Differential Geometry III — Lorentzian Geometry

This picture arose from computer calculations using basic properties of Lorentzian manifolds. It represents a black hole.
The image was obtained from the web page linked here
Picture created by Alain Riazuelo, IAP/UPMC/CNRS under the license CC-BY-SA 3.0.

Content

In this lecture we want to deepen our knowledge about semi-Riemannian and in particular Lorentzian manifolds. The precise content will be fixed a bit later, but it is likely to be a choice out of the following:

Where do the Einstein equations come from? The Einstein equations as stationary points of a variational problem
gravitational waves
special solutions of Einstein's equations, e.g. the Kerr solution for rotating black holes
sagemath/python tools to do calculations in general relativity
the positive mass theorem
wave equations on Lorentzian manifolds, leading to quatization of fields
solving the Einstein equations as a pde
more about causality, Cauchy hypersurfaces, global hyperbolicity
The Yamabe problem

Prerequisites

Analysis I-IV, Lineare Algebra I+II, differential geometry I. Helpful is the lecture differential geometry II, but as the topics are pretty disjoint, the gaps could be compensated by reading some literature.

Time and Location

Monday and Wednesday 8.15-10.00
Monday 8.15-10.00 in M102 and Wednesday 8-15-10.00 in M101
Please register on GRIPS to get the latest news and information about the exercises.

Lecture Notes

Partial lecture Notes are available here. Please assume that they will not be continued. In parts of the lecture I use excerpts from the the Ammann-Bär lecture about the Yamabe problem (German language)

Exercise Sheets

(some links are not active yet)

All Exercise sheets in one file

Related Websites

Literature

Literature directly associated to the lecture

C. Bär. Vorlesungsskript "Lorentzgeometrie", SS 2004, English version: Lecture Notes "Lorentzian geometry", Summer term 2004,
M. Kriele. Spacetime, Foundations of General Relativity and Differential Geometry. Springer 1999
B. O'Neill. Semi-Riemannian geometry. With applications to relativity. Pure and Applied Mathematics, 103. Academic Press
R. Wald. General Relativity. University of Chicago Press
C. W. Misner, K. S. Thorne, and J. A. Wheeler. Gravitation, Freeman New York, 2003
S. W. Hawking and G. F. R. Ellis. The large scale structure of space-time, Cambridge Monographs on Mathematical Physics, 1973

Literature about mathematical foundations

Scripts whose knowledge is required for the lecture

B. Ammann, Lineare Algebra I, WS 2007/08
B. Ammann, Analysis I+II, 2018/19
B. Ammann, Analysis III, WS 2019/20
B. Ammann, Analysis IV, SS 2020
C. Löh, Differential Geometry I, Lecture Notes, Regensburg Winter term 2020/21
B. Ammann, Differential Geometry II/Lorentzian geometry, SS 2021 (partial notes)

Alternative scripts

C. Bär, Differential Geometry (Unpublished Lecture Notes), Summer Term 2013.
C. Bär, Differentialgeometrie (Vorlesungsskript in deutsch), Summer Term 2006.
W. M. Boothby. Introduction to differential manifolds and Riemannian geometry, Academic Press 1986

Literature about the physics associated to the subject

C. Bär, Script to the lecture 'Relativity Theory', Summer Term 2013
Helmut Fischer und Helmut Kaul. Mathematik für Physiker, Band 3. Teubner, 2003.
R. d'Inverno. Einführung in die Relativitätstheorie, deutsche Ausgabe, (Ed. G. Schäfer, übers. O. Richter), VCH Weinheim, 1995
H. Stephani. Relativity. An Introduction to Special and General Relativity, Cambridge University Press, 2004
N. M. J. Woodhouse, Special Relativity, Springer 1992
Chrusciel, Piotr. Lectures on Mathematical Relativity
E. Gourgoulhon, Jamarillo, New theoretical approaches to black holes
Gourgoulhon, Talk about "Black holes: from event horisons to trapping horizons"

Books

M. do Carmo, Riemannian Geometry, Birkhäuser
Cheeger, Ebin, Comparison theorems in Riemannian Geometry
F. Warner, Foundations of differentiable manifolds and Lie groups, Springer
T. Sakai, Riemannian Geometry, Transl. Math. Monogr., AMS
W. Kühnel, Differentialgeometrie, Vieweg
J. Lee, Introduction to topological manifolds, Springer
J. Lee, Introduction to smooth manifolds, Springer
J. Lee, Riemannian manifolds, Springer

Some interesting links at the borderline of the lecture

This is a collection of links which are only loosely connected to the lecture, but might might be of interest to the audience.

Some calculations of high scientific quality about gravitational lensing effects close to black holes, and thus about geodescis on Lorentzian manifolds in connection to the movie "Interstallar" are available here.

Bernd Ammann, 28.01.2022
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