I will introduce the celebrated black hole stability conjecture according to which the Kerr family of metrics are stable as solutions to the Einstein vacuum equations of general relativity. I will then discuss the history of this problem, including a recent work on the resolution of the black hole stability conjecture for small angular momentum.
This is an introductory course on the wave equation, to be delivered in weekly two-hour sessions. Topics to be covered include Fourier and physical space methods, existence and uniqueness of solutions to linear equations with constant or variable coefficients, energy estimates, vector fields, Strichartz estimates and local/global well posedness for nonlinear wave equations. Time permitting, singularity formation and geometric optics may also be addressed.
While Einstein’s theory of gravity is formulated in a smooth setting, the celebrated singularity theorems of Hawking and Penrose describe many physical situations in which this smoothness must eventually break down. In positive-definite signature, there is a highly successful theory of metric and metric-measure geometry which includes Riemannian manifolds as a special case, but permits the extraction of nonsmooth limits under dimension and curvature bounds analogous to the energy conditions in relativity: here sectional curvature is reformulated through triangle comparison, while Ricci curvature is reformulated using entropic convexity along geodesics of probability measures. This lecture explores recent progress in the development of an analogous theory in Lorentzian signature, whose ultimate goal is to provide a nonsmooth theory of gravity.
This is an introductory course on the wave equation, to be delivered in weekly two-hour sessions. Topics to be covered include Fourier and physical space methods, existence and uniqueness of solutions to linear equations with constant or variable coefficients, energy estimates, vector fields, Strichartz estimates and local/global well posedness for nonlinear wave equations. Time permitting, singularity formation and geometric optics may also be addressed.
This is an introductory course on the wave equation, to be delivered in weekly two-hour sessions. Topics to be covered include Fourier and physical space methods, existence and uniqueness of solutions to linear equations with constant or variable coefficients, energy estimates, vector fields, Strichartz estimates and local/global well posedness for nonlinear wave equations. Time permitting, singularity formation and geometric optics may also be addressed.
This is an introductory course on the wave equation, to be delivered in weekly two-hour sessions. Topics to be covered include Fourier and physical space methods, existence and uniqueness of solutions to linear equations with constant or variable coefficients, energy estimates, vector fields, Strichartz estimates and local/global well posedness for nonlinear wave equations. Time permitting, singularity formation and geometric optics may also be addressed.
This is an introductory course on the wave equation, to be delivered in weekly two-hour sessions. Topics to be covered include Fourier and physical space methods, existence and uniqueness of solutions to linear equations with constant or variable coefficients, energy estimates, vector fields, Strichartz estimates and local/global well posedness for nonlinear wave equations. Time permitting, singularity formation and geometric optics may also be addressed.
In 2013, Vasy proved that solutions to linear wave equations in Kerr-de Sitter spacetimes have asymptotic expansions in quasinormal modes up to an exponentially decaying term, assuming the angular momentum of the black hole satisfies certain bounds. This was the first step towards the proof of non-linear stability for slowly rotating Kerr-de Sitter black holes by Hintz and Vasy in 2018. In this talk, we extend Vasy’s result to the full subextremal range of Kerr-de Sitter spacetimes, by removing the restrictions on the angular momentum of the black hole. The proof is based on a new Fredholm setup and a new analysis of the trapping of photons around a Kerr-de Sitter black hole. This is joint work with Andras Vasy.
This is an introductory course on the wave equation, to be delivered in weekly two-hour sessions. Topics to be covered include Fourier and physical space methods, existence and uniqueness of solutions to linear equations with constant or variable coefficients, energy estimates, vector fields, Strichartz estimates and local/global well posedness for nonlinear wave equations. Time permitting, singularity formation and geometric optics may also be addressed.
In the '80s, Klainerman and Sarnak obtained explicit solutions for the wave equation in homogeneous and isotropic universes filled with dust. We generalize their work to several Friedmann-Lemaître-Robertson-Walker (FLRW) spacetimes of hyperbolic, flat and spherical spatial curvature and discuss decay rates and the validity of the Huygens principle. The method consists in finding differential operators that map solutions of the wave equation in these FLRW spacetimes to solutions of the conformally invariant wave equation in simpler, ultra-static spacetimes, for which spherical mean formulas are available. Conjectures for the general decay rates in flat and hyperbolic FLRW spacetimes, inspired by numerical results, will also be discussed. The talk is based on joint work with J. Natário and A. Vañó-Viñuales.
A definition for `bounded Lorentzian metric space’ is presented and discussed. This is an abstract notion of Lorentzian metric space that is sufficiently general to comprise compact causally convex subsets of globally hyperbolic (smooth) spacetimes, and causets.
It is shown that a generalization of the Gromov-Hausdorff distance and convergence can be applied to these spaces. Furthermore, two additional axioms of timelike connectedness and existence of maximizers, which are stable under GH-convergence, lead to suitable notions of Lorentzian pre-length and length spaces. Similarly, sectional curvature bounds stable under GH-convergence can be introduced. A (pre)compactness theorem is also mentioned and its limitations are discussed. Talk based on joint work with Stefan Suhr (Bochum).
This is an introductory course on the wave equation, to be delivered in weekly two-hour sessions. Topics to be covered include Fourier and physical space methods, existence and uniqueness of solutions to linear equations with constant or variable coefficients, energy estimates, vector fields, Strichartz estimates and local/global well posedness for nonlinear wave equations. Time permitting, singularity formation and geometric optics may also be addressed.
This is an introductory course on the wave equation, to be delivered in weekly two-hour sessions. Topics to be covered include Fourier and physical space methods, existence and uniqueness of solutions to linear equations with constant or variable coefficients, energy estimates, vector fields, Strichartz estimates and local/global well posedness for nonlinear wave equations. Time permitting, singularity formation and geometric optics may also be addressed.
This is an introductory course on the wave equation, to be delivered in weekly two-hour sessions. Topics to be covered include Fourier and physical space methods, existence and uniqueness of solutions to linear equations with constant or variable coefficients, energy estimates, vector fields, Strichartz estimates and local/global well posedness for nonlinear wave equations. Time permitting, singularity formation and geometric optics may also be addressed.
This is an introductory course on the wave equation, to be delivered in weekly two-hour sessions. Topics to be covered include Fourier and physical space methods, existence and uniqueness of solutions to linear equations with constant or variable coefficients, energy estimates, vector fields, Strichartz estimates and local/global well posedness for nonlinear wave equations. Time permitting, singularity formation and geometric optics may also be addressed.
In this talk, we will explore the role of harmonic and p-harmonic functions in establishing fundamental geometric inequalities in mathematical relativity, such as the positive mass theorem and the Penrose inequality. We will introduce new monotonicity formulas along the level flow of these functions, which provide simple proofs for these inequalities. Additionally, we will discuss how these formulas can be used to characterize the nonnegativity of the scalar curvature, suggesting a synthetic notion of this concept applicable in non-smooth contexts.
This is an introductory course on the wave equation, to be delivered in weekly two-hour sessions. Topics to be covered include Fourier and physical space methods, existence and uniqueness of solutions to linear equations with constant or variable coefficients, energy estimates, vector fields, Strichartz estimates and local/global well posedness for nonlinear wave equations. Time permitting, singularity formation and geometric optics may also be addressed.
This is an introductory course on the wave equation, to be delivered in weekly two-hour sessions. Topics to be covered include Fourier and physical space methods, existence and uniqueness of solutions to linear equations with constant or variable coefficients, energy estimates, vector fields, Strichartz estimates and local/global well posedness for nonlinear wave equations. Time permitting, singularity formation and geometric optics may also be addressed.
This is an introductory course on the wave equation, to be delivered in weekly two-hour sessions. Topics to be covered include Fourier and physical space methods, existence and uniqueness of solutions to linear equations with constant or variable coefficients, energy estimates, vector fields, Strichartz estimates and local/global well posedness for nonlinear wave equations. Time permitting, singularity formation and geometric optics may also be addressed.
The Teukolsky master equations describe the gauge invariant degrees of freedom in linear gravitational perturbations of the Kerr black hole family. In this talk, we focus on separable solutions to the Teukolsky equations, for which superradiance and trapping phenomena can be understood. Particular emphasis will be given to mode stability, that is non-existence of exponentially growing or bounded but non-decaying separable solutions. A corollary of our analysis is that general solutions to the Teukolsky equations are bounded and decay in time; this is a key first step in establishing (linear) stability of Kerr black holes. This talk contains joint work with Marc Casals (Leipzig/CBPF/UC Dublin) and Yakov Shlapentokh-Rothman (Toronto).
This seminar is joint with CENTRA, and will take place on the Physics Department (seminar room, 2nd floor).
This is an introductory course on the wave equation, to be delivered in weekly two-hour sessions. Topics to be covered include Fourier and physical space methods, existence and uniqueness of solutions to linear equations with constant or variable coefficients, energy estimates, vector fields, Strichartz estimates and local/global well posedness for nonlinear wave equations. Time permitting, singularity formation and geometric optics may also be addressed.
In this talk, I will review some works in collaboration with Jonathan Luk on the behaviour of high frequency solutions to the Einstein equations. In the eighties, the physicist Burnett conjectured that the adherence of the solutions to the Einstein equations, converging strongly at the level of the metric, and with boudedness assumptions at the level of derivative of the metric, are the solutions to the Einstein-massless Vlasov equations. I will focus on the resolution of this conjecture with the additionnal assumption of a translation symmetry, and explain how to approximate a solution to the Einstein-Vlasov equations by vacuum solutions.
This is an introductory course on the wave equation, to be delivered in weekly two-hour sessions. Topics to be covered include Fourier and physical space methods, existence and uniqueness of solutions to linear equations with constant or variable coefficients, energy estimates, vector fields, Strichartz estimates and local/global well posedness for nonlinear wave equations. Time permitting, singularity formation and geometric optics may also be addressed.
This is an introductory course on the wave equation, to be delivered in weekly two-hour sessions. Topics to be covered include Fourier and physical space methods, existence and uniqueness of solutions to linear equations with constant or variable coefficients, energy estimates, vector fields, Strichartz estimates and local/global well posedness for nonlinear wave equations. Time permitting, singularity formation and geometric optics may also be addressed.
This is an introductory course on the wave equation, to be delivered in weekly two-hour sessions. Topics to be covered include Fourier and physical space methods, existence and uniqueness of solutions to linear equations with constant or variable coefficients, energy estimates, vector fields, Strichartz estimates and local/global well posedness for nonlinear wave equations. Time permitting, singularity formation and geometric optics may also be addressed.
We discuss the problem of global uniqueness for the Einstein equations with a positive cosmological constant. After reviewing the modern formulations of strong cosmic censorship (SCCC), we prove a stability result suggesting a potential failure of SCCC in the spherically symmetric framework. In particular, we study the characteristic initial value problem (IVP) for the spherically symmetric Einstein-Maxwell-charged-Klein-Gordon system, with initial data as expected to arise from charged gravitational collapse. The solution to the IVP describes the interior of a black hole asymptotically approaching a sub-extremal Reissner-Nordström-de Sitter spacetime. By using bootstrap methods in the black hole interior, we establish the $H^1$ non-linear stability of the Cauchy horizon for a large set of initial data.
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Seminar room (2.8.3), Physics Building
Mokdad Mokdad, University of Burgundy-Franche-Comté
Scattering theories are of large importance for many problems in General Relativity, and scattering in the interior of a black hole is particularly relevant in the context of the cosmic censorship conjecture and the related Cauchy horizon instability problem. The Cauchy horizon instability is thought to be directly linked to a notion of gravitational blue-shift at the horizon, which manifests itself as a blow-up in some observed quantity. By constructing the scattering channels, one aims to recover information about the behavior of the field near the horizons, where the instabilities might be seen as the unboundedness of the scattering operators (energy blow-up) or from the lack of regularity at the horizon of the propagating field (C^1-blow-up). In this talk I will present the recent development of the scattering theories in the interior black holes between the Cauchy horizon and the event horizon, and the various blow-up results. From a mathematical point of view, different fields exhibit contrasting scattering phenomena. For example, the scattering of linear waves is delicate and surprising breakdowns of scattering happens in generic situations.
This seminar is joint with CENTRA, and will take place on the Physics Department (seminar room, 2nd floor).
The lecture examines older and newer concepts of quasi-local mass for one domain relative to another as well as concepts of quasi-local radius in relation to the hoop-conjecture.
Some theorems in relativity work directly only in low dimensions because of the possibility of singularities in area minimizing hypersurfaces and MOTS. These include both Riemannian and spacetime positive mass theorems as well as the Riemannian Penrose inequality. In some cases the singularities are relatively easy to work around and in others they present significant problems to overcome. This will be a general lecture which will attempt to give perspective and more clarity to this issue.
