Abstract
Talk
Introduction to general relativity
We study Lorentzian manifolds in different spacetime model. (e.g. Minkowski model)
The spaceteimes satisfy the Vacuum Einstein equations. They are a Evolution equation (whatever that means).
We study some perturbations. This gives us some notion of stability. There are variant notions:
- Orbital and symptotic non-linear stability of the spacetime
- Linear stability
- Toy stability
They are likely related to Stable solution and Asymptotically stable solution but for PDEs. The stability of Minkowski space has been described by Christodoulou-Klainerman ‘91. PDEs are very often mentioned.
Geometry of Anti de Sitter space times
Next, we define the Anti-de Sitter space. These are spacetimes, where the Ricci metric fulfills some equation. Once again we study a PDE “the conformal wave equation” on anti de Sitter space.
We state facts about boundary conditions to have a “well-posed evolution”, they are the Dirichlet, Neumann and dissipative boundary condition.
(In)Stability of AdS
Theorem
Given a negativ spacetime and given some initial data in a Weighted Sobolev space. Imposing some boundary conditions induces instability
We describe a Schwarzschild metric for Anti-de Sitter space using a Penrose diagram. This somehow also gives a description of black hole properties.
We repeatedly talk about solutions decaying exponentially. What does that mean? An important info is that the decay depends qualitatively on the spacetime constant. Positive spacetime curvature leads to especially fast decay (of energy?)
Ah, I think we are studying the effects of black holes depending on the spacetime constant. How do objects fall into black holes? All of this talks about Energy decay but I still don’t understand what energy were talking about!
Theorem (Graf-Holzegel '24)
Schwarzschild-AdS for negative curvature is linearly stable. (In a nutshell)
The next part of the talk, describes the proof of the theorem. It uses some foliation of the system. It also causes a “gauge”