Physics 231: General Relativity
shortcut to the homework assignments
Time: Lectures on Mon and Fri, 11:10am-12:30pm.
Discussion sessions on Fri, 2:40-3:30pm.
Place: 402 Le Conte Hall.
Office: 401 Le Conte Hall. Office hours: Wed, 2-3pm.
Ammar Husain (email: email@example.com)
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
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
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).
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
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).