GENERAL RELATIVITY (MARCO PELOSO)

1. Preliminaries
Lorentz transformations and addition of velocities in special relativity. The geometry of flat spacetime. Time dilatation, length contraction, and the relativity of simultaneity. Four-vectors and special relativistic kinematics. Special relativistic dynamics and the energy-momentum tensor. Variational principle for Newtonian mechanics and for a free motion in special relativity. Light rays and Doppler shift. Observers and Observations.

2. Space, Time, and Gravity in Newtonian Physics.
Inertial frames. The principle of relativity. Newtonian Gravity. Gravitational and Inertial Mass.

3. Gravity as Geometry
The equivalence principle. Clocks in a gravitational field and gravitational redshift. Coordinates, line element, and the metric. Light cones and world lines. Length, area, volume computations. Vectors in curved spacetime. Hypersurfaces. Newtonian gravity in spacetime terms (weak field approximation).

4. The Einstein equations
Parallel transport and curvature. Covariant derivative, Riemann, Ricci, and Einstein tensor.
The source of curvature. Einstein equations and weak field approximation.

5. Geodesics
The geodesic equation. Symmetries and Killing vectors. Local inertial frames and freely falling frames.

6. Schwarzschild Geometry
Gravitational redshift. Particle orbits: the precession of the perihelion. Light ray orbits: the deflection and time delay of light. Solar system tests of general relativity.

7. Horizons and Coordinate Systems
Minkowski spacetime in Rindler coordinates. Schwarzschild black-holes. Eddington-Finkelstein, and Kruskal-Szekeres coordinates, Kruskal and Penrose diagrams.

8. Rotations and Kerr Geometry
Geodetic precession around a non-rotating, and a slowly rotating body. Kerr metric and the ergosphere.

9. Cosmology
FLRW geometry. Spatial curvature. Evolution in presence of matter, radiation, and a cosmological constant. Cosmological redshift. Luminosity and angular distance.


10 Gravitational waves (if time permits)