UNIVERSITY OF CALIFORNIA, SANTA BARBARA
Department of Physics
Physics 231B
Winter 2015
Prof. Gary Horowitz
Grader: Philip Saad
ASSIGNMENT #5
Due Wednesday, Feb. 11 at 5pm (in Broida 6239)
1. Show that the vector
= /t + H /
with H = a/2M r+ , is null on t
UNIVERSITY OF CALIFORNIA, SANTA BARBARA
Department of Physics
Physics 231B
Winter 2015
Prof. Gary Horowitz
Grader: Philip Saad
ASSIGNMENT #6
Due Wednesday, Feb. 25 at 5pm (in Broida 6239)
1. Show that the stress energy tensor for any isotropic spacetime m
UNIVERSITY OF CALIFORNIA, SANTA BARBARA
Department of Physics
Physics 231B
Winter 2015
Prof. Gary Horowitz
Grader: Philip Saad
ASSIGNMENT #2
Due Wednesday, Jan. 21 at 5pm (in Broida 6239)
1. Show that r is an ane parameter along radial outgoing and ingoin
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
UNIVERSITY OF CALIFORNIA, SANTA BARBARA
Department of Physics
Physics 231B
Winter 2015
Prof. Gary Horowitz
Grader: Philip Saad
ASSIGNMENT #4
Due Wednesday, Feb. 4 at 5pm (in Broida 6239)
1. Practice with Penrose diagrams:
a. Consider the Reissner-Nordstrm
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
(73L +
UNIVERSITY OF CALIFORNIA, SANTA BARBARA
Department of Physics
Physics 231B
Winter 2015
Prof. Gary Horowitz
Grader: Philip Saad
ASSIGNMENT #3
Due Wednesday, Jan. 28 at 5pm (in Broida 6239)
1. Compute the proper acceleration of a static observer at constant
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 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
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
UNIVERSITY OF CALIFORNIA, SANTA BARBARA
Department of Physics
Physics 231B
Winter 2015
Prof. Gary Horowitz
Grader: Philip Saad
ASSIGNMENT #1
Due Wednesday, Jan. 14 at 5pm (in Broida 6239)
1. (a) Show that the Schwarzschild metric in ingoing radial null co