Duration: 21 weeks.
Number of lectures per week: 3
End-of-year Examination: One 3-hour examination
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.
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].
Nov 8, 2000