General Relativity

The program contains

1. Elements of differential geometrical framework of GR

2. Geometry of Special Relativity

3. The action and equations of the gravitational field according to Einstein, Hilbert and including Palatini

4. Cosmological spacetimes

5. Black hole spacetimes

6. Gravitational waves and radiated energy

7. The relevance of a cosmological constant

1. Elements of differential geometrical framework of GR:

manifolds, tangent and cotangent bundles, natural operations on tensor fields, covariant derivatives, curvature, metric tensors.

2. Geometry of Special Relativity:

Minkowski spacetime, proper time and distance, 4-velocity, acceleration, inertial frames, accelerated and rotating frames, 4-momentum and mechanics.

3. The action and equations of the gravitational field according to Einstein, Hilbert and inc luding Palatini: variational principles for Einstein's general relativity in terms of metric tensors and in terms of tetrads and connections, the algebraic and differential identities satisfied by the Riemann, Ricci and Einstein tensors.

4. Cosmological spacetimes:

the Einstein equations with matter, the Friedman-Lemaître-Robertson-Walker solutions, other homogeneous universes, initial singularity, observer's horizon, the singularity theorem.

5. Black hole spacetimes:

Schwarzschild solution, horizon, Eddington–Finkelstein coordinates, Kruskal–Szekeres coordinates, black hole interior and exterior, geodesic curves, orbits, surface gravity, adding electric charge, rotation, topological non-triviality, extraction of rotational energy.

6. Gravitational waves and radiated energy: linear perturbations of spacetime, gravitational radiation, its energy, the quadrupole formula, the Trautman-Bondi energy.

7. The relevance of a cosmological constant: de Sitter and anti de Sitter spacetimes, cosmological horizons, black holes with cosmological constant, gravitational radiation at the presence of cosmological constant