Universität Wien

250133 VO Lorentzian Geometry (2023S)

6.00 ECTS (4.00 SWS), SPL 25 - Mathematik
MIXED
Moodle

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Details

Language: English

Examination dates

Lecturers

Classes (iCal) - next class is marked with N

  • Wednesday 01.03. 13:15 - 14:45 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 02.03. 09:45 - 11:15 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 08.03. 13:15 - 14:45 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 09.03. 09:45 - 11:15 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 15.03. 13:15 - 14:45 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 16.03. 09:45 - 11:15 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 22.03. 13:15 - 14:45 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 23.03. 09:45 - 11:15 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 29.03. 13:15 - 14:45 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 30.03. 09:45 - 11:15 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 19.04. 13:15 - 14:45 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 20.04. 09:45 - 11:15 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 26.04. 13:15 - 14:45 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 27.04. 09:45 - 11:15 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 03.05. 13:15 - 14:45 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 04.05. 09:45 - 11:15 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 10.05. 13:15 - 14:45 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 11.05. 09:45 - 11:15 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 17.05. 13:15 - 14:45 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 24.05. 13:15 - 14:45 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 25.05. 09:45 - 11:15 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 31.05. 13:15 - 14:45 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 01.06. 09:45 - 11:15 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 07.06. 13:15 - 14:45 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 14.06. 13:15 - 14:45 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 15.06. 09:45 - 11:15 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 21.06. 13:15 - 14:45 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 22.06. 09:45 - 11:15 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 28.06. 13:15 - 14:45 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 29.06. 09:45 - 11:15 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.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://journals.aps.org/prl/abstract/10.1103/PhysRevLett.14.57, and here (https://arxiv.org/pdf/1410.5226.pdf) you can find an appraisal written on the accassion 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

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

Barrett O'Neill, Semi-Riemannnian Geometry (With Applications to Relativity) (Volume 103 of Pure and Applied Mathematics, Academic Press, San Diego, 1983), chapters 10 and 14.
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

Last modified: We 27.11.2024 00:17