Differential Geometry III — Lorentzian Geometry
Prof. Bernd Ammann, Office no. 119
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.
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 This lecture will be held in the summer term 2021. It will be held via the video software zoom. The access data for the zoom conference are available on the GRIPS system after you have registered there for the lecture.
- The Yamabe problem
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.
(some links are not active yet)
All Exercise sheets in one file
- Exercise Sheet No. 1,
- Exercise Sheet No. 2,
- Exercise Sheet No. 3,
- Exercise Sheet No. 4,
- Exercise Sheet No. 5,
- Exercise Sheet No. 6,
- Exercise Sheet No. 7,
- Exercise Sheet No. 8,
- Exercise Sheet No. 9,
- Exercise Sheet No. 10,
- Exercise Sheet No. 11,
- Exercise Sheet No. 12,
- Exercise Sheet No. 13 (?),
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
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"
- 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, 17.10.2021
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