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Monthly Seminar

Laboratoire Jacques-Louis Lions

Sorbonne Université, Paris

Organizers

 Cécile Huneau (Ecole Polytechnique, Palaiseau)

 Philippe G. LeFloch (Sorbonne Université, Paris)

 Jacques Smulevici (Sorbonne Université, Paris)

Jérémie Szeftel (Sorbonne Université, Paris)


Lectures given during the Academic year 2022–2023


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) ——-> Lecture–Shen-Dawei–Sorbonne-October-2022

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”.


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.)

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.



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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.

15h30 Allen Fang (Paris)

Nonlinear stability of Kerr-de Sitter

Abstract. The nonlinear stability of the slowly-rotating Kerr-de Sitter family was first proven by HIntz-Vasy in 2018 using microlocal techniques. I will present a novel proof of the nonlinear stability of Kerr-de Sitter that avoids frequency-space techniques outside of a neighborhood of the trapped set. Similar to the original work of Hintz-Vasy, the critical step is to prove exponential decay for solutions of the linearized problem, which is done by using a high-frequency ILED estimate, and a mode stability result.


_____________________________________________________________________________________________________________

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.

 

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 so-called U(1) standard model of electroweak interactions. This amounts to establishing a global-in-time theory for the initial value problem for a nonlinear wave-Klein-Gordon 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 log-growth. This is a joint work with P.G. LeFloch (Paris) and Z. Wyatt (Edinburgh).

 



 

Monday March 18, 2019

lecture room 15/16-101

 

14h Maria-Caterina 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 non-linéaire 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 Vlasov-Maxwell system with small data

Abstract. The Vlasov-Maxwell 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 time-like). 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 15/16-309

14h Erwann Delay (Avignon)

Le théorème d’énergie positive hyperbolique

Abstract. Le théorème d’énergie positive hyperbolique affirme que toute variété rieman-nienne complète, asymptotique à l’espace hyperbolique réel, et dont la courbure scalaire est minorée par celle du modèle, possède un vecteur énergie-impulsion 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 spacelike-characteristic 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 15/16-309

14h Oscar J. C. Campos-Dias (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 fine-tuned 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 Jacques-Louis 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 15/16-309

 

 14h  Dietrich Häfner (Grenoble)

Scattering for Dirac and Klein-Gordon fields on the (De Sitter) Kerr metric and the Hawking effect

Abstract. We consider Dirac and Klein-Gordon 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 Klein-Gordon 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 Klein-Gordon 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 Jean-Philippe 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 15/16-101

 

14h Joe Keir (Cambridge)

The weak null condition and the p-weighted 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 Lindblad-Rodnianski’s proof of global existence and in any attempt to apply other techniques to these equations. One such technique is the “p-weighted 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 p-weighted 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 small-data 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 semi-linear wave equations

Abstract. I will discuss the construction of global solutions from scattering data (at null infinity) for various semi-linear 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 15/16-309

14h Adam Layne (Stockholm)

Stability within T2-symmetric expanding spacetimes

Abstract. We present a recently completed, non-polarized analogue of the asymptotic characterization of T2-symmetric 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 T2-symmetric 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 T2-symmetric 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 near-Schwarzschild-black 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 non-linear partial results in the near-Schwarzschild case, both backwards and forwards in time, with and without symmetries.




 

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Seminar at the

Laboratoire Jacques-Louis Lions

Université Pierre et Marie Curie, Paris

Organizers

 Philippe G. LeFloch (Paris)

 Jacques Smulevici (Orsay)

Jérémie Szeftel (Paris)


Dates of the Seminar for this Academic year: 

October 9, November 20, December 4, February 12, March 12

+ Conference from May 28 to June 1rst


Monday March 12, 2018

room 15-16 309

 14h  Carla Cederbaum (Tubingen)

On foliations related to the center of mass in general relativity

Abstract.  We will discuss new developments in the analysis of asymptotic foliations by prescribed curvature in relativistic initial data sets with prescribed asymptotic decay, generalizing results by Huisken and Yau. We will relate these foliations to the definition of the center of mass of the initial data sets under consideration. The results presented are joint work with Cortier–Sakovich and with Nerz.

 15h30  Maxime Van de Moortel (Stanford)

Stability and instability in spherical symmetry of Reissner-Nordström black holes for the Einstein-Maxwell-Klein-Gordon model

Abstract. Penrose’s Strong Cosmic Censorship Conjecture is one of the central problems of Mathematical General Relativity. Its proof for the Einstein-Maxwell-Uncharged-Scalar-Field (EMSF) model in spherical symmetry relies on the formation of a Cauchy horizon that is C0 regular but C2 singular for generic Cauchy data. EMSF model however only admits two-ended black holes, unlike its charged analogue that allow for one-ended black holes, relevant to the study of charged gravitational collapse in spherical symmetry. In this talk I will present my work about spherically symmetric charged and massive scalar fields on black holes. This includes a study of the black hole interior, that relates the behavior of fields on the event horizon to the formation of a C0 regular and C2 singular Cauchy horizon. I will also mention my more recent work on the black hole exterior stability, for weakly charged massless scalar fields.

 


Monday February 12, 2018

room 15-16 309

 14h Shadi Tahvildar-Zadeh (Rutgers)

General relativity at the subatomic scale

Abstract. The idea that General Relativity (GR) may have something to say about the subatomic world is about as old as GR itself, but very few physicists have taken it seriously, and little is known rigorously about it. In this talk I use the problem of the “general- relativistic Dirac spectrum of Hydrogen” to convey the conceptual and technical issues one is up against, and survey recent results obtained in collaboration with my colleague Michael Kiessling and by some of our students and postdocs.

 15h30 Thomas W. Johnson  (Cambridge)

Abstract. I shall discuss the linear stability of the Schwarzschild family of black holes as solutions to the Einstein vacuum equations when expressed in a generalised wave gauge, a result which complements the recent work of Dafermos, Holzegel and Rodnianski in a similar vein as the pioneering result of Lindblad and Rodnianski complemented the monumental achievement of Christodoulou and Klainerman in establishing the global nonlinear stability of the Minkowski space. The proof relies on classical insights regarding the linearised Einstein equations about the Schwarzschild family, in particular the decoupling of certain gauge-invariant scalars into the Regge—Wheeler and Zerilli equations, and recent advances for the linear wave equation on the Schwarzschild exterior, both of which shall be reviewed.

 


Monday December 4, 2017

room 16-26 113

 14h Siyuan Ma (Potsdam)

On Maxwell field and linearized gravity in Kerr

Abstract.  I will consider both Maxwell field and linearized gravity on Kerr backgrounds, and present recent results in obtaining energy and Morawetz estimates for the extreme Newman-Penrose components.

 15h30 Claudio Paganini (Potsdam)

Mode stability on the real axis

Abstract.  I will discuss a generalization of the mode stability result of Whiting (1989) for the Teukolsky equation for the case of real frequencies. The main result states that a separated solution of the Teukolsky equation governing massless test fields on the Kerr spacetime, which is purely outgoing at infinity, and purely ingoing at the horizon, must vanish.

 


Monday November 20, 2017

room 16-26 113

 14h Frederico Pasqualotto (Princeton)

Nonlinear stability for the Maxwell–Born–Infeld system on a Schwarzschild background

Abstract. The Maxwell–Born–Infeld (MBI) theory is a hyperbolic system of PDEs which describes nonlinear electromagnetism. Due to its tensorial and quasilinear nature, this system can be seen as a nonlinear model problem to study the stability properties of solutions to the Einstein vacuum equations. In this talk, I will present a nonlinear stability result for the MBI system on a fixed Schwarzschild background, when the initial data are constrained to be small. A crucial element of the proof is the observation that some null components of the MBI field satisfy “good” Fackerell–Ipser equations, as in the linear Maxwell case. However, in the MBI case, these equations are coupled through cubic nonlinear right hand sides, which contain all components of the MBI field. In order to resolve the coupling, we prove high-order energy decay and, subsequently, pointwise decay for all the components of the MBI field. This is achieved through the application of many ideas developed in recent years, regarding the decay of linear fields.

 15h30 Volker Schlue (Paris)

On `hard stars’ in general relativity

Abstract. I will review the classical picture of gravitational collapse in spherical symmetry, from the Oppenheimer-Snyder model (1939) to Christodoulou’s two phase model (1995). I will then turn to the possible end states of gravitational collapse, in particular discuss non-trivial static solutions to the two-phase model, which are idealized models of neutron stars. The main results concern a variational characterization of hard stars, and I will outline their relevance for the orbital stability problem of neutron stars in spherical symmetry. I hope to conclude with a discussion of the various remaining problems that have to be overcome for a global in time result, in particular related to possible phase transitions in this model.

 


Monday October 9, 2017

room 15/16-309

 14h Daniel Monclair (Orsay)

Attractors in spacetimes and time functions

Abstract.  A time function on a Lorentzian manifold is a continuous real valued function which is increasing along all future directed causal curves. A result of Hawking states that the existence of a time function is equivalent to stable causality, i.e. the impossibility of generating timelike loops even after small perturbations of the metric. We will discuss a construction of time functions which is quite different from Hawking’s construction, in the sense that it produces functions that still have interesting properties for non stably causal spacetimes (while Hawking’s time functions fail to be continuous without stable causality). Our approach is based on a notion of attracting sets in spacetimes, following the work of Conley on Lyapunov functions.

 15h30 The-Cang Nguyen (Paris)

Global dispersion of self-gravitating massive matter in spherical symmetry

Abstract.  We study massive matter fields evolving under their own gravitational field and we generalize results established by Christodoulou for the spherically symmetric evolution of massless scalar fields governed by the Einstein equations. We encompass both Einstein’s theory and the f(R)-theory of modified gravity defined from a generalized Hilbert-Einstein functional depending on a nonlinear function f(R) of the spacetime scalar curvature R. This is a joint work with P.G. LeFloch and F. Mena.



 

Philippe LeFloch, DIRECTOR OF RESEARCH AT CNRS Email address: pglefloch [at] gmail.com

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