Physics/Astronomy 226, Problem set 9, Due 3/17?
Solutions
1. Letting u = (t z)/ 2 and v = (t + z)/ 2 transforms the usual Minkowski metric
ds2 = dt2 + dx2 + dy 2 + dz 2 into ds2 = (dudv + dvdu) + dx2 + dy 2 . This suggests
considering the metric ansatz:
d
Physics/Astronomy 226, Problem set 1, Due 1/19
Reading: Carroll, Ch. 1
1. Consider a Euclidean space with Cartesian coordinates xi , i.e. distances are given by
(s)2 = ij xi xj .
0
(a) By Taylor expanding xi (xi ), argue that the same formula will hold wh
Physics/Astronomy 226, Problem set 7, Due 2/28
Solutions
1. Show that for a Killing vector K , and with no torsion (as usual),
K
= R K and from this,
K
= R K .
(Hint: use the identity for R in which the sum of three permutations vanishes).
Use this, the
Physics/Astronomy 226, 2014, Final Exam
Solutions
1. True/False section. (30 Points; 2 each) For each of the following statements, say
whether the statement is true or false. If false, provide a true statement that entails
fairly minimal modication of the
Physics/Astronomy 226, Problem set 8, Due 3/10
Solutions
1. There is extremely strong astrophysical evidence that black holes of mass 106 108 M
reside in the centers of galaxies, and our own galaxy probably hosts a (probably Kerr)
black hole of 106 M . As
Physics/Astronomy 226, Problem set 6, Due 2/21
Solutions
1. In at spacetime, Maxwells equations can be written
F = J ,
where F = 2[ A] . In going to curved spacetime according to the Equivalence
Principle, we would like to replace partial derivatives suc
Physics/Astronomy 226, Problem set 2, Due 1/27
Solutions
1. Consider a distribution f (p , x ) of particles, so that the total number of particles of
all positions and momenta is
d4 p d4 xf (p , x ).
N=
(a) Write the expression for the total number of par
Physics/Astronomy 226, Problem set 5, Due 2/14
Solutions
1. Prove that R + R + R = 0.
Solution:
We can evaluate this in a local inertial frame, in which the Christoel symbols (but
not their derivatives) vanish, so that:
R = g ( )
1
= ( g g g + g ).
2
(1)
Physics/Astronomy 226, Problem set 1, Due 1/17
Solutions
1. Consider a Euclidean space with Cartesian coordinates xi , i.e. distances are given by
(s)2 = ij xi xj .
(a) By Taylor expanding xi (xi ), argue that the same formula will hold when
xi xi if and
Physics/Astronomy 226, Problem set 4, Due 2/7
Solutions
1. Derive the explicit expression for the components of the commutator (a.k.a. Lie
bracket):
[X, Y ]u = X Y Y X .
Solution:
Our vectors consist of components and basis vectors:
X = X .
We begin by br
Physics/Astronomy 226, Problem set 3, Due 1/31
Solutions
1. A light beam is emitted in vacuo from a height of 10 m and in a direction parallel to
the surface of the Earth. Assuming for present purposes that Earth is at, what is
the light beams distance fr