# SCIENTIFIC PROGRAMS AND ACTIVITIES

June 17, 2021

THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES

## International Conference on Black Holes June 1-5, 2015

 Organizing Committee 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

Abstracts

Ivan Booth, Memorial University
Lessons from distorted black holes

This talk will review some of my work with distorted black hole spacetimes and consider the implications for the mathematical and physical characterization of black holes, the range of possible horizon geometries and black hole mechanics. The distorted spacetimes are drawn from the Weyl and Ernst families of exact solutions and include Schwarzschild, Reissner-Nordström, and Kerr-Newman black holes distorted by gravitational and electromagnetic fields. While the boundaries of these black holes continue to obey the theorems constraining extremal and near-extremal horizons, they need be neither event horizons nor marginally trapped surfaces and may be highly distorted compared to the asymptotically flat seed solutions from which they are generated. Some physical quantities (horizon area, charge and angular momentum) remain well-defined however others (mass, surface gravity, Coulomb potential and angular velocity) do not. I will examine the implications for black hole mechanics.

Robert Brandenberger, McGill University
Towards an Effective Energy Momentum Tensor for Black Hole Evaporation

In an accelerating cosmological space-time the growth of fluctuations on super-Hubble scales drains energy for the cosmological background and acts to neutralize the agent which is providing the accelerated expansion. I will discuss the effective energy-momentum tensor for cosmological perturbations which can be used to study this back-reaction effect. I will then suggest that a similar formalism can be used to study black hole evaporation. There will also be a short "teaser" related to the mystery of the origin of super-massive high redshift black holes.

Ramin Daghigh, Metropolitan State University
High Overtone Quasinormal Modes of Analog Black Holes and the Small Scale Structure of the Background Fluid

The goal of this work is to build a foundation for, and explore the possibility of, using high overtone quasinormal modes of analog black holes to probe the small scale (microscopic) structure of a background fluid in which an analog black hole is formed. This may provide a tool to study the small scale structure of some interesting quantum systems such as Bose-Einstein condensates.

Grigorios Fournodavlos, University of Toronto
Partial stability of a (real) Schwarzschild singularity

We will discuss how one can produce non-spherically symmetric Einstein vacuum spacetimes containing a singularity of Schwarzschild type by realizing a backward construction plan.

Valeri Frolov, University of Alberta
Point charge near a black hole: Bi-conformal symmetry and bi-conformal anomaly

We study a field of a point charge at rest near a higher dimensional static black hole. It is shown that the field equations possess, so called, biconformal symmetry. This symmetry allows one to relate this problem to a similar problem in the Bertotti-Robertson metric, which has enhanced symmetry. Using this approach we obtained a useful representation for a static Green function for a point charge near higher-dimensional Tangherlini and Reissner-Norstrom black holes. Exact expressions for scalar massless and electric fields of point charges in the spacetime of static 5 dimensional black holes are obtained. We also discuss applications of the obtained results to the problem of self-energy and self-force, bi-conformal anomalies, and fields and self-force of charges in the homogeneous gravitational field.

Daniel Green, University of Toronto
The Present and Future of Cosmology

The past decade has seen incredible progress in observational cosmology. Primarily from the cosmic microwave background, we have measured our many properties of our cosmic history with a sub-percent level precision. I will review our current understanding of the universe that has been provided by these observations and discuss the future directions of the field over the next decade.

Stephen Green, Perimeter Institute for Theoretical Physics

We consider the dynamics of a spherically symmetric massless scalar field coupled to general relativity in anti–de Sitter spacetime in the small-amplitude limit. We first develop the "two time framework" (TTF) approximation to study the leading self-gravitating effects of the scalar field. This framework allows us to rapidly obtain approximate solutions to the Einstein-scalar system. Within TTF, we also uncover the presence of 3 conserved quantities: the energy E, the particle number N, and the Hamiltonian H. Simultaneous conservation of E and N implies that weakly turbulent processes undergo dual cascades (direct cascade of E and inverse cascade of N or vice versa), and it rules out energy equipartition for generic initial data. Finally, TTF leads us to uncover a large class of quasi-periodic solutions, and we discuss their role as possible islands of stability.

Coauthors: Alex Buchel, Luis Lehner, Steven L. Liebling, Antoine Maillard

Gary Horowitz, University of California, Santa Barbara
Hovering Black Holes

I describe a new class of asymptotically anti-de Sitter charged black holes that have recently been constructed numerically. These black holes exhibit some surprising universality properties which are not yet understood analytically. Using the remarkable gauge/gravity duality, they have applications to localized defects in special condensed matter systems.
Coauthors: Nabil Iqbal, Jorge Santos and Benson Way.

Uzair Hussain, Memorial University
Master equations and the boundary stress tensor for Ads4-Schwarzschild blackholes

We revisit the problem of perturbations of Schwarzschild-AdS$_4$ black holes by using a combination of the Martel-Poisson gauge-invariant formalism for perturbations of Schwarzschild [gr-qc/0502028] and the Kodama-Ishibashi formalism [hep-th/0305147]. We clarify the relationship between both formalisms and calculate the boundary stress tensor, $T^\mu_\nu$, on a constant-\emph{r} surface purely in terms of the even and odd master functions. Invoking the conservation equations $\nabla_\mu T^\mu_\nu =0$ on such a constant-\emph{r} surface we find that the wave equations for both the even and odd master functions are equivalent to the conservation of stress energy along directions tangent to spherical symmetry. This direct comparison of the master equations and the conservation equations is presented explicitly for the first time. We also calculate the renormalized stress tensor at the boundary $\frac{r}{L} \lim_{r \rightarrow \infty} T_{\mu\nu}$ by using our expressions for the stress tensor in terms of the master function and demonstrate the fluid/gravity duality for large black holes. We also investigate the possibility of a Cotton tensor/stress tensor duality, on a constant-$r$ surface, motivated by the duality being held at $r\rightarrow\infty$ [arXiv:0809.4852v2].

Coauthors: Ivan Booth (Memorial University) Hari Kunduri (Memorial University)

Alexandru Ionescu, Princeton University
On the stability of the wave-map equation in Kerr spaces

I will discuss some recent work on the stability of the wave-map equation in Kerr spaces with small angular momentum. This problem should be viewed in the context of the larger stability/rigidity problem for the family of Kerr spaces. The talk is based on joint work with Alexakis and Klainerman.

Soichiro Isoyama, University of Guelph
Hamiltonian Dynamics of Self-Forced Motion in Kerr Spacetime

Post-geodesic motion of a point particle in Kerr spacetime subjected to the back reaction of its own perturbation field, namely self-force, is an important subject to model an inspiral of stellar-mass compact object into a massive black hole, with application to gravitational astronomy. To establish an efficient scheme to compute such binary dynamics in the extreme mass ratio limit, we develop a Hamiltonian formulation of the self-force dynamics in Kerr spacetime by describing them as geodesic motion in a certain locally defined effective spacetime. In this talk, focusing on the conservative dynamics, we show that the perturbed Hamiltonian system is effectively "integrable" for most generic stable bound orbits; there exists the perturbed version of action variables for geodesic motion in Kerr spacetime, which are conserved along the orbit. Based on theintegrable'' Hamiltonian, we also sketch to compute the frequency shift of the inner most stable inclined circular orbit, which provides a potentially observable effect of the conservative self-force effect, and present its numerical result in the equatorial circular limit.

Coauthors: Ryuichi Fujita, Alexandre Le Tiec, Hiroyuki Nakano, Norichika Sago and Takahiro Tanaka

David Kubiznak, Perimeter Institute for Theoretical Physics
Ultraspinning limits and super-entropic black holes

By employing the new ultraspinning limit we construct novel classes of black holes in four and higher dimensions with non-compact event horizons and finite horizon area. Our ultraspinning limit can be understood as a simple generating technique that consists of three steps: i) transforming the known rotating AdS black hole solution to a special coordinate system that rotates (in a given 2-plane) at infinity ii) boosting this rotation to the speed of light iii) compactifying the corresponding azimuthal direction. In so doing we qualitatively change the structure of the spacetime since it is no longer possible to return to a frame that does not rotate at infinity. The obtained black holes have non-compact horizons with topology of a sphere with two punctures. The entropy of some of these exceeds the maximal bound implied by the reverse isoperimetric inequality, such black holes are super-entropic.

Coauthors: Robie Hennigar, Robert B. Mann, Nathan Musoke

Hari Kunduri, Memorial University
New Black Holes in Five Dimensions

We will discuss a new asymptotically flat, supersymmetric black hole solution to five-dimensional supergravity. This solution is regular on and outside an event horizon of lens-space topology L(2,1) and is the first example of an asymptotically flat black hole with lens-space topology. The geometry is characterized by a charge, two angular momenta, and a magnetic flux though a non-contractible disc region ending on the horizon. In addition, we will discuss a second new family of solutions describing a spherical black hole with a two-cycle in the exterior region. We show there are black holes in this family with identical conserved changes to the BMPV black hole, thereby demonstrating black hole non-uniqueness in this context.

Coauthors: James Lucietti

Philippe Landry, University of Guelph
Tidal Deformation of a Slowly Rotating Compact Body

The deformation of a compact body subject to weak, slowly varying tidal forces is characterized by a set of dimensionless, equation-of-state-dependent constants known as tidal Love numbers. If the body in question is non-rotating, its gravitational response to a generic quadrupolar tidal field is encoded in two of these constants: the gravitoelectric Love number k&#8322;&#7497;&#737; and the gravitomagnetic Love number k&#8322;&#7504;&#7491;&#7501;. If, however, the body is rotating – as is typically the case in astrophysical scenarios – coupling between its angular momentum vector and the tidal field complicates its response. The problem is tractable in the slow-rotation limit, and four additional quantities, designated tidal-rotational Love numbers, are required to give a complete description of the body's deformation. These new Love numbers are shown to vanish when the body is a black hole.

Coauthors: Eric Poisson

Geoffrey Lovelace, California State University, Fullerton
Nearly extremal apparent horizons in numerical simulations of colliding black holes

The spin $S$ of a Kerr black hole is bounded by the surface area $A$ of its apparent horizon: $8 \pi S \leq A$. In this talk, we will present recent results (arXiv:1411.7297) for the extremality of apparent horizons for merging, rapidly rotating black holes with equal masses and equal spins aligned with the orbital angular momentum. Measuring the area and (using approximate Killing vectors) the spin on the individual and common apparent horizons, we find that the inequality $8 \pi S < A$ is satisfied but is very close to equality on the common apparent horizon at the instant it first appears---even for initial black-hole spins as large as $S/M^2=0.994$. We introduce a gauge-invariant lower bound $e_0$ on the extremality by computing the smallest value that Booth and Fairhurst's extremality parameter can take for any scaling of the horizon's null normal vectors, concluding that the common horizons are at least moderately close to extremal just after they appear. We construct binary-black-hole initial data with marginally trapped surfaces with $8 \pi S > A$ and $e_0 > 1$, but these surfaces are always surrounded by apparent horizons with $8 \pi S < A$ and $e_0 < 1$.

Coauthors: Mark A. Scheel, Robert Owen, Matthew Giesler, Reza Katebi, Bela Szilagyi, Tony Chu, Nicholas Demos, Daniel A. Hemberger, Lawrence E. Kidder, Harald P. Pfeiffer, Nousha Afshari

Jonathan Luk, University of Cambridge
The stability of the Kerr Cauchy horizon and the strong cosmic censorship conjecture in general relativity

I will discuss recent work on the structure of black hole interiors for dynamical vacuum spacetimes (without any symmetry) and what this means for the question of the nature of generic singularities in general relativity and the celebrated strong cosmic censorship of Penrose. This is joint work with Mihalis Dafermos.

Raissa Mendes, University of Guelph
On the possibility of setting a new constraint to scalar-tensor theories

Scalar-tensor theories are a widely studied alternative to general relativity in which gravity is endowed with an additional scalar degree of freedom. Although severely constrained by solar system and pulsar timing experiments, there remains a large set of scalar-tensor theories which are consistent with all present day observations. In this poster, I discuss a recent result [PRD 91, 064024 (2015)] on the possibility of probing a yet unconstrained region of the parameter space of scalar-tensor theories based on the fact that stability properties of highly compact neutron stars in these theories may radically differ from those in general relativity.

Jonas Mureika, Loyola Marymount University
Sub-Planckian Black Holes and the Generalized Uncertainty Principle

The generalized uncertainty principle suggests there is a minimum mass for a black hole, or alternatively a maximum particle mass ($M = M_{\rm Pl}$). The prospect of sub-Planckian black holes ($M \ll M_{\rm Pl}$) is explored in the context of a new self-dual, Schwarzschild-like metric that encodes features of the GUP. It is shown that not only do sub-Planckian black holes exist in this scenario, but that they are governed by what appears to be an effectively (1+1)-dimensional gravitational theory. This adds support to the notion that dimensional reduction is an expected feature of quantum gravity.

Coauthors: Bernard Carr, Piero Nicolini

Maria Okounkova, California Institute of Technology
Numerical Tests of the Cosmic Censorship Conjecture via Event-Horizon Finding

We present the current state of our research on the possibility of naked singularity formation in gravitational collapse, numerically testing both the cosmic censorship conjecture and the hoop conjecture. The former of these posits that all singularities lie behind an event horizon, while the later conjectures that this is true if collapse occurs from an initial configuration with all circumferences C = 4 pi M. We reconsider the classical Shapiro & Teukolsky (1991) prolate spheroid naked singularity scenario. Using the exponentially error-convergent Spectral Einstein Code (SpEC) we simulate the collapse of collisionless matter and probe for apparent horizons. We propose a new method to probe for the existence of an event horizon by following characteristic from regions near the singularity, using methods commonly employed in Cauchy characteristic extraction.

Coauthors: Mark Scheel, Yanbei Chen

Amanda Peet, University of Toronto
String modelling of black holes and holography

The string theory approach to quantum gravity forms the context for our discussion of two interrelated themes: microscopic string modelling of black holes, and AdS/CFT holography.

A 2009 result of S.Mathur showed that the black hole information paradox cannot be resolved via semiclassical perturbation theory. Starting from the microscopic SCFT of the prototype D1-D5-brane system in a top-down approach, we use conformal perturbation theory to explore aspects of deforming towards the classical black hole geometry.

Our second theme recruits a more bottom-up motivation to explore holographic setups with increasing amounts of broken symmetry. We discuss modelling disorder holographically for the charged case, using perturbation theory in disorder strength to construct solutions including gravitational backreaction, and find the conductivity.

This talk will be pitched at colloquium level and will assume no prior knowledge of string theory.

Paolo Pani, Sapienza University of Rome
Tidal deformations of a spinning compact object

The deformability of a compact object induced by a perturbing tidal field is encoded in the tidal Love numbers, which depend sensibly on the object's internal structure. Tidal Love numbers are known only for static, spherically-symmetric objects. As a first step to compute the tidal Love numbers of a spinning compact star, here we extend powerful perturbative techniques to compute the geometry of a tidally-distorted spinning object to second order in the angular momentum. The spin of the object introduces couplings between electric and magnetic deformations and new classes of induced Love numbers emerge. For example, a spinning object immersed in a quadrupolar, electric tidal field can acquire some induced mass, spin, quadrupole, octupole and hexadecapole moments to second order in the spin. The deformations are encoded in a set of inhomogeneous differential equations which, remarkably, can be solved analytically in vacuum. We prove that the tidal Love numbers of a Kerr black hole are zero to second order in the spin and provide the explicit solution for a slowly-rotating, tidally-deformed Kerr black hole.

Coauthors: Leonardo Gualtieri, Andrea Maselli, Valeria Ferrari

Point particle perturbations in Kerr spacetime: reconstruction, completion, and self-force

Computing the metric perturbation produced by a point particle moving around a Kerr black hole, and finding the back-reaction of that perturbation on the particle's motion, is an important problem in GR, with applications to both gravitational wave astronomy and fundamental physics. Since the 1970s, there has been a standard, efficient method of obtaining metric perturbations in a Kerr background by reconstructing them from a Weyl curvature scalar. However, when a point particle is introduced into this reconstruction procedure, gauge singularities arise that extend away from the particle, significantly complicating the definition and calculation of the gravitational self-force (i.e., the back-reaction). Furthermore, the reconstruction procedure does not uniquely determine the metric perturbation, instead leaving the freedom to add "trivial" perturbations that shift the spacetime's mass and angular momentum; because the particle's orbit divides the spacetime into two regions, one must "complete" the reconstructed perturbation by finding the correct mass and angular momentum perturbations to add inside and outside the orbit. In this talk I describe recent work that establishes the correct "completion" terms and a rigorous method of computing the self-force (and related quantities) from the reconstructed and completed metric.

Coauthors: Cesar Merlin Gonzalez, Leor Barack, Amos Ori, Abhay Shah, and Maarten van de Meent

Enthalpy and Geometric Volume for Van der Waals Black Hole \

Interpreting the negative cosmological constant as a dynamical pressure and the volume is its thermodynamically conjugate variable then the gravitational mass could be expressed as the total gravitational enthalpy rather than the energy. A new phenomena then emerges in the context of extended phase space thermodynamics. We \emph{examine} here these features for recently discovered Van der Waal black hole which is analogous to the Van der Waals fluid. We show that the thermodynamic volume is \emph{greater} than the naive geometric volume. We also show that the Smarr-Gibbs-Duhem relation is \emph{satisfied} for this black hole. Furthermore, by computing the specific heat we find the stability criterion for this black hole. It has been observed that under certain condition the black hole displays the \emph{second order phase transition}.

Frans Pretorius, Princeton University
Eccentric Mergers

Binary compact object mergers are among the primary gravitational wave sources expected to be observed by the next generation of ground-based gravitational wave detectors. Early success of this endeavour will to a large extent depend on how well we can model these events. I will discuss a class of compact object binaries that arise from dynamical capture in dense cluster environments. The typical merger from this class of event will differ from a traditional primordial binary in that the initial orbit will be highly eccentric, and when neutron stars are involved the neutron stars could have high spin. Even if, as expected, such events are rare, they could offer exceptional laboratories to learn about general relativity in the dynamical strong-field regime, and with neutron stars could host unusually bright electromagnetic counterparts. However, to realize this potential of eccentric systems will requiring overcoming some significant challenges in modelling them.

Volker Schlue, University of Toronto
Non-existence of time-periodic dynamics in general relativity

In general relativity, a self-gravitating system is not expected to display time-periodic behavior due to the emission of gravitational waves. We show that any asymptotically flat solution to the Einstein vacuum equations, which is assumed to be time-periodic, is in fact stationary near infinity. Thus genuinely time-periodic vacuum space-times do not exist, at least far away from the sources. The proof applies under physically relevant smoothness assumptions, and employs a uniqueness theorem for linear waves obtained jointly with S. Alexakis and A. Shao.

Coauthors: Spyros Alexakis

Saul Teukolsky, Cornell University
Simulations of Black Holes and Neutron Stars

Advanced LIGO will conduct its first science run this summer. One of the prime scientific goals is to detect waves from the coalescence and merger of black holes and neutron stars in binary systems. Confronting such signals with the predictions of Einstein's General Theory of Relativity will be the first real strong-field test of the theory. I will describe the status of numerical simulations of such systems, which have set things up for an epic confrontation between theory and experiment. I will also describe the limitations of current codes for computational astrophysics and the ingredients of a next-generation code for upcoming exascale machines.

William Unruh, University of British Columbia
Partners

Black holes evaporate sending out thermal incoherent radiation which can be detected by detectors. But those particles each have a partner, which purifies the state. Ie, for each detected particle mode there is another one which is maximally entangled with it. We derive this mode, show that the relation is not linear, use it to ask about the partner of detector excited by the vacuum, and argue that in some cases, the partner can be a vacuum fluctuation. This may have some relation to the question of black hole "Unitarity".

Robert Wald, University of Chicago
Dynamic and Thermodynamic Stability of Black Holes and Black Branes

I describe work with Stefan Hollands that establishes a new criterion for the dynamical stability of black holes and black branes with respect to axisymmetric perturbations. Our analysis is done in vacuum general relativity without a cosmological constant in $D \geq 4$ spacetime dimensions, but our approach is applicable to much more general situations. We show that the positivity of the canonical energy, $\mathcal E$, on a subspace of linearized solutions that have vanishing linearized ADM mass and angular momentum implies mode stability. Conversely, failure of positivity of canonical energy on this subspace implies instability in the sense that there exist perturbations that cannot asymptotically approach a stationary perturbation. We further show that the canonical energy is related to the second order variations of mass, angular momentum, and horizon area by $\mathcal E = \delta^2 M - \sum_i \Omega_i \delta^2 J_i - (\kappa/8\pi) \delta^2 A$. This establishes that dynamic stability of a black hole is equivalent to its thermodynamic stability (i.e., its area, $A$, being a maximum at fixed state parameters'' $M$, $J_i$). For a black brane, we further show that a sufficient condition for instability is the failure of the Hessian of $A$ with respect to $M$, $J_i$ to be negative, thus proving a conjecture of Gubser and Mitra. We also prove that positivity of $\mathcal E$ is equivalent to the satisfaction of a local Penrose inequality,'' thus showing that satisfaction of this local Penrose inequality is necessary and sufficient for dynamical stability.

S.T. Yau, Harvard University
General Relativity and Mathematics

Exactly one century ago, Einstein wrote down his famous equation that governs gravity and the dynamical spacetime. I will describe the mathematics behind Einstein's work and its influence to modern development of geometry, analysis and string theory. I will talk about our recent work on defining quasi-local mass in general relativity, which has been developed in collaboration with Mu-Tao Wang and Po-Ning Chen.