Phys 231A Fall 2014
Assignment 3 Solutions
Problem 1
We are to show four properties of the vector commutator,
[X, Y ]f = X(Y (f ) Y (X(f ),
namely
(a) linearity,
(b) the Leibnitz rule,
(c) the component formula,
(d) transformation as a vector eld .
Let a,
Phys 231A Fall 2014
Assignment 6 Solutions
Problem 1 (Carroll Chapter 4, problem 6)
If Aa is a Killing vector then
Fab =
a Ab
b Aa
= 2
We must show the vacuum Maxwell equations
satised.
To show
aF
ab
a Ab
aF
ab
.
= 0 and
[a Fbc]
= 0 are
= 0 we evaluate
aF
Phys 231A Fall 2014
Assignment 7 Solutions
Carroll, Chapter 7, Problem 2
(a) In Lorenz gauge, the linearized Einsteins equations look like (Carroll (7.125)
h = 16T
We can recover Maxwells equations1 A = J from this if we make the sub
stitutions h0 A as in
Phys 231A Fall 2014
Assignment 5 Solutions
Problem 1 (Carroll Chapter 3, problem 12)
(a) Consider:
[
,
]K
=
K
K
= R K .
Using the Killing equation we can also write this as:
K
+
K
= R K .
(1)
Now, we will write two more versions of equation (1) with
Phys 231A Fall 2014
Assignment 4 Solutions
Problem 1
Lets do this the fast way - to set up the variation principle, let
L d
I=
g
dx dx
d =
d d
2 (x2 t2 ).
Here dots denote derivatives with respect to . Varying L with respect to x we nd
L
= 2(x ) (x2 t2 )
Phys 231A Fall 2014
Assignment 1 Solutions
Problem 1 (Tensors)
(a) Let cfw_vi with i = 1, . . . , n be a basis of V. We must nd a basis of V and show
that it also has n elements. Dene the n dual vectors i V with i = 1, . . . , n by
i (vj ) = i j . We sh
Phys 231B Winter 2015
Assignment 2 Solutions
1. Using r as the parameter, the (t, r) components of the tangent to a radial null geodesic
are = (1/f, 1). It follows that (/t) = 1. Since this is a constant (and
not zero), r must be an ane parameter. We are
Phys 231B Winter 2015
Assignment 1 Solutions
Problem 1
(a) Lets begin with the metric
2M
ds = 1
r
2
1
2M
dt + 1
r
2
dr2 + r2 d2 .
For geodesics that are radial (d2 = 0) and null (ds2 = 0), this gives
1
2M
dt = dr 1
r
where we chose the overall sign for
Phys 231B Winter 2015
Assignment 7 Solutions
1. We rst want to nd the redshift z such that for an object of angular size (z) and
physical length L, the ratio (z)/L is minimized. We begin by noting a couple of
equations from Carroll. Equation (8.125) denes
Phys 231B Winter 2015
Homework 8 Solutions
Problem 1
True or false: If the futures of two points do not intersect then neither do their pasts, i.e.
?
I + (p) I + (q) =
I (p) I (q) =
=
False. Here is a counter example in Schwarzschild spacetime. Consider
Physics 231B Winter 2014
Prof. Gary Horowitz
Assignment 4 Solutions '
1. Here are the Penrose diagrams 2. The metric of the hypersurface t : constant, 7' = r+ 2 M + \/ M 2 a2 is given by
setting dt 2 d7 = 0 and 7 = r+ in the Kerr metric. This gives
(7
Phys 231B Winter 2015
Assignment 3 Solutions
1. The Reissner-Nordstrom metric is
ds2 = 1
2M
Q2
+ 2
r
r
dt2 + 1
2M
Q2
+ 2
r
r
1
dr2 + r2 d2 .
(1)
The acceleration of an observer with 4-velocity ua is ab = ua a ub . Since a = (/t)a =
a
t is a Killing vect
Phys 231B Winter 2015
Assignment 6 Solutions
1. An isotropic spacetime is lled by a congruence of timelike curves with tangents ua .
Isotropy implies that it is impossible to construct a geometrically preferred tangent
vector orthogonal to ua . Thus, the