Physics 6553: Problem Set 12 Solutions
by Jolyon Bloomeld
December 2012
1
Problem 1
From Question 3 of Problem Set #7, the isotropic form of the Schwarzschild metric is
2
ds =
1
1+
M
2
r
M
2
r
2
dt2 + 1 +
4
M
2
r
dr2 + r2 d2 + r2 sin2 d2
where
2
M
.
2
r
Physics 6553: Problem Set 1 Solutions
by Jolyon Bloomeld
September 5, 2012
1
Problem 1 [10 points]
V is the set of linear maps from V R. We have a mapping t : V V dened by
t(v ) = Fv , Fv V , where Fv (w) = w(v ) w V .
We need to show that this mapping is
Physics 6553: Problem Set 11 Solutions
by Jolyon Bloomeld
November 2012
1
Problem 1
We begin with the Schwarzschild metric:
ds2 = w(r)dt2 +
1
dr2 + r2 d2
w(r)
where w(r) = 1 2M/r as per usual. We are interested in the point r = 2M, t = = = 0. The rst
coor
Physics 6553: Problem Set 2 Solutions
by Jolyon Bloomeld
Sept 2012
1
1.1
Problem 1
Part a
The aim of this problem is to show that P transforms as a tensor. To do this, we want to
show that P e = P e , or alternatively, that P = P . To do this problem, wel
Physics 445: Problem Set 3
Due Thursday, Sept 23, 2004
1. Reading:
Read chapters 5 and 6 of Hartle.
2. Relative velocity of two particles: Two particles have 3-velocities v1 and v2 as measured in the lab
frame. Show, without using Lorentz transformations,
Physics 445: Problem Set 8
Due Thursday, Nov 4, 2004
1. Reading:
Read section 13.1 and chapter 20 of Hartle.
2. Hartle, chapter 12, problem 14.
3. Hartle, chapter 12, problem 15.
4. Hartle, chapter 12, problem 24.
5. Hartle, chapter 12, problem 25.
6. Pai
Physics 445: Problem Set 5
Due Thursday, Oct 7, 2004
1. Reading:
Read chapter 9 of Hartle.
2. Hartle, chapter 8, problem 2.
3. Hartle, chapter 8, problem 4.
4. Hartle, chapter 8, problem 14.
5. Hartle, chapter 9, problem 1.
6. Rindler spacetime: Consider
Physics 445: Problem Set 2
Due Thursday, Sept 16, 2004
1. Reading:
Read chapters 3 and 4 of Hartle.
2. Hartle, chapter 3, problem 5
3. Hartle, chapter 3, problem 6
4. Hartle, chapter 4, problem 3
5. Hartle, chapter 4, problem 13
6. Example of Local Inerti
Physics 445: Problem Set 10
Due Thursday, Nov 18, 2004
1. Reading:
Read chapter 22 of Hartle.
2. The Riemann Tensor:
a. Derive from the definition given in class the formula
R =
, , + .
b. For a given point P in spacetime, specialize to a coordinate
1
Physics 445
Cornell University
Solution for homework 7
(39 points)
I.
Fall 2004
Steve Drasco
HARTLE CHAPTER 10, PROBLEM 6 (10 POINTS)
The values for the Parametrized-Post-Newtonian (PPN) parameters and are measured to be
= 1 2 103 ,
= 1 3 103 .
(1.1)
1
Physics 445
Cornell University
Solution for homework 5
(41 points)
Fall 2004
Steve Drasco
NOTE From here on, unless otherwise indicated we will use the same conventions as in the last two solutions:
four-vectors ~v are denoted by an arrow, three-vectors
1
Physics 445
Cornell University
Solution for homework 8
(41 points)
I.
Fall 2004
Steve Drasco
HARTLE CHAPTER 12, PROBLEM 14 (8 POINTS)
Proper time integrals are functionals which take worldlines as arguments. The local maxima of these functionals
are tim
1
Physics 445
Cornell University
Solution for homework 10
(41 points)
I.
Fall 2004
Steve Drasco
THE RIEMANN TENSOR (10 POINTS)
WARNING! Be careful how you use equations from this problem. The equations in Hartles supplement, along
with Eqs. (1.1), (1.2),
Physics 445: Problem Set 11
Due Thursday, Dec 2, 2004
1. Reading:
Read chapters 16 and 23 of Hartle.
2. Hartle, chapter 22, problem 6.
3. Derive from the Bianchi identity Rcfw_; = 0 that the Einstein tensor satisfies G = 0.
Deduce the local law of conserv
Physics 445: Problem Set 6
Due Thursday, Oct 21, 2004
1. Reading:
Read chapters 10 and 11 of Hartle.
2. The Newtonian limit of general relativity. The Newtonian limit is described by a metric of the form
ds2 =
1
1 + 22 (t, x) + O(4 ) dt2 + O(2 )dxi dt +
Physics 445: Problem Set 9
Due Thursday, Nov 11, 2004
1. Reading:
Read chapter 21 of Hartle.
2. Observations of black holes in binaries: Consider a binary system consisting of a black hole of mass M2
together with a companion star of mass M1 . The frequen
Physics 6553: Problem Set 3 Solutions
by Jolyon Bloomeld
Sept 2012
1
1.1
Problem 1
Part a
We have an arbitrary vectorial basis cfw_e (P ) of the tangent space TP (M ). Let us begin with
an arbitrary coordinate system cfw_y . As we have a coordinate system
Physics 6553: Problem Set 4 Solutions
by Jolyon Bloomeld
October 3, 2012
1
1.1
Problem 1
Part a
This problem doesnt assume a metric on the sphere, just a set of geodesics. While nding the connection
coecients assuming a metric is relatively trivial, to th
Physics 6553: Problem Set 5 Solutions
by Jolyon Bloomeld
October 10, 2012
1
1.1
Problem 1
Part a
c e
(e e e e )
=
1.2
[e , e ]
=
c
=
( )e
=
Part b
There are a couple of ways of doing this. The rst is to use the denition of the covariant derivative
Physics 6553: Problem Set 7 Solutions
by Jolyon Bloomeld
October 2012
1
Problem 1
1.1
Part a)
We want to construct a coordinate system adapted to the spherical symmetry. We start by picking a point
P . Introduce coordinates and on the 2-sphere on which th
Physics 6553: Problem Set 8 Solutions
by Jolyon Bloomeld
November 2012
1
Problem 1
This problem uses some simple relations repeatedly, and so we describe them here in some detail.
By denition, we have the following relations, where is the ane parameter of
Physics 6553: Problem Set 9 Solutions
by Jolyon Bloomeld
November 2012
1
Problem 1
In this problem, we will use the following three equations repeatedly. If you are interested in seeing where
these equations come from, see the section Brief Derivation of
Physics 6553: Problem Set 10 Solutions
by Jolyon Bloomeld and Justin Vines
November 2012
1
Problem 1
1.1
Part a)
We make use of some equations of geodesic motion derived in Problem Set 8, Question 1. See equations 1
and 3 in particular, which are repeated
Physics 445: Problem Set 4
Due Thursday, Sept 30, 2004
1. Reading:
Read chapters 7 and 8 of Hartle.
2. Existence of Local Lorentz frames: In this exercise you will prove that, given a point P, it is always
possible to find a coordinate system which achiev
Physics 445: Problem Set 7
Due Thursday, Oct 28, 2004
1. Reading:
Read chapter 12 of Hartle.
2. Hartle, chapter 10, problem 6.
3. Hartle, chapter 10, problem 10.
4. Hartle, chapter 12, problem 5.
5. Hartle, chapter 12, problem 13.
6. Ingoing Eddington-Fin
1
Physics 445
Cornell University
I.
Solution for homework 6
(38 points)
Fall 2004
Steve Drasco
THE NEWTONIAN LIMIT OF GENERAL RELATIVITY (8 POINTS)
Write the Newtonian limit line element as
ds2 = (2 2)dt2 + ij dxi dxj + O(2 ),
(1.1)
where = (x ), is the u