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## MATHEMATICAL METHODS IN GENERAL RELATIVITY

CODE 90700 2017/2018 5 credits during the 1st year of 9011 Mathematics (LM-40) GENOVA MAT/07 Italian GENOVA (Mathematics) 1° Semester This unit is a module of:

## 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

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 the study of causal structure in pseudo-riemannian geometry.

In presence

### 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.
• Robertson-Walker metric: cosmology and the Big-Bang.

• Causal structure.
• Singularity.

"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

### Exam Board

CLAUDIO BARTOCCI (President)

PIERRE OLIVIER MARTINETTI (President)

NICOLA PINAMONTI (President)

## LESSONS

In presence

### LESSONS START

The class will start according to the academic calendar.

### Class schedule

MATHEMATICAL METHODS IN GENERAL RELATIVITY

## EXAMS

Oral

### Exam schedule

Date Time Location Type Notes
19/01/2018 10:00 GENOVA Esame su appuntamento
19/01/2018 10:00 GENOVA Orale
23/01/2018 10:00 GENOVA Esame su appuntamento
23/01/2018 10:00 GENOVA Orale
26/01/2018 10:00 GENOVA Esame su appuntamento
26/01/2018 10:00 GENOVA Orale
14/02/2018 10:00 GENOVA Orale

### 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.