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Monthly Seminar
Laboratoire JacquesLouis Lions
Sorbonne Université
Organizers
Cécile Huneau (i) Philippe G. LeFloch (ii)
Jacques Smulevici (ii) Jérémie Szeftel (ii)
(i) Ecole Polytechnique, Palaiseau
(ii) Sorbonne Université, Paris
Lectures given during the Academic year 2022–2023
Wednesday May 10, 2023
lecture room 1516309 (Jussieu)
14h Arick SHAO (Queen Mary, London)
Bulkboundary correspondence for vacuum asymptotically Antide Sitter spacetimes
Abstract: The AdS/CFT conjecture in physics posits the existence of a correspondence between gravitational theories in asymptotically Antide Sitter (aAdS) spacetimes and field theories on their conformal boundary. In this presentation, we prove a rigorous mathematical statement toward this conjecture in the classical relativistic setting. In particular, we show there is a onetoone correspondence between aAdS solutions of the Einsteinvacuum equations and a suitable space of data on the conformal boundary (consisting of the boundary metric and the boundary stressenergy tensor), provided the boundary satisfies a geometric condition. We also discuss applications of this result to symmetry extension, as well as its connection to unique continuation problems. This is joint work with Gustav Holzegel, and refers to joint works with Athanasios Chatzikaleas, Simon Guisset, and Alex McGill.
15h30 Christof KEHLE (ETH, Zürich)
TBA
Abstract: TBA
Wednesday March 29, 2023
lecture room 1516309 (Jussieu)
14h Pascal MILLET (Institut Fourier, Grenoble)
Optimal decay for the Teukolsky equation on subextremal Kerr black holes.
Abstract: The study of wave propagation on black hole spacetimes has been an intense field of research in the last decades. This interest has been driven by the stability problem for black holes and by scattering questions. For Maxwell equations and the equations of linearized gravity, it is possible to base the analysis on the study of the Teukolsky equation, which has the advantage of being scalar in nature. I will present a result providing the large time leadingorder term for initially localized and regular solutions and valid for the full subextremal range of black hole parameters. I will also present some aspects of the proof which relies on spectral and microlocal methods.
15h30 AnneSophie DE SUZZONI (Ecole Polytechnique, Palaiseau)
Strichartz estimates for the Dirac equation on asymptotically flat manifolds
Abstract: We will discuss Strichartz estimates for the Dirac equation on asymptotically flat manifolds. We will present the Dirac equation in a curved setting and some of its symmetries. To derive Strichartz estimates, we see the Dirac equation as a perturbation of the KleinGordon or wave equation and we combine weak dispersive estimates with Strichartz and smoothing estimates for the wave and KleinGordon flows, exploiting previous results in the same geometrical setting.
Thursday January 12, 2023
lecture room 1626113 (Jussieu)
14h Ioannis ANGELOPOULOS (CalTech)
Linear and nonlinear problems in general relativity
Abstract: I will discuss two different topics: a) the derivation of precise asymptotics for linear waves on black hole spacetimes, and b) the construction of spacetimes containing curvature singularities. If time permits, I will try to make connections with more general problems for quasilinear wave equations (for both topics).
15h30 Jacek JENDREJ (Sorbonne ParisNord)
Soliton resolution for the energycritical wave maps equation in the equivariant case
Abstract: I will present a joint work with Andrew Lawrie (MIT) on the wave maps equation from the (1+2)dimensional space to the 2dimensional sphere, in the case of initial data having the equivariant symmetry. We prove that every solution of finite energy converges in large time to a superposition of harmonic maps (solitons) and radiation. It was proved by Côte, and Jia and Kenig, that such a decomposition is true for a sequence of times. Combining the study of the dynamics of multisolitons by the modulation technique with the concentrationcompactness method, we prove a “nonreturn lemma”, which allows to improve the convergence for a sequence of times to convergence in continuous time. Here, the PDF file of this lecture
Thursday December 15, 2022
lecture room 1516309 (Jussieu)
14h Siyuan MA (Albert Einstein Institute)
Revisiting the strong cosmic censorship for the scalar field in Kerr interior
Abstract: I will show the precise latetime asymptotics for the scalar field (and its derivatives) globally in the interior of a nonstatic subextremal Kerr black hole based on recent advances in deriving the asymptotics in Kerr exterior, which then provides a new proof of the generic H^1(loc) inextendibility of the Kerr Cauchy horizon against scalar perturbations. A similar result holds also for ReissnerNordstrom. We expect this result to be extended to the linearized gravity model and the approach to be useful in nonlinear evolution in the black hole interior. This is a joint work with Lin Zhang. Here, the PDF file of this lecture.
15h30 Renato Velozo RUIZ (Sorbonne Université)
Linear and nonlinear stability of collisionless manyparticle systems on black hole exteriors
Abstract: I will present upcoming linear and nonlinear stability results concerning the asymptotic behavior of collisionless manyparticle systems on black hole exteriors. On the one hand, I will discuss decay properties for solutions to the massive Vlasov equation on Schwarzschild spacetime. On the other hand, I will discuss an asymptotic stability result for the exterior of Schwarzschild as a solution to the Einstein–massless Vlasov system assuming spherical symmetry. I will explain the use of hyperbolic dynamics to obtain time decay of the stress energy momentum tensor by considering a linear Vlasov equation with an unstable trapping potential. Here, the PDF file of this lecture.
Thursday November 17, 2022
lecture room 1516201 (Jussieu)
14h Dietrich HÄFNER (Université Grenoble Alpes)
On the linear stability of Kerr black holes
Abstract: I will explain a result obtained in collaboration with P. Hintz and A. Vasy on the linear stability of slowly rotating Kerr black holes as solutions of the Einstein vacuum equations: linearized perturbations of a Kerr metric decay at an inverse polynomial rate to a linearized Kerr metric plus a pure gauge term. We work in a natural generalized wave map/DeTurck gauge and show that the pure gauge term can be taken to lie in a fixed finite dimensional space with a simple geometric interpretation. Our proof rests on a robust general framework, based on recent advances in microlocal analysis and nonelliptic Fredholm theory. The restriction to small angular momentum mainly comes from the analysis of mode solutions and I will explain at the end of the talk how this analysis can be carried out also in the case of large angular momentum of the black hole. (This last part is based on joint work with L. Andersson and B. Whiting.) Here, the PDF file of this lecture.
15h30 Nicolas MARQUE (Université de Lorraine)
Energie pour la gravité du quatrième ordre
Abstract: J’aborderai un travail mené en collaboration avec R. Avalos, P. Laurain et J. Lira. En considérant l’espacetemps comme point critique de courbures élastiques quadratiques (type LovelockBach) généralisant l’énergie d’EinsteinHilbert, nous obtenons des équations de courbure d’ordre 4 dont les espacestemps d’Einstein sont des solutions naturelles. L’objectif de ce travail est d’étudier ces métriques de Lorentz d’ordre quatre via une analyse de quantités conservées inspirées de la masse ADM. Nous nous appuierons sur ces quantités conservées et leurs liens avec la Qcourbure pour établir des théorèmes de rigidité pour des feuilles Riemanniennes de tels espacestemps. Here, the PDF file of this lecture.
Wednesday October 19, 2022
lecture room 1516309 (Jussieu)
14h Annalaura STINGO (Ecole Polytechnique, Palaiseau)
Global stability of KaluzaKlein theories: a toy model
Abstract. The KaluzaKlein theories represent the classical mathematical approach to the unification of general relativity with electromagnetism and more generally with gauge fields. In these theories, general relativity is considered in 1+3+d dimensions and in the simplest case d=1 dimensional gravity is compactified on a circle to obtain at low energies a (3+1)dimensional EinsteinMaxwellScalar systems. In this talk I will discuss the problem of the classical global stability of KaluzaKlein theories when d=1 and present a toy model we studied in collaboration with C. Huneau.
15h30 Dawei SHEN (Sorbonne University, Paris)
General covariant modulated (GCM) procedure
Abstract. I will start by introducing the main idea of the proof of the “Kerr stability for small angular momentum” by Klainerman and Szeftel, as our motivation to introduce the General Covariant Modulated (GCM) procedure. Then, I will present the paper “Constructions of GCM spheres in perturbations of Kerr” of Klainerman and Szeftel concerning the construction of GCM spheres. Finally, by applying the GCM spheres I will explain the main result concerning the “Constructions of GCM hypersurfaces in perturbations of Kerr”. Here, the PDF file of this lecture.
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Monthly Seminar taking place at the
Laboratoire JacquesLouis Lions
Sorbonne Université, Paris
Organizers
Philippe G. LeFloch (Sorbonne, Paris)
Jacques Smulevici (Sorbonne, Paris)
Jérémie Szeftel (Sorbonne, Paris)
Lectures given during the Academic year 2021–2022
Wednesday December 8, 2021
lecture room 1516–309
14h Renato Velozo Ruiz (Cambridge, UK)
Stability of Schwarzschild for the spherically symmetric Einsteinmassless Vlasov system
Abstract. The Einstein–massless Vlasov system is a relevant model in the study of collisionless many particle systems in general relativity. In this talk, I will present a stability result for the exterior of Schwarzschild as a solution of this system assuming spherical symmetry. We exploit the hyperbolicity of the geodesic flow around the black hole to obtain decay of the energy momentum tensor, despite the presence of trapped null geodesics. The main result requires a precise understanding of radial derivatives of the energy momentum tensor, which we estimate using Jacobi fields on the tangent bundle in terms of the Sasaki metric.
15h30 Arthur Touati (Ecole Polytechnique, Palaiseau)
Construction of highfrequency spacetimes
Abstract. I will present a recent work on highfrequency solutions of Einstein’s vacuum equations. The motivation behind the study of such solutions comes from physical and mathematical questions. These solutions model the propagation of highfrequency gravitational waves, which enjoy some polarizationrelated properties. From a mathematical point of view, they partially answer Burnett’s conjecture in general relativity, which concerns the lack of compactness of a family of solutions to Einstein’s vacuum equations. I will start by reviewing the existing literature, and then discuss my results for a toy model. I will then sketch the proof of the local wellposedness in harmonic gauge for highfrequency solutions.
Wednesday November 10, 2021
lecture room 1525104
14h José Luis Jaramillo (Université de Bourgogne)
On the stability of black hole quasinormal modes: a pseudospectrum approach
Abstract. Black hole (BH) quasinormal modes (QNM) encode the resonant response of black holes under linear perturbations, their associated complex frequencies providing an invariant probe into the background spacetime geometry. In the late nineties, Nollert and Price found evidence of a BH QNM instability phenomenon, according to which perturbed QNMs of Schwarzschild spacetime migrate to new perturbed QNM branches of different qualitative behavior and asymptotics. Here we revisit this BH QNM instability issue by adopting a pseudospectrum approach. Specifically, rather than starting from the formulation of QNMs in scattering resonance theory, we cast the QNM problem as an eigenvalue problem for a nonselfadjoint operator by adopting a hyperboloidal formulation of spacetime. Nonselfadjoint (more generally nonnormal) operators suffer potentially of spectral instabilities, the notion of pseudospectrum providing a tool suitable for their study. We explore this problem in a numerical methodology based on pseudospectral methods. As a result, we find evidence that perturbed Nollert & Price BH QNM branches track the pseudospectrum contour lines, therefore probing the analytic structure of the resolvent. Specifically, we find strong support to claim: i) the stability of the slowest decaying (fundamental) mode, and ii) the instability of all QNM ‘overtones’. But numerical evidence is not a proof. Or goal in this talk is to boost the interaction between physicists and analysts to fully assess this BH QNM instability problem.
15h30 Allen Fang (Paris)
Nonlinear stability of Kerrde Sitter
Abstract. The nonlinear stability of the slowlyrotating Kerrde Sitter family was first proven by HIntzVasy in 2018 using microlocal techniques. I will present a novel proof of the nonlinear stability of Kerrde Sitter that avoids frequencyspace techniques outside of a neighborhood of the trapped set. Similar to the original work of HintzVasy, the critical step is to prove exponential decay for solutions of the linearized problem, which is done by using a highfrequency ILED estimate, and a mode stability result.
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International Conference
MATHEMATICAL GENERAL RELATIVITY
June 2nd to 5th, 2020
Unfortunately, we have to postpone this event,
and we will reschedule it in a few months.
Institut Henri Poincaré
11 rue Pierre et Marie Curie, Paris
Invited Speakers

Xinliang An (Singapore)

Mihalis Dafermos (Cambridge/Princeton)

David Fajman (Vienna)

Grigorios Fournodavlos (Sorbonne)

Elena Giorgi (Princeton)

Mahir Hadzic (London)

Gustav Holzegel (London)

Cécile Huneau (Palaiseau)

Sergiu Klainerman (Princeton)

Philippe G. LeFloch (Sorbonne)

Jonathan Luk (Stanford)

Siyuan Ma (Sorbonne)

Yue Ma (Xi’an)

Maxime Van De Moortel (Princeton)

Georgios Moschidis (Princeton)

Hans Ringström (Stockholm)

Jared Speck (Cambridge, USA)

Shiwu Yang (Beijing)

Pin Yu (Beijing)
Schedule of the conference: TBA
Schedule for TUESDAY
Schedule for WEDNESDAY
Schedule for THURSDAY
Schedule for FRIDAY
Titles and abstracts of the lectures: TBA
Organizers
Philippe G. LeFloch (Sorbonne), Jacques Smulevici (Sorbonne), Jérémie Szeftel (Sorbonne)
Funding
 GEOWAKI
“The analysis of geometric nonlinear 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
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Seminar at the
Laboratoire JacquesLouis Lions
Sorbonne Université, Paris
Organizers
Philippe G. LeFloch (Sorbonne, Paris)
Jacques Smulevici (Sorbonne, Paris)
Jérémie Szeftel (Sorbonne, Paris)
Seminar organized during the WinterSpring 2019
Tuesday February 19, 2019
lecture room 15/16309
14h João Costa (Lisbon)
Strong cosmic censorship, linear waves, and quasinormal modes
Abstract. I will present some recent results concerning the Strong Cosmic Censorship Conjecture (SCCC) in the presence of a positive cosmological constant. I will start by reviewing some of the progress made in the context of the EinsteinMaxwellscalar field system in spherical symmetry and the linear wave equation in the black hole interior of ReissnerNordström de Sitter. These results show that the validity of the SCCC hinges on the precise decay rates of perturbations along the event horizon, which are known to be determined by the black hole’s quasinormal spectrum. I will also discuss recent numerical computations of quasinormal modes that suggest the failure of the SCCC in a near extremal regime of charged de Sitter black holes.
15h30 Shijie Dong (Paris)
Evolution of the U(1) Higgs Boson: global nonlinear stability with energy bounds
Abstract. Relying on the hyperboloidal foliation method, we establish the nonlinear stability of the ground state for the socalled U(1) standard model of electroweak interactions. This amounts to establishing a globalintime theory for the initial value problem for a nonlinear waveKleinGordon system coupling massive (Dirac, scalar, gauge) equations together. In particular, we need to investigate here for the Dirac operator and the properties of energy functionals associated with the hyperboloidal foliation of Minkowski spacetime. We also provide a unified decay result for the Dirac equation when the mass coefficient can be arbitrarily small. Our energy bounds are uniform with respect to our (hyperboloidal) time variable, except for a mild loggrowth. This is a joint work with P.G. LeFloch (Paris) and Z. Wyatt (Edinburgh).
Monday March 18, 2019
lecture room 15/16101
14h MariaCaterina Valcu (Lyon)
Des équations de contrainte en relativité générale
Abstract. On s’intéresse à la caractérisation des données initiales en relativité générale, c’est à dire aux solutions des équations de contrainte. On s’appuie sur une version modifiée de la méthode conforme, introduite cette fois par David Maxwell, qui semble mieux adaptée à l’étude du système dans le cas où la courbure moyenne n’est pas constante. Par contre, le système devient bien plus compliqué du point de vue analytique. On travaille sous des conditions de petitesse sur nos paramètres, en petite dimension (n=3,4,5) et en présence d’un champ scalaire avec potentiel positif, ce qui mène à un terme nonlinéaire dominant focalisant. L’analyse est assez fine et implique une série d’outils différents, dont des résultats de compacité et un théorème du point fixe.
15h30 Léo Bigorgne (Orsay)
Sharp asymptotic behavior of solutions of the 3d VlasovMaxwell system with small data
Abstract. The VlasovMaxwell system is a classical model in plasma physics. Glassey and Strauss proved global existence for the small data solutions of this system under a compact support assumption on the initial data. They used in particular that under these hypotheses, the velocity support of the particle density remains compact. This allows a clean separation between the characteristics of the wave equations (which are null geodesics) and those of the transport equation (which are timelike). We will see how vector field methods can be applied to revisit this problem. In particular, it allows to remove all compact support assumptions on the initial data and obtain sharp asymptotics on the solutions and their derivatives. We will also study the null structure of the system, which constitutes a crucial element of the proof and allows us to deal with high velocities.
Monday May 6, 2019
lecture room 15/16309
14h Erwann Delay (Avignon)
Le théorème d’énergie positive hyperbolique
Abstract. Le théorème d’énergie positive hyperbolique affirme que toute variété riemannienne complète, asymptotique à l’espace hyperbolique réel, et dont la courbure scalaire est minorée par celle du modèle, possède un vecteur énergieimpulsion de genre temps dirigé vers le futur, ce vecteur étant nul seulement pour le modèle. Nous verrons une preuve de ce résultat en toutes dimensions et sans condition spin. Il s’agit d’un travail en collaboration avec Piotr Chrusciel.
15h30 Olivier Graf (Sorbonne)
The spacelikecharacteristic Cauchy problem with L2 bounded curvature
Abstract. The bounded L2 curvature theorem by Klainerman, Rodnianski, and Szeftel states that the time of existence of a solution to Einstein’s vacuum equations is controled by the L2 norm of its curvature on spacelike Cauchy hypersurfaces. I will present a version of this result where the curvature is bounded in L2 on null hypersurfaces. This provides a first breakdown criterion on characteristic hypersurfaces at this level of regularity. The proof relies on an extension procedure, as well as on the existence and control at low regularity of a new parabolic foliation of null hypersurfaces. This is a joint work with Stefan Czimek (Toronto).
Monday June 24, 2018
lecture room 15/16309
14h Oscar J. C. CamposDias (Southampton)
Strong cosmic censorship (in de Sitter backgrounds)
Abstract. Generically, strong cosmic censorship (SCC) is the statement that physics within general relativity should be predicted from initial data prescribed on a Cauchy hypersurface. In this talk I will review how finetuned versions of SCC have been formulated and evolved along the last decades up to the point where we believe that Christodoulou’s version is true in asymptotically flat spacetimes. However, I will also describe that in the last 2 years it was found that this is no longer necessarily true for some other backgrounds, namely in de Sitter (with a positive cosmological) spacetimes.
15h30 Shiyuan Ma (Sorbonne)
Linear stability for the Kerr spacetime
Abstract. The Teukolsky Master Equation governs the dynamics of linearized gravity on the Kerr rotating black hole spacetime. In this talk, based on recent works on basic energy and Morawetz estimates for solutions of the Teukolsky equation, I shall show how to derive improved decay estimates for the Teukolsky equation and explain how such results can be used to prove linear stability for the Kerr spacetime. The proof relies on using a radiation gauge. This is joint work with Lars Andersson, Thomas Bäckdahl, and Pieter Blue.
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Seminar at the
Laboratoire JacquesLouis Lions
Sorbonne Université, Paris
Organizers
Philippe G. LeFloch (Sorbonne, Paris)
Jacques Smulevici (Sorbonne, Paris)
Jérémie Szeftel (Sorbonne, Paris)
Dates of the Seminar this Fall 2018
September 24, October 29, November 19
Monday September 24, 2018
room 15/16309
14h Dietrich Häfner (Grenoble)
Scattering for Dirac and KleinGordon fields on the (De Sitter) Kerr metric and the Hawking effect
Abstract. We consider Dirac and KleinGordon fields on the (De Sitter) Kerr metric which describes rotating black holes. Whereas there exists a conserved L2 norm for the Dirac field, no positive conserved quantity exists for the KleinGordon field, which makes the analysis more difficult for the latter. We obtain asymptotic completeness results for the Dirac field on the Kerr and for the KleinGordon field on the De Sitter Kerr metric. We then present a rigorous result about the Hawking effect for fermions in the setting of a collapse of a rotating charged star. This effect predicts the creation of particles by black holes.
15h30 JeanPhilippe Nicolas (Brest)
Peeling for scalar fields on the Kerr metric
Abstract. The peeling is an asymptotic behavior of massless fields along outgoing null geodesics in asymptotically flat spacetimes, initially observed by Sachs at the beginning of the 1960’s, then reformulated in very simple terms by Penrose in 1965 using conformal geometry. The question of its genericity, especially when talking about the peeling of the Weyl tensor of an Einstein spacetime, was controversial for several decades after Penrose’s paper. For Einstein’s equations, the question is now essentially settled, but given an Einstein spacetime, it is not clear whether there is a large class of Cauchy data giving rise to solutions with a good peeling. Lionel Mason and the speaker answered the question for fields of spin 0, 1/2 and 1 on Schwarzschild’s spacetime in 2009 and 2012. We extended recently the results to linear and non linear scalar fields on the Kerr geometry in a joint work with Pham Truong Xuan. We shall recall the history of the subject, describe the principles of the approach developed with Lionel Mason and talk about the specific features of our work for Kerr metrics.
Monday October 29, 2018
room 15/16101
14h Joe Keir (Cambridge)
The weak null condition and the pweighted energy method
Abstract. The Einstein equations in wave coordinates are an example of a system which does not obey Klainerman’s “null condition”. Their failure to satisfy this condition leads to many difficulties, both in LindbladRodnianski’s proof of global existence and in any attempt to apply other techniques to these equations. One such technique is the “pweighted energy method” of Dafermos Rodnianski, which is a very powerful and robust method that can easily be adapted to understand the behavior of waves in many interesting situations, including black holes. In this talk I will explain how to modify this method to systems which only obey the “weak null condition”, including the Einstein equations. This involves adapting the pweighted energy method, and combining it with the many of the geometric methods used by Christodoulou and Klainerman. Among other things, this allows us to enlarge the class of wave equations which are known to admit smalldata global solutions, and it also yields a detailed description of null infinity. In particular, in some situations we can understand the geometric origin of the slow decay towards null infinity exhibited by these systems: it is due to the formation of “shocks at infinity”.
15h30 Volker Schlue (Paris)
Scattering from infinity for semilinear wave equations
Abstract. I will discuss the construction of global solutions from scattering data (at null infinity) for various semilinear wave equations on Minkowski space satisfying the (weak) null condition. I will elaborate on the proof which relies, i) on a fractional Morawetz estimate, and (ii) on the construction of suitable approximate solutions from the scattering data. Finally I will outline the application of these results to Einstein’s equations in harmonic coordinates. This is joint work with Hans Lindblad.
Monday November 19, 2018
room 15/16309
14h Adam Layne (Stockholm)
Stability within T2symmetric expanding spacetimes
Abstract. We present a recently completed, nonpolarized analogue of the asymptotic characterization of T2symmetric Einstein flow solutions by P. LeFloch and J. Smulevici. We impose a far weaker condition, but obtain similar rates of decay for the normalized energy and associated quantities. Critical to this work have been novel numerical simulations which indicate that there is locally attractive behavior for those T2symmetric solutions not subject to this weakened condition. This local attractor is distinct from the local attractor in our main theorem, thereby indicating that the polarized asymptotics are on one hand stable within a larger class than merely polarized solutions, but unstable within all T2symmetric solutions.
15h30 Grigorios Fournodavlos (Sorbonne)
Dynamics of the Einstein vacuum equations about the Schwarzschild black hole interior
Abstract. We will talk about the dynamical behavior of the Schwarzschild black hole singularity, in the context of the Einstein equations in vacuum, from the point of view of the Cauchy problem in general relativity. As it is well known, the Schwarzschild singularity is highly unstable under arbitrarily small perturbations, which makes the study of its dynamics in full generality a difficult problem. We will begin by giving an overview of the current status of the nearSchwarzschildblack hole interior problem and we will compare it to the dynamics observed near other singularity models, in black hole interiors or Big Bangs. Then we will discuss linear and nonlinear partial results in the nearSchwarzschild case, both backwards and forwards in time, with and without symmetries.