OVERVIEW

This course offers an advanced study of the curved space-time of general relativity. We will see how, under very general condition, such a space-time necessarily has singularities.

This means that black holes and the Big-Bang are not pathologies of general relativity, but are intrinsic to the theory.

AIMS AND CONTENT

LEARNING OUTCOMES

The purpose of this course is to show Hawking's and Penrose's singularity theorems in general relativity (for which Penrose won the Nobel Prize in Physics in 2020). To get there, we will first study the notion of completeness and extensibility for pseudo-Riemannian varieties, then the cuasale struttsara of this variety. A key notion will be that of globally hyperbolic space, the starting point of numerous advanced arguments in general relativity.

AIMS AND LEARNING OUTCOMES

Ability to confront well known mathematical concepts (completeness, metric space) in a new context (lorentzian geometry).

Knowledge of the mathematical tools required to study the causal structure of spacetime (global iperbolicity).

Developing a multidisciplinary scientific culture, at the cutting edge of recent advances in the field (study of black holes is currenbtly under a revolution, due to the discovery of black holes and the photographies of Event Horizon Telesecope).

PREREQUISITES

Previous course of differential geometry and/or general relativity

TEACHING METHODS

traditional

SYLLABUS/CONTENT

• Completeness and extendibilty
  • manifold vs metric space
  • geodetic completeness
  • completeness of pseudo-riemannian geometry
• Singular spaces
  • cartesian and deformed product
  • Rindler space and constant acceleration
  • Kruskal extension and white/black holes
  • FRLW space and the Big-Bang
• Singularity theorems
  • causal structure in lorentzian geometry
  • variation and geodesic congruence
  • singularity theorem

RECOMMENDED READING/BIBLIOGRAPHY

O'Neill  "Semi-riemannian geometry"

Hawking & Ellis "The large scale structure of spacetime"

Wald "General relativity"

Lectures notes

TEACHERS AND EXAM BOARD

Exam Board

PIERRE OLIVIER MARTINETTI (President)

MARCO BENINI

CLAUDIO BARTOCCI (President Substitute)

LESSONS

LESSONS START

According to the academic calendar

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

Traditonal oral exam or preparation of a seminar on a subject related to the course

ASSESSMENT METHODS

Tradiontal oral evaluation, or oral presentation of the seminar

FURTHER INFORMATION

Students with DSA certification ("specific learning disabilities"), disability or other special educational needs are advised to contact the teacher at the beginning of the course to agree on teaching and examination methods that, in compliance with the teaching objectives, take account of individual learning arrangements and provide appropriate compensatory tools.