Mathematical general relativity, compressible fluids, and more

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Monthly Seminar

Laboratoire Jacques-Louis 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 15-16-309 (Jussieu)

14h Arick SHAO (Queen Mary, London)

Bulk-boundary correspondence for vacuum asymptotically Anti-de Sitter spacetimes

Abstract: The AdS/CFT conjecture in physics posits the existence of a correspondence between gravitational theories in asymptotically Anti-de 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 one-to-one correspondence between aAdS solutions of the Einstein-vacuum equations and a suitable space of data on the conformal boundary (consisting of the boundary metric and the boundary stress-energy 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 15-16-309 (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 leading-order term for initially localized and regular solutions and valid for the full sub-extremal range of black hole parameters. I will also present some aspects of the proof which relies on spectral and microlocal methods.

15h30 Anne-Sophie 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 Klein-Gordon or wave equation and we combine weak dispersive estimates with Strichartz and smoothing estimates for the wave and Klein-Gordon flows, exploiting previous results in the same geometrical setting.

Thursday January 12, 2023

lecture room 16-26-113 (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 Paris-Nord)

Soliton resolution for the energy-critical 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 2-dimensional 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 multi-solitons by the modulation technique with the concentration-compactness method, we prove a “non-return 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 15-16-309 (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 late-time asymptotics for the scalar field (and its derivatives) globally in the interior of a non-static sub-extremal 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 Reissner-Nordstrom. 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 non-linear stability of collisionless many-particle systems on black hole exteriors

 Abstract: I will present upcoming linear and non-linear stability results concerning the asymptotic behavior of collisionless many-particle 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 15-16-201 (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 micro-local analysis and non-elliptic 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’espace-temps comme point critique de courbures élastiques quadratiques (type Lovelock-Bach) généralisant l’énergie d’Einstein-Hilbert, nous obtenons des équations de courbure d’ordre 4 dont les espaces-temps 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 Q-courbure pour établir des théorèmes de rigidité pour des feuilles Riemanniennes de tels espaces-temps. Here, the PDF file of this lecture.

Wednesday October 19, 2022

lecture room 15-16-309 (Jussieu)

14h Annalaura STINGO (Ecole Polytechnique, Palaiseau)

Global stability of Kaluza-Klein theories: a toy model

Abstract. The Kaluza-Klein 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 Einstein-Maxwell-Scalar systems. In this talk I will discuss the problem of the classical global stability of Kaluza-Klein 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 Jacques-Louis 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 15-16–309

14h Renato Velozo Ruiz (Cambridge, UK)

Stability of Schwarzschild for the spherically symmetric Einstein-massless 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 high-frequency spacetimes

Abstract. I will present a recent work on high-frequency 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 high-frequency gravitational waves, which enjoy some polarization-related 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 well-posedness in harmonic gauge for high-frequency solutions.

Wednesday November 10, 2021

lecture room 15-25-104

 14h José Luis Jaramillo (Université de Bourgogne)

On the stability of black hole quasi-normal modes: a pseudo-spectrum approach

Abstract. Black hole (BH) quasi-normal 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 pseudo-spectrum 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 non-self-adjoint operator by adopting a hyperboloidal formulation of spacetime. Non-selfadjoint (more generally non-normal) operators suffer potentially of spectral instabilities, the notion of pseudo-spectrum providing a tool suitable for their study. We explore this problem in a numerical methodology based on pseudo-spectral methods. As a result, we find evidence that perturbed Nollert & Price BH QNM branches track the pseudo-spectrum 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.

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Seminar at the

Laboratoire Jacques-Louis Lions

Sorbonne Université, Paris

Organizers

 Philippe G. LeFloch (Sorbonne, Paris)

 Jacques Smulevici (Sorbonne, Paris)

Jérémie Szeftel (Sorbonne, Paris)

Seminar organized during the Winter-Spring 2019

Tuesday February 19, 2019

lecture room 15/16-309

 14h João Costa (Lisbon)

Strong cosmic censorship, linear waves, and quasi-normal 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 Einstein-Maxwell-scalar field system in spherical symmetry and the linear wave equation in the black hole interior of Reissner-Nordström 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 quasi-normal spectrum. I will also discuss recent numerical computations of quasi-normal modes that suggest the failure of the SCCC in a near extremal regime of charged de Sitter black holes.