GENERAL RELATIVITY

This class represents an introduction to the theory of general relativity (GR). It starts with an introduction to differential geometry, the language in which GR is written. After that, the Einstein field equations are derived heuristically, and are finally solved in certain contexts, such as spherical symmetry (leading to the Schwarzschild solution), gravity waves and cosmology

At the end of the course the student is expected to have the following skills:
1. Profound knowledge of differential geometry;
2. Knows the Einstein field equations and their Newtonian limit;
3. Is able to solve the Einstein equations in a context with enough symmetry;
4. Knows the physics of the Schwarzschild solution and the classical tests of GR;
5. Knowledge in modern cosmology.

0. Introduction

• The equivalence principle;

•  Elements of Special Relativity: inertial observers, Lorentz transformations, the Minkowski metric and the causal structure of the space-time, the Lorentz group, 4-vectors, the Lagrangian of the free particle, covariant formulation of E&M, 
• Tests of General relativity

• An introduction to differential geometry: Differential manifolds, tangent and cotangent space, tensor analysis, differential forms, (pseudo-)Riemannian manifolds, linear connections, curvature, geodetic deviation;

2. Einstein Field Equations 
Einstein's Equations: Euristic derivation, Einstein-Hilbert Action, Bianchi Identitities and diffeomorphism;
 Conserved quantities in GR: Killing Vectors
 Gravitational waves (GW): Einstein's equations to first order, gauge fixing, propagation of
GW in empy space. Production of GW, the quadrupole formula and the radiated power, binary systems 
 Schwarzschild's solution and black holes: Birkhoff's theorem, physics at the event horizon, Kruskal diagrams
 Cosmology (briefly): the FLRW metric and some simple solutions of the cosmological standard model  .

• R. Wald, "General Relativity"

• S. Weinberg, "Gravitation and Cosmology"

• S. Carroll, Spacetime and Geometry: An Introduction to General Relativity 

• Misner &  Thorne, Gravitation