R ELATIVITY AND S PACETIME
S PRING 2014
Problem Set 3
1. Riemann curvature in low dimensions (LPPT 9.7)
In space of fewer than 4 dimensions, simple expressions can be given for the Riemann
curvature tensor.
(a) What is the Riemann tensor in a 1-dimensional space?
(b) Express the Riemann tensor for a 2-dimensional space in terms of the metric and
the Ricci scalar.
(c) Express the Riemann tensor for a 3-dimensional space in terms of the metric and
the Ricci tensor.
2. Killing vector and curvature (LPPT 10.7)
If X a is a Killing vector, prove that rb rc Xd = R a bcd Xa .
3. Energy-momentum tensor
The energy-momentum tensor Tab for a matter field can be computed by
Tab =
p
2
d p
gL M ,
g dg ab
where L M is the Lagrangian density for the matter field. Compute Tab for
(a) a scalar field f with the action,
S=
Z
4
d x
p
g
✓
1 a
(r f)(r a f)
2
1 2 2
m f
2
(b) a vector field A a with the action (Fab = ∂ a Ab ∂b A a ),
✓
◆
Z
p
1 ab
4
S= d x
g
F Fab .
4
◆
.
Verify that r a Tab = 0 in each example, provided that the equation of motion for the
matter field holds (dS/df = 0 or dS/dA a = 0).
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R ELATIVITY AND S PACETIME
P ROBLEM S ET 3
4. Spherically symmetric, static geometry
Consider the following ansatz for a spherically symmetric, static geometry:
ds2 =
e2F(r) dt2 + e
2Y(r )
dr2 + r2 (dq 2 + sin2 qdf2 ).
(a) In the coordinate basis, compute the components of the Levi-Civita connection
a ), the Riemann tensor ( R a
(Gbc
bcd ), the Ricci tensor (R ab ) and the Ricci scalar (R).
(b) Show that the energy-momentum conservation for perfect fluid in this geometry
implies (Tab = (r + p)Ua Ub + pgab )
(r + p)
dF
=
dr
dp
,
dr
and gives no further restrictions.
(c) Consider the vacuum Schwarzschild geometry: e2F = e2Y = 1 2GM/r.
Find a new coordinate r̃ (r ) which brings the metric to the form
⇣
⌘
2
2A(r̃ ) 2
2B(r̃ )
2
2
2
2
2
ds = e
dt + e
, dr̃ + r̃ (dq + sin qdf ) .
5. Angular momentum in Schwarzschild geometry (LPPT 15.1)
Prove that the total angular momentum squared, L2 = p2q + p2f / sin2 q, is a constant of
motion along any Schwarzschild geodesic.
6. Correction to Kepler’s third law
Recall that the period of an elliptic Kepler orbit is given by
2pa3/2
,
( GM)1/2
T=
where a is the semi-major axis of ellipse.
(a) Compute the leading post-Newtonian correction to this period.
Express the answer in a and the eccentricity # of ellipse.
(b) Estimate the numerical value of the correction for Mercury.
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R ELATIVITY AND S PACETIME
P ROBLEM S ET 3
7. Einstein ring
A special case of gravitational lensing (bending of light due to a massive spherical
body) occurs when the source, lens and observer are all exactly aligned. Let ds and
dl be the distance from the observer to the source and the lens, respectively. Assume
that ds and dl are both much larger than the ‘size’ of the lens, so that a small angle
approximation is valid; this is almost always true for astronomical objects. Assume
also that the source, lens and observer are all static in a common rest frame in an
asymptotically Minkowski spacetime. Show that the size of the Einstein ring in terms
of the angle measured by the observer is
s
4GM (ds dl )
qo =
.
ds dl
8. Signal from a particle falling into a black hole (Schutz 11.21)
A particle of mass m falls radially toward the horizon of a Schwarzschild black hole of
mass M. The geodesic it follows has the conserved ‘energy per mass’ e = 0.95.
(a) Find the proper time required to reach r = 2GM from r = 3GM.
(b) Find the proper time required to reach r = 0 from r = 2GM.
(c) Find its four-velocity components at r = 2.02GM.
(d) As it passes r = 2.02GM, it sends a photon out radially to a distant stationary
observer. Compute the redshift of the photon when it reaches the observer. Don’t
forget to allow for the Doppler part of the redshift caused by the particle’s velocity.
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