# MATHEMATICAL METHODS IN GENERAL RELATIVITY

5 credits during the 3nd year of 8760 Mathematics (L-35) GENOVA

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.

AIMS AND LEARNING OUTCOMES

The course will provide the student with all the mathematical tools necessary to formulate the conceptual basis of general relativity.

More generally, the aim is to show how mathematics and physics harmoniously speak to each other:

- on the one side, we will see why, and how, differential geometry is the right tool to give a rigourous formulation of Einstein's intuition (in particular: the equivalence principle);

- on the other side, we will see how the final formulation of the theory (Einstein equation of general relativity) yields new mathematical developments (like singularity).

Teaching methods

In presence

**Covid upgrade**: all the lectures will be accessible online, via Teams live session (that can be registrated by the students).

Hopefully, depending on the epidemic situation, there will be some encounters (non obligatory) for discussion, explanantions etc

SYLLABUS/CONTENT

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

1. Fundations of General Relativity

- Special Relativity: Minkowski space, four-vectors, Lorentz group.
- Pseudo-Riemannian geometry: manifolds, vector fields, connection, tensor, curvature, metric.
- Fundations of General Relativity: trajectories in a curved space-time, Einstein equations.

2. Solutions and applications

- Linearizzed theory: Newton approximation, gravitational wave.
- Schwarzschild metric: gravitational redshift, precession of the perihelion, bending of the light and gravitational lensing, singularity and black hole.

RECOMMENDED READING/BIBLIOGRAPHY

"General Relativity", R. M. Wald, The University of Chicago Press (1984) [THE reference, but more suitable for advanced topics].

“Geometry, topology and physics”, M. Nakahara, IOP (1990) [for differential geometry].

"Gravitation and cosmology: principle and applications of the general theory of relativity", S. Weinberg, J. Wiley & Sons (1972) [the best reference for tensor calculus, but this book adopts an anti-geometric point of view that will not be the one of the course].

"Relativity: special, general and cosmological". W. Rindler, Oxford University Press (2006) [excellent book, especially regarding physical discussion].

"Introduction to General Relativity", L. P. Hughston and K. P. Tod, Cambridge University Press (1990) [good introductory text].

"Corso di fisica teorica vol. 2: teoria dei campi", L. Landau, E. Lifchitz, MIR Moscva (1989) [Synthetic presentation of the thoery, maybe not suitalbe for a first approach, but some arguments are explained in an extremly clear and concise way].

## TEACHERS AND EXAM BOARD

**Ricevimento:** On appointment

Exam Board

PIERRE OLIVIER MARTINETTI (President)

NICOLA PINAMONTI

MARCO BENINI (President Substitute)

CLAUDIO BARTOCCI (President Substitute)

## LESSONS

Teaching methods

In presence

**Covid upgrade**: all the lectures will be accessible online, via Teams live session (that can be registrated by the students).

Hopefully, depending on the epidemic situation, there will be some encounters (non obligatory) for discussion, explanantions etc

LESSONS START

The class will start according to the academic calendar.

## EXAMS

Exam description

Written problem to solve at home. Not marked, but obligaotry to attend the oral exam.

Oral exam.

Assessment methods

Traditional

## FURTHER INFORMATION

Previous knowledge of differential geometry and special relativity will help, but these are not necessary.

All the tools of differential geoemetry needed to the undersatanding of General Relativity will be carefully introduced and explained.

As well, some basics of Special Relativity will be given.