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 . Find the trajectory which extremizes the time dif
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 . Find the Lorentz
transformation between frames 1 and 3,
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. Please write neatly.
Late exams will not be accepted. B
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,
giving p = /3. Better, each photon has E = |p|, and averagi
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 rest mass (e.g. photons) and imagine that one
smooths
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 parallel-transport
it once around a circle of constant latit
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 of E and |P |?
2. Hartle 9.7. Two particles fall radial
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 denote velocity or anything
like that), with the line ele
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. constant v . The
second gives the top of the light cone, wit
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 and Ricci scalar. 2. Prove the geodesic deviation e
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 ; - V ; = V , - V , . Assume no torsion. 5. For
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 that dx /ds transforms as a tensor. Find the tran
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 coordinates. 2. Convert the one-form = x dy - y dx to polar
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 ) .
2 2 1b. For your friends path, ds2 = H(r2 )dt2 + r2 sin