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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 2023–2024
Thursday June 13, 2024
lecture room 15-16-309 (Jussieu)
14h TBA
15h30 TBA
Thursday May 2nd, 2024
lecture room 15-16-309 (Jussieu)
11h Greg Fournodavlos (Heraklion, Crete)
Global stability of cosmological fluids with extreme tilt
Abstract. In cosmology, the equation of state of a perfect fluid is considered to be p = a rho, where a is the squared of the speed of sound. The simplest solution to the Einstein-Euler system, known as FLRW, representing a cosmological fluid, was discovered by Friedmann already in 1922. There is an extensive literature in physics concerning the dynamics of cosmological fluids. However, rigorous mathematical works proving the stability of homogeneous backgrounds are so far restricted to small sound speeds, up to the radiation threshold. Interesting bifurcation phenomena and instabilities are predicted for larger sound speeds. I will discuss joint work with E. Marshall and T. A. Oliynyk proving the global stability of homogeneous solutions with so-called extreme tilt, whose fluid vector field becomes asymptotically null, beyond the radiation case.
14h Marios Apetroaie (Münster)
On the Linear (In)stability of Extremal Reissner-Nordström
Abstract. The Reissner-Nordström spacetime, as a solution to the Einstein-Maxwell equations, has been shown to be linearly stable for the full sub-extremal range, |Q|<M, by Elena Giorgi. We address the aforementioned problem for the extremal case, |Q|=M, which contrary to the subextremal one we show instability results manifesting along the future event horizon of the black hole. In particular, depending on the number of translation invariant derivatives of derived gauge-invariant quantities, we obtain decay, non-decay, and polynomial blow-up estimates asymptotically along the horizon . In this presentation, we motivate the main ideas showing that solutions to the generalized Teukolsky system of positive and negative spin satisfy analogous estimates as well. Stronger and unprecedented instabilities are realised for the negative spin solutions, with one of the extreme curvature component not decaying asymptotically along the event horizon.
Thursday March 28, 2024
lecture room 15-16-309 (Jussieu)
14h Pin Yu (Tsinghua University, Beijing)
Sur la stabilité non-linéaire des ondes de raréfaction multidimensionnelles
Abstract. En 1860, Riemann a introduit le problème de Riemann et l’a résolu pour les gaz isentropiques en terme d’ondes de choc et d’ondes de raréfaction. Ce travail a fourni la fondation de la théorie des lois de conservation en dimension un développée au 20e siècle. Dans un travail en collaboration, Tian-Wen Luo et moi-même prouvons la stabilité structurelle non-linéaire du problème de Riemann pour les équations d’Euler isentropiques multidimensionnelles dans le régime de deux familles d’ondes de raréfaction.
15h30 Volker Schlue (Melbourne)
Scattering for wave equations with sources close to the light cone
Abstract. I will describe the scattering problem for wave equations with sources in the wave zone. These arise for example for the Einstein equations in harmonic coordinates, and introduce the problem of slowly decaying solutions in the interior. The asymptotics of solutions are captured by homogeneous solutions at timelike and spacelike infinity, and a radiation field at null infinity. For the scattering problem this introduces matching conditions, which relate the tails of the radiation field to interior and exterior asymptotics. This is joint work with Hans Lindblad.
Thursday February 29, 2024
lecture room 15-16-309 (Jussieu)
14h Simon Guisset (Queen Mary University, London)
Counterexamples to unique continuation for critically singular wave operators:
application to anti-de Sitter spacetimes
Abstract. I will show how one can extend the classical result of Alinhac and Baouendi to critically singular wave operators. In particular, we will consider wave operators blowing up critically on a spacelike or null hypersurface and will show that the unique continuation property from such a hypersurface does not hold by constructing counterexamples, provided there exists a family of trapped null geodesics. As an application to relativity and holography, I will also show how one can apply this non-uniqueness result to obtain counterexamples to unique continuation for some Klein- Gordon equations from the conformal boundaries of asymptotically Anti-de Sitter spacetimes. This work is in collaboration with Arick Shao.
15h30 An Xinliang (Singapore)
Dynamics of apparent horizon and progress toward weak cosmic censorship
Abstract. I will report some recent results on dynamics of apparent horizon and progress toward weak cosmic censorship within and without spherical symmetry.
Thursday January 25, 2024
lecture room 15-16-309 (Jussieu)
14h Dawei Shen (Sorbonne Université)
Global stability of Minkowski spacetime with minimal decay
Abstract. The global stability of Minkowski spacetime has been proven in the celebrated work of Christodoulou-Klainerman in 1993. In 2007, Bieri has extended the result of Christodoulou-Klainerman under lower decay and regularity assumptions on the initial data. In this talk, I will introduce a recent work, which extends the result of Bieri to minimal decay assumptions.
15h30 Viet Dang Nguyen (Sorbonne Université)
The Phi43 theory on curved backgrounds
Abstract. I will describe a joint work with Bailleul, Ferdinand and To in which we construct the Phi43 quantum field theory measure on a compact Riemannian three-manifold, as an invariant measure of a stochastic partial differential equation. I will try to motivate the approach and show many examples. If time permits, I will discuss work in progress where we try to implement rigorously the Wick rotation to extend our construction to the de Sitter space. This should give the first example of a non-perturbative, interacting, non-topological quantum field theory constructed on a Lorentzian three-manifold.
Thursday December 21, 2023
lecture room 15-16-309 (Jussieu)
11h Maxime Van De Moortel (Rutgers University)
Comportement en temps long de l’équation de Klein-Gordon sur le trou noir de Schwarzschild
Abstract. Il est facile de voir que les solutions de l’équation de Klein-Gordon sur un trou noir de Schwarzschild/Reissner-Nordstrom sont bornées en temps. Cependant, le comportement asymptotique en temps long, en particulier le taux de dispersion, sont resté élusifs. Nous discuterons notre récente résolution du problème pour des données initiales localisées. Travail en collaboration avec Federico Pasqualotto et Yakov Shlapentokh-Rothman.
14h Filipe Mena (Lisbon University)
Global solutions to the spherically symmetric Einstein-scalar field system
Abstract. The Einstein-scalar field system has applications in models of compact objects in astrophysics and in cosmology, being an important framework for numerical relativity. In this talk we consider a characteristic initial value problem, with initial data given on a future null cone, for the Einstein massless scalar field system with a positive cosmological constant. We will outline the prove that, for small data, this system has a unique global classical solution which is causally geodesically complete to the future and decays polynomially in radius and exponentially in time, approaching the de Sitter solution. We will then make some remarks about the case with a massive scalar field and large data as well as possible generalisations to higher order theories of gravity.
15h30 Marica Minucci (Queen Mary University, London)
On the non-linear stability of the Cosmological region of the Schwarzschild-de Sitter spacetime
Abstract. I will present a joint work with Juan Antonio Valiente Kroon (QMUL) on the non-linear stability of the sub-extremal Schwarzschild-de Sitter spacetime in the stationary region near the conformal boundary. Our strategy relies on the use of a more general formulation of the conformal Einstein field equations (CEFE) and a gauge based on the properties of conformal geodesics. We observe that the Cosmological stationary region of the Schwarzschild-de Sitter spacetime can be covered by a non-intersecting congruence of conformal geodesics. Thus, the future domain of dependence of spacelike hypersurfaces in this region of the spacetime can be expressed in terms of a conformal Gaussian gauge system. This allows us to recast the conformal evolution equations in a symmetric hyperbolic form. Then, we use a perturbative argument to prove existence and stability results near the conformal boundary and away from the asymptotic points. This analysis is the first step towards a stability argument for perturbation data on the cosmological horizon.
Thursday November 16, 2023
lecture room 15-16-309 (Jussieu)
14h Olivier Graf (Univ. Grenoble-Alpes)
An L2 curvature pinching result for the Euclidean 3-disk
Abstract. When studying the Cauchy problem of general relativity we typically obtain L2 bounds on the (Ricci) curvature tensor of spacelike hypersurfaces and its derivatives. In many situations it is useful to deduce from these H^k bounds that there exists coordinates on the spacelike hypersurface with optimal H^{k+2} bounds on the components of the induced Riemannian metric. The general idea is that this can be achieved using harmonic coordinates –in which the principal terms of the Ricci curvature tensor are the Laplace-Beltrami operators of the metric components– and standard elliptic regularity results. In this talk, I will make this idea concrete in the case of Riemannian 3-manifolds with boundary, with Ricci curvature in L2 and second fundamental form of the boundary in H^{1/2} both close to their respective Euclidean unit 3-disk values. The crux of the proof is a refined Bochner identity with boundary for harmonic functions. The cherry on the cake is that this result does not require any topology assumption on the Riemannian 3-manifold (apart from its boundary), and that we obtain –as a conclusion– that it must be diffeomorphic to the 3-disk. This talk is based on a result that I obtained in “Global nonlinear stability of Minkowski space for spacelike-characteristic initial data’’.
15h30 Grigalius Taujanskas (Cambridge University, UK)
Low regularity wave maps on the Einstein cylinder
Abstract. Nonlinear wave equations of wave maps type are expected to be well-posed for initial data just above scaling critical regularity. For wave maps from Minkowski space, this was proven by Klainerman and Machedon in the 90s by using their famous, sharp, null form estimates which capture Fourier space cancellations between parallel waves. I will introduce a new approach to computing analogous estimates on the Einstein cylinder, where traditional Fourier theory is unavailable, which uses the Lie group structure of SU(2) and Peter-Weyl theory. Time permitting, I will outline how they may be used to show almost optimal well-posedness of wave maps equations from the Einstein cylinder.
Thursday October 5, 2023
lecture room 15-16-309 (Jussieu)
14h Christopher Kauffman (Munster University)
Stability of perturbed wave equations on Kerr black hole spacetimes
Abstract. I will discuss a recent work with Gustav Holzegel, in which we prove integrated decay bounds for solutions of the geometric wave equation with small linear perturbations on Kerr black hole spacetimes. Our proof adapts the framework introduced by Dafermos, Rodnianski, and Shlapentokh-Rothman for the homogeneous wave equation on Kerr spacetimes. When adding the perturbative term one must also compensate for obstructions caused by the necessary degeneration of Morawetz-type estimates in these spacetimes, which is due to the presence of trapped null geodesics. Mathematically, the key mechanism to our approach is the construction of a pseudodifferential commutator W, such that for the commuted equation one may obtain a nondegenerate Morawetz-type estimate.
15h30 Stefan Czimek (Leipzig University)
Obstruction-free gluing for the Einstein Equations
Abstract. We present a new approach to the gluing problem in General Relativity, that is, the problem of matching two solutions of the Einstein Equations along a spacelike or characteristic (null) hypersurface. In contrast to previous constructions, the new perspective actively utilizes the nonlinearity of the constraint equations. As a result, we are able to remove the 10-dimensional spaces of obstructions to gluing present in the literature. As application, we show that any asymptotically flat spacelike initial data set can be glued to Schwarzschild initial data of sufficiently large mass. This is joint work with I. Rodnianski.
Mathematical general relativity, compressible fluids, and more 