# SCIENTIFIC PROGRAMS AND ACTIVITIES

July 23, 2018

THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES

## Singularities in General Relativity June 15-18, 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 Jim Isenberg, University of Oregon Jonathan Luk, University of Cambridge

 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.nts 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

Details to come.

Participants as of June 9, 2015
* Indicates not yet confirmed

 Full Name University/Affiliation Spyros Alexakis University of Toronto Ellery Ames Chalmers University Xinliang An Rutgers University Florian Beyer University of Otago Wilson Brenna University of Waterloo * Vitor Cardoso Universidade Tecnica de Lisboa João Costa University Institute of Lisbon Stefan Czimek Laboratoire Jacques-Louis Lions, Universite Paris 6 Mihalis Dafermos Princeton University Dominic Dold Cambridge University Grigorios Fournodavlos University of Toronto Dejan Gajic University of Cambridge David Garfinkle Oakland University * Gary Gibbons University of Cambridge Cécile Huneau Ecole Normale Superieure Jim Isenberg University of Oregon Orchidea Maria Lecian Sapienza University of Rome Luis Lehner Perimeter Institute Adam Lewis University of Toronto Jonathan Luk University of Cambridge * Brian Markle Sung-Jin Oh University of California, Berkeley * Amos Ori Technion * Harvey Reall University of Cambridge Jan Sbierski University of Cambridge Volker Schlue University of Toronto Arick Shao Imperial College London Yakov Shlapentokh-Rothman Massachusetts Institute of Technology Matteo Smerlak Perimeter Institute * Jacques Smulevici Université Paris-Sud * Jared Speck Massachusetts Institute of Technology * Oh Sung-jin University of California, Berkeley Norihiro Tanahashi University of Cambridge Martin Taylor Cambridge University Aaron Zimmerman Canadian Institute for Theoretical Astrophysics

 Monday, June 15 2:00-2;45 Mihalis Dafermos, Princeton University 3:00-3:45 Jan Sbierski, Magdalene College, Cambridge. The C^0 inextendibility of the Schwarzschild spacetime 4:00-4:30 Coffee break 4:30-5:15 Grigorios Fournodavlos, University of Toronto On the backward stability of the Schwarzschild black hole singularity Tuesday, June 16 10:00-10:45 João Costa, University Institute of Lisbon On strong cosmic censorship with a cosmological constant 11:00-11:45 Luis Lehner, Perimeter Institute Black hole instabilities in higher dimensions and naked singularities 12:00-2:00 Lunch 2:00-2:45 Jonathan Luk, University of Cambridge 3:00-3:45 Sung-Jin Oh, University of California, Berkeley Linear instability of the Reissner-Nordström Cauchy horizon under scalar perturbations 4:00-4:30 Coffee break 4:30-5:00 Discussion of Weak Null Singularities and Strong Cosmic Censorship Wednesday, June 17 10:00-10:45 Yakov Shlapentokh-Rothman, Massachusetts Institute of Technology Stability and Instability of Scalar Fields on Kerr Spacetimes 11:00-11:45 Huan Yang, Perimeter Institute Holographic Insights into Black Hole Spacetimes 12:00-2:00 Lunch break 2:00-2:45 Norihiro Tanahashi, University of Cambridge Causality, Hyperbolicity and Shock Formation in Lovelock Theories 3:00-3:45 Jacques Smulevici, Université Paris-Sud On the future asymptotics of polarized T2 symmetric spacetime Thursday, June 18 10:00-10:45 Jim Isenberg, University of Oregon 11:00-11:45 Ellery Ames, Chalmers University The Asymptotic Value Problem for the Einstein Field Equations and AVTD Solutions 12:00-2:00 Lunch break 2:00-2:45 David Garfinkle, Oakland University 3:00-3:30 Discussion of AVTD, Mixmaster, and Strong Cosmic Censorship

Ellery Ames, Chalmers University
The Asymptotic Value Problem for the Einstein Field Equations and AVTD Solutions

The notion of an asymptotic value problem can be introduced in cases where one is interested in solutions to PDE with prescribed asymptotics. The problem of finding AVTD solutions is naturally formulated as an asymptotic value problem for the Einstein field equations. In this case the prescribed asymptotics are obtained from a certain set of equations called the VTD system, which are in turn derived from the Einstein equations. In this talk I will present the asymptotic value problem for the vacuum Einstein field equations in wave gauges. The reason for using wave gauges is to obtain a symmetric hyperbolic formulation, a structural condition which is key in obtaining non-analytic solutions. The wave gauge formulation also allows one to consider different coordinate systems. As an example I will discuss results for AVTD solutions in spacetimes with Gowdy symmetry. This work provides tools with which to investigate the coordinate-dependence of the VTD property, as well as establishing smooth AVTD solutions with U(1)-symmetry.

João Costa, University Institute of Lisbon
On strong cosmic censorship with a cosmological constant

Motivated by the Strong Cosmic Censorship Conjecture (SCCC) we consider the problem of global uniqueness for the Einstein-Maxwell-scalar field system with a cosmological constant, for spherically symmetry characteristic initial data. First we consider the situation where the outgoing data is stationary (i.e., prescribed by a subextremal Reissner Nordstroem black hole event horizon) and the remaining data is otherwise free. In that case, one can find an open set of free data for which it is possible to construct regular extensions of the maximal (globally hyperbolic) development. This provides indirect evidence for the failure of the SCCC in the case of a positive cosmological constant.

To go from indirect evidence to results applying unequivocally to the conjecture at hand we present some preliminary results concerning the case where the outgoing data, instead of stationary, satisfies Price's law.

Grigorios Fournodavlos, University of Toronto
On the backward stability of the Schwarzschild black hole singularity

We study the backwards-in-time stability of the Schwarzschild singularity from a dynamical PDE point of view. More precisely, considering a spacelike hypersurface $\Sigma_0$ in the interior of the black hole region, tangent to the singular hypersurface $\{r=0\}$ at a single sphere, we study the problem of perturbing the Schwarzschild data on $\Sigma_0$ and solving the Einstein vacuum equations backwards in time. We obtain a local well-posedness result for small perturbations lying in certain weighted Sobolev spaces, without any symmetry assumptions. The perturbed spacetimes all have a singularity at a collapsed'' sphere on $\Sigma_0$, where the leading asymptotics of all geometric components match those of their Schwarzschild counterparts to a suitably high order. This result thus yields a class of non-symmetric vacuum spacetimes, evolving forwards-in-time from smooth initial data, which form a Schwarzschild type singularity at a collapsed sphere. We rely on a precise asymptotic analysis of the Schwarzschild geometry near the singularity which appears to be just borderline for our method to be applicable.

Sung-Jin Oh, University of California, Berkeley
Linear instability of the Reissner-Nordström Cauchy horizon under scalar perturbations

Consider the linear scalar wave equation on a fixed subextremal Reissner-Nordström spacetime with non-vanishing charge. In this talk, I will present a proof that generic smooth and compactly supported initial data on a Cauchy hypersurface give rise to solutions with infinite nondegenerate energy near the Cauchy horizon in the interior of the black hole. This is a joint work with J. Luk.

This linear instability of the Cauchy horizon is related to the celebrated blue shift effect in the interior of the black hole. The problem is motivated by the strong cosmic censorship conjecture, and it is expected that for the full nonlinear Einstein-Maxwell system this instability leads to a singular Cauchy horizon for generic small perturbations of Reissner-Nordström spacetime.

Jan Sbierski, Magdalene College, Cambridge.
The C^0 inextendibility of the Schwarzschild spacetime

The maximal analytic Schwarzschild spacetime is manifestly inextendible as a Lorentzian manifold with a C^2 regular metric. In this talk I will describe how one proves the stronger statement that the maximal analytic Schwarzschild spacetime is inextendible as a Lorentzian manifold with a continuous metric. The investigation of low-regularity inextendibility criteria is motivated by the strong cosmic censorship conjecture.

Yakov Shlapentokh-Rothman, Massachusetts Institute of Technology
Stability and Instability of Scalar Fields on Kerr Spacetimes

I will discuss some stability and instability results for wave and Klein-Gordon equations on sub-extremal Kerr exterior backgrounds. More specifcally, for the wave equation we will see that general finite energy solutions have a uniformly bounded energy and satisfy an integrated local energy decay estimate. In contrast, for the Klein-Gordon equation we will see that there exist finite energy solutions which grow exponentially. We will also discuss the implications of these results for black hole stability. Some of this work is joint with Mihalis Dafermos and Igor Rodnianski.

Norohiro Tanahashi, University of Cambridge
Causality, Hyperbolicity and Shock Formation in Lovelock Theories

We study gravitational wave propagation in Lovelock theories, which are extensions of Einstein's theory by higher-curvature corrections, to examine if these theories have good properties such as causality and hyperbolicity. We study the propagation on various background spacetime, and find that initial value problem may cease to be well-posed in some case. We also show that the sound speed of gravitational wave depends on the background and it may cause shock formation in these theories. We discuss implications of these phenomena.

Huan Yang, Perimeter Institute
Holographic Insights into Black Hole Spacetimes

Motivated by the gravity/fluid correspondence, I will first discuss a new type of nonlinear instability of rapidly-spinning black holes, which display a inverse-cascading turbulent-like phenomenon. After that I will generalize the analysis and introduce a new method to characterizing nonlinear gravitational interaction. Namely the nonlinear perturbative form of Einstein equation is mapped to the equation of motion of a collection of nonlinearly-coupled harmonic oscillators. These oscillators correspond to the quasinormal or normal modes of the background spacetime. In spacetimes with gravity/fluid correspondence, such as AdS black branes, this formalism gives equivalent equations of motion as the Navier-Stokes equation of the boundary fluid. It is also expected to remain valid in more general spacetimes.