HOMEWORK 2 SOLUTIONS
PHIL SAAD
1. Carroll 1.4
1.1. A quasar, a distance D from an observer on Earth, emits a jet of gas at a speed v and an angle
from the line of sight of the observer. The apparent speed of this jet is the perceived velocity in
the dire
UNIVERSITY OF CALIFORNIA, SANTA BARBARA
Department of Physics
Physics 231A
Fall 2014
Prof. Gary Horowitz
Grader: Philip Saad
ASSIGNMENT #1
Due Wednesday, October 15 at 5pm (in room 6239)
1. Tensors
a. Show that the linear vector spaces V and its dual V ha
UNIVERSITY OF CALIFORNIA, SANTA BARBARA
Department of Physics
Physics 231A
Fall 2014
Prof. Gary Horowitz
Grader: Philip Saad
FINAL
EXAM
You may use class notes, homework solutions, and the books by Carroll and Wald as
references while completing this exam
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 )
UNIVERSITY OF CALIFORNIA, SANTA BARBARA
Department of Physics
Physics 231A
Fall 2014
Prof. Gary Horowitz
Grader: Philip Saad
ASSIGNMENT #4
Due Wednesday, November 12 at 5pm (in room 6239)
1. The general two dimensional spacetime can be written locally as
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 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 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
UNIVERSITY OF CALIFORNIA, SANTA BARBARA
Department of Physics
Physics 231A
Fall 2014
Prof. Gary Horowitz
Grader: Philip Saad
ASSIGNMENT #5
Due Wednesday, November 19 at 5pm (in room 6239)
1. Carroll, Chapter 3, problem 12.
2. Carroll, Chapter 3, problem 1
UNIVERSITY OF CALIFORNIA, SANTA BARBARA
Department of Physics
Physics 231A
Fall 2014
Prof. Gary Horowitz
Grader: Philip Saad
ASSIGNMENT #3
Due Wednesday, October 29 at 5pm (in room 6239)
1. Problem 4, page 91 of Carroll.
2. Problem 8, page 91 of Carroll.
UNIVERSITY OF CALIFORNIA, SANTA BARBARA
Department of Physics
Physics 231A
Fall 2014
Prof. Gary Horowitz
Grader: Philip Saad
ASSIGNMENT #2
Due Wednesday, October 22 at 5pm (in room 6239)
1. Problem 4, page 46 of Carroll. (For this problem, ignore the expa