The course will include an exposition of the basic elements of differential geometry and general relativity. To this extent it will overlap with courses such as one standardly finds in a math or physics department. But the emphasis throughout will be on foundational issues. There will be no study of techniques for solving Einstein's equation, nor of astrophysical applications. But there will be careful consideration of the "logical structure" of the theory, its relation to Newtonian gravitation theory, the geometric interpretation of Einstein's equation, the "causal structure of spacetime", and other such topics. The comparison with Newtonian theory will exploit the possibility (first discovered by Elie Cartan) of giving the earlier theory a "geometrized" four-dimensional formulation.
Instructor: David Malament, Social Science Tower (SST), 757, 824-7374. I can be reached most reliably by e-mail (dmalamen@uci.edu).
Office hours:
by appointment.
Reading:
My lecture notes will serve as a text for the course. They can be downloaded here.
(Please read the notes with caution. They are still in
preliminary form. Corrections will be much appreciated.) A
number of recommended books are
also
listed for those wishing to do additional reading.
Requirements:
Auditors are welcome. But students wanting a quality grade will have to
submit written work. The two quarters will have different requirements.
In the first, students will be asked to submit solutions to assigned
problem sets. In the second quarter, students will be asked to write a
paper on some topic related to the subject matter of the course. (A few
examples of topics
are listed below.) Papers
must be submitted by the friday of 11th week. Students are strongly
urged to discuss their papers with me in advance of final submission
and, if possible, to submit a preliminary draft or detailed outline for
comments.
Tentative Course Outline
(What follows is table of contents for the notes as they stand at present. I do not expect to work through all of this material. In particular, I do not expect to reach the latter sections of part 4.)
Part 1: Differential Geometry (8 weeks?)
1.1 Manifolds
1.2 Tangent vectors
1.3 Vector fields, integral curves, and
flows
1.4 Tensors and tensor fields on
manifolds
1.5 Lie derivatives
1.6 Derivative operators and geodesics
1.7 Curvature
1.8 Metrics
1.9 Hypersurfaces
1.10 Volume elements
Part 2: General Relativity (8 weeks?)
2.1 Relativistic Spacetimes2.2 Temporal Orientation
2.3 Proper Time
2.4 Space/Time Decomposition at a Point and Particle Dynamics
2.5 The Energy-Momentum Field
2.6 Electromagnetic Fields
2.7 Einstein's equation
2.8 Fluid Flow - Rotation and Expansion
2.9 Killing Fields and Conserved Quantities
2.10 The Initial Value Formulation
2.11 Friedmann Spacetimes
Part 3: Special Topics
(The notes for this part are not yet ready.)
4.1 Classical Spacetimes
4.2 Geometrized Newtonian Theory -- First Version
4.3 Interpreting the Curvature Conditions
4.4 A Solution to an Old Problem About Newtonian Cosmology
4.5 Geometrized Newtonian Theory -- Second Version
Background
M. Spivak, Calculus on Manifolds, Benjamin, 1965 (paperback)
E. Taylor and J. Wheeler, Spacetime Physics, Freeman, 1966 (paperback)
Differential Geometry
R. L. Bishop and S. I. Goldberg, Tensor Analysis on Manifolds, Dover, 1980 (paperback)
W. M . Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press, 1986 (paperback)
B. O' Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, 1983
General Relativity
S. Hawking and G. F. R. Ellis, The Large Scale Structure of
Space-Time, Cambridge, 1973 (paperback)
M. Ludvigsen, General Relativity, Cambridge, 1999 (paperback)
B. O'Neill (cited above)
R. K. Sachs and H. Wu, General Relativity for Mathematicians, Springer, 1977
N. Straumann, General Relativity, Springer, 2004
R. Wald, General Relativity, Chicago, 1984 (paperback)
Works by Philosophers
J. Earman, World Enough and Space-Time, MIT, 1989
J. Earman, Bangs, Crunches, Whimpers, and Shrieks, Oxford, 1995
M. Friedman, Foundations of Space-Time Theories, Princeton, 1983 (paperback)
R. Torretti, Relativity and Geometry, Pergamon, 1983
(1) The "Paradox of Centrifugal Force"
The cover story of the March 1993 issue of the Scientific American,
"Black Holes and the Centrifugal Force Paradox", by Marek Artur
Abramowicz, begins by claiming: "An object orbiting close to
a black
hole feels a centrifugal force pushing inward rather than outward."
Make
the assertion
precise using our notation and prove it. (You may, if you
wish,
consult Abramowicz' more technical papers for guidance. Some
are cited
in this one.) Also, discuss Abramowicz' claims about the
significance
of this result.
(2) How Big Was "Space" Just After the Big Bang?
This
question is
ambiguous. Explain why, making use of the distinction between "private
space" and "public space". For guidance, see a paper by Don
Page on a
closely related question: "How Big is the Universe Today", General Relativity and
Gravitation 15
(1983), 181-185.
(3)
Different Notions of "Rotation" in General Relativity
I
discuss two criteria of rotation (for a ring about an axis) in "A
No Go Theorem About Rotation in Relativity Theory"
-- the "zero angular momentum criterion" and the "compass of inertia on
the axis criterion" -- and point out that they are not
equivalent, i.e., not equivalent in all spacetimes. Formulate other
criteria of rotation and discuss their relation to these two.
For
guidance, see another paper by Don Page: "Maximal
Acceleration is
Non-Rotating", Classical
and Quantum Gravity 15
(1998), 1669-1719.
To
be continued ... .