AIMS AND LEARNING OUTCOMES

The lecture aims at providing the students whith advanced tools of mathematical physics in order to  study curved spacetime.

The first part of the lecture adresses the question of the completion and extension of a space, first in the riemannian then in the pseudo-riemannian cases. This leads to a precise definition of "singular space" as a geodesically incomplete and unextendable semi-riemannian manifold.

We then study various examples of such singular spaces. We will encounter both "coordinate singularities", that can be removed by a suitable extension, and "real singularities" in the sense above. 

Tipically these are of two kinds:  Big-Bang singularity (in Friedmann-Lemaître-Robertson-Walker space) and black hole singularity  (Schwarzschild space and its Kruskal extension).

Finally we will study the causal structure of spacetime, which culminates into the definition of "globally hyperbolic space". This yields to Hawking and Penrose singularity theorems. 

.

Teaching methods

Traditional

SYLLABUS/CONTENT

1. Completion and extendibility

1.1 Manifold vs metric space 

1.2 Geodesic completion (Hopf-Rinow theorem)

1.3 Completion of a pseudo-riemannian manifold.

2.  Spazi singolari

2.1 Cartesian product and deformed product

2.2 Rindler space and constant acceleration

2.3 Schwarzschild space and Kruskal extension

2.3 Estensione di Kruskal

2.4 Friedmann-Lemaître-Robertson-Walker and the Big-Bang

3. Singularity theorems

3.1 Causal structure in lorentzian geometry

3.2 Geodesic  congruence and variation

3.3 Hawking and Penrose theorems

RECOMMENDED READING/BIBLIOGRAPHY

Detailed notes available on aulaweb. For further reading:

Semi-Riemannian Geometry, Barrett O'Neill (Academic Press 1983).

The large scale structure of space-time, S. W. Hawking, G. F. R. Ellis (Cambrige Univ. Press 1973).

Exam Board

PIERRE OLIVIER MARTINETTI (President)

NICOLA PINAMONTI

MARCO BENINI

CLAUDIO BARTOCCI

Teaching methods

Traditional