Physics 231: General Relativity

Spring 2010

The lecture on Monday, April 19, was cancelled due to a volcanic eruption!
UPDATE (April 20):
Starting with the Wed., April 21 lecture, the standard schedule of classes and discussions resumes!

shortcut to the homework assignments

Basic Info

Time: lectures are now scheduled for Mon and Wed, 9:40-11:00am.
(I was hoping to move them in order to avoid conflicts with other graduate courses, but as it turns out, any change would generate additional conflicts.)
The discussion sessions are scheduled for Wed 2:10-3:00pm.
Place: 402 Le Conte Hall

Instructor: Petr Hořava (email:
Office: 401 Le Conte Hall.

General relativity represents one of the two main paradigms in modern theoretical physics, and an essential ingredient in our current understanding of the Universe. This is true not only on macroscopic scales (from the solar system to the cosmological), but increasingly also at the microscopic scales: The attempts to reconcile general relativity with the other leading paradigm, quantum mechanics, have dominated the landscape of theoretical physics in recent decades. As a result, having a good understanding of the basic structure of general relativity is now indispensable not only to astrophysicists and cosmologists, but also to particle physicists and phenomenologists, string theorists and (based on the most recent developments in the area of AdS/CFT correspondence) even to condensed matter theorists!

This course will provide the introduction to the principles of general relativity, as a geometric theory of gravity. The course will be loosely divided into the following six chapters:

I. Overview of special relativity: The physics and geometry in Minkowski spacetime.
II. Differential geometry of curved spacetimes.
III. Dynamics of gravity and spacetime geometry.
IV. Weak gravitational fields; gravitational waves/radiation.
V. Black holes.
VI. Cosmological solutions in general relativity.

Each chapter will occupy approximately two weeks of the course. This plan should leave some time, in the final week or two, to cover VII. Selected special topics: These might include aspects of quantum gravity, quantum field theory in curved spacetimes, quantum aspects of black holes, connections to string theory, or alternative theories of quantum gravity. The precise selection of the Special topics will be chosen interactively, based on the interests expressed by the students registered for the class.

The main textbook is going to be

Sean M. Carroll, Spacetime and Geometry: An Introduction to General Relativity (Addison Wesley, 2004).

This is a modern text, with just the right amount of information to cover the basics of general relativity. More information about the book (including errata) can be found here. The layout of the book is very compatible with the outline of the course as mentioned above. In addition, we might occasionally use other resources to complement Carroll, including some review papers from the arXiv. Students interested in a more detailed, in-depth presentation of some aspects of general relativity might also find it useful to consult Bob Wald's book

Robert M. Wald, General Relativity (U. Chicago Press, 1984).

While reading parts of Wald would certainly enhance the experience from this course, it is not essential for the material covered in the lectures, homeworks or the final exam.

Basics of modern differential geometry of manifolds, with a particular focus on differential forms and their integration on manifolds, can be found in an old (but cute!) little book by Spivak,

Michael Spivak, Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus (Perseus, 1965).


The prerequisites for this course are as explained in Carroll's book: No prior knowledge of differential geometry of manifolds is required, we will develop the mathematical apparatus as needed during the semester. On the other hand, a decent background in special relativity will be useful, as we will review the physics in flat Minkowski space only briefly at the beginning of the course.

Homeworks and Grading Policy

There will be weekly homework assignments, posted on this website. The grading policy will depend on whether or not we have enough students registered to get a GSI.

The final grade will be based 40% on homeworks, 40% on the final exam, and 20% on participation in discussions (the "two-out-of-three" rule applies, as explained in class).

Homework Assignments

The homework assignments will be posted here on Mondays (noonish), and will be due in one week, on Monday in class. The solutions will then be discussed in the discussion session on Wednesday following the due date. Until or unless we get a GSI, the grading will be on the crude scale of + (for good effort) and - (for no homework or little effort).

HW1 (due Mon, Feb 1): Problems 2, 3, 5 and 7 from Chapter 1 of [Carroll] (on page 46).

HW2 (due Mon, Feb 8): Problems 10 and 13 from Chapter 1 (on page 47), and Problems 1, 3 and 6 from Chapter 2 (on pages 90-91) of [Carroll].

HW3 (due IN TWO WEEKS, Mon, Feb 22): Problems 4, 5, 7, 8 and 9 from Chapter 2 (pages 91-92), Problems 1 and 4 from Chapter 3 (pages 146-147) of [Carroll].

HW4 (due Mon, March 1): Problems 5, 8 and 11 of Chapter 3 (pages 147-149) of [Carroll].

HW5 (due Mon, March 8): Problem 13(b) from Chapter 3 (page 149), Problems 1(a) and 2 from Chapter 4 (pp. 190-191) of [Carroll]. In addition, after solving Problem 1(a), you can choose one of two problems: Either continue to 1(b), or check the part of Problem 4 (of Chapter 4) that asks about the properties of the energy-momentum tensor of electromagnetism.

HW6 (due IN THREE WEEKS, Mon, March 29): This homework consists of two parts. Part one: Problems 4, 9 and 10 from Chapter 7 (pages 321-2) of [Carroll]. Part two consists of one additional problem with gravitational waves: Problem 5 of Chapter 7 (p. 321), and two black-hole problems: Problem 3 and 4(a) of Chapter 5 (p. 237) of [Carroll].

HW7 (due Mon, April 5): Problems 1 and 3(a) of Appendix G (on pages 469-70 of [Carroll]); Problem 4 of Chapter 6 (on page 273).
NOTE ADDED on Wed March 31: Because we did not get to the notions of a Killing horizon and surface gravity in lectures yet, Problem 4 of Ch. 6 in HW 7 is now optional (but it will become mandatory as part of HW8).

HW8 (due IN TWO WEEKS, Mon, April 19): Problems 1 and 4 of Chapter 6 (on pages 272-3), Problems 2 and 5 of Chapter 8 (p. 375).
UPDATE (April 21): Due to the volcanic eruption, this homework is now due Mon., April 26, in class.

HW9 (due Mon, May 3): Problem 2 of Appendix J (p. 494), and the following problem: Assuming the validity of the second law of thermodynamics for black holes, what is the minimum mass M of a Schwarzschild black hole that results from the collision of two Kerr black holes of equal mass m and opposite specific angular momentum a? Suppose further that the Kerr black holes are nearly extremal; what fraction of the original mass of the system can be radiated away?

Final Exam will be a take-home exam, due on Wednesday, May 5. Students with excellent results on homework problems will be exempt from the exam. All students should receive an email notification from me by Tue April 27, either informing them that they are exempt from the exam, or containing the pdf file with the assignment of the exam.