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#### Institut Henri Poincaré

11 rue Pierre et Marie Curie, Paris

#### Invited Speakers

• Spyros Alexakis (Univ. of Toronto)
• Xinliang An (Univ. Toronto)
• Lars Andersson (Einstein Inst., Potsdam)
• Stefanos Aretakis (Princeton)
• Grigorios Fournodavlos (Univ. of Cambridge)
• Dejan Gajic (Cambridge University)
• Peter Hintz (Univ. of California, Berkeley)
• Gustav Holzegel (Imperial College, London)
• Cécile Huneau (Ecole Polytechnique, Palaiseau)
• Jérémie Joudioux (Univ. of Vienna)
• Jonathan Luk (Stanford Univ.)
• Sun-jin Oh (Korea Inst. Advanced Study)
• Andrzej Rostworowski (Univ. Krakow)
• Jan Sbierski (Oxford Univ.)
• Yakov Shlapentokh-Rothman (Princeton)
• Volker Schlue (Sorbonne Univ.)
• Jérémie Szeftel (Sorbonne Univ.)

#### Schedule of the conference

Schedule for MONDAY

• 9:00am coffee
• 9:30am L. Andersson
• 10:30am coffee break.
• 11:00am S. Alexakis
• 2:30pm P. Hintz
• 3:30pm coffee break
• 4:00pm V. Schlue

Schedule for TUESDAY

• 9:00am: coffee
• 9:30am J Joudioux
• 10:30am coffee break
• 11:00am G. Fournodavlos
• 2:30pm Sun-jin Oh
• 3:30pm coffee break
• 4:00pm D Gajic

Schedule for WEDNESDAY

• 9:00am coffee
• 9:30am G Holzegel
• 10:30am coffee break
• 11:00am Y. Shlapentokh-Rothman
• noon:  C. Huneau
• 7:00pm Reception at Sorbonne Univ.
Central Tower, Jussieu campus, 4 Place Jussieu

Schedule for THURSDAY

• 9:00am coffee
• 9:30am S. Aretakis
• 10:30am coffee break
• 11:00am J. Sbierski
• 2:30pm A. Rostworowski
• 3:30pm coffee break
• 4:00pm  X. An

Schedule for FRIDAY

• 9:00am: coffee
• 9:30am  J. Szeftel
• 10:30am coffee break
• 11:00am J Luk

#### Titles and attracts of the lectures

• Spyros Alexakis (Univ. of Toronto): Singularity formation in black hole interiors: Polarized perturbations of SchwarzschildAbstract: We consider the stability of the Schwarzschild singularity in vacuum under polarized and axially symmetric perturbations. We find that the space-like singularities persist under such perturbations, but their dynamics exhibit a great richness, consistent with the asymptotically velocity term dominated behavior. The result relies crucially on a new approach for the Einstein equations in axial symmetry. Joint work with G. Fournodavlos.
• Xinliang An (Univ. Toronto):  On apparent horizon formation.

Abstract:Combining both hyperbolic and elliptic techniques, we study the formation of a marginally outer trapped tube (apparent horizon) in gravitational collapse. Analytic and geometric properties of this apparent horizon will also be discussed.

• Lars Andersson (Einstein Inst., Potsdam): Linear stability for the Kerr spacetime.

Abstract: The Teukolsky Master Equation governs the dynamics of linearized gravity on the Kerr rotating black hole spacetime. Recent work of Ma, and of Dafermos, Holzegel and Rodnianski provides energy, Morawetz, and pointwise decay estimates for solutions of the Teukolsky equation. In this talk I shall show how to derive improved decay estimates for the Teukolsky equation and explain how such results can be used to prove linearized stability for the Kerr spacetime, including energy, Morawetz, and pointwise estimates for the linearized metric. The proof relies on using a radiation gauge. This is based on ongoing joint work with Thomas Bäckdahl, Pieter Blue, and Siyuan Ma.

• Stefanos Aretakis (Princeton): Asymptotics for the wave equation on black hole backgrounds

Abstract: We will present asymptotic results for solutions to the wave equation for the full Reissner-Nostrom family of black holes. These spacetimes are spherically symmetric asymptotically flat solutions to the Einstein-Maxwell system. We will consider both the sub-extremal and the extremal cases. We will see that conservation laws on null hypersurfaces play an important role in the precise late-time asymptotics for solutions to the wave equation. For the extremal Reissner-Nordstrom the situation is more subtle given that there are two independent conservation laws (in contrast to the sub- extremal where this is only one such conservation law). We will also present a scattering theory in the extremal case which in particular allows us to construct exponentially decaying smooth solutions. This work is joint with Gajic (Cambridge) and Angelopoulos (UCLA).

• Grigorios Fournodavlos (Univ. of Cambridge): On ‘hard stars’ in general relativity

Abstract: After a brief review of the classical results on gravitational collapse in spherical symmetry, from the Oppenheimer-Snyder model (1939) to Christodoulou’s two-phase model (1995), I will discuss one possible end state in the latter model: hard stars. These are idealized models of neutron stars. I will present a variational characterization and discuss its relevance to the orbital stability problem in spherical symmetry. Various obstacles to a global in time result are outlined, in particular the absence of a dispersion mechanism, the trapped surface formation scenario due to reflecting boundary conditions (cf. AdS-scalar field) and the possibility of phase transitions within the two phase model to avoid Rayleigh-Taylor instabilities. This is a joint work with Volker Schlue.

• Dejan Gajic (Cambridge University): Conservation laws and late-time tails of waves on Schwarzschild for all angular momenta

Abstract: In 1972, Price suggested that inverse polynomial tails should be present in the late-time behaviour of scalar fields on Schwarzschild black holes with fixed angular momentum and the decay rates should depend in a precise manner on the angular momentum. In the decades since, many features of these tails have been explored both numerically and heuristically. The presence of polynomial tails along event horizons has important implications for the nature of singularities inside dynamical black holes. In this talk I will discuss work done in collaboration with Y. Angelopoulos and S. Aretakis that establishes rigorously the existence of these polynomial late-time tails in Schwarzschild spacetimes. I will give a sketch of how the decay rates of Price can be derived using only physical space methods and how the coefficients in the late-time asymptotics of the scalar field are related to the existence of conservation laws.

• Peter Hintz (Univ. of California, Berkeley): Global stability problems

Abstract: I will discuss the problem of proving the stability of (families of) exact spacetimes (M,g) such as Minkowski space or the family of Kerr-de Sitter (KdS) black holes as solutions of Einstein’s vacuum equation, focussing on geometric aspects of this problem: how to compactify M for the purpose of analyzing the underlying nonlinear wave equation; how to choose a gauge to break the diffeomorphism invariance of Einstein’s equation; and the role of constraint damping.

• Gustav Holzegel (Imperial College, London): Boundedness and Decay for Solutions to the Teukolsky Equation on slowly rotating Kerr spacetimesAbstract.

Abstract: I will outline a proof (joint work with M. Dafermos and I. Rodnianski) of boundedness and polynomial decay statements for solutions of the spin ±2 Teukolsky equations on a Kerr exterior background with parameters satisfying |a| ​<< M. The estimates are obtained through natural generalisations of the higher order quantities P and \underline{P} introduced in our previous work on the linear stability of the Schwarzschild metric.

• Cécile Huneau (Ecole Polytechnique, Palaiseau): High frequency limit for Einstein equations with U(1) symmetry.

Abstract: I will present the construction of a family of solutions to vacuum Einstein equations with U(1) symmetry which consist of an arbitrary number of high frequency waves travelling in different directions. In the high frequency limit, our family of solutions converges to a solution of Einstein equations coupled to null dusts. This construction is an illustration of the so called backreaction, studied by physicists (Isaacson, Burnet, Green, Wald…) : the small scale inhomogeneities have an effect on the large scale dynamics in the form of an energy impulsion tensor in the right-hand side of Einstein equations. This is a joint work with Jonathan Luk (Stanford).

• Jérémie Joudioux (Univ. of Vienna): The vector-field method for the transport equation with application to the Einstein-Vlasov system.

Abstract: The vector-field method, developed by Klainerman, was a key tool to understand the global existence of solutions to quasi-linear wave equations. In a series of work in collaboration with D. Fajman (Vienna), and J. Smulevici (Orsay), the vector- field method is extended to the relativistic transport equation where it is used to derive decay estimates for velocity averages for solutions to the relativistic Vlasov equation. An important application of this method is the proof of the stability of Minkowski space as a solution to the Einstein-Vlasov system. I will present in this talk this commutator technique for the transport equation, and describe the decay estimates for velocity averages, and sketch the key steps of the stability proof.Joint work with D. Fajman (Vienna) and J. Smulevici (Orsay).

• Jonathan Luk (Stanford Univ.): The interior of extremal black holes

Abstract: I will contrast the interior regions of subextremal and extremal black holes and present a recent result regarding the interior of dynamical extremal black holes for the Einstein-Maxwell-charged scalar field system in spherical symmetry. This is a joint work with Dejan Gajic.

• Sun-jin Oh (Korea Inst. Advanced Study): Strong cosmic censorship and generic mass inflation for charged black holes in spherical symmetry.

Abstract: I will first review a recent joint work with J. Luk, in which the C2-formulation of the strong cosmic censorship is proved for the Einstein-Maxwell-(real)-Scalar Field system in spherical symmetry for two-ended asymptotically flat data. More precisely, it was shown that a “generic” class of data for this model gives rise to maximal future developments which are future C2-inextendible. In the second part of the talk, I will present a new, complementary theorem (also joint with J. Luk) that for a further “generic” subclass of such data, the Hawking mass blows up identically along the Cauchy horizon. This result confirms, rigorously and unconditionally, the mass inflation scenario of Poisson-Israel and Dafermos for the model at hand.

• Andrzej Rostworowski (Univ. Krakow):  New insights into nonlinear perturbations of vacuum spacetimes.

Abstract: I will present a systematic and robust approach to nonlinear gravitational perturbations of maximally symmetric black holes. In particular, I will show that at each order of perturbation expansion, the system of perturbative Einstein equations can be reduced to two (for each gravitational mode in 3+1 dimensions on which the study will be focused) scalar wave equations, and the metric perturbations can be explicitly obtained, once the solutions to these scalar wave equations are known. That is, this approach extends the field of gravitational
perturbations of black holes, initiated in the seminal Regge & Wheeler 57′ paper beyond linear order. The talk will be partially based on a recent work Phys. Rev. D96, 124026 (2017).

• Jan Sbierski (Oxford Univ.): On the unique evolution of solutions to wave equations

Abstract: The well-known theorem of Choquet-Bruhat and Geroch states that for given smooth initial data for the Einstein equations there exists a unique maximal globally hyperbolic development. In particular, the time-evolution of globally hyperbolic solutions is unique. This talk investigates whether the same results hold for quasilinear wave equations defined on a fixed background. We first present an example of a quasilinear wave equation for which unique evolution of smooth globally hyperbolic solutions in fact fails and contrast this case with the Einstein equations. We then proceed by presenting conditions which guarantee unique evolution. This talk is based on joint work with Felicity Eperon and Harvey Reall.

• Yakov Shlapentokh-Rothman (Princeton Univ.): The asymptotically self-similar regime for the Einstein vacuum equations.

Abstract: We will dynamically construct singular solutions to the Einstein vacuum equations which are asymptotically self-similar in that successive rescalings around the singularity converge to a self-similar solution. Connections both to Christodoulou’s bounded variation solutions of the spherically symmetric Einstein-scalar field system and to the ambient metric construction of Fefferman and Graham will be elaborated on. This is joint work with Igor Rodnianski.

• Volker Schlue (Sorbonne Univ.): Expanding black hole space-times: Towards the stability of the cosmological region

Abstract: The Schwarzschild de Sitter space-time is the simplest model of a black hole in the expanding universe. I will discuss the challenges arising in the stability problem for this solution of the Einstein vacuum equations with positive cosmological constant, and focus in particular on the evolution in the expanding region beyond the cosmological horizon of the black hole. I will present a result for the decay of the conformal Weyl curvature, and discuss its relation to the existence of asymptotic degrees of freedom in this problem. Moreover, I will discuss the behavior of solutions to the Eikonal equation in de Sitter, and questions related to the definition of asymptotic quantities.

• Jérémie Szeftel (Sorbonne Univ.):  The nonlinear stability of Schwarzschild

Abstract: I will discuss a joint work with Sergiu Klainerman on the stability of Schwarzschild as a solution to the Einstein vacuum equations with initial data  subject to a certain symmetry class.

#### Organizers

Philippe G. LeFloch (Paris), Jacques Smulevici (Orsay), Jérémie Szeftel (Paris)

#### Funding

• GEOWAKI
“The analysis of geometric non-linear wave and kinetic equations”
Principal investigator: Jacques Smulevici
ERC Starting Grant 2016

• EPGR
“The Evolution Problem in General Relativity”
Principal investigator: Jérémie Szeftel
ERC Consolidator Grant 2016

• PERSU Sorbonne Université
Principal investigator: Philippe LeFloch

#### List of hotels

(in the neighborhood of Jussieu, IHP, etc.)

## ANR project MATH-GR

### Laboratoire Jacques-Louis Lions

Université Pierre et Marie Curie

175, rue du Chevaleret, Paris.

http://www.ann.jussieu.fr/acces.php3

Lecture Room 1C18. Exceptionally the lectures will take place in the afternoon only.

Organizers Luc Blanchet (IAP), Eric Gourgoulhon (LUTH), Philippe G. LeFloch (Univ. P.M. Curie).

## ANR project MATH-GR

Wednesday January 27, 2010

This seminar takes place at the Laboratoire Jacques-Louis Lions

175, rue du Chevaleret, 75013 Paris

Lecture room 2E01

I gave recently some lectures (Berlin, Rutgers, College Park, Miami) about the construction of local canonical foliations of observers in Einstein spacetimes of general relativity when the curvature is solely assumed to be bounded and no assumption on its derivatives is made. In this joint work with B.-L. Chen, under geometric bounds on the curvature and injectivity radius near the observer, I proved that there exist a CMC (constant mean curvature) foliation as well as CMC–harmonic coordinates. These objects are defined in geodesic balls with definite size depending only on the assumed bounds, and the components of the Lorentzian metric has optimal regularity in these coordinates. The proof combines geometric estimates (Jacobi field, comparison theorems) and quantitative estimates for nonlinear elliptic equations with low regularity.

The lectures are based on the following two papers:

B.-L. Chen and P.G. LeFloch, Injectivity radius estimates for Lorentzian manifolds, Commun. Math. Phys. 278 (2008), 679–713.

B.-L. Chen and P.G. LeFloch, Local foliations and optimal regularity of Einstein spacetimes, J. Geom. Phys. 59 (2009), 913–-941.

In this post, I review the book Partial Differential Equations in General Relativity, recently published by A.D. Rendall, in: Graduate Texts in Mathematics, vol. 16, Oxford University Press, 2008. I recommend it as an excellent source of inspiration for problems arising in general relativity. The text below has appeared in the “Gazette des Mathématiciens” of the SMF (Sociéte Française de Mathématiques) and is, therefore,… in French.

Ce livre est consacré aux aspects mathématiques de la relativité générale et en présente les développements les plus récents de manière concise. Il porte une attention particulière à la formulation des équations d’Einstein en un système hyperbolique symétrique avec contraintes couplé avec les équations d’évolution de la matière. Il est destiné aux étudiants de Mastère, tant mathématiciens que physiciens, mais sera aussi apprécié des spécialistes de la relativité générale.

Rappelons que l’inconnue principale de la théorie est une variété lorentzienne de dimension quatre, ${(M,g)}$, satisfaisant aux équations

$\displaystyle G_{\alpha \beta} + \Lambda \, g_{\alpha \beta} = 8 \pi \, T_{\alpha\beta},$

reliant le tenseur de courbure d’Einstein ${G_{\alpha \beta}}$ au tenseur de moment-énergie de la matière ${T_{\alpha\beta}}$. Le scalaire ${\Lambda}$ représente la constante cosmologique de l’espace-temps, et les indices ${\alpha, \beta}$ varient de ${0}$ à ${3}$. (Par exemple, dans le vide le tenseur de courbure de Ricci d’une telle variété est identiquement nul.) Pour formuler le problème, on se donne une variété riemannienne de dimension trois ${(N,h)}$ munie d’un champ de ${2}$-tenseurs symétriques ${K}$. On cherche alors une variété lorentzienne (un développement maximal) satisfaisant aux équations d’Einstein avec la contrainte que ${(N,h)}$ est isométriquement plongée dans ${(M,g)}$ et admet ${K}$ comme deuxième forme fondamentale.

Les deux premiers chapitres présentent les bases incontournables de la relativité générale d’un point de vue à la fois physique et mathématique: notions de géométrie lorentzienne, théorèmes d’incomplétude de Penrose et de Hawking, feuilletages ${3+1}$, et décompositions des équations d’Einstein.

L’auteur entre dans le vif de son sujet en mettant en parallèle différents modèles de matière: équations d’ondes pour les champs scalaires (à valeurs réelles ou à valeurs dans une variété), équations de Maxwell de l’électro\-magné\-tisme, équations de Yang-Mills, équations d’Euler des fluides compressibles et équation de Vlasov de la théorie cinétique des fluides raréfiés. Pour chacun de ces modèles, un Lagrangien détermine l’expression du tenseur de moment-énergie ${T_{\alpha \beta}}$ en fonction de la métrique lorentzienne et des variables physiques décrivant l’état de la matière.

Peu de résultats sont disponibles dans la littérature à ce niveau de généralité, et, le plus souvent, des hypothèses sur les symétries de l’espace-temps sont nécessaires. L’auteur introduit ici les espaces-temps admettant un ou plusieurs champs de Killing: statiques, stationnaires, spatialement homogènes, etc.

Il consacre ensuite un chapitre à l’étude des espaces spatialement homogènes pour lesquels les équations d’Einstein se réduisent à des équations différentielles non-linéaires. En dépit de leur simplicité apparente, ces modèles sont intéressants pour l’interprétation physique de la théorie. Par ailleurs, leur étude mathé\-matique est très délicate et fait appel à toutes les facettes de la théorie des équations différentielles: variété centrale, systèmes dynamiques, théorie des bifurcations, etc.

La deuxième moitié de l’ouvrage traite directement la résolution du problème de Cauchy pour les équations d’Einstein. L’auteur présente d’abord les résultats principaux d’existence sans hypothèse de symétrie; ceux-ci sont centrés sur le théorème de Christodoulou et de Klainerman (stabilité de l’espace-temps de Minkowski). Il explique aussi les techniques d’analyse (harmonique) intervenant dans les démonstrations. Les derniers chapitres concernent les modèles possédant deux champs de Killing et l’auteur étudie en détail l’existence globale des solutions des équations d’Einstein et la nature géométrique de leurs singularités, ce qui lui permet de déterminer le comportement asymptotique des espace-temps construits.

En conclusion, il s’agit d’un ouvrage particulièrement bien organisé et documenté, dont la lecture est vivement recommandéee et permet d’accéder à l’état de l’art sur le sujet.

Philippe G. LeFloch (Paris).