Classical Electrodynamics — PHY5347

HOMEWORK 16

(January 16, 2013)

Due on Thursday, February 14, 2013

PROBLEM 46: The Energy-Momentum Tensor

We have shown in class that the force per unit volume fµ transforms as a Lorentzvector defined as follows:

fµ =∂Pµ

∂t=

1

cF µνJν .

In this problem you will show that fµ may be written as the derivative of the energy-momentum tensor T µν . That is,

fµ =1

cF µνJν = −∂νT µν .

(a) Use the pair of inhomogeneous Maxwell’s equations in Lorentz covariant form toobtain the following expression for fµ:

fµ =1

4π

[∂α (F µνFαν)− (∂αF µν)Fαν

].

(b) Show that the second term on the right-hand side of the previous expressionmay be written as:

(∂αF µν)Fαν =1

2

(∂αF µν + ∂νFαµ

)Fαν .

(c) Show that the pair of homogeneous Maxwell’s equations may also be written inthe following Lorentz covariant form:

∂αF µν + ∂νFαµ + ∂µF να = 0 .

(d) Use the results from (b) and (c) to prove the following relation:

(∂αF µν)Fαν =1

4∂µ(FανFαν

).

(e) Given that fµ = −∂νT µν , show that the energy-momentum tensor is given by

T µν = − 1

4π

(F µαF ν

α −1

4gµνFαβFαβ

).

PROBLEM 47

You have shown in the previous problem that the energy-momentum tensor is givenby the following expression:

T µν = − 1

4π

(F µαF ν

α −1

4gµνFαβFαβ

).

In this problem you will compute all the elements of T µν by using standard matrixmultiplication techniques.

(a) Show that the energy-momentum tensor may be written exclusively in termsof F µν (i.e., the purely contravariant form of the electromagnetic tensor) asfollows:

T µν = − 1

4π

[F µαF νβgβα −

1

4gµν(FαβgαλF

λσgσβ

)].

(b) Treating F µν and gµν as elements of an antisymmetric matrix and a symmetricmatrix respectively, show that the energy-momentum tensor may be computedusing standard matrix multiplication techniques as follows:

T µν =1

4π

[(F gF

)µν− 1

4gµνTr

(F gF g

)].

(c) Using Mathematica, Maple, or any method of your choice, compute all theelements of T µν in terms of the electromagnetic fields E and B.

PROBLEM 48

Assume that the electric and magnetic fields in some region of space are uniform andorthogonal, with E=Ex and B=By.

(a) Find all the components of the energy-momentum tensor T µν .

(b) Find all the components of the energy-momentum tensor T ′µν in a frame that ismoving relativity to the first one with constant velocity v=vz.

(c) Show explicitly that the trace of the energy momentum tensor T µµ remains in-

variant.