Homepage for Ph236 (2006-2007): General Relativity

Instructor:   Marc Kamionkowski
                  Bridge Annex 120
                   [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.

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  it 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.

    Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, by Steven Weinberg.  A remarkable book, written by a particle theorist, essentially in his spare time, well before general relativity became an essential ingredient in the particle theorists' arsenal.  Many subjects are presented very nicely here, but some parts of the book are a bit outdated.

    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!!!

Additional books for third quarter:

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 7 (due 14 November 2006; we'll extend the due date until 16 November, but you should try to finish by 14 Nov if possible)

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