , (6.3)

to write J(x, t) asJ(x, t) =

dx0(n)

n

q(n)

x0 x0(n)

3

x x(n)(t) dx(n)(t)

dx0(n)

/c

= c

cdt

n

q(n) 4

x x(n)(t) dx(n)(t)

cdt

= J(xi, t) = c

d

n

q(n) 4

x x(n)()

dx(n)()d

. (6.4)

As the four-dimensional delta function 4

x x(n)

()

is a scalar, and so is the proper time variable ,

whereas dx(n) is a four vector, J is a four vector. One may easily check from the above expression (6.2) for

J that its temporal and spatial components are proportional to and jjj, given by Eqs. (6.1), respectively:

J(xi, t) =

c

xi, t

, jjj

xi, t

. (6.5)

One may also verify, by using the continuity equation, that the four current density is covariantly conserved,i.e., its four divergence is zero:

J = t

+ jjj = 0 . (6.6)

6.2 Covariant Lorentz Force Equation

In three dimensions, the electromagnetic force on a particle moving with instantaneous velocity v, and carryinga charge q is given in terms of the well-known 3-dimensional Lorentz force equation:

FFFL

= q

EEE +

1

cv BBB

. (6.7)

In a covariant formalism, since the four force is given by Eq.( 5.28), we have the covariant Lorentz force equation

FL

=

FFFL v

c, FFF

L

=

q

c

EEE v,

c EEE+ v BBB

=

q

c

EEE u,

u0 EEE+ u BBB

, (6.8)

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6 SPECIAL RELATIVISTIC ELECTRODYNAMICS

u0 and u being the temporal and spatial components of the four velocity u =

u0, u

= (c, v).One may verify that in compact form the above equation can be expressed as ( q/c) times the covariant

dot product of a second rank antisymmetric tensor F, constructed out of the electric and the magnetic fieldvectors, and the four velocity vector u:

FL =

q

c F

u . (6.9)

Explicitly, the form of F, which is referred to as the electromagnetic field tensor, is given by

F : F00 = 0 , F0i = Fi0 = Ei , Fik = Fki = ijk Bj (6.10)

= F = F =

0 E1 E2 E3E1 0 B3 B2E2 B3 0 B1E3 B2 B1 0

0 Ex Ey EzEx 0 Bz ByEy Bz 0 BxEz By Bx 0

. (6.11)

To see that F is indeed a tensor of rank 2, i.e., Eq. (6.9) is a tensorial equation which encodes Lorentz force

equations in every Lorentz frame, we suppose Eq. (6.9) to hold in a Lorentz frame . In another Lorentz frame, we have the transformed equation

FL

=q

cF u . (6.12)

Now,

FL

=dP

d=

x

xdP

d=

x

xFL

. (6.13)

Also

u =x

xu or u =

x

xu . (6.14)

Substituting this in Eq. (6.13), and using Eqs. (6.9) and (6.12), we have

FL

=q

cF u =

x

xFL

=x

x

qc

Fu

=

x

x

q

cF

x

xu

=

F x

xx

xF

u = 0 = F =x

xx

xF , (6.15)

as u is arbitrary. Thus F transforms like a rank-2 tensor.

6.3 Covariant formulation of Maxwells Equations

Consider a homogeneous, isotropic and non-dispersive medium in which the Maxwells equations for the electricand magnetic fields

EEE(x, t), BBB(x, t)

, produced by a charge density (x, t) and a current density jjj(x, t), are:

(i) EEE = 4

(ii) BBB EEE

c=

4

cjjj

(iii) BBB = 0

(iv) EEE +BBB

c= 0

or, in index notation:

(i) iEi =

4

cJ0

(ii) ijk iBj + 0Ek = 4

cJk

(iii) iBi = 0

(iv) ijk iEj 0Bk = 0

, (6.16)

where the overhead dot {} denotes /t and 0 = (1/c)/t.

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6.3 Covariant formulation of Maxwells Equations

The first two of the Maxwells equations (6.16), that involve the sources J0 = c and Jk =

jjjk

, k = 1, 2, 3,

can be expressed in terms of the components of the electromagnetic field tensor F [Eq. (6.10)] as

(i) iFi0 =

4

cJ0 , (ii) iF

ik + 0F0k =

4

cJk . (6.17)

Similarly, the last two of the Maxwells equations (6.16), that do not involve the sources, can be expressedin terms of the components of a second rank antisymmetric tensor F (which like the electromagnetic field

tensor F, is also constructed out of EEE and BBB components), as

(i) iFi0 = 0 , (ii) iF

ik + 0F0k = 0 . (6.18)

The components of F are given explicitly by

F : F00 = 0 , F0i = Fi0 = Bi , Fik = Fki = ijk Ej (6.19)

= F = F =

0

B1

B2

B3

B1 0 E3 E2B2 E3 0 E1B3 E2 E1 0

0

Bx

By

Bz

Bx 0 Ez EyBy Ez 0 ExBz Ey Ex 0

. (6.20)

The covariant tensors F and F can be obtained respectively from F and F as:

F = F = F : FEEE EEE, BBB BBB F , (6.21)

F = F = F : FEEE EEE, BBB BBB F . (6.22)

Using these, and the explicit forms of F and F, given by Eqs. (6.11) and (6.20), one may verify that F

is indeed identical to the dual of F:

F F = 12

F or F = 2 F = 2

F . (6.23)

Moreover, the above two Maxwells equations involving the sources, viz., Eqs. (6.17 i, ii), whose l.h.s. containthe three-divergence or(and) time-derivative of the components of F, can be encoded in a single tensorialequation whose l.h.s. is the four-divergence of F. Similarly, the two source-free Maxwells equations, viz.,Eqs. (6.18 i, ii), can be encoded in a single tensorial equation whose l.h.s. is the four-divergence of F. Thatis, in the covariant formalism the Maxwells equations are expressed as:

(i) F =

4

cJ

(ii) F = 0

, (6.24)

where J =

c, jjj

is the four current density. Eq. (6.24 ii) is in fact an identity, known as the Maxwell-Bianchi

identity, and can also be expressed like a Jacobi identity:

F =

1

2 F = 0 = F + F + F = 0 . (6.25)

In absence of sources ( = 0,jjj = 0), the above two Maxwells equations (6.24) in covariant form, are indistin-guishable from each other under the transformation:

F F i.e. EEE BBB, BBB EEE , (6.26)which is therefore called the electromagnetic duality transformation.

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6.4 Transformation of Electromagnetic Fields

In Special Relativity, F and F being rank-2 tensors, follow the transformation rules

F =

F , F =

F , (6.27)

where

are the Lorentz transformation matrix elements, which for a generic Lorentz boost, in an arbitrary

direction, are given by (see 2.5):

00 = , i

0 = i , 0i = ikk = i ,ij =

ij +

( 1)||2

jk ik = ij +

( 1)||2

ij. (6.28)

Using theseFi0 = Ei = i

0 F

= i0 0

k F0k + ij

00 F

j0 + ij 0

k Fjk

=

EEE + BBB

i

1

|

|2

EEE

i , (6.29)

Fi0 = Bi = i 0

F = i0

0k

F0k + ij 0

0Fj0 + ij

0k

Fjk

=

BBB EEEi

1

||2

BBB

i . (6.30)

As2 =

1

1 ||2= ||2 =

2 12

, we have 1||2

=

+ 1. (6.31)

Hence, the expressions (6.29) for the transformed electric and magnetic fields reduce to

EEE

=

EEE + BBB

2

+ 1 EEE

BBB

=

BBB EEE

2+ 1

BBB

. (6.32)

For a single Lorentz boost along, say, the x1 x-axis, i.e., = x, we have from the above equations (6.32)

Ex = Ex

Ey = (Ey Bz)Ez = (Ez + By)

and

Bx = Bx

By = (By + Ez)

Bz = (Bz Ey). (6.33)

So, a purely electric or a purely magnetic field in one reference frame, is in general a combination of electricand magnetic fields in another.

6.5 Electromagnetic Scalar invariants

In principle, one can construct three independent four-scalar quantities using the electromagnetic field tensoror(and) its dual, viz., FF

, FF. However, only two of these are actually linearly independent:

(i) FF = F F = 2

BBB2 EEE2 , (ii) F F = FF = 4 EEE BBB . (6.34)Therefore,

I. If EEE = BBB in one Lorentz frame , then EEE

= BBB

in every other Lorentz frame .II. If EEE BBB in one Lorentz frame , then EEE BBB in every other Lorentz frame , unless EEE or BBB vanishesidentically.

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6.6 Potential formulation of Electrodynamics


6.6.1 Electromagnetic Four Potential

As pointed out in 6.3, the Maxwell-Bianchi identity, viz. Eq. (6.24 ii), is equivalent to vanishing of the sumof all the cyclic permutations of F :

F = 0 = F + F + F = 0 [cf. Eq. (6.25)] .

This has a simple solution of the form

F = A A , (6.35)

where A :=

, A

is referred to as the electromagnetic four vector potential. Its temporal part is called the

electromagnetic scalar potential , whereas the spatial part is the electromagnetic three-vector potential A. The

covariant vector A has components

, A

.

Verification:

F + F + F= A A + A A + A A = 0 .

In three-vectorial notation, the relations between the electric and magnetic vectors (E and B) and the scalarand three-vector potentials ( and A), can be obtained from the above equation (6.35) as follows:

F0i = Ei = 0Ai iA0 = E = A

c, (6.36)

F0i = Bi =1

20ijk Fjk =

1

20ijk (j Ak kAj) = 0ijk jAk = B = A . (6.37)

Now, substituting Eq. (6.35) in the Maxwells equation (6.24 i) [see

6.3], one obtains the evolution equationfor the four vector potential:

F =

A A

=

4

cJ

= A A = 4c

J

= 2A (A) = 4c

J . (6.38)

6.6.2 Gauge Transformations

Under the transformation

A A = A + a , (x, t) := Arbitrary scalar function ofx and t , (6.39)

the r.h.s. of Eq. (6.38) is unaffected, whereas the l.h.s. transforms to

2A A = 2A + 2 (A + )

= 2A + 2 A 2 = 2A A ,

i.e., the l.h.s., and hence the entire equation (6.38), is invariant. This implies that there is a freedom to choosethe gauge (i.e., to impose a condition on the hitherto arbitrary scalar function ) while solving the equation(6.38) for the four vector potential A, and hence to determine the measurable quantities electric and

magnetic fieldsE,

B using the Eqs. (6.36) and (6.37). As such, the above transformation (6.39) is referredto as the Gauge transformation, the potential A as the Gauge potential, the fields E, B as the Gauge fields,

the function as the Gauge function, and the condition on (required to be imposed) as the Gauge condition.

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6.6.3 Gauge conditions: Lorentz and Coulomb gauges

In the following, we will discuss two popular gauge conditions which are of most importance in describingphysical phenomena associated with the gauge fields.

Lorentz gauge condition:

If we choose to be such that

2 = A , (6.40)then

A

AA+A

+ 2 = 0 . (6.41)

The above condition (6.40) is therefore equivalent to

A = 0 i.e., A +

c= 0 . (6.42)

This is known as the Lorentz gauge condition. In this gauge, the above equation (6.38) reduces to

2

A

=

4

c J

. (6.43)

Recalling that J :=

c,jjj

, the temporal and spatial components of the above equation (6.43) are given as

(a) 2

1

c22

t2 2

= 4

(b) 2 A

1

c22

t2 2

A =

4

cjjj

. (6.44)

These are the inhomogeneous wave equations satisfied by the scalar potential and the components of thethree-vector potential A in the Lorentz gauge.

Coulomb gauge condition:

If we now choose to be such that

2 = A i.e., ii = iAi , (6.45)then

iAi

AiAi+iiA

i + ii = A 2 = 0 . (6.46)

The above condition (6.45) is therefore equivalent to

iAi = 0 i.e., A = 0 . (6.47)

This is known as the Coulomb gauge condition. In this gauge, Eq. (6.38) reduces to

2A

c

=

4

cJ . (6.48)

It is easy to verify that the temporal and spatial components of this equation are given by

(a) 2 = 4

(b) 2 A +

c=

4

cjjj

. (6.49)

Eq. (6.49 a) is the time-dependent Poisson equation satisfied by the scalar potential and Eq. (6.49 b) is a

coupled second order differential equation in and the components of the three-vector potential

A.Or, more appropriately the Lorentz-Lorentz gauge condition, after Hendrik Antoon Lorentz who, in the course of demonstrating

the covariance of Maxwells equations in 1903, rediscovered this condition already found in 1867 by Ludvig Valentin Lorentz.

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One way to work out the solution for A is to solve first the equation (6.49 a) for , and then to substitute thatsolution in (6.49 b) and solve for A. However, this is a tedious and cumbersome procedure. There is in fact aeasier way out:

Split the three-current density jjj in longitudinal and transverse parts (jjj andjjj) as

jjj = jjj + jjj , jjj = 0 , jjj = 0 . (6.50)

Now, by the charge-current continuity equation (6.6)

jjj = jjj = =

t

1

42

by Eq. (6.49 a)

= jjj =1

42 = 1

4

=

jjj 1

4

= 0 . (6.51)

Moreover,

jjj

1

4 = 0 . (6.52)

Hence = 4 jjj , (6.53)substituting which in Eq. (6.49 b) we finally obtain the inhomogeneous wave equation satisfied by the three-vector potential A in the Coulomb gauge:

2 A

1

c22

t2 2

A =

4

cjjj . (6.54)

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7 WAVE SOLUTIONS IN ELECTRODYNAMICS THE RETARDED POTENTIALS

7 Wave Solutions in Electrodynamics the Retarded Potentials

In this section, we will focus on determining the solutions of the generic wave equations for the gauge potentials and A in Lorentz and Coulomb gauges. We will first discuss the non-covariant approach of obtaining thesolutions for and the individual components of A, in these gauges. The wave equation for in the Coulomb

gauge being simply the time-dependent Poissons equation (6.49 a), it is easier to obtain its solution, thanin the Lorentz gauge in which satisfies the inhomogeneous wa

sr lectures.dvi

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