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 lecture is also open to interested persons outside of Regensburg, however the possibilty to take an exam is only available for registered students from Regensburg. If you are interested in the lecture, but have no access to our GRIPS system, please send an email to Bernd Ammann.

- The lecture will start on Monday, April 12, at 8.15.
- The zoom login data are available on the GRIPS system.
- Rough structure of the lecture on Monday April 12
- The lecture will start with some general comments about organization and how we will proceed
- Then I will give some overview over the area, of what I intend to do in the lecture. In this part, I will write down online on my computer, and you will see my shared screen.

Every page that is finished, will be saved and put online on this www-page (there might be a delay of at most 60 seconds). - In the next part, I will follow a prepared script which will be available here. Unfortunately, you will not be able to download it before the lecture.

- If you want to know more about the more physically oriented background, you may also register at Christian Bär's lecture "Relativity" in Potsdam, which probably covers aspects close to the script with the same title, which takes place on Tuesday 14-16 and Thursday 14-16 via zoom.

## ContentIn this lecture we will study semi-Riemannian manifolds, mainly concentrating on Lorentzian manifolds, assuming basic knowledge about Riemannian manifolds as provided e.g. in the lecture "Differential geometry I" by Clara Löh.Lorentzian manifolds arise when one combines n-dimensional space and time to an (n+1)-dimensional manifold. An understanding of Lorentzian manifold is the key ingredient to understand the theoretical aspects of general relativity.
A Lorentzian metric is a symmetric (0,2)-tensor g on a manifold of
dimension n+1, such that in every p∈M there is a basis (e Many aspects that you know from Riemannian geometry also hold for Lorentzian manifolds, we just have to add some signs at some places. These manifolds may be curved, and important notions of curvature are sectional curvature, Ricci curvature and scalar curvature. The famous Einstein equations are a statement about the Ricci curvature of the Lorentzian manifold describing our universe, e.g. vacuum spacetime is simply a Lorentzian manifold with vanishing Ricci curvature. This allows to study important examples, as e.g. the Schwarzschild solution which is a (3+1)-dimensional manifold with vanishing Ricci-curvature, but non-zero sectional curvature. Other examples are so-called Robertson-Walker spacetimes which are used to model the evolution of the universe. |
This picture arose from computer calculations using basic properties
of Lorentzian manifolds. It represents a black hole. |

This schematic picture represents the formation of a black hole as predicted by the Penrose singularity theorem. Permission by Eric Gourgoulhon, March 2021
| When these examples were discovered, scientists were astonished by their predictions, e.g. black holes and a big bang. However the drawback is these explicit examples is, that they all have a high degree of symmetry, and thus physicists were convinced for many decades that they do not emerge in real world: this high degree of symmetry is physically not realistic. In the 1960s Hawking and Penrose proved two singularity theorems. Roughly speaking, they state that under suitable assumptions -- which need not be of high symmetry -- a black hole type singularity, respectively a big bang type singularity necessarily has to exist. The goal of the lecture is to lay the mathematical foundations to understand this and to finally prove these singularity theorems. In principle, all our statements are mathematical statements, though we will mention their physical interpretation regularly as a motivation. We will spend few time in experimental verification and astronomical observation. If time permits we will study gravitational waves at the end of the lecture. Depending on the interest of the students, and interest for bachelor and master theses, there will be a seminar in the winter term building on this lecture. The precise subject will be clarified later, but it will probably be associated to wave type equations on Lorentzian manifolds which are e.g. an essential ingredient to provide models for quantum field theory on curved spacetimes. The lecture also allows to continue with an interdisciplinary seminar, potentially in collaboration with physicists. Here, the ongoing digitalization allows new types of collaborations. A further continuation might lead to concrete calculations using the SageMath package, see e.g. this summary.. |

If you have not heard this lecture, you may also compensate this by reading in the references below.

I will often refer to my lectures Analysis I to IV for which scripts are available below.

As long as we are in pandemy mode, the lecture will be held via zoom.

Please register on GRIPS to get the access data.

There will be two exercise groups

- Group 1: Thursday 14-16
- Group 2: Friday 10-12

The first homework sheet should be completed until Tuesday, April 20.

- Online Exercise Sheet (for the week April 12 to 16),
- 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 (Bonus points),

- 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

- 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

- 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 hier. 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, 26.07.2021

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