picture of Christoph KehleTitle: Retiring the third law of black hole thermodynamics

Abstract: Extremal black holes are special types of black holes which have exactly zero temperature. I will present a proof that extremal black holes form in finite time in gravitational collapse of charged matter. In particular, this construction provides a definitive disproof of the “third law” of black hole thermodynamics. We also show that extremal black holes take on a central role in gravitational collapse, giving rise to a new conjectural picture of “extremal critical collapse.” This is joint work with Ryan Unger (Princeton).


Title: The equality case of the positive mass theorem

Abstract: We will review the equality case of the positive mass theorem for initial data sets and discuss its proof in the general (not necessarily spin) setting. This is joint work with Lan-Hsuan Huang.


Title: The Teukolsky equation on Kerr in the full subextremal range

Abstract: The Teukolsky equation is one of the fundamental equations governing linear gravitational perturbations of the Kerr black hole family as solutions to the vacuum Einstein equations. I will discuss joint work with Yakov Shlapentokh-Rothman (Toronto), where we show that solutions arising from suitably regular initial data decay inverse polynomially in time. Our proof holds for the entire subextremal range of Kerr black hole parameters (|a| < M).

No video available

Title: Squeezing a fixed amount of gravitational energy to arbitrarily small scales, in U(1) symmetry

Abstract: We discuss a construction of polarized U(1) space-times of a “fixed size”, where the initial data contains a given amount η>0 of incoming gravitational energy which is allowed to be supported at arbitrarily small scales, without affecting the time of existence. More specialized examples can be constructed where any chosen fraction of this initial incoming energy can stay focused along a null geodesic, in the U(1) picture. We discuss the relationship of this result with one direction of Thorne’s hoop conjecture. The work relies on a special structure of the Einstein equations in this symmetry class, a new re-scaling of the equations and large-time control, along with a new local Klainerman-Sobolev estimate. Time permitting, some comparisons of the result and the methods with large-data constructions for the Einstein equations over the last decade will also be presented. This is joint work with Nathan Carruth.


Title: How minimal hypersurface and MOTS singularities affect relativity theorems

Abstract: Some theorems in relativity work directly only in low dimensions because of the possibility of singularities in area minimizing hypersurfaces and MOTS. These include both Riemannian and spacetime positive mass theorems as well as the Riemannian Penrose inequality. In some cases the singularities are relatively easy to work around and in others they present significant problems to overcome. This will be a general lecture which will attempt to give perspective and more clarity to this issue.

This was a special lecture streamed from the ESI conference “Mathematical Relativity: Past, Present, Future” on the occasion of the 100th birthday of Yvonne Choquet-Bruhat on the 29th of December 2023.  Watch recording.

Title: Some observations on quasi-local mass and quasi-local radius

Abstract: The lecture examines older and newer concepts of quasi-local mass for one domain relative to another as well as concepts of quasi-local radius in relation to the hoop-conjecture.

No video available


Title: High frequency solutions in general relativity and the Burnett’s conjecture

Abstract: In this talk, I will review some works in collaboration with Jonathan Luk on the behaviour of high frequency solutions to the Einstein equations. In the eighties, the physicist Burnett conjectured that the adherence of the solutions to the Einstein equations, converging strongly at the level of the metric, and with boudedness assumptions at the level of derivative of the metric, are the solutions to the Einstein-massless Vlasov equations. I will focus on the resolution of this conjecture with the additionnal assumption of a translation symmetry, and explain how to approximate a solution to the Einstein-Vlasov equations by vacuum solutions.

No video available

Title: A Green’s function proof of the positive mass theorem and related topics

Abstract: In this talk, we will explore the role of harmonic and p-harmonic functions in establishing fundamental geometric inequalities in mathematical relativity, such as the positive mass theorem and the Penrose inequality. We will introduce new monotonicity formulas along the level flow of these functions, which provide simple proofs for these inequalities. Additionally, we will discuss how these formulas can be used to characterize the nonnegativity of the scalar curvature, suggesting a synthetic notion of this concept applicable in non-smooth contexts.


Title: Abstract Lorentzian metric spaces and their Gromov-Hausdorff convergence

Abstract: A definition for `bounded Lorentzian metric space’ is presented and discussed. This is an abstract notion of Lorentzian metric space that is sufficiently general to comprise compact causally convex subsets of globally hyperbolic (smooth) spacetimes, and causets.
It is shown that a generalization of the Gromov-Hausdorff distance and convergence can be applied to these spaces. Furthermore, two additional axioms of timelike connectedness and existence of maximizers, which are stable under GH-convergence, lead to suitable notions of Lorentzian pre-length and length spaces. Similarly, sectional curvature bounds stable under GH-convergence can be introduced. A (pre)compactness theorem is also mentioned and its limitations are discussed. Talk based on joint work with Stefan Suhr (Bochum).


Title: Wave equations in subextremal Kerr-de Sitter spacetimes

Abstract: In 2013, Vasy proved that solutions to linear wave equations in Kerr-de Sitter spacetimes have asymptotic expansions in quasinormal modes up to an exponentially decaying term, assuming the angular momentum of the black hole satisfies certain bounds. This was the first step towards the proof of non-linear stability for slowly rotating Kerr-de Sitter black holes by Hintz and Vasy in 2018. In this talk, we extend Vasy’s result to the full subextremal range of Kerr-de Sitter spacetimes, by removing the restrictions on the angular momentum of the black hole. The proof is based on a new Fredholm setup and a new analysis of the trapping of photons around a Kerr-de Sitter black hole. This is joint work with Andras Vasy.


Title: A nonsmooth approach to Einstein’s theory of gravity

Abstract: While Einstein’s theory of gravity is formulated in a smooth setting, the celebrated singularity theorems of Hawking and Penrose describe many physical situations in which this smoothness must eventually breakdown. In positive-definite signature, there is a highly successful theory of metric and metric-measure geometry which includes Riemannian manifolds as a special case, but permits the extraction of nonsmooth limits under dimension and curvature bounds analogous to the energy conditions in relativity: here sectional curvature is reformulated through triangle comparison, while Ricci curvature is reformulated using entropic convexity along geodesics of probability measures. This lecture explores recent progress in the development of an analogous theory in Lorentzian signature, whose ultimate goal is to provide a nonsmooth theory of gravity.


Title: The nonlinear stability of Kerr for small angular momentum

Abstract: I will introduce the celebrated black hole stability conjecture according to which the Kerr family of metrics are stable as solutions to the Einstein vacuum equations of general relativity. I will then discuss the history of this problem, including a recent work on the resolution of the black hole stability conjecture for small angular momentum.

Title: Physical-space estimates on black hole perturbations
Abstract: Most works on the analysis of the wave equation on Kerr black holes rely on a combination of the vector field method and Fourier decomposition, with the notable exception of a generalized vector field method introduced by Andersson-Blue. Their method allows for commutation with second order differential operators entirely in physical-space by supplementing the Killing vector fields with the Carter operator of Kerr to obtain a local energy decay identity at the level of three derivatives of the solution for sufficiently small |a|. I this talk I will describe the main ideas of Andersson-Blue’s method and explain its advantages in two recent applications where physical space-estimates have been crucial: the linear stability of Kerr-Newman black hole to coupled gravitational-electromagnetic perturbations and our proof of the non-linear stability of the slowly rotating Kerr family with Klainerman-Szeftel.

Title: Relativistic elasticity and compactness bounds

Abstract: After reviewing the basics of relativistic elasticity theory, I will introduce a general framework to study spherically symmetric self-gravitating elastic bodies systematically within general relativity, and apply it to investigate compactness bounds in this context.


Title: Level Set Methods in Mathematical GR

Abstract: We will review recent developments concerning the use of level set techniques associated with solutions to elliptic equations, in particular spacetime harmonic functions, and their application to positive mass theorems and comparison geometry.


Title: A quasi-local view of black hole mergers

Abstract: In this talk I will summarize recent progress and open mathematical problems in understanding black hole mergers quasi-locally, i.e. by using marginally trapped surfaces instead of event horizons.



Title: Uniqueness of large area-constrained Willmore spheres in initial data sets

Abstract: The Hawking mass of an area-constrained Willmore sphere is a useful quasi-local measure for the strength of the gravitational field of an initial data set for the Einstein field equations. If the scalar curvature satisfies certain asymptotic assumptions, the asymptotic region of such an initial data set is foliated by large area-constrained Willmore spheres with non-negative Hawking mass. It has been conjectured that these are the only such spheres. In this talk, I will present a proof of this conjecture for all large area-constrained Willmore spheres with non-negative Hawking mass and outer radius less than the logarithm of the inner radius. This is joint work with Michael Eichmair, Jan Metzger, and Felix Schulze.


Title: Stability of AVTD Behavior for Polarized T^2-Symmetric Space-Times

Abstract: One of the most effective tools for proving that Strong Cosmic Censorship holds for a family of cosmological solutions of Einstein’s equations is to verify that these solutions all exhibit Asymptotically Velocity Term Dominated (AVTD) behavior in a neighborhood of their Big Bang singularities. After presenting some background history of known results and conjectures concerning the behavior of the gravitational field near the Big Bang in cosmological solutions of Einstein’s equations, we discuss recent work with Ellery Ames, Florian Beyer, and Todd Oliynyk in which we prove that polarized T^2-Symmetric vacuum solutions in a neighborhood of Kasner solutions all exhibit AVTD behavior close to the initial singularity. We also discuss our very recent extension of these results to the case of non-vanishing cosmological constant.


Title: Towards a Spacetime Intrinsic Flat Convergence

Abstract: In order to define a spacetime intrinsc flat convergence, Carlos Vega and I defined the null distance to convert spacetimes endowed with regular cosmological times into metric spaces. Currently Anna Sakovich and I have been exploring the properties of these metric spaces.  We can prove that one can recover the causal structure from the null distance and the cosmological time.  We can
prove that a distance-preserving time-preserving bijection between the spacetimes endowed with a null distance is in fact a Lorentzian isometry under suitable conditions.  Next we will prove there are
biLipschitz charts so that we may view the spacetimes endowed with the null distance as integral current spaces.  This allows us to rigorously define the spacetime intrinsic flat convergence for
spacetimes that arise as the future maximal developments of initial
data sets.   For more information about intrinsic flat convergence see


Title: The Memory Effect and Infrared Divergences

Abstract: The “memory effect” is the permanent relative displacement of test particles after the passage of gravitational radiation. It is associated with both the propagation of massive bodies out to timelike infinity (“ordinary memory”) or the propagation of radiation out to null infinity (“null memory”). The memory effect can be characterized by the failure of the shear tensor at order 1/r to return to zero at late times, even though it is “pure gauge.” Closely analogous effects occur in electromagnetism, where the vector potential at order 1/r fails to return to zero even though it is “pure gauge.” In both cases, the Fourier transform of the radiative field has divergent behavior at low frequencies. This gives rise to infrared divergences (i.e., infinite numbers of “soft” gravitons/photons) in the quantum field theory description if one attempts to describe these states as vectors in the usual Fock Hilbert space representation. To obtain a mathematically sensible quantum scattering theory, one must allow states with nonvanishing memory in the “in” and “out” Hilbert spaces. An elegant solution to this problem in massive quantum electrodynamics was given by Kulish and Fadeev, who constructed a Hilbert space of incoming/outgoing charged particle states that are “dressed” with radiative fields of corresponding memory, so as to yield vanishing large gauge charges at spatial infinity. However, we show that this type of construction fails in quantum gravity. The primary underlying reason is that the “dressing” contributes to null memory, thereby invalidating the construction of eigenstates of large gauge charges. In quantum gravity, there does not appear to be any choice of (separable) Hilbert space of incoming/outgoing states that can accommodate all scattering states. Thus, we argue that scattering should be described at the level of algebraic incoming/outgoing states rather than attempting to artificially restrict states to a particular Hilbert space.


Title: Global Stability of Spacetimes with Supersymmetric Compactifications

Abstract: Spacetimes with compact directions which have special holonomy, such as Calabi-Yau spaces, play an important role in supergravity and string theory. In this talk I will discuss a recent work with Lars Andersson, Pieter Blue and Shing-Tung Yau, where we show the global, nonlinear stability a spacetime which is a cartesian product of a high dimensional Minkowski space with a compact Ricci flat internal space with special holonomy. This stability result is related to a conjecture of Penrose concerning the validity of string theory. Our proof uses the intersection of methods for quasilinear wave and Klein-Gordon equations, and so towards the end of the talk I will also comment more generally on coupled wave–Klein-Gordon equations.


Title: On the discrete Dirac spectrum of a point electron in the zero-gravity Kerr-Newman spacetime

Abstract: In relativistic quantum mechanics, the discrete spectrum of the Dirac hamiltonian with a Coulomb potential famously agrees with Sommerfeld’s fine structure formula for the hydrogen atom. In the Coulomb approximation, the proton is assumed to only have a positive electric charge. However, the physical proton also appears to have a magnetic moment which yields a hyperfine structure of the hydrogen atom that’s normally computed perturbatively. Aiming towards a non-perturbative approach, Pekeris in 1987 proposed taking the Kerr-Newman spacetime with its ring singularity as a source for the proton’s electric charge and magnetic moment. Given the proton’s mass and electric charge, the resulting Kerr-Newman spacetime lies well within the naked singularity sector which possess closed timelike loops. In 2014 Tahvildar-Zadeh showed that the zero-gravity limit of the Kerr-Newman spacetime (zGKN) produces a flat but topologically nontrivial spacetime that’s no longer plagued by closed timelike loops. In 2015 Tahvildar-Zadeh and Kiessling studied the hydrogen problem with Dirac’s equation on the zGKN spacetime and found that the hamiltonian is essentially self-adjoint and contains a nonempty discrete spectrum. In this talk, we show how their ideas can be extended to classify the discrete spectrum completely and relate it back to the known hydrogenic Dirac spectrum but yielding hyperfine-like and Lamb shift-like effects.  


Title: The characteristic gluing problem of general relativity

Abstract: In this talk we introduce and solve the characteristic gluing problem for the Einstein vacuum equations. We prove that obstructions to characteristic gluing come from an infinite-dimensional space of conservation laws along null hypersurfaces for the linearized equations at Minkowski. We show that this obstruction space splits into an infinite-dimensional space of gauge-dependent charges and a 10-dimensional space of gauge-invariant charges. We identify the 10 gauge-invariant charges to be related to the energy, linear momentum, angular momentum and center-of-mass of the spacetime. Based on this identification, we explain how to characteristically glue a given spacetime to a suitably chosen Kerr black hole spacetime. As corollary we get an alternative proof of the Corvino-Schoen spacelike gluing to Kerr. Moreover, we apply our characteristic gluing method to localise characteristic initial data along null hypersurfaces. In particular, this yields a new proof of the Carlotto-Schoen spacelike localization where our method yields no loss of decay, thus resolving an open problem in this direction. We also outline further applications. This is joint work with S. Aretakis (Toronto) and I. Rodnianski (Princeton).


Title: Singularity theorems in low regularity

Abstract: The singularity theorems of R. Penrose and S. Hawking from the 1960s show that a spacetime satisfying certain physically reasonable curvature and causality conditions cannot be causal geodesically complete. Despite their great success these classical theorems still have some drawbacks, one of them being that they require smoothness of the metric while in many physical models the metric is less regular. In my talk I will present work on singularity theorems based on distributional energy conditions for metrics that are merely continuously differentiable – a regularity where one still has existence but not uniqueness for solutions of the geodesic equation. We will see that an approximation-based approach to the low-regularity issue is closely linked to establishing singularity theorems under weakened energy conditions and while the improvements necessary are still entirely straightforward in the present case, attempting to lower the regularity further would require some new methods.


Title: The black hole stability problem in general relativity

Abstract: I will review the current status of the black hole stability problem in general relativity and discuss joint work Holzegel, Rodnianski and Taylor.



Title: The instability of Anti-de Sitter spacetime for the Einstein-scalar field system

Abstract: The AdS instability conjecture provides an example of weak turbulence appearing in the dynamics of the Einstein equations in the presence of a negative cosmological constant. The conjecture claims the existence of arbitrarily small perturbations to the initial data of Anti-de Sitter spacetime which, under evolution by the vacuum Einstein equations with reflecting boundary conditions at conformal infinity, lead to the formation of black holes after sufficiently long time. In this talk, I will present a rigorous proof of the AdS instability conjecture in the setting of the spherically symmetric Einstein-scalar field system. The construction of the unstable initial data will require carefully designing a family of initial configurations of localized matter beams and estimating the exchange of energy taking place between interacting beams over long periods of time, as well as estimating the decoherence rate of those beams. I will also discuss possible paths for extending these ideas to the vacuum case.


Title: The hyperbolic positive energy theorem

Abstract: I will review the notion of mass of asymptotically locally

hyperbolic manifolds, and sketch the proof of positivity of the energy for manifolds with spherical conformal infinity.

Talk based on joint work with Erwann Delay in arXiv: 1901.05263 [math.DG]


Title: Existence of static vacuum extensions

Abstract: The study of static vacuum Riemannian metrics arises naturally in general relativity and differential geometry. A static vacuum metric produces a static spacetime by a warped product, and it is related to scalar curvature deformation and gluing. The well-known Uniqueness Theorem of Static Black Holes says that an asymptotically flat, static vacuum metric with black hole boundary must belong to the Schwarzschild family. In contrast to the rigidity phenomenon, R. Bartnik conjectured that there are asymptotically flat, static vacuum metric realizing certain arbitrarily specified boundary data. I will discuss recent progress toward this conjecture. It is based on joint work with Zhongshan An. 

In the second part of the talk (if time permits), I will discuss related topics in the non-time-symmetric case. In many ways, a stationary vacuum initial data set is a generalization of a static vacuum metric. However, from the point of view of deforming the dominant energy condition, we discover that a “ground state” initial data set need not be stationary vacuum but can sit in a null dust spacetime with a global Killing vector field, such as the pp-waves. This part is based on joint work with Dan Lee. 

Title: New results on compact Cauchy horizons of smooth vacuum spacetimes

Abstract: Cauchy horizons are rather unique and peculiar objects that have been studied for decades. In this talk I will first review old and new breakthroughs on the subject by Isenberg and Moncrief, and by Petersen and Rácz, and then discuss joint recent work together with I. Bustamante where it is proved that non-degenerate Compact Cauchy horizons on smooth vacuum spacetimes (shortly CHs) have indeed constant non-zero temperature.  It then follows by the work of Petersen and Petersen-Rácz that CHs are Killing horizons, and by the work of  Beig-Chruściel-Schoen that such objects are non-generic on the initial data, (in agreement with the Cosmic Censorship conjecture). A null-orbital and topological classification of CHs will also be commented.


Title: On naked singularities in Einstein equations

Abstract: I will describe recent and ongoing work with Y. Shlapentokh-Rothman on a construction of solutions to the Einstein vacuum equations  corresponding to a naked singularity forming from a regular past. The talk will focus on a new geometric phenomenon, corresponding to twisting of null geodesics, new type of self-similarity, dynamical approach to the problem, and on the comparisons with the naked singularity solutions constructed by Christodoulou for the spherically symmetric Einstein-scalar field model.


Title: Well-posed formulation of Lovelock and Horndeski theories

Abstract: Lovelock theories are the most general diffeomorphism invariant theories of gravity in higher dimensions with second order equations of motion. Horndeski theories are the most general diffeomorphism invariant theories of gravity coupled to a scalar field in four dimensions with second order equations of motion. In this talk I will discuss well-posedness of the initial value problem for these theories. Previous work has shown that (generalised) harmonic gauge does not give a well-posed initial value problem. I will describe recent work with Aron Kovacs in which we introdued a modification of harmonic gauge that does give a well-posed initial value problem provided that the theory remains “weakly coupled”. Our modified harmonic gauge may also have applications in conventional GR.


Title: New Structures in Gravitational Radiation 

Abstract: Gravitational waves are transporting information from faraway regions of the Universe. A new era began with the first detection of gravitational waves by Advanced LIGO in September 2015, and since then several events have been recorded by the LIGO/VIRGO collaboration. New challenges await us to unravel the interesting interplay between physics, astrophysics and mathematics. Most studies so far have been devoted to sources like binary black hole mergers or neutron star mergers, or generally to sources that are stationary outside of a compact set. We describe these systems by asymptotically-flat manifolds solving the Einstein equations. These sources have in common that far away their gravitational field decays fast enough towards Minkowski spacetime. In particular, far away from the source, the decay behavior can be described by a term that is homogeneous of degree -1 and lower order terms. I will present new results on gravitational radiation for sources that are not stationary outside of a compact set, but whose gravitational fields decay more slowly towards infinity. A panorama of new gravitational effects opens up when delving deeper into these more general spacetimes. In particular, whereas the former sources produce memory effects that are of purely electric parity (permanent displacement only), the latter in addition generate memory of magnetic type, thus allowing for rotation in the system. These new effects emerge naturally from the Einstein equations. 


Title: Initial data rigidity results

Abstract: We present several rigidity results for initial data sets motivated by the positive mass theorem. An important step in our proofs is to establish conditions that ensure that a marginally outer trapped surface is “weakly outermost”. A rigidity result for Riemannian manifolds with a lower bound on their scalar curvature is included as a special case. Relevant background on marginally outer trapped surfaces will be discussed. This talk is based on joint work with Michael Eichmair and Abraão Mendes.


Title: On highly anisotropic big bang singularities. 

Abstract: In cosmology, the universe is typically modelled by spatially homogeneous and isotropic solutions to Einstein’s equations. However, for large classes of matter models, such solutions are unstable in the direction of the singularity. For this reason, it is of interest to study the anisotropic setting.
The purpose of the talk is to describe a framework for studying highly anisotropic singularities. In particular, for analysing the asymptotics of solutions to linear systems of wave equations on the corresponding backgrounds and deducing information concerning the geometry.
The talk will begin with an overview of existing results. This will serve as a background and motivation for the problem considered, but also as a justification for the assumptions defining the framework we develop.
Following this overview, the talk will conclude with a rough description of the results.