Physics 236a assignment, Week 1:
(October 1, 2015. Due on October 8, 2015)
1. Proof of invariance of s2 . [10 Points]
We will prove that s2 is invariant with respect to coordinate
transformations between Lorentz frames, without assuming the
explicit forms
Physics 236a assignment, Week 8:
(November 19, 2015. Due on December 01, 2015)
1. More on the Euler equation [15 points]
(a) If is a timelike Killing vector and u = /| |1/2 is a
4-velocity, show that
a=
uu
=
1
2
log | |
(1)
(b) Hydrostatic equilibrium mea
Physics 236a assignment 9:
(Dec 01, 2015. Due on December 10, 2015)
1. Dragging of inertial frames [20 points]
A rigid innitesimally-thin spherical shell of (uniformly-distributed)
mass M and radius R rotates with angular velocity with respect to observer
Ph 236 Homework 5
Due: Wednesday, November 7, 2012
1. Circumference of a circle. [18 points]
Weve provided several descriptions in class of what information the Riemann tensor contains. In this exercise
you will work through one more. For this problem, yo
Ph 236 Homework 4
Due: Wednesday, October 31, 2012
1. Gravitational elds and the equivalence principle. [18 points]
Consider a spacetime whose metric is given by
g = + h ,
(1)
where h are perturbations 1. This is a small perturbation on Minkowski space, a
Ph 236 Homework 6
Due: Wednesday, November 14, 2012
1. Cosmological constant. [16 points]
Consider the highly symmetrical spacetime with line element given by
ds2 =
1
(d 2 + dx2 + dy 2 + dz 2 ),
H 2 2
(1)
where H > 0 is a constant, and in the domain < 0.
Ph 236 Homework 1
Due: Wednesday, October 10, 2012
1. Superluminal motion. [20 points]
One of the most spectacular observations from long-baseline radio interferometry is apparent superluminal
motion. This occurs when the transverse motion of an object (a
Ph 236 Homework 2
Due: Wednesday, October 17, 2012
1. Basis independence of contractions. [6 points]
Suppose that S is a rank 2 tensor. Show that the contraction T of S, dened as the rank
1
1
0
tensor:
3
T(k)
S( , k, e )
(1)
=0
is independent of the ch
Ph 236 Homework 8
Due: Wednesday, December 6, 2012
Note: This will be a half-homework.
1. Variational approach to universes with matter. [18 points]
This problem works through the Lagrangian derivation of the Friedmann equations, and then considers some
i
Ph 236 Homework 7
Due: Wednesday, November 28, 2012
1. Gravitomagnetic precession. [18 points]
In class, I mentioned that objects with angular momentum create a gravitomagnetic eld around them
described by the potential h0i . In this problem, you will eva
Lecture I: Vectors, tensors, and forms in at spacetime
Christopher M. Hirata
Caltech M/C 350-17, Pasadena CA 91125, USA
(Dated: September 28, 2011)
I.
OVERVIEW
The mathematical description of curved spacetime is an involved subject and we will spend much
Lecture XV: Gravitational energy and orbital decay by gravitational radiation
Christopher M. Hirata
Caltech M/C 350-17, Pasadena CA 91125, USA
(Dated: December 2, 2011)
I.
OVERVIEW
The previous discussions have taken place in the context of linearized GR,
Physics 236a assignment, Week 7:
(November 12, 2015. Due on November 19, 2015)
1. Einstein tensor [10 points]
(a) Given some time coordinate, show that the components of
the Einstein tensor with at least one time index, G0 , involves only rst time derivat
Physics 236a assignment, Week 2:
(October 8, 2015. Due on October 15, 2015)
1. Equation of motion for a spin in a magnetic field. [10
points]
We will obtain the relativistic generalization of the nonrelativistic
(i.e. rest frame) equation of motion for th
Physics 236a assignment, Week 3:
(October 15, 2015. Due on October 22, 2015)
1. Manipulations of dierential forms [10 points]
(a) For 1-forms, we dened the wedge product as
u v = u v v u.
(1)
If 1 is a p-form and 2 is a q-form, show that
1 2 = (1)pq 2 1
Physics 236a assignment, Week 4:
(October 22, 2015. Due on October 29, 2015)
1. First Law [10 points]
For this problem, assume special relativity (or equivalently, assume that you are in a local Lorentz frame).
The first law of thermodynamics for a relati
Physics 236a assignment, Week 5:
(October 29, 2015. Due on November 5, 2015)
1. Spherical polar coordinates yet again [20 points]
Consider 3-dimensional Euclidean space. In Cartesian coordinates (x, y, z) the metric is gij = ij . We dene the usual spheric
Physics 236a assignment, Week 6:
(November 5, 2015. Due on November 12, 2015)
1. A 2-sphere [10 points]
Consider a sphere with xed radius r. This is a 2-dimensional
manifold, and in spherical coordinates , , the metric is
ds2 = r2 (d2 + sin2 d2 ).
(1)
(a)
Lecture XI: Weak eld tests of GR: the gravitational redshift, deection of light, and
Shapiro delay
Christopher M. Hirata
Caltech M/C 350-17, Pasadena CA 91125, USA
(Dated: November 11, 2011)
I.
OVERVIEW
General relativity is supposed to be a theory of nat
Lecture XIV: The energy of gravitational waves
Christopher M. Hirata
Caltech M/C 350-17, Pasadena CA 91125, USA
(Dated: November 23, 2011)
I.
OVERVIEW
The previous discussions have taken place in the context of linearized GR, which is not a fully consiste
Lecture X: Linearized gravity
Christopher M. Hirata
Caltech M/C 350-17, Pasadena CA 91125, USA
(Dated: November 9, 2011)
I.
OVERVIEW
We are now ready to consider the solutions of GR for the case of weak gravitational elds. This encompasses
Newtonian gravi