A.V. Manohar
Ph225A: General Relativity
Problem Set 1
1. A particle moves along a vertical trajectory in a uniform gravitational field, starting at h = 0 at t = 0, and returning to h = 0 at t = T .
Physics 225, Homework 1, Due Monday April 4.
1. Relative to frame 1, frame 2 is boosted along the +x axis by velocity v12 . Relative
to frame 2, frame 3 is boosted along the +x axis by velocity v23 .
Physics 225 Final Exam. Due Wednesday June 8, 5pm
Do your own work no collaborations.
You may use your class notes and one textbook of your choice, no internet
You must show all work for full credit.
Physics 225, Homework 2 solutions.
m2 3
n
n En (x
1. T =
xn (t), using p pn = m2 . So, for particles of zero mass
n
n
T = 0. Smoothing over the delta functions, its still zero. So T = + 3p = 0,
givi
Physics 225, Homework 2, Due Monday April 11.
1. As discussed in lecture, for a collection of particles (labeled by n) we have
p p 3
nn
(x xn (t).
En
T (x, t) =
n
Suppose that the particles have zero
Physics 225, Homework 5, Due Wednesday June 1.
1. Consider the 2-sphere with coordinates xA = (, ) and metric
dS 2 = d2 + sin2 d2 .
Take a vector with components V A = (1, 0) (i.e. V =
d
d )
and paral
Physics 225, Homework 4, Due Wednesday May 18.
1. Hartle 9.3. (a) How are a protons E and |P | related in the Schwarzschild geometry?
(b) What are the p components in the Schwarzschild basis in terms
Physics 225, Homework 3, Due Monday April 25.
1. (Taken from Hartle 7.5) Consider the 2d spacetime spanned by coordinates (v, x) (the
coordinate v here is like time; v is just its name, it does not de
Physics 225, Homework 3 solutions.
1. Light rays move along ds2 = 0, which has two possibilities:
dv
= 0,
dx
and
dv
1
=
.
dx
2x
The rst gives the bottom of the light cone, which has dv = 0, i.e. const
A.V. Manohar
Ph225A: General Relativity
Problem Set 5
1. Compute the Riemann tensor for the metric ds2 = e2A(r) dr2 + r2 d2 - e2B(r) dt2 using the Cartan method, and then compute the Ricci tensor a
A.V. Manohar Show that: 1.
Ph225A: General Relativity
Problem Set 4
ln det M = Tr M -1 M for any square matrix M . 2. = 3. V ; = 1 |g| x |g|V 1 |g| |g| x
4. V; - V; = V, - V, but that V ;
A.V. Manohar
Ph225A: General Relativity
Problem Set 3
1. Show that -g d4 x is invariant under a change of coordinates, where g = det g and d4 x = dx0 dx1 dx2 dx3 . 2. Let x (s) be a curve. Show
A.V. Manohar
Ph225A: General Relativity
Problem Set 2
1. You are given a vector field (Ax , Ay , Az ) in Cartesian coordinates. Compute (Ar , A , A ) and (Ar , A , A ) in spherical polar coordinate
Physics 225, Final exam solutions. 1a. The time on your watch in the proper time = ds2 . For the given path, H(r1 )t = 1 hour. So
dr = d = d = 0, so ds2 = H(r1 )dt2 and thus you = t = 1 hour H(r1 ) .