Titles and Abstracts

 

  • Sergiu Klainerman

    • Title: Nonlinear stability of slowly rotating Kerr black holes

    • Abstract: This talk is about a series of papers by Sergiu Klainerman, Jérémie Szeftel, Elena Giorgi and Dawei Shen, presenting the proof of the nonlinear stability of slowly rotating Kerr black holes.

 

  • Jan Sbierski

    • Title: Instability of the Kerr Cauchy horizon under linearised gravity

    • Abstract:  One can paraphrase Penrose’s strong cosmic censorship conjecture as stating that general relativity is generically a deterministic theory. While the full conjecture remains wide open there has long been evidence pointing towards its validity at least for small perturbations of exact rotating Kerr black holes. In this talk I will give a brief historical account of this evidence, discuss the analytic as well as the geometric aspects of this conjecture, and conclude by presenting work on the linear instability of the Kerr Cauchy horizon.

 

  • Jean-Philippe Nicolas

    • Title: Peeling and extreme horizons

    • Abstract: This talk is about the peeling of fields at an extreme horizon instead of at infinity as it is usually considered. This is a joint project with Jack Borthwick and Eric Gourgoulhon. The starting point was the existence of an intriguing similarity between the behaviour of fields near a degenerate horizon and near the infinity of an asymptotically flat spacetime, as revealed by the scattering theory for Dirac fields in the “exterior” region of the extreme Kerr - de Sitter black hole (which was studied by Jack). However, in that situation, the comparison was somewhat clouded by some of the analytical techniques used in intermediate steps of the proof. Our aim was to clarify the comparison further by studying the peeling behaviour of solutions to the wave equation at an extremal horizon. We focused on the extreme Reissner-Nordström black hole first, for which the Couch-Torrence inversion (a global conformal isometry that exchanges the horizon and infinity) makes the analogy explicit. Then, we explored more general spherically symmetric situations using the Couch-Torrence inversion outside of its natural context.
       

  • Thomas Bäckdahl

    • Title: Perturbations of Kerr -- Gauge and structure of equations

    • Abstract: In this talk I will discuss different kinds of gauge issues for the study of vacuum perturbations of a Kerr spacetime. This includes the GHP formalism and radiation gauges. Also the choice of variables and the structure of the equation systems they satisfy will be discussed. Most of the talk will be about the linearized equations, but I will also briefly mention the nonlinear case.

 

  • Pieter Blue

    • Title: Linear Stability of Kerr Black Holes

    • Abstract: This talk will discuss decay of solutions to the linearised Einstein equation around a Kerr black hole. The nonlinear stability of black holes was a major open problem in mathematical relativity for many decades, and it has now been proven in the slowly rotating case in a series of papers by Klainerman, Szeftel, and coauthors. This talk will present our somewhat different approach from 2019 to the linear problem. In 1972, Teukolsky discovered equations governing certain components of the linearised curvature that are invariant under linearised gauge transformations. In 1975, Chrzanowski introduced the "outgoing radiation gauge", a condition on the linearised metric that allows for the construction of the linearised metric from the linearised curvature. We begin from an assumption of a basic energy and Morawetz estimate for the Teukolsky curvature components, then improve this to more rapid decay, and then prove decay for some connection coefficients and for all metric components.
      This is joint work with Lars Andersson, Thomas Backdahl, and Siyuan Ma. This talk is a companion talk to that of Thomas Backdahl.

 

  • Marc Casals

    • Title: Mode stability analysis of rotating black hole spacetimes

    • Abstract: In this talk we will discuss some mode stability properties under classical matter field perturbations of several rotating black hole spacetimes. In particular, we will present results on some of the mode stability properties of the following: (i) a maximally-rotating (extremal) Kerr black hole, whose event horizon suffers from an instability; (ii) Kerr-de Sitter spacetime, which possesses a cosmological horizon as well as an event horizon; and (iii) the inner (Cauchy) horizon of Kerr-Newman-de Sitter spacetime, where we find evidence that the matter stress-energy tensor is locally integrable at the Cauchy horizon, thus violating the linear version of the strong Cosmic Censorship conjecture.

 

  • Sari Ghanem

    • Title: Stability of Minkowski space-time governed by the Einstein-Yang-Mills equations
    • Abstract: I shall start by presenting the Einstein-Yang-Mills system and by writing it in the Lorenz gauge and in wave coordinates as a coupled system of non-linear hyperbolic partial differential equations, and I will then show how one constructs the initial data for a Cauchy hyperbolic formulation of the problem. Thereafter, I will present the idea behind the proof of the non-linear stability of the Minkowski space-time, solution to the Einstein-Yang-Mills equations, in the Lorenz gauge and in wave coordinates, in all space dimensions greater or equal to three, based on a continuity argument for a higher order weighted energy norm. In the critical case of three space-dimensions, we use a null frame decomposition, that was first used by Lindblad and Rodnianski for the Einstein vacuum equations. We then deal with new difficulties that do not exist for Einstein vacuum nor for Einstein-Maxwell fields. In particular, we treat new terms that have a different structure in the non-linearities, and we derive a more refined formula to estimate the commutator term. This provides a new independent proof of the result by Mondal and Yau, that I posted on arXiv a few months ago in a series of three papers that build up on each other, which cover all space dimensions greater or equal to three.

 

  • Christoph Kehle

    • Title: Strong Cosmic Censorship for Λ<0 and its connection to Diophantine approximation

    • Abstract: I will present my results on the behavior of solutions to the conformal wave equation on the interior of Reissner-Nordstöm-AdS and Kerr-AdS black holes. Despite the very slow inverse logarithmic decay on the exterior, I show that linear waves arising from smooth Cauchy data with Dirichlet boundary conditions at infinity remain bounded at the Cauchy horizon for Reissner-Nordström-AdS. For Kerr-AdS, however, the situation is far more delicate: If the black hole parameters satisfy a certain non-Diophantine condition, then generic linear waves blow up in amplitude at the Cauchy horizon of Kerr-AdS. In particular, this proves the linear scalar version of the $C^0$-formulation of Strong Cosmic Censorship for Λ<0 if genericity is imposed in the sense of Baire category.

 

  • João Lopes Costa

    • Title: Strong Cosmic Censorship in the context of de Sitter Black Holes - a perspective on the current state of affairs.

    • Abstract: In this talk, I will assess the current status of the Strong Cosmic Censorship (SCC) Conjecture in the context of black hole spacetimes with a positive cosmological constant. After a brief overview of the origins and formulations of SCC, I will address the significant impact that introducing a cosmological term has on the conjecture. Notably, in this case, the validity of SCC is intricately linked to the precise decay rates of perturbations in the exterior region. Additionally, I will present a set of results suggesting the potential failure of SCC, in certain near-extremal regimes, and will discuss proposals aimed at preserving the conjecture. 

 

  • Pascal Millet

    • Title:  Leading-order term expansion for the Teukolsky equation on subextremal Kerr black holes.

    • Abstract: The study of wave propagation on black hole spacetimes has been an intense field of research in the past decades. This interest has been driven by the stability problem for black holes and by questions related to scattering theory. In the analysis of Maxwell's equations and the equations of linearized gravity, the focus often shifts to the study of the Teukolsky equation, which offers the advantage of being scalar in nature. I will present a result providing the large time leading-order term for initially localized and regular solutions and valid for the full subextremal range of black hole parameters. I will also discuss some aspects of the proof which relies on recent advances in spectral and microlocal analysis.
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