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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.