Minicourse

13:00-14:00	O. Petersen — Formation of quiescent big bang singularities	HFG 707
14:15-16:00	A. Vasy — Minicourse 1: The black hole stability problem — an introduction and results	HFG 611

11:00-12:00	A. Vasy — Minicourse 2: Analysis and geometry in the black hole stability problem	HFG 611
13:00-14:00	A. Vasy — Minicourse 3: Stability of the black hole exterior and cosmological spacetimes	HFG 611
14:15-15:00	T. Stucker — Quasinormal modes for the Kerr black hole	HFG 611
15:15-16:00	G. Hood — A scattering construction for nonlinear wave equations on Kerr-Anti de Sitter spacetimes	HFG 611

Abstract Minicourse

The minicourse consists of three lectures:

1. The black hole stability problem — an introduction and results
2. Analysis and geometry in the black hole stability problem
3. Stability of the black hole exterior and cosmological spacetimes

In the first lecture I will explain the black hole stability problem in classical general relativity and some of the recent results on it — these involve a fascinating combination of geometry and the analysis of partial differential equations. I will also give at least some indication of some of the tools that went into proving this. In the second lecture, I will discuss in more detail the analytic and geometric tools that lead to the understanding of black hole stability, especially with a positive cosmological constant (Kerr-de Sitter spacetimes), though also mentioning aspects of the vanishing cosmological constant case (Kerr). The third lecture will discuss the stability of the expanding (cosmological) region of Kerr-de Sitter spacetimes. I will also discuss the smoothness of the metric up to the future conformal boundary, with a Fefferman–Graham type asymptotic expansion, which is already of interest in de Sitter spaces, for which the global stability theorem of Friedrich in the 1980s was the first general stability result! This is based on joint works with Dietrich Häfner, Peter Hintz and Oliver Petersen.

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Accompanying lectures

Three talks on related topics will be given by junior researchers in the field.

A scattering construction for nonlinear wave equations on Kerr-Anti de Sitter spacetimes
by Gemma Hood (University of Leipzig)
 
Abstract: Given the sharp logarithmic decay of linear waves on the Kerr-AdS black hole (Holzegel, Smulevici '13), it is expected that the Kerr-AdS spacetime is unstable as a solution of the Einstein vacuum equations. However, the scattering construction presented here for exponentially decaying nonlinear waves on a fixed Kerr-AdS background serves as a first step to confronting the scattering problem for the full Einstein system. In this context, one may hope to derive a class of perturbations of Kerr-AdS which remain 'close' and dissipate sufficiently fast.

Formation of quiescent big bang singularities
by Oliver Petersen (Stockholm University)
 
Abstract: Hawking's singularity theorem says that cosmological solutions arising from initial data with positive mean curvature have a past singularity, in the sense that all inextendable causal geodesics eventually cease to exist or leave the spacetime. In this talk I will present a general condition on initial data ensuring in addition that the curvature blows up at the singularity (meaning that a big bang is formed) in the Einstein-nonlinear scalar field setting. This is joint work with Hans Oude Groeniger and Hans Ringström.

Quasinormal modes for the Kerr black hole
by Thomas Stucker (ETH Zürich)
 
Abstract: The late-time behavior of solutions to the wave equation on Kerr spacetime is governed by inverse polynomial decay. However, at earlier time-scales, numerical simulations are found to be dominated by quasinormal modes (QNMs). These are exponentially damped oscillatory solutions with complex frequencies characteristic of the system. In this talk, I will present a rigorous characterization of QNMs for the scalar wave equation on Kerr. They are obtained as the discrete set of poles of the meromorphically continued cutoff resolvent. The construction combines the method of complex scaling near asymptotically flat infinity with microlocal methods near the black hole horizon. I will also discuss the distribution of QNMs in both the high and low energy regimes. In particular, I will present uniform low energy resolvent estimates, which exclude the accumulation of QNMs at zero energy.