Prof J.J.

Binney 4th year: Option C6

Classical Fields III

1. The standard covariant derivative, ∇µ pν = ∂µ pν + Γνλµ pλ , acts on 4-vectors that inhabit the four-
dimensional “tangent space” of the space-time manifold. In particle physics other vector spaces are
associated with each event. For example, a complex scalar field ψ associates with each event x a point
in the complex plane – a two-dimensional vector space. Let e1 and e2 be two unimodular complex
numbers. Show that we can write ψ = ψ 1 e1 + ψ 2 e2 , where the ψ a are real numbers.
If we make a different choice of basis numbers ea at each event x, ∂µ ψ a will not vanish even if ψ is
the same everywhere. To detect this hidden equality we define a connection
Dµ ψ a = ∂µ ψ a + Γabµ ψ b ,
where Γµ is a 2 × 2 matrix.
In quantum mechanics an e.m. field affects the dynamics through the replacement of the usual
momentum operator by pµ = −ih̄{∂µ − i(q/h̄)Aµ }. Show that for an appropriate choice of Γµ this can
be written pµ = −ih̄Dµ .
2. The curvature tensor is most conveniently defined by (∇µ ∇ν − ∇ν ∇µ )Z α = Rα βµν Z β , which holds
for any field Z. From this definition derive an expression for R in terms of the Christoffel symbols.
In the notation of the previous problem, we define the curvature tensor for a scalar complex field
through (Dµ Dν − Dν Dµ )ψ a = Ra bµν ψ b . Assume that, as in the previous problem, summation over the
index of ψ can be absorbed into complex multiplication, so we can write simply (D µ Dν − Dν Dµ )ψ =
Rµν ψ. Show that Rµν = −i(q/h̄)Fµν , where Fµν = ∂µ Aν − ∂ν Aµ is the Maxwell field tensor.
3. With coordinates xµ = (t, r, θ, φ) the Schwarzschild metric may be written
 2
−c D 0 0 0

 D ≡ 1 − rs ,

 0 D−1 0 0 r
gµν =   where

0 0 r2 0  r ≡ 2GM/c2 .
2 2 s
0 0 0 r sin θ
Show that the only non-vanishing Christoffel symbols of the form Γtµν are
D0
Γtrt = Γttr = .
2D
From the equation of motion of a photon of momentum h̄k, show that in the Schwarzschild metric
the time component ω ≡ k 0 of a photon’s 4-vector obeys
d(ωD) dxµ
= 0 where the photon’s path xµ (s) satisfies kµ = ,
ds ds
and give a physical interpretation of this equation.
4. Derive the form of the energy-momentum tensor associated with a uniform magnetic field of strength
B parallel to the x-axis. In which direction or directions does the field exert pressure?
5. A rope made of nylon of density ρ and cross-section A lies along the x-axis under tension F . Write
down the form of the energy-momentum tensor inside the rope. Show that requiring that the energy
density in the rope be positive for all observers, limits the permissible tension F .
6. A metric for the interior of a cosmic string is
ds2 = −c2 dt2 + r02 (dθ2 + sin2 θdφ2 ) + dz 2 ,
where r0 is a constant. Show that the only non-vanishing Christoffel symbols are
Γθφφ = − 21 sin 2θ and Γφθφ = Γφφθ = cot θ.
Given that the only non-vanishing components of the Ricci tensor are Rθθ and Rφφ and that the edge
of the string is at θ = θm , show that the tension in the string is F = c4 (1 − cos θm )/(4G).
 Classical Fields III 2

7. With (t, x, y, z) having their usual meanings, double-null coordinates for space-time are defined by

u = ct − x y0 = y
v = ct + x z0 = z .

Write down the Minkowski line element in double-null coordinates.

Consider the line element
ds2 = −du dv + f 2 dy 2 + g 2 dz 2 ,
where f (u) and g(u). Show that the only non-vanishing Cristoffel symbols are

Γvyy = 2f f 0 , Γvzz = 2gg 0 , Γyyu = Γyuy = f 0 /f , Γzzu = Γzuz = g 0 /g .

Hence, or otherwise, show that trajectories on which the spatial coordinates x, y, z are constant are
geodesics.
The metric’s Ricci tensor vanishes provided

f 00 g 00
+ = 0,
f g

where a prime denotes differentiation with respect to u. Show that this equation is satisfied by the
choice
u u
f (u) = 1 + Θ(u) , g(u) = 1 − Θ(u),
L L
where L is a constant and Θ(u) is the Heaviside step function that vanishes for u < 0 and is unity for
u > 0.
For the above choice of f and g, determine as a function of time the invariant distance between
particles that move on x = 0, y = 0, z = ±a, and similarly the distance between particles that move on
x = 0, y = ±a, z = 0.
Interpret your results physically.

8. The Robertson-Walker metric may be written

dr2
· ¸
2
ds2 = −dt2 + a2 + r 2
(dθ 2
+ sin θdφ 2
) .
1 − Kr 2

Explain the significance of the quantities a and K, and of the world-lines (r, θ, φ) = constant.
Show that photons can travel down curves (θ = constant, φ = constant).
Given that a = (t/t0 )2/3 for K = 0, find the distance now (t0 ) in the case K = 0 between us and a
galaxy from which we are currently receiving photons emitted at t1 .
Suppose the Universe is closed with the Earth at the point r = 0. A distant galaxy of radius R is
currently distance D from us with its centre on the line θ = 0. Show that its rim is at angular coordinate
√
(1 + z)R K
θ= √ .
sin(D K)

where z is the galaxy’s redshift. Simplify this formula for the case z ¿ 1 and discuss the difference
between the general result and this case.

9. Show that for any two vectors u, v we have

(uα ∇α )v β − (v α ∇α )uβ = [u, v]β ,

where the vector [u, v] is defined by

[u, v]β ≡ uα ∂α v β − v α ∂α uβ .
 Classical Fields III 3

For each fixed ², xα (τ, ²) defines a geodesic, with τ the affine parameter. Show that
· ¸β
dx dx
, = 0.
dτ d²

Show further that

dxγ dxν
(ẋα ∇α )(ẋβ ∇β ) = Rγ λµν ẋλ ẋµ ,
d² d²
where ẋ ≡ dx/dτ and the curvature tensor R can be taken to be defined by
³ ´
(uα ∇α )(v β ∇β ) − (v α ∇α )(uβ ∇β ) − [u, v]α ∇α wγ = Rγ λµν uµ v ν wλ ,

with u, v and w arbitrary vectors.

Two masses are dropped from points a small height ² apart. Show that just after they are released,
the separation δx between them satisfies

D2 δxγ
= c2 Rγ 00ν δxν ,
Dt2

where x0 ≡ ct. Hence show that the gravitational field at the Earth’s surface has the curvature compo-
nent
Rz 00z = 2g/(c2 R)
where z is an upwards directed coordinate, g is the usual acceleration due to gravity, and R is the
Earth’s radius.