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.