# Math 993 Section 2 --- General Relativity

Spring 07

Tuesday & Thursday 10:40 -- 12 noon.

Room A-204A Wells Hall

Prof. T. Parker

Mathematics Department

A346 Wells Hall

(517) 353-8493

## Office hours:

Mondays 2:45 -- 3:45

Tuesday & Thursday 2-3.

and by appointment

parker@math.msu.edu

This is a course in General Relativity from a geometric perspective. The goal
is to understand gravitational radiation ( the next-to-last topic listed below)
in terms of geometric analysis. One key focus will be understanding the Peeling
Theorem without recourse to conformal compactifications, and to understand
the relation between ADM and Bondi mass. \\

Students should be familiar with the objects of Riemannian geometry: manifolds,
vector and tensor fields, differential forms, metrics and curvature, etc. and
have a solid knowledge of linear algebra.

**Tentative Outline:**

- Review of Riemannian geometry.
- Special relativity, the Lorentz group, group representations.
- The Einstein equation and invariant theory; variational principles and
stress-energy tensors.
- Spinors, Clifford algebras and the (classical) Dirac equation.
- Asymptotically flat spacetimes and ADM mass. A summary of elliptic theory.
Witten's proof of the Positive Energy Theorem.
- Graviational radiation geometric effects of gravitational waves, curvature
decompositions using spinors (``Newman-Penrose formalism''), the Peeling
Theorem, Bondi mass.
- Conformal geometry of spacetime.

**Text:** There is no required textbook for the course, but the
following books are helpful references.

*Spacetime and Geometry: an Introduction to General Relativity * by
Sean Carroll
*General Relativity* by Robert M. Wald
*Advanced General Relativity * by John Stewart (no -- not *Jon* Stewart!)