School of Mathematics
Course 442 - General Relativity 1998-99 (Optional JS & SS Theoretical Physics, JS & SS Mathematics )
Lecturer: P.S. Florides
Requirements/prerequisites: 211, 221, 241

Duration: 21 weeks.

Number of lectures per week: 3

Assessment:

End-of-year Examination: One 3-hour examination

Description: Mathematical Background
Linear and Multilinear algebra, general tensors, tensor products, tests for tensor character, inner product, associated tensors (overview). Differentiable manifolds (M1, M¢,¼), differentiable functions, differentiable mappings f\colon M ® M¢, tangent vectors, tangent spaces and their duals (briefly).

The differential (f*) of a map f\colon M ® M¢ and its dual (f*). Vector fields and their commutator product, 1-form fields, tensor fields, moving frames. Differentiable curves, and vector fields on such curves, integral curves, r-surfaces.

Affine manifolds, connection coefficients, covariant derivatives of vector, 1-form and tensor fields, parallel propagation along a differentiable curve, intrinsic derivatives , Geodesics. The Riemann tensor and the torsion tensor.

Riemannian manifolds, the Riemann connection, properties of the Riemann tensor; the Ricci and Einstein tensors. The necessary and sufficient condition for a Riemannian manifold to be flat.

Equation of geodesic deviation, stationary properties of geodesics, Riemannian curvature, spaces of constant curvature, Schurs theorem.

General Relativity (G.R.)
Physical foundations of G.R., space-time as a Riemannian manifold, Einstein's field equations. The linearized Einstein field equations, Newton's theory as a first approximation, further (exact) analogies between Newton's and Einstein's theories.
(Digression: Lie derivatives, groups of motions, killing vectors. General form of the metric with spherical symmetry).

Exact solutions of Einstein's equations. The Schwarzschild exterior solution, Birkhoff's theorem, the three ``crucial tests'', the generalized red-shift formula. Study of the Schwarzschild radius, the Eddington - Finkelstein coordinates, Kruskal coordinates, black holes.

Dynamics of perfect fluids, the Oppenheimer-Volkov equation, maximum mass, the interior Schwarzschild solution; the Einstein and de Sitter Universes.

Electromagnetic Theory
The electromagnetic tensor, the electromagnetic energy tensor, the combined Einstein-Maxwell field equations, motion of charged particles in a electromagnetic field (in curved space-time), the Reissner-Nordström solution.

[Selected topics (time permitting): variational principles and conservation laws, (linearized) gravitational radiation, standard cosmology].

Jun 10, 1998