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.

TEACHING METHODS

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.

3. Advanced topics (time permetting)

• Causal structure.
• Singularity.

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].