**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 (M_{1}, 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