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.*

- 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

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.

- 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,

- Link to the lecture's G.R.I.P.S. page / Link zur G.R.I.P.S.-Seite
- Page in the "KVV" / Seite im kommentierten Vorlesungsverzeichnis
- Differential geometry II in the summer term 2021

- 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

- 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)

- 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Differential Geometry (Unpublished Lecture Notes), Summer Term 2013.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

- 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

- An animation of a black hole is here. More on the background of this animation can be found here.
- web site of Eric Gourgoulhon

- 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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