Lecturer: Lashi Bandara

Lorentzian geometries are a special class of pseudo-Riemannian geometries which form the basis for general relativity. They are the natural analogue to describe curved space-times in the way that Riemannian geometry describes curved spaces. This course aims to be an introduction to the topic and it will be based on lecture notes by Christian Bär. There will also be an English version of these notes available before the commencement of the course. The two aims of the subject are (1) to treat the singularity theorems of Hawking and Penrose, and (2) understand the topological structure of globally hyperbolic Lorentzian manifolds through the classic theorem of Geroch's and its geometry through the recent theorem of Bernal and Sánchez.

The course will commence by recalling the basics concepts of differential geometry in the first lectures, before treating the important examples of Lorentzian geometry to help gain intuition for the general aspects of the subject. Then, we will treat causality, the soul of the subject, before looking at Cauchy hypersurfaces and global hyperbolicity. This is required to phrase and prove the singularity theorems of Hawking and Penrose, which are also of physical relevance. Then, we will consider the recent theorem due to Bernal and Sánchez which gives a product type decomposition for the metric of a globally hyperbolic manifold. The latter is a recent significant development in Lorentzian geometry which paves way for the use of mathematical analysis in studying physical problems.

Further information and enrolment:
Click here or email lashi.bandara@uni-potsdam.de with the subject line "Lorentzian Geometry 2019"

When and where:
Monday 10:15-11:45 in Campus Golm, Building 9, Room 0.14
Tuesday 12:15-13:45 in Campus Golm, Building 9, Room 0.14
The first lecture will take place on Monday October 14.

Tutorial class:
Tuesday 10:15-11:45 in Campus Golm, Building 9, Room 0.13 (Rubens Longhi)

Moodle link:
Lecture notes and exercise sheets are now available in this Moodle. Participants with no account at University of Potsdam need to create a Moodle account for external students on this page.

Semester (recommended):
5. Semester or higher

Module numbers:
81j, 771, 772, 781, MATVMD611, MATVMD612, MATVMD811, MATVMD812, MATVMD814, MATVMD815, MATVMD911, MATVMD912, PHY-775

Necessary prerequesites:

• Basic differential geometry (manifolds, connections, curvature, metrics)
• Linear Algebra
• Analysis

Literature:

• B. O'Neill: Semi-Riemannian Geometry - With Applications to Relativity, Academic Press, 1983
• C. Bär: Lorentzgeometrie, Vorlesungsskript, Universität Potsdam, 2006

Poster:
Click here to download the pdf.