Homework 4
Ted Jacobson and William D. Linch III
March 28, 2005
1.a The null generators are radial and are specied by r() and t(), with xed angles. As
1
they form a null geodesic congruence, they extremize the functional I = 2 (t2 r2 ). (The
= = 0.) The
HW#5 Phys776Spring2005
Due Thursday, April 14, before class.
Prof. Ted Jacobson
Room 4115, (301)405-6020
jacobson@physics.umd.edu
1. Consider a three-dimensional spherical box of circumferential radius R, that is held
at temperature T (as measured by a st
Homework 4
Ted Jacobson and William D. Linch III
March 28, 2005
1.a The null generators are radial and are specied by r() and t(), with xed angles. As
1
they form a null geodesic congruence, they extremize the functional I = 2 (t2 r2 ). (The
= = 0.) The
HW#4 Phys776Spring2005
Due Thursday, March 17, before class.
Prof. Ted Jacobson
Room 4115, (301)405-6020
jacobson@physics.umd.edu
1. The past light cone of a point in Minkowski spacetime is a null hypersurface whose
generators form a null geodesic congrue
Homework 3
William D. Linch III
March 11, 2005
1.a The rate of change of a vector v a owing along the integral curves of ua is given by
v a = ub b v a = v b b ua = B a b v b = 1 v a + a b v b + a b v b for B a b := b ua where we have used
3
the fact that
HW#3 Phys776Spring2005
Due Tuesday, March 8, before class.
Prof. Ted Jacobson
Room 4115, (301)405-6020
jacobson@physics.umd.edu
1. Raychaudhuri equation & cosmology
The Raychaudhuri equation for a timelike geodesic congruence parameterized by
proper time
Homework 2
William D. Linch III
March 2, 2005
1. We work in a coordinate system adapted to and compute
=(
)g = (, + )g .
(1)
Since the coordinates cfw_x are adapted to , the components = for some coordinate
direction . In particular, , 0 and
2
= g (g
Homework 2, Physics 776, Spring 2005
Due Thursday Feb. 24, at the beginning of class.
1. If the metric components are independent of a particular coordinate x
in a particular coordinate system, then the corresponding vector eld
a = ( )a is called a Killi
Homework 1 solution, Physics 776, Spring 2005
Painlev-Gullstrand coordinates
e
The line element for the unique spherically symmetric, vacuum solution
to the Einstein equation can be written as
2M
dT )2 r2 d2
r
ds2 = dT 2 (dr +
(1)
Note that a surface of c
Homework 1, Physics 776, Spring 2005
Due Tuesday Feb. 8, at the beginning of class.
Painlev-Gullstrand coordinates
e
The line element for the unique spherically symmetric, vacuum solution
to the Einstein equation can be written as
2M
dT )2 r2 d2
r
ds2 = d
HW#5 Phys776Spring2005
Due Thursday, April 14, before class.
Prof. Ted Jacobson
Room 4115, (301)405-6020
jacobson@physics.umd.edu
1. Consider a three-dimensional spherical box of circumferential radius R, that is held
at temperature T (as measured by a st