# SCIENTIFIC PROGRAMS AND ACTIVITIES

May 20, 2022

THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES

## Black Hole Stability June 8 - 12, 2015

 Organizing Committee Focus Week Organizers Spyros Alexakis, University of Toronto Mihalis Dafermos, Princeton University Luis Lehner, Perimeter Institute for Theoretical Physics and University of Guelph Harald Pfeiffer, Canadian Institute for Theoretical Astrophysics (CITA) Eric Poisson, University of Guelph Gustav Holzegel, ICL Steven Liebling, Long Island University

 On-line Registration open to May 31 also on-site during Focus Weeks $100 registration fees, students and PDF$50 *Please note that a nominal registration fee is required of all participants for this Focus Program. Your contributions allow us to provide the Program with refreshments and social events for each of the Focus Weeks. Application for Participant support Deadline to apply April 30, 2015 Accommodation in Toronto Information for speakers Reimbursement information for funded participants Map to Fields

Overview

Black holes are one of the most celebrated predictions of general relativity. It is widely believed that space-time in the vicinity of these black holes can be described to a suitable approximation by a Kerr metric, a remarkable family of explicit solutions of the Einstein vacuum equations discovered in 1963. Yet even the most basic mathematical questions about the dynamics of the Einstein equations in a neighbourhood of these solutions remain to this day unanswered. Are the Kerr metrics stable as solutions of the Einstein equations? Does gravitational collapse generically lead to black holes, or can so-called "naked singularities" form instead? What happens to observers who enter black hole regions? The latter two questions probe the very limits of the theory and are tied to the more general issue of singularities and the celebrated cosmic censorship conjectures of Penrose.

The last few years has seen intense activity which has brought us to the threshold of a denitive resolution of some of these issues. After intense work by several groups, and using important insights from the physics literature, the decay properties of linear scalar fields on the Kerr exterior backgrounds are now denitively mathematically understood. Concerning the problem of gravitational collapse, a breakthrough recent theorem of Christodoulou proves that trapped surfaces can from from initial data which are arbitrarily dispersed. This work introduces methods for rigorously understanding the Einstein equations in the large field regime, and already has given several applications to other problems involving singularities. Regarding the black hole interior, heuristic, numerical, and now rigorous mathematical theorems have shed light on the singular boundary, denitively showing that it always has a null piece-in contrast to older expectations that singularities should generically be everywhere space-like.

Participants as of June 3, 2015
* Indicates not yet confirmed

 Full Name University/Affiliation Spyros Alexakis University of Toronto Stefanos Aretakis Princeton University Piotr Bizon Jagiellonian University Pieter Blue University of Edinburgh Wilson Brenna University of Waterloo Alex Buchel University of Western Ontario * Vitor Cardoso Universidade Tecnica de Lisboa * Paul Chesler Harvard University * Demetrios Christodolou ETH Zurich Dominic Dold Cambridge University Semyon Dyatlov Massachusetts Institute of Technology Roberto Emparan Universitat de Barcelona Grigorios Fournodavlos University of Toronto Valeri Frolov University of Alberta Dejan Gajic University of Cambridge David Garfinkle Oakland University Gary Gibbons University of Cambridge Stephen Green Perimeter Institute for Theoretical Physics Hafner Häfner Universite de Grenoble Stefan Hollands Universitaet Leipzig Gustav Holzegel Imperial College London Jim Isenberg University of Oregon Soichiro Isoyama University of Guelph Thomas Johnson Imperial College London Jordan Keller Columbia University Gabor Kunstatter University of Winnipeg Luis Lehner Perimeter Institute Adam Lewis University of Toronto Steven Liebling Long Island University Hans Lindblad Johns Hopkins University Jonathan Luk University of Cambridge Georgios Moschidis Princeton University Oscar Reula Universidad Nacional de Cordoba * Igor Rodnianski Princeton University Volker Schlue University of Toronto Jacques Smulevici Université Paris-Sud Martin Taylor Cambridge University Robert Wald University of Chicago Claude Warnick Warwick University Helvi Witek University of Cambridge

 Monday, June 8 8:30-9:15 On site registration and morning coffee 9:15-9:30 Welcoming remarks 9:30-10:30 Stefanos Aretakis, Princeton University On linear and non-linear wave equations on black holes 10:30-11:00 Coffee break 11:00-12:00 Dietrich Häfner, Universite de Grenoble Scattering theory for Dirac and Klein-Gordon fields on the (De Sitter) Kerr metric 12:00-2:00 Lunch break 2:00-3:00 Helvi Witek, University of Cambridge The Black-hole bomb'' mechanism in astrophysical environments 3:00-3:30 Coffee break 3:30-4:30 -open- Tuesday, June 9 9:30-10:30 Semyon Dyatlov, Massachusetts Institute of Technology Quasi-normal modes: the spectrum of Kerr-de Sitter black holes 10:30-11:00 Coffee break 11:00-12:00 Roberto Emparan, Universitat de Barcelona Black hole stability: large-D approach 12:00-2:00 Lunch break 2:00-3:00 Dejan Gajic, University of Cambridge 3:00-3:30 Coffee break 3:30-4:30 -open- Wednesday, June 10 9:30-10:30 Stefan Hollands, Universitaet Leipzig Dynamical vs. Thermodynamical Instabilities of Black Objects 10:30-11:00 Coffee break 11:00-12:00 Valeri Frolov, University of Alberta Small mass collapse in the ghost-free gravity 12:00-2:00 Lunch break 2:00-3:00 -open- 3:00-3:30 Coffee break 3:30-4:30 -open- Thursday, June 11 09:30-10:30 Claude Warnick, Warwick University Stability problems in anti-de Sitter space times (Overview) 10:30-11:00 Coffee break 11:00-12:00 Stephen Green, Perimeter Institute for Theoretical Physics Two-timescale analysis of AdS (in)stability: Conserved 12:00-2:00 Lunch break 2:00-3:00 Piotr Bizon, Jagiellonian University New evidence for the instability of AdS 3:00-3:30 Coffee break 3:30-4:30 Nils Deppe, Cornell University Two-Mode Initial Data and Massive Scalar Fields in AdS 5:00-6:00 Xinliang An, Rutgers University Formation of Trapped Surfaces in General Relativity Friday, June 12 09:30-10:30 Alex Buchel, Perimeter Institute for Theoretical Physics Black hole spectra in holography: consequences for equilibration of dual gauge theories 10:30-11:00 Coffee break 11:00-12:00 Gabor Kunstatter, University of Winnipeg Stability of AdS in Einstein Gauss Bonnet Gravity 12:00-2:00 Lunch break 2:00-3:00 Arick Shao, Imperial College London Unique Continuation in Asymptotically Anti-de Sitter Spacetimes 3:00-3:30 Coffee break 3:30-4:30 Jacques Smulevici, Université Paris-Sud Trapping, decay and stability problems in asymptotically AdS spacetimes

Abstracts

Xinliang An, Rutgers University
Formation of Trapped Surfaces in General Relativity

In this talk, I will present two results regarding the formation of trapped surfaces in general relativity.

The first is a simplified approach to Christodoulou’s monumental result which showed that trapped surfaces canform dynamically by the focusing of gravitational radiation from past null infinity. We extend the methods of Klainerman-Rodnianski, who gave a simplified proof of this result in a finite region.

The second result extends the theorem of Christodoulou by allowing for weaker initial data but still guaranteeing that a trapped surface forms in the causal domain. In particular, we show that a trapped surface can form dynamically from initial data which is merely large in a scale-invariant way. The second result is obtained jointly with Luk.

Stefanos Aretakis, Princeton University
On linear and non-linear wave equations on black holes

I will present some recent results on linear and non-linear wave equations on extremal and sub-extremal black hole backgrounds.

Piotr Bizon, Jagiellonian University
New evidence for the instability of AdS

Four years ago Andrzej Rostworowski and I conjectured that AdS is unstable under arbitrarily small perturbations. In my talk (based on yet unpublished joint work with Maciej Maliborski and Andrzej Rostworowski) I will present a new piece of evidence supporting our conjecture.

Alex Buchel, Perimeter Institute for Theoretical Physics
Black hole spectra in holography: consequences for equilibration of dual gauge theories

For a closed system to equilibrate from a given initial conditionthere must exist an equilibrium state with the energy equal to theinitial one. Equilibrium states of a strongly coupled gauge theorywith a gravitational holographic dual are represented by black holes.We study the spectrum of black holes in Pilch-Warner geometry. Theseblack holes are holographically dual to equilibrium states of stronglycoupled SU(N) N=2^* gauge theory plasma on S^3 in theplanar limit. We find that there is no energy gap in the black holespectrum. Thus, there is a priory no obstruction for equilibration ofarbitrary low-energy states in the theory via a small black holegravitational collapse. The latter is contrasted withphenomenological examples of holography with dual four-dimensionalCFTs having non-equal central charges in the stress-energy tensortrace anomaly.

Nils Deppe, Cornell University
Two-Mode Initial Data and Massive Scalar Fields in AdS

It has been argued that anti-de Sitter spacetime in general relativity is unstable against the formation of black holes for arbitrarily small perturbations, at least for a large class of initial data. Stable evolution has been observed for initial data of the form of a single Gaussian within a range of widths, boson stars, specially constructed time-periodic solutions, and certain superpositions of multiple Gaussian wavepackets. We perform a detailed study of the single Gaussian, multiple Gaussian and two eigenmode initial data using the energy per mode to quantify the dynamics. We find interesting and unexpected chaotic behaviour, as well as the previously predicted inverse cascade of energy to lower modes. Additionally, we study massive scalar field perturbations over a range of masses and different forms of initial data, finding qualitatively similar results to the massless case for smaller masses, but different behaviour for extremely massive fields.

Semyon Dyatlov, Massachusetts Institute of Technology
Quasi-normal modes: the spectrum of Kerr-de Sitter black holes

Consider linear waves on the Kerr-de Sitter spacetime, which models a rotating black hole with a positive cosmological constant. In contrast with the Kerr solution, solutions to the wave equation decay exponentially up to a finite dimensional subspace. This makes it possible to expand waves asymptotically in terms of quasi-normal modes, which are the complex characteristic frequencies associated to the spacetime. I present several recent results, giving a rigorous definition of quasi-normal modes and describing their asymptotic behavior in the high frequency limit. The high frequency picture relies on the normally hyperbolic structure of the set of trapped light rays.

Roberto Emparan, Universitat de Barcelona
Black hole stability: large-D approach

After introducing the main elements of the large D approach to black hole physics, I will focus on its use in the analysis of mode stability of black holes. This is greatly simplified, by isolating the modes that are potentially stable, and by drastically simplifying their equations.

Valeri Frolov, University of Alberta
Small mass collapse in the ghost-free gravity

We discuss a problem of a black hole formation in the ghost-free gravity. We demonstrate how a non-local modification of gravity equations regularizes static and dynamical solutions. We focus on the problem of a collapse of small masses in the ghost-free gravity, and demonstrate that there exists a mass gap for mini-black-hole formation in this model.

Stephen Green, Perimeter Institute for Theoretical Physics
Two-timescale analysis of AdS (in)stability: Conserved

We consider the dynamics of a spherically symmetric massless scalarfield coupled to general relativity in anti–de Sitter spacetime in thesmall-amplitude limit. We first develop the "two time framework" (TTF)approximation to study the leading self-gravitating effects of thescalar field. Within this context, we uncover the presence of 3conserved quantities: the energy E, the particle number N, and theHamiltonian H. Simultaneous conservation of E and N implies thatweakly turbulent processes undergo dual cascades (direct cascade of Eand inverse cascade of N or vice versa), and it rules out energyequipartition for generic initial data. Furthermore, TTF admits alarge class of quasi-periodic (QP) solutions that extremize H. Weperform a linear stability analysis of QP solutions within TTF, and weshow that there exist several families of stable solutions. We arguethat certain spacetime solutions that avoid collapse (for long times)are perturbations about QP solutions, and we use the stabilityanalysis to calculate approximate recurrence times that have beenobserved in numerical simulations. We also discuss how collapsingsolutions can be understood within TTF.

Dietrich Häfner, Universite de Grenoble
Scattering theory for Dirac and Klein-Gordon fields on the (De Sitter) Kerr metric

I will discuss scattering theory for Dirac and Klein-Gordon fields on a (perturbed) Kerr resp. De-Sitter Kerr metric. Asymptotic completeness results are obtained for both the Dirac field and the Klein-Gordon field, where for the Klein-Gordon field the angular momentum of the field has to be fixed. For the Dirac field I will explain the equivalence between the classical formulation in terms of direct and inverse wave operators and the interpretation as an existence and uniqueness result for the Goursat problem at null infinity. For the Klein-Gordon field I will explain how superradiance can be understood in terms of spectral theory using an appropriate functional calculus. If there is enough time I will also discuss the link between the Hawking effect and scattering theory on a space-time describing a collapsing star. The talk is based on joint work with Jean-Philippe Nicolas (Dirac fields) and Vladimir Georgescu, Christian Gérard (Klein-Gordon fields).

Stefan Hollands, Universitaet Leipzig
Dynamical vs. Thermodynamical Instabilities of Black Objects

Black holes are well known to have properties that are strikingly similar to the ordinary laws of phenomenological thermodynamics. These properties are therefore often referred to as the "laws of black hole mechanics," and play a key role in many attempts to quantize gravity. However, as has been appreciated more recently, appropriate extensions of these laws can also be used to understand the *dynamical properties* of *classical* black holes as well as their higher dimensional counterparts. Such ideas suggest that simple, and standard, criteria for thermodynamical instability as formulated in ordinary phenomenological thermodynamics ('negative heat capacity') are informative also for the -- technically very challenging -- analysis of the stability properties of black holes or even stars. In this talk, I will review these kinds of ideas, their motivation, and applications to several interesting examples including a) (near) extremal, rotating BHs in higher dimensions, b) super radiant-type instabilities for AdS BHs. In particular, I will argue that the famous laws of black hole mechanics should be supplemented by a further analogy relating thermodynamical and dynamical instability.

Gabor Kunstatter, University of Winnipeg
Stability of AdS in Einstein Gauss Bonnet Gravity

Authors: Nils Deppe, Allison Kolly, Andrew Frey, and Gabor Kunstatter

Recently it has been argued that in Einstein gravity Anti-de Sitter spacetime is unstable againstthe formation of black holes for a large class of arbitrarily small perturbations. We have recently examined the effects of a change in the small scale gravitational dynamics on stability by adding a Gauss Bonnet term to the action. In five dimensions, spherically symmetric Einstein-Gauss-Bonnet (EGB) gravity has two key features: Choptuik scaling exhibits a radius gap, and the mass function goes to a finite value as the horizon radius vanishes. These suggest that black holes will not form dynamically if the total mass/energy content of the spacetime is too small, thereby restoring the stability of AdS space-time for large families of generic initial data. After a brief review, I will present numerical evidence to support this claim. Our numerical simulations have uncovered a rich structure in horizon radii and formation times as a function of perturbation amplitude. Although our calculations were specific to 5D EGB, I will argue that the qualitative behaviour we observed is likely to exist in a large class of theories in which the microscopic dynamics is governed by a new length scale.

Arick Shao, Imperial College London
Unique Continuation in Asymptotically Anti-de Sitter Spacetimes

In this talk, we consider the problem of unique continuation from infinity for Anti-de Sitter (AdS) and asymptotically AdS spacetimes. We show, roughly, that given a solution $\phi$ of a linear (massive or massless) wave equation on AdS spacetime, if $\phi$ and its first derivative vanish to high enough order (depending on the mass) on a sufficiently large but finite portion of infinity, then $\phi$ must also necessarily vanish in a small neighborhood of infinity. In particular, this establishes a correspondence between data for $\phi$ at infinity and the value of $\phi$ in the interior.

When available, we also connect our results to the well-posedness theory: we show that trivial Dirichet and Neumann data at (a large enough portion of) infinity along with sufficient regularity implies vanishing in the interior. Furthermore, all these results generalize to a large class of asymptotically AdS spacetimes, as well as to tensor-valued waves. These techniques are also viable for studying nonlinear wave equations; one application is to study corresponding uniqueness properties for the Einstein-vacuum equations with negative cosmological constant.

This is joint work with Gustav Holzegel.

Claude Warnick, Warwick University
Stability problems in anti-de Sitter space times (Overview)

Over recent years there has been considerable progress in understanding solutions of Einstein’s equations with negative cosmological constant. The presence of a timelike conformal boundary for these spacetimes introduces many novel features to the classical evolution problem. In particular, questions of dynamical stability are intimately tied to the structure of infinity. In this talk I will give an overview of recent work on the stability problem for AdS spacetimes, including numerical and analytic results. I will in particular discuss the connection of the stability problem with the structure of null infinity.

Helvi Witek, University of Cambridge
The Black-hole bomb'' mechanism in astrophysical environments

Many fundamental questions concerning the (non-linear) stability of black holes (BHs) even in four-dimensional spacetimes are still unanswered. In particular, rotating black holes may suffer from the superradiant or BH-bomb'' type instability in the presence of massive fields.These fields appear naturally in modifications of General Relativity, where interactions with the environment bestow them with an effective mass, or in extensions of the standard model predicting additional ultra-light bosonic field, such as axion-like particles or dark matter candidates.

In this talk, I will discuss phenomena resulting from the interaction between BHs and massive fields (both in the linear and non-linear regime),such as the development of long-lived bosonic condensates whose presence may lead to gaps in the BH Regge-plane or the induction of characteristic gravitational wave signals, that may transform astrophysical BHs into laboratories'' to hunt for beyond standard model physics.