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 book, Topics in the Foundations of General Relativity and Newtonian Gravitation Theory, will serve as a text for the course. It can be downloaded here.  (It will appear in 2012 in the "Chicago Lectures in Physics" series published by the University of Chicago Press.) 

Here, in addition, is a list of  recommended books  for those wishing to do further reading. 

Requirements: Auditors are welcome. But students wanting a 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. 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, submit a preliminary draft or detailed outline for comments.

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Tentative Course Outline

What follows is the table of contents for my book.   I will certainly not reach all of this material. My tentative plan is to work through parts 1 and 2 as well as sections 4.1 and 4.2 of part 4.     

Part 1: Differential Geometry  

    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 The Action of Smooth Maps on Tensor Fields
    1.6 Lie derivatives
    1.7 Derivative operators and geodesics
    1.8 Curvature
    1.9 Metrics
    1.10 Hypersurfaces
    1.11 Volume elements

Part 2:  General Relativity   

    2.1 Relativistic Spacetimes
    2.2 Temporal Orientation and "Causal Connectibility"
    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
 
    3.1  Gödel Spacetime
    3.2  Two Criteria of Orbital (Non-)Rotation
    3.3  A No-Go Theorem about Orbital (Non-)Rotation 
   

    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  

       

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Selected References (for additional reading)

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