# Differential Geometry II — Lorentzian Geometry

Prof. Bernd Ammann, Office no. 119

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.

# Old announcements

• 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
1. The lecture will start with some general comments about organization and how we will proceed
2. 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).
3. 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.

## Content

In 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 (e0,...,en) with g(eij)=0 for i ≠ j, g(e0,e0)=-1, and g(ei,ei)=1 for i>0. In other words, up to the minus sign, the definition coincides with the one of a Riemannian manifold.

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.
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.

 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..

## Prerequisites

In order to follow the lecture, it is advised to have a basic knowledge about Riemannian geometry, as explained, e.g. in the lecture Differential Geometry I by Clara Löh held during the winter term 2020/21.
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.

## Time and Location

Monday and Wednesday 8.15-10.00
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 groups will already take place during the first week, i.e. on April 15 and 16. There the online exercise sheet will be discussed. You may just go to the group of your choice. It the distribution is not equilibrated automatically, we will read just later. The access data for the exercises is on GRIPS as well.
The first homework sheet should be completed until Tuesday, April 20.

### Exercise Sheets

(some links are not active yet)
All Exercise sheets in one file

## Literature

### Literature about the physics associated to the subject

#### 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