Physics 231: General Relativity
The lecture on Monday, April 19, was cancelled due to a volcanic
UPDATE (April 20):
Starting with the Wed., April
21 lecture, the standard schedule of classes and discussions resumes!
shortcut to the homework assignments
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
The discussion sessions are scheduled for Wed 2:10-3:00pm.
Place: 402 Le Conte Hall
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
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).
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 (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.