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 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 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 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 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 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 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 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 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 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 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