Physics 231: General Relativity

Spring 2012

shortcut to the homework assignments

Basic Info

Time: Lectures on Mon and Fri, 11:10am-12:30pm.
Discussion sessions on Fri, 2:40-3:30pm.
Place: 402 Le Conte Hall.

Lecturer: Petr Hořava (email:
Office: 401 Le Conte Hall. Office hours: Wed, 2-3pm.
GSI: Ammar Husain (email:
Office: 405 Birge Hall. Office hours: Wed, 1-2pm.

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 between a week and a half and 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.

The more mathematically oriented student, with interest in the global geometric aspects of general relativity, will appreciate

S.W. Hawking and G.F.R. Ellis, The Large Scale Structure of Space-time (Cambridge U.P., 1973).

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 opposite extreme, with nearly 700 pages, is the more recent book

Theodore Frankel, The Geometry of Physics. An Introduction (2nd edition, Cambridge U.P., 2004).

In addition, many other books both on gravity and on differential geometry exist; if you wish to know my opinion of a specific text, please feel free to ask.


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 1/3 on homeworks, 1/3 on the final exam, and 1/3 on participation in discussions (the "two-out-of-three" rule applies, as explained in class).

Homework Assignments

The homework assignments will be posted here weekly, typically on Fridays afternoon; and will be due in six days, on Thursdays by 5pm, in Ammar's box. The solutions will then be discussed in the discussion sessions that same week right after their due date. The precise style of grading for the homeworks will be determined by Ammar.

The Final Exam will be a take-home exam. Students with excellent results on homework problems will be exempt from the exam. All students should receive an email notification from me by the end of April, either informing them that they are exempt from the exam, or containing the pdf file with the assignment of the exam. Additional details may appear here as we get closer to the end of the semester.

Unless stated otherwise, the problems are from [Carroll].

HW1 (due on Fri, Feb 3, in class): Problems 2 and 7 in Chapter 1 of [Carroll] (on page 46). In Problem 7, assume that the space on which the tensors are defined is equipped with the standard Minkowski metric.

HW2 (due on Thu, Feb 9): Problem 6 of Chapter 1 (p. 46), Problems 1, 3, 6 and 7 of Chapter 2 (pp. 90-91).

HW3 (due on Thu, Feb 16): Problems 4, 8, 9 and 10 of Chapter 2 (pp. 91-92).

HW4 (due on Thu, Feb 23): Problems 1, 4 and 8(a,b) of Chapter 3 (pp. 146-149).

HW5 (due on Thu, March 1): Problems 5 and 6 of Chapter 3 (p. 147), and Problem 1(a) of Chapter 4 (p. 190). Part (b) of Problem 1 in Ch. 4 is optional.

HW6 (due on Thu, March 8): Problems 2 and 4 of Chapter 4 (pp. 190-191), and Problem 1 of Chapter 7 (p. 320).

HW7 (due on Thu, March 15): Problems 12, 14 and 15 of Chapter 3 (pp. 149-150), Problem 5 of Chapter 4 (pp. 191-192), and Problem 1 of Appendix B (p. 437).

HW8 (due on Thu, March 22): Problems 7, 9 and 10 of Chapter 7 (pp. 321-322), Problem 2 of Appendix J (p. 494).

HW9 (due on Thu, April 5): Problems 3 and 5 of Chapter 5 (p. 237).

HW10 (due on Thu, April 12): Problem 1 of Appendix G (p. 469), Problems 1 and 4 of Chapter 6 (pp.272-3).

HW11 (due on Thu, April 19): Problems 1(a), 1(b) and 2 of Chapter 8 (p. 375). In addition, if you are interested in a problem related to the Kerr black holes, then Problem 6 of Chapter 6 (p. 273) is very interesting (and realistic: this is indeed how astrophysicists measure the characteristics of rotating black holes). However, since this Problem 6 is rather advanced, I declare it to be optional: a bonus problem.

HW11 was the last homework assignment of the semester. All students who are being asked to take the final take-home exam have already been informed by email. If you did not receive any such email, this means that your work so far already qualifies for an A (or the "pass" grade if you signed up on the S/F basis) and that no more action on your part is needed.
Since we still have one discussion section on Friday April 27, I suggest several Problems to think about before next Fri (but you do not have to turn in this assignement by Thursday as usual, since this assignment does not count towards the final grade; instead, just bring your notes with you to the discussion next Fri): Problem 2 of Chapter 5 (p.236), Problem 5 of Chapter 8 (p. 375).