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.
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 (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.
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..
I will often refer to my lectures Analysis I to IV for which scripts are available below.
There will be two exercise groups