# MATHEMATICAL METHODS IN GENERAL RELATIVITY

OVERVIEW

These lectures will give an extended presentation of General Relativity, that is the relativistic theory of gravitation published by Einstein in 1916. Besides the classical applications ot physics (cosmology, gravitational lensing, black-hole), one will stress the mathematical framework required to formulate the theory in a rigourous way (that is, pseudo-Riemannian differential geometry), as well as some further mathematical developments inspired by theory.

## AIMS AND CONTENT

LEARNING OUTCOMES

During these lectures, the various elements of differential geometry needed to formulate General Relativity in a rigourous way will be studied. More precisely, one will introduce the notions of connection and curvature on a pseudo-Riemannian manifold. Then Einstein equations will be discussed, as well as some of their solutions. These include the linearized solutions of gravitational waves, and those with spherical symmetry, used to describe the gravitational attraction of spherical objects.

LEARNING OUTCOMES (FURTHER INFO)

Besides the learning outcomes described in the general section, some more advanced mathematical topics will be studied as well, such as Hawking-Penrose singularity theorem, or some more advanced mathematical technics to solve Einstein equations.

Teaching methods

Teaching style: In presence

SYLLABUS/CONTENT

**Mathematical methods for General Relativity**

0. Scientific and historical introduction to the theory of General Relativity.

1. **Formalism**

- Special Relativity: Minkowski space, four-vectors, Lorents group.
- Pseudo-Riemannian manifolds and vector fields.
- Curvatura, parallel transport and geodesics.
- Einstein equations.
- Linear approximation: Newtonian gravity, gravitational wave.

2. **Solutions and applications**

- Robertson-Walker metric: cosmology and the Big-Bang.
- Schwarzschild metric: gravitational redshift, precession of the perihelion, bending of the light and gravitational lensing.
- Kerr metric: black holes.

3. **Advanced topics**

- Methods for solving Einstein equation: stationary solution and Killing fields, homogenous cosmology, perturbation.
- Causal structure.
- Singolarity.

RECOMMENDED READING/BIBLIOGRAPHY

"General Relativity", R. M. Wald, The University of Chicago Press (1984).

## TEACHERS AND EXAM BOARD

**Ricevimento:** On appointment

Exam Board

NICOLA PINAMONTI (President)

PIERRE OLIVIER MARTINETTI (President)

CLAUDIO BARTOCCI (President)

## LESSONS

Teaching methods

Teaching style: In presence

LESSONS START

February 27, 2017

## EXAMS

Exam description

to be defined later