Mathematical general relativity, compressible fluids, and more

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Seminar on Mathematical General Relativity

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

Academic year 2025–2026

Thursday March 19, 2026

lecture room 15-16–309 (Jussieu)

14h Xuantao CHEN (LJLL, Sorbonne)

Formation of trapped surfaces from characteristic or spacelike data

Abstract.  I will discuss the formation of trapped surfaces in general relativity, starting from a brief overview of the pioneering results on the characteristic initial value problem using a double null foliation. I will then explain a new proof built upon the non-integrable PT (Principal Temporal) frame from the recent development on the proof of the Kerr stability conjecture. This gauge yields transport equations for all Ricci coefficients along the incoming null direction and avoids any loss of derivatives. I will next propose a formulation on spacelike initial hypersurfaces, where we characterize the short-pulse data in the spacelike setting. This is based on joint work with Sergiu Klainerman.

15h30 Georgios MOSCHIDIS (EPFL, Lausanne)

Naked singularities with finite blue-shift for the Einstein–massless Vlasov system

Abstract. In his celebrated proof of the weak cosmic censorship conjecture for the spherically symmetric Einstein-scalar field system, Christodoulou exploited the following property of that specific matter model: Naked singularities, when they arise, exhibit infinite blue-shift along the null geodesics terminating at the singularity. This behaviour is consistent with self-similarity: Even for more general spherically symmetric matter models, it can be shown that self-similar naked singularities must exhibit infinite blue-shift. Whether, for these more general models, all naked singularities have the infinite blue shift property (and hence are potentially subject to an instability mechanism analogous to that introduced by Christodoulou) still remains an open question. In this talk, I will present the construction of a spherically symmetric solution to the Einstein-massless Vlasov system which contains a locally naked singularity with finite total blue-shift along its past null cone. The initial data giving rise to this solution have limited differentiability, but belong to a regularity class above the scale invariant threshold.

Thursday February 19, 2026

lecture room 15-25-322 (nouvelle salle)

14h Jack BORTHWICK (IMJ, Sorbonne)

Geometry at time-like infinity and massive fields

Abstract. This talk explores recent geometric constructions in the setting of projectively compact Ricci-flat Einstein manifolds, with an emphasis on the role of projective geometry. I will discuss how these constructions encode aspects of the geometry at infinity, and present some initial ideas and results on their potential relevance for the analysis of massive classical particle fields a « timelike » infinity.

15h30 Itsvan KADAR (ETH, Zürich)

Matching conditions for scattering solutions of scalar wave equations on extremal black holes

Abstract. The existence of spacetimes describing multiple black holes and their asymptotic properties—such as their late-timebehavior—is an exciting open area of mathematical research. In this talk, I will report on some preliminary steps towards addressing this problem. As a toy model, we study a nonlinear scalar wave equation on a multi–extremal- black-hole spacetime with a prescribed, polynomially decaying, smooth radiation field at null infinity. We construct smooth solutions in the spacetime by identifying appropriate initial data on a spacelike hypersurface inside the black hole regions. This is joint ongoing work with Yannis Angelopoulos.

Thursday December 18, 2025

lecture room 15-25-322 (nouvelle salle)

14h Flavio ROSSETTI (L’Aquila)

Strong cosmic censorship for de Sitter black holes

Abstract. We will discuss modern formulations of the strong cosmic censorship conjecture (SCCC) and possible resolutions supported by rigorous non-linear results for the spherically symmetric Einstein-Maxwell-scalar field system. We will show that the presence of a positive cosmological constant suggests a violation of the SCCC at a fundamental level of regularity. Indeed, the blueshift mechanism occurring at the Cauchy horizon can be counter-balanced by the dispersive effects encoded in the exponential Price law along (cosmological) black hole event horizons. On the other hand, we show that, if non-smooth black hole solutions are allowed, then the aforementioned violations are non-generic in a positive co-dimension sense.

15h30 Gemma HOOD (Leipzig)

A scattering construction for nonlinear wave equations on Kerr-Anti de Sitter spacetimes

Abstract. Given the sharp logarithmic decay of linear waves on the Kerr-AdS black hole (Holzegel, Smulevici, 2013), it is expected that the Kerr-AdS spacetime is unstable as a solution of the Einstein vacuum equations. However, the scattering construction presented here for exponentially decaying nonlinear waves on a fixed Kerr-AdS background serves as a first step to confronting the scattering problem for the full Einstein system. In this context, one may hope to derive a class of perturbations of Kerr-AdS which remain ‘close’ and dissipate sufficiently fast.

Thursday November 27, 2025

lecture room 15-25-101 (Jussieu)

14h Mahdi HAGHSHENAS (Imperial College, London)       

Boundedness and decay of waves on decelerated FLRW spacetimes

Abstract. After outlining the stability problem for Friedmann–Lemaître–Robertson–Walker (FLRW) spacetimes, we study the wave equation —as a proxy for the Einstein equations— on decelerated FLRW spacetimes with non-compact, flat spatial sections. We demonstrate how dispersion and expansion affect the long-time behavior of waves. In particular, we present uniform energy bounds and integrated local energy decay estimates across the full decelerated expansion range. Furthermore, we describe a hierarchy of r-weighted energy estimates, in the spirit of the Dafermos–Rodnianski method, which lead to energy decay estimates. 

15h30 Pau FIGUERAS (Queen Mary, London)

The initial value problem for higher derivative theories of gravity

Abstract. General relativity can be thought of as a low energy (classical) effective field theory (EFT) of gravity. As such, on general grounds, it is expected that it should receive higher derivative corrections. However, the equations of motion of such higher derivative theories are higher than second order; in particular, they have more than two time derivatives and hence they are plagued with runaway solutions that are unphysical. Furthermore, being higher than second order, it is not clear how to formulate the initial value problem and thus extract their predictions consistently with the EFT expansion. In this talk, I will review the various approaches to this old problem and I will present our recent proposal called “regularization”. As I will show, regularisation allows to formulate the initial value problem for a very general class of higher derivative theories in a manifestly well-posed way, it is covariant and it does not require any fine tuning.

Thursday October 16, 2025

lecture room 15-16-309 (Jussieu)

14h Ludovic SOUETRE (Sorbonne)

Geometric reflective boundary conditions for asymptotically Anti-de Sitter spaces

Abstract. Modeled on the Anti-de Sitter space, asymptotically Anti-de Sitter spaces are defined as Lorentzian manifolds that possess a timelike conformal boundary. As a result, they are not globally hyperbolic. In order to find such spaces that also solve the Einstein equations (with a negative cosmological constant), it is therefore necessary to consider the Cauchy problem as an initial boundary value problem. In this talk, I will discuss the geometric boundary conditions that can be prescribed on the conformal boundary to ensure local existence and uniqueness of solutions in dimension 4. The first one, introduced by Friedrich in his pioneering 1995 work, consists in imposing the boundary conformal class and is known as the Dirichlet boundary condition. The second is a new family of geometric reflective boundary conditions involving both the boundary conformal class and the boundary stress-energy tensor. It can be regarded as the homogeneous Robin boundary conditions.

15h30 Taoran HE (IHES)

Stability of Big Bang singularity for the Einstein-Maxwell-scalar field-Vlasov system in the full strong sub-critical regime

Abstract. I will present our recent work on the stability of Kasner solutions for the Einstein-Maxwell-scalar field-Vlasov system in 1+3 dimensions. This system incorporates gravity, electromagnetic, weak and strong interactions for the initial stage of our universe. The inclusion of the Vlasov field introduces several new challenges. By observing detailed mathematical structures and designing new delicate arguments, we identify a new strong sub-critical regime and prove the nonlinear stability with Kasner exponents lying in this entire regime. Our results extend the work of Fournodavlos, Rodnianski, and Speck from the Einstein-scalar field system to the physically more complex system with the Vlasov field. This is joint work with Xinliang An and Dawei Shen.

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Seminar on Mathematical General Relativity

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

Academic year 2024–2025

Thursday July 3, 2025

lecture room 15-16-309 (Jussieu)

14h Armand Coudray (Franche-Comté)

 Peeling-off behavior of the wave equation in the Vaidya spacetime

Abstract. I will discuss the asymptotic regularity of the wave equation in Vaidya spacetime, which is a spherical solution to Einstein’s equations in which the metric depends on time. In other words, the aim is to prove that the asymptotic regularity of the field is equivalent to the regularity of the initial data. The analysis is based on two methods. First, I will characterize the asymptotic behavior of the wave equation using the conformal construction developed by Penrose in the 1960’s. Secondly, I will derive the equivalence between the asymptotic and initial data using energy estimates. In view of  the structure of the conformal boundary of the black hole solution’s, a particular focus is put on the regularity at spacelike infinity, a singularity of the boundary. Finally, I will prove that the class of initial data ensuring the field’s regularity on the boundary is the same as the one in flat spacetime.

15h30 Tony Salvi (Palaiseau)

Semi-classical limit for the Klein-Gordon and Klein-Gordon-Maxwell equations

Abstract. Quantum mechanics is well-approximated by classical physics when the Planck constant is very small, i.e. in the semi-classical limit. Typically, one studies an observable associated with a particle, such as its momentum or position, and shows that its dynamics is given by classical dynamics at first order, with corrections of order of Planck’s constant. In this talk, I will present the concept of semi-classical limit, the standard mathematical results known for non-relativistic quantum mechanics, and my results concerning the semi-classical limit in the context of relativistic quantum mechanics. I will show how to adapt the modulated energy method that was developed for the Schrödinger equation, to the Klein- Gordon and Klein-Gordon-Maxwell equations, and how to recover relativistic mechanics (instead of classical mechanics) in the semi-classical limit.

Thursday March 20, 2025

lecture room 15-16-309 (Jussieu)

14h Hans Ringström (Stockholm) 

Formation of quiescent big bang singularities

Abstract. A basic question in general relativity is if big bang singularities appear naturally in solutions to Einstein’s equations. In the early days of general relativity, explicit solutions with such singularities were found. Later, the singularity theorem of Hawking was demonstrated. It provides an indication that big bang singularities should appear naturally but does not say much about their nature. Recently, many results concerning stable big bang formation have appeared. Most of the results concern stability of spatially homogeneous and isotropic solutions. However, a recent result of Fournodavlos, Rodnianski and Speck covers the full regime in which stability is to be expected. On the other hand, it is restricted to the stability of spatially homogeneous and spatially flat solutions. In this talk, I will present a new result (joint work with Hans Oude, Groeniger and Oliver Petersen) in which we identify a general condition on initial data ensuring big bang formation. The solutions need, in this case, not be close to symmetric background solutions. Moreover, the result reproduces previous results in the Einstein-scalar field and Einstein-vacuum settings. Finally, the result is in the Einstein-non-linear scalar field setting, and therefore yields future and past global non-linear stability of large classes of spatially locally homogeneous solutions.

15h30 Arthur Touati (Bordeaux)

Spacelike initial data for black hole stability

Abstract. I will present a joint work with Allen Juntao Fang and Jérémie Szeftel on the resolution of the constraint equations in the context of black hole stability. After having presented the motivations coming from the evolution problem and reviewing the literature on the constraint equations, I will show how a bit of Fredholm theory and the careful construction of a boundary condition for the elliptic system can help avoiding linear obstructions (the famous KIDS) and thus produce initial black hole perturbations with arbitrary decay.

Thursday March 13, 2025

lecture room 15-16-309 (Jussieu)

14h Sam Collingbourne (Edinburgh)

The Gregory–Laflamme Instability of the 5D Schwarzschild black string exterior

Abstract. Gregory–Laflamme type instabilities seem to plague black holes in dimensions greater than 4. Gregory–Laflamme instabilities are mode instabilities at the level of the linearised Einstein vacuum equation and have been identified heuristically and numerically for black holes that have an event horizon that has one direction that is large compared to all others. In this talk, I will discuss a direct rigorous mathematical proof of the Gregory–Laflamme instability for the 5D Schwarzschild black string. Under a choice of ansatz for the perturbation and a gauge choice, the linearised vacuum Einstein equation can be reduced to a Schrödinger eigenvalue equation to which an energy functional is assigned. It is then shown by direct variational methods that the lowest eigenfunction gives rise to an exponentially growing mode solution which has admissible behaviour at the future event horizon and spacelike infinity in harmonic/transverse-traceless gauge.

15h30 Allen Fang (Munster, Germany)

Wave behavior in the vanishing cosmological constant limit

Abstract. Black hole stability is a central topic in mathematical relativity that has seen numerous advancements in recent years. Both the Kerr-de Sitter and the Kerr black hole spacetimes have been proven to be stable in the slowly-rotating regime. However, the methods used have been markedly different, as well as the decay rates proven. Perturbations of Kerr-de Sitter converge exponentially back to a nearby Kerr-de Sitter black hole, while perturbations of Kerr only converge polynomially back to the family. In this talk, I will speak about wave behavior that is uniform in the cosmological constant by considering solutions to the model Regge-Wheeler equations in Kerr(-de Sitter). The main point is a careful handling of the relevant estimates on the region of the spacetime far from the black hole. This provides a first step into understanding the uniform (in the cosmological constant) stability of black hole spacetimes. This is joint work with Jeremie Szeftel and Arthur Touati.

Thursday January 23, 2025

lecture room 15-16-309 (Jussieu)

14h Volker Schlue (Melbourne) 

Nonlinear stability of the expanding region of Kerr de Sitter spacetimes

Abstract. The topic of this lecture is the global nonlinear stability problem for Kerr de Sitter spacetimes. Following a discussion of the results of P Hintz and A Vasy on the nonlinear stability of the stationary black hole exterior, and those of H Friedrich for de Sitter spacetimes, we turn to the so-called cosmological, or expanding region of Kerr de Sitter spacetimes. I will present recent work with G Fournodavlos, which establishes global existence and asymptotics in this region, compatible with de Sitter like degrees of freedom. Our work can be combined with the results of P Hintz and A Vasy, to yield the global nonlinear stability of Kerr de Sitter spacetimes, featuring both aspects of an asymptotic and orbital stability result.

15h30 Sebastian Gurriaran (Sorbonne)  

On the linear instability of the Kerr Cauchy horizon

Abstract. I will describe a recent work on the linear instability of the Kerr Cauchy horizon, first proven by Sbierski. In the context of a linearized gravitational perturbation of a Kerr black hole, I will present the precise asymptotics of the spin +2 Teukolsky scalar in the interior region of the black hole, proving its exponential blow-up on the Cauchy horizon, as well as an oscillatory behavior caused by the non-zero angular momentum of the black hole. The result supports Strong Cosmic Censorship in Kerr spacetimes. I will give a sketch of the proof, which is done purely in physical space and which is based on Price’s law results in Kerr proven by Ma and Zhang.

Thursday December 19, 2024

lecture room 15-16-309 (Jussieu)

14h Filip Ficek (Vienna)

Time-periodic solutions to nonlinear wave equations

Abstract. I will discuss time-periodic solutions for nonlinear wave equations on an interval with Dirichlet boundary conditions. The classic works prove the existence of small amplitude periodic solutions with frequencies belonging to nowhere dense sets. I will show numerical evidences suggesting that these gaps in frequencies are filled with large amplitude solutions. This leads to intricate bifurcation structures forming fractal-like patterns. I will propose a systematic approach of analyzing these complex structures. This talk is based on joint work with Maciej Maliborski.

15h30 Robert Sansom (Queen Mary, London)

The null gluing problem for the Einstein equations

Abstract. The null gluing problem for the Einstein equations is the problem of matching two solutions of the Einstein equations along a null hypersurface. I will present a solution to the third-order null gluing problem, that is, where up to third-order derivatives of the metric are to be glued. In the regime where the characteristic data is close to Minkowski data, this third-order null gluing problem is solvable up to a 20-dimensional space of obstructions. The obstructions correspond to 20 linearly conserved quantities: 10 of which are already present in the second-order null gluing problem analyzed by Aretakis-Czimek-Rodnianski, and 10 are novel obstructions. The 10 novel obstructions are linearly conserved charges calculated from third-order derivatives of the metric.

Thursday November 21, 2024

lecture room 15-16–201 (Jussieu)

14h Louie Bernhardt (Melbourne) 

Scattering of linear waves on the expanding region of Schwarzschild-de Sitter

Abstract. I will discuss several new results relating to the linear wave equation on Schwarzschild-de Sitter spacetime. We establish a finite-order asymptotic expansion for solutions in the expanding region. This is accomplished by introducing new higher-order weighted energy estimates that capture the higher-order asymptotics of solutions. We also prove the existence and uniqueness of scattering solutions from data at infinity which possess asymptotics consistent with the forward problem. I will explain how this is achieved via the construction of approximate solutions that capture the desired asymptotics, as well as a new weighted energy estimate suitable for the backward problem. Time permitting, I will discuss how these results generalise to a class of expanding spacetimes which do not necessarily converge back to Schwarzschild-de Sitter at infinity.

15h30 Warren Li (Princeton)

BKL bounces outside homogeneity

Abstract. In the latter half of the 20th century, physicists Belinski, Khalatnikov and Lifshitz (BKL) proposed a general Ansatz for solutions to the Einstein equations possessing a (spacelike) singularity. They suggest that, near the singularity, the evolution of the spacetime geometry at different spatial points decouples and is well-approximated by a system of autonomous nonlinear ODEs, and further that general orbits of these ODEs resemble a (chaotic) cascade of heteroclinic orbits called “BKL bounces”. In this talk, we present recent work verifying the validity of BKL’s heuristics in a large class of symmetric, but spatially inhomogeneous, spacetimes which exhibit (up to one) BKL bounce on causal curves reaching the singularity. In particular, we prove AVTD behavior (i.e. decoupling) even in the presence of inhomogeneous BKL bounces. The proof uses nonlinear ODE analysis coupled to hyperbolic energy estimates, and one hopes our methods may be applied more generally.

Thursday October 17, 2024

lecture room 15-16-309 (Jussieu)

14h Liam Urban (Vienna)

On the past maximal development of near-FLRW data for the Einstein scalar-field Vlasov system

Abstract. I will discuss recent results on the past nonlinear stability of FLRW solutions to the Einstein scalar-field Vlasov system in three and four spacetime dimensions without symmetry assumptions, the latter based on joint work with David Fajman. Such near-FLRW solutions are shown to be asymptotically velocity term dominated toward the past, resulting in a quiescent Big Bang singularity. The Vlasov distribution, however, becomes highly anisotropic towards the Big Bang singularity, concentrating in preferred momentum eigen-directions associated with the second fundamental form. To ensure that the scalar field sufficiently mitigates this behavior, one crucially exploits a scaling hierarchy between horizontal and vertical derivatives in the expansion normalized Vlasov equation. In 3+1 dimensions, this is combined with Bel-Robinson variable approach to control the quiescent variables, and similar ideas apply in a significantly simpler proof in 2+1 dimensions.

15h30 Matthew Schrecker (Bath) 

Stability of gravitational collapse

Abstract. In the Newtonian setting, a star is modelled as a spherically symmetric gas obeying the compressible Euler-Poisson system. In certain regimes, smooth initial data may give rise to blow-up solutions, corresponding to the collapse of a star under its own gravity. In recent years, certain families of self-similar gravitational collapse solutions have been rigorously constructed. In this talk, I will first survey some of these constructions, and then present the nonlinear stability of the Larson-Penston solution, which is conjectured in the astrophysical literature to act as a ground state in gravitational collapse. This is based on joint works with Yan Guo, Mahir Hadzic, and Juhi Jang.