Homepage for Ph236 (2006-2007): General Relativity

Instructor: Marc Kamionkowski

Bridge Annex 120

x2563

[email protected]

**TA:**
** Tristan Smith** ([email protected]; 122 Bridge Annex; x4259)

** **(TA emeritus: Donal O'Connell)

**Class times:** Tues and Thurs 1:00-2:30pm in Downs 107

Problem
Session: We will have weekly problem sessions every Friday at 4pm, at which time
we will preview some of the homework problems for the week, or work
through some similar problems. Until further notice, the
classroom for that will be Downs 107.

**
Summary: **The purpose of this class will be
to learn the theory of general relativity, Einstein's theory of
relativistic gravity, as well as some applications. Over the
course of the year, we will probably cover applications in cosmology,
black holes, relativistic stars, gravitational waves, solar system
tests of gravitational theories, high-energy astrophysics, alternative
gravity theories, quantum fields in curved space, etc. The first
quarter will focus primarily on the formalism and basic structure of
the theory. Most applications will come later, but we'll try to
throw in some interesting physics all along the way.

**Prerequisites:** I'm assuming you have a
good foundation in undergraduate physics, particularly in classical
mechanics (including Lagrangians and Hamiltonians), electromagnetism,
and special relativity. I'm also assuming you know quantum
mechanics at the undergraduate level, although we won't necessarily be
using it too much. The mathematics background I'm assuming is
advanced calculus, differential equations, and some complex analysis
(although we won't need too much). The mathematics of general
relativity is differential geometry, but I am not assuming you have had
any---we will spend a good fraction of the first quarter learning the
relevant differential geometry. If you've already had
differential geometry, then things may be a bit easier for you the
first quarter. Some background in quantum field theory might help
out later in the year, but again, I am not assuming that you've studied
it before.

**Homework: ** There
will be problem sets assigned weekly. If you are serious about
understanding general relavity in any depth, then i*t is
imperative*
*that you do these problem sets
regularly. *
Since class time is limited, there will be many essential aspects of
general relativity that
you
will work out on your own in the problem sets. I will try to find
problem
sets
that stress concepts and interesting physics, and I will
try
to avoid problems that involve excessive algebra. You will learn
best if you first
try to do the problems by yourself. If you run into
trouble,
you may consult with classmates and attempt to work out the problems
together. I suppose you can also find solutions to many of these
problems in books or online, but you'll learn the subject better if you
try to first figure them out for yourself. Either way, when you
have figured out the solution, you should go your
separate
ways and then each write up the solutions from scratch. Please
write
your solutions as clearly as possible in order to make life easier for
the TAs---besides, learning to write clearly is important if you're
gonna get anywhere in science. I anticipate that there will be a
broad range of students in the class---from those who are planning to
become professional general relativists to those who are just looking
to obtain some working familiarity with the subject---as
well as a range of levels, from (ambitious) undergrads to advanced
graduate students; I'm also guessing some of you may have already had
some introduction to general relativity and/or differential
geometry. I will therefore assign quite lengthy problem sets so
that the more ambitious students in the class can go into considerable
depth if they choose to do so. I will try to describe and/or
prioritize the problems in each problem set, so that those who can
spend only a limited amount of time on the class can begin with the
most interesting/important problems first.

**Homework policy**:
Homeworks are due in class. Homeworks turned in after the
deadline but up to a week after the deadline will have 50% of the
points deducted. Anything over a week late automatically gets
zero. Exceptions are made if a student is sick (and hands in their work
with a medical note). The TAs will allow reasonable extensions,
and extension requests should go directly to the TA.

Solutions:
Will be provided regularly when the problem sets are turned in.

**Grade:
**The
grade will be based entirely on the problem sets. I am guessing
that most students in this class are motivated by an interest in the
subject, rather than the need to get a good grade. The TAs will
determine whether the student has made
a reasonable effort to make regular progress on the subject, and they
will also determine (in consultation with the professor) letter grades
for those students who require a letter grade.

Algebraic-manipulation
software for general
relativity is available. Many applications and problems in
general relativity often require considerable algebraic
manipulation. It may therefore behoove you to become familiar
with some of the software and perhaps to develop some facility using
it. As with all such things, its imperative to understand the
mathematical principles behind the software, but once you do, facility
with the software may help you cover more territory in the same
time. No problem sets will require use of this software, but some
of the homework problems may be facilitated with it. I refer you
to Lee Lindblom's page on software for GR
for more information.

**Required Text:**
Spacetime
and Geometry, an Introduction to General Relativity*, *by Sean
Carroll. The discussion in lectures will most closely
resemble
the approach taken in this book. Although its mathematically
sophisticated, it is quite pedagogical and discusses much of the math
in plain English. Its quite recent and goes into a number
of applications and subjects relevant for current research in
cosmology, string theory, and astrophysics, some of which we will get
to in quarters b and c. My guess is that this will be the
preferred book for most students.

**Some other recommended books (on
reserve
in the library):**

*
General
Relativity, *by Robert M.
Wald is
also very highly recommended. It provides perhaps the most
elegant presentation of general relativity. The only drawback is
that it is very formal and mathematical and sometimes terse.
Students with a strong mathematics background and/or those interested
in formal developments will probably like this book a lot. Those
with less mathematical sophistication may prefer Carroll's book.
Many of the problems, especially
during the first quarter, will come from this book.

Gravitation by Misner, Thorne, and Wheeler (MTW) is a classic, extremely comprehensive, detailed, with tons of examples, problems, and applications. Some of the subjects in the later chapters are out of date, but there are also lots of things worked out here that aren't available elsewhere. Drawback: may cause lower back problems.

A
First Course in General Relativity, by
Bernard F. Schutz is an excellent book with a pedagogical
approach, but
it is geared more toward undergraduate students and goes too slow for
this class.

Classical
Theory of Fields,
by Landau and
Lifschitz. This provides an excellent, clear, and
economical
introduction to general relativity. However, some students may
not find it sufficiently pedagogical, and the approach is a little out
of date. Some of the applications (e.g., cosmological
perturbations and gravitational-wave theory) are way ahead of their
time.....no surprise.

Gravity,
an Introduction to Einstein's General Relativity, by James B. Hartle. What
a great book! It succeeds
in
presenting the physics of general relativity before the
mathematics.
But its an undergraduate level book, and goes too slow for this
class.
Still, its worth owning.

Was Einstein Right? by Clifford M. Will is a spectacular book. This is not a technical book---it's a popular book, no equations. Still its remarkably well written, explains the physics very clearly and accurately, and tells the story of experimental tests of GR. Its short and an easy read, and I very strongly recommend it.

**Syllabus
and Reading Assignments!!!**

Modern Cosmology, by Scott Dodelson. This book covers basic homogeneous cosmology and then really focuses on cosmological perturbations, with a particular emphasis on the CMB.

Physical Foundations of Cosmology, by V. Mukhanov. This is another book that begins with homogeneous cosmology, and then discusses in depth inflation and cosmological perturbations.

Principles of Physical Cosmology, by P. J. E. Peebles. This is an excellent book that covers a very large set of topics. There are very nice short introductions to a number of topics in physical cosmology and early-Universe cosmology.

Cosmological Inflation and Large-Scale Structure, by A. Liddle and D. Lyth. This book focusses primarily on inflation and the production and evolution of cosmological perturbations.

Useful articles for third quarter:

"Cosmological dynamics," E. Bertschinger, astro-ph/9503125.

"Theory of cosmic microwave background polarization," P. Cabella and M. Kamionkowski, astro-ph/0403392.

"Anisotropies in the cosmic microwave background: an analytic approach," W. Hu and N. Sugiyama, astro-ph/9407093.

Problem Set 1 (due 3 October 2006)

Problem Set 2 (due 10 October 2006) (typo
fixed, 4 October)

Problem Set 3 (due 17 October 2006)

Problem Set 4 (due 24 October 2006)

Problem Set 5 (due 31 October 2006)

Problem Set 6 (due 7 November 2006)

Problem Set 8 (due 21
November 2006)

Problem Set
9 (due 5 December 2006)

Winter Quarter (Ph236b) problem
sets:

Problem Set 1 (due 16
January 2007)

Problem Set 2 (due 23 January 2007)

Problem Set 3 (due 30 January 2007)Problem Set 4 (due 8 February 2007; due date extended)

Problem Set 5 (due 20 February 2007)

Problem Set 6 (due 1 March 2007; due date extended)

Problem Set 7 (due 6 March 2007)

Problem Set 8 (due 13 March 2007; typo fixed 3/11)

Brief note on finite-temperature field theory (Re. Problems 3-4 in problem set 8)

Spring Quarter (236c) problem sets:Problem Set 1 (due 10 April 2007)

Problem Set 2 (due 17 April 2007)

Problem Set 3 (due 24 April 2007)

Problem Set 4 (due 1 May 2007; typo fixed 30 April)

Problem Set 5 (due 15 May 2007; due date extended one week)

Problem Set 6 (due 29 May 2007)

Last updated 21 May 2007