250115 VO Lorentzian Geometry (2025W)

6.00 ECTS (4.00 SWS), SPL 25 - Mathematik

Moodle

Registration/Deregistration

Note: The time of your registration within the registration period has no effect on the allocation of places (no first come, first served).

Details

Language: English

Examination dates

Lecturers

Clemens Sämann

Classes (iCal) - next class is marked with N

Wednesday 01.10. 13:15 - 14:45 Seminarraum 3 Oskar-Morgenstern-Platz 1 1.Stock
Thursday 02.10. 11:30 - 13:00 Seminarraum 3 Oskar-Morgenstern-Platz 1 1.Stock
Wednesday 08.10. 13:15 - 14:45 Seminarraum 3 Oskar-Morgenstern-Platz 1 1.Stock
Thursday 09.10. 11:30 - 13:00 Seminarraum 3 Oskar-Morgenstern-Platz 1 1.Stock
Wednesday 15.10. 13:15 - 14:45 Seminarraum 3 Oskar-Morgenstern-Platz 1 1.Stock
Thursday 16.10. 11:30 - 13:00 Seminarraum 3 Oskar-Morgenstern-Platz 1 1.Stock
Wednesday 22.10. 13:15 - 14:45 Seminarraum 3 Oskar-Morgenstern-Platz 1 1.Stock
Thursday 23.10. 11:30 - 13:00 Seminarraum 3 Oskar-Morgenstern-Platz 1 1.Stock
Wednesday 29.10. 13:15 - 14:45 Seminarraum 3 Oskar-Morgenstern-Platz 1 1.Stock
Thursday 30.10. 11:30 - 13:00 Seminarraum 3 Oskar-Morgenstern-Platz 1 1.Stock
Wednesday 05.11. 13:15 - 14:45 Seminarraum 3 Oskar-Morgenstern-Platz 1 1.Stock
Thursday 06.11. 11:30 - 13:00 Seminarraum 3 Oskar-Morgenstern-Platz 1 1.Stock
Wednesday 12.11. 13:15 - 14:45 Seminarraum 3 Oskar-Morgenstern-Platz 1 1.Stock
Thursday 13.11. 11:30 - 13:00 Seminarraum 3 Oskar-Morgenstern-Platz 1 1.Stock
Wednesday 19.11. 13:15 - 14:45 Seminarraum 3 Oskar-Morgenstern-Platz 1 1.Stock
Thursday 20.11. 11:30 - 13:00 Seminarraum 3 Oskar-Morgenstern-Platz 1 1.Stock
Wednesday 26.11. 13:15 - 14:45 Seminarraum 3 Oskar-Morgenstern-Platz 1 1.Stock
Thursday 27.11. 11:30 - 13:00 Seminarraum 3 Oskar-Morgenstern-Platz 1 1.Stock
Wednesday 03.12. 13:15 - 14:45 Seminarraum 3 Oskar-Morgenstern-Platz 1 1.Stock
Thursday 04.12. 11:30 - 13:00 Seminarraum 3 Oskar-Morgenstern-Platz 1 1.Stock
Wednesday 10.12. 13:15 - 14:45 Seminarraum 3 Oskar-Morgenstern-Platz 1 1.Stock
Thursday 11.12. 11:30 - 13:00 Seminarraum 3 Oskar-Morgenstern-Platz 1 1.Stock
Wednesday 17.12. 13:15 - 14:45 Seminarraum 3 Oskar-Morgenstern-Platz 1 1.Stock
Thursday 18.12. 11:30 - 13:00 Seminarraum 3 Oskar-Morgenstern-Platz 1 1.Stock
Wednesday 07.01. 13:15 - 14:45 Seminarraum 3 Oskar-Morgenstern-Platz 1 1.Stock
Thursday 08.01. 11:30 - 13:00 Seminarraum 3 Oskar-Morgenstern-Platz 1 1.Stock
Wednesday 14.01. 13:15 - 14:45 Seminarraum 3 Oskar-Morgenstern-Platz 1 1.Stock
Thursday 15.01. 11:30 - 13:00 Seminarraum 3 Oskar-Morgenstern-Platz 1 1.Stock
Wednesday 21.01. 13:15 - 14:45 Seminarraum 3 Oskar-Morgenstern-Platz 1 1.Stock
Thursday 22.01. 11:30 - 13:00 Seminarraum 3 Oskar-Morgenstern-Platz 1 1.Stock
Wednesday 28.01. 13:15 - 14:45 Seminarraum 3 Oskar-Morgenstern-Platz 1 1.Stock
Thursday 29.01. 11:30 - 13:00 Seminarraum 3 Oskar-Morgenstern-Platz 1 1.Stock

Information

Aims, contents and method of the course

Lorentzian geometry is the differential geometry of Lorentzian manifolds, that is semi-Riemannian manifolds with an indefinite metric of index 1. Its particular importance comes from the fact that Lorentzian manifolds, also called spacetimes, act as the stage for General Relativity (GR), Albert Einstein's celebrated theory of space, time and gravity. In fact GR's fundamental idea is contained in its field equations, also called Einstein's equations, which state that the gravitational field is a property of spacetime and that its energy matter content is proportional to its curvature. In this sense GR is but the study of 4-dimensional Lorentzian manifolds which satisfy the Einstein equations.

The decisive difference between Riemannian and Lorentzian manifolds is that a Riemannian metric encodes the topological structure of the manifolds (as eg. seen from the Hopf-Rinow theorem) while a Lorentzian metric does not even induce a metric in the toplogical sense via its length functional. Instead it gives rise to the causal structure: the vectors in each tangent space fall into one of the distinct classes of timelike, null and spacelike vectors according to the sign of their norm.

In this course we study Lorentzian manifolds, and, in particular, their local and global causal structure with the goal of covering the famous singularity theorems of Penrose and Hawking. These are milestones in the development of GR, in particular the 1965-paper by Roger Penrose which won him the 2020-Nobel Price in physics. (You can find the paper here: https://doi.org/10.1103/PhysRevLett.14.57 and here (https://doi.org/10.48550/arXiv.1410.5226 ) you can find an appraisal written on the occasion of the centennial of GR in 2015). The theorems assert that under physically realistic conditions spacetimes generically become singular in the sense that they contain an incomplete causal geodesic. Moreover, they do not make any use of the field equations but only suppose a condition on the Ricci curvature along with some causality conditions, and so they are actually purely geometric results. If time permits we will cover as a further highlight a rather recent result by Bernal and Sanchez on the structure of globally hyperbolic spacetimes.

More specifically the topics of the course will be

Prerequisites: sectional curvature, semi-Riemannian submanifolds
Basic examples of spacetimes (Minkowski, (anti-)de Sitter, and Robertson-Walker spaces, Schwarzschild half-plane)
Basic causality theory (local causality, causality conditions)
Calculus of variations (Jacobi fields, focal and conjugate points)
Global hyperbolicity (Cauchy hypersurfaces, developments, and horizons)
The singularity theorms of Penrose and Hawking
The stucture of globally hyperbolic spacetimes

The prerequisites for following the course are a solid working knowledge in analysis on manifolds and some basics of Riemannian geometry.

Assessment and permitted materials

Oral exam by personal appointment.

Minimum requirements and assessment criteria

For a successful exam, a thorough understanding of the definitions, results, and proofs has to be shown in detailed answers to questions.

Examination topics

Content of the lecture notes.

Reading list

An updated version of the lecture notes of Kunzinger and Steinbauer will be provided http://www.mat.univie.ac.at/~mike/teaching/ss23/lorentzian.pdf
Further reading:
Barrett O'Neill, Semi-Riemannnian Geometry (With Applications to Relativity) (Volume 103 of Pure and Applied Mathematics, Academic Press, San Diego, 1983).
Christian Bär, Lorentzian geometry: https://www.math.uni-potsdam.de/fileadmin/user_upload/Prof-Geometrie/Dokumente/Lehre/Veranstaltungen/WS0405-SS08/LorentzianGeometryEnglish13Jan2020.pdf

Association in the course directory

MGEV; ML2; MEL

Last modified: We 11.02.2026 15:27