Homework # 4

Introduction to GENERAL RELATIVITY
Phys 469/569
Prof. Fulvio Melia
Problem Set 4
Problem 1: Derive the following explicit expressions for the components of the electromagnetic
energy-momentum tensor T
αβ
E
:
00
TEM = 1 (
E 2 + B 2)
8π
(the energy density of the electromagnetic field),
0i
TEM = 1
4π N i ,
where N = E × B is the Poynting vector, and
ij
TEM = 1 [
E i E j + B i B j − 1 4π
2 δ (
ij E 2 + B 2)] ,
the (three-dimensional) Maxwell stress tensor.
Problem 2: (a) Show that it is always possible to add a four-divergence to the Lagrangian
density,
L → L ′ = L + ∂ β F β ,
where F β is an arbitrary vector function, without altering the Euler-Lagrange equations
for the fields.
(b) Cast the Lagrangian density
− 1 4 ϵ 0c 2 F αβ F αβ
for the free electromagnetic field into the form
− 1 2 ϵ 0c 2 A α ,β
(
A
α ,β − A β ,α)
and, using the theorem in part (a), together with the Lorentz condition A α ,α = 0, show
that an acceptable form for the Lagrangian density is
− 1 2 ϵ 0c 2 A α ,β A α ,β

(the Fermi form, often used in quantum field theory). Verify that the Euler-Lagrange
equations for this modified Lagrangian density lead to the free-field equations for the
potential: (
⃗∇ 2 − 1 ∂ 2 )
c 2 ∂t 2 A α = 0 .
Problem 3: Show that in the case of a slow particle in a weak uniform static gravitational
field K pointing in the positive z-direction, the equation of motion
d 2 x α
dτ 2
reduces to the Newtonian free-fall equation
+ dx β dx γ
Γα βγ
dτ dτ = 0
d 2 z
dt 2 = K .
2