Category Archive

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Seminar on

Mathematical General Relativity

Organizers: 

 S. Klainerman (Princeton)

P.G. LeFloch (Paris)

A. Zeghib (Lyon)

Fondations des Sciences Mathématiques de Paris

ANR Project

“Mathematical General Relativity. Analysis and geometry of spacetimes with low regularity”

Thursday January 17, 2013

Laboratoire J-L Lions

Université Pierre et Marie Curie, Paris

Lecture room (see below)

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 11h (Room  15-25- 104)    Sergiu Klainerman (Princeton)   On  the formation of trapped surfaces

Abstract. I will talk about a new result obtained in collaboration with J. Luk and I. Rodnianski in which we relax significantly Christodoulou’s main condition for the formation of trapped surfaces in vacuum.

 14h (Room 15-25-326) Chung-Tse Arick Shao (Toronto)   Null cones to infinity, curvature flux, and Bondi mass

Abstract. In general relativity, the Bondi mass in an asymptotically flat spacetime represents, roughly, the mass remaining in the system after some has radiated away. In this talk, we make sense of and control the Bondi mass for a single null cone in an Einstein-vacuum spacetime under minimal assumptions. In terms of regularity, we assume only small weighted curvature flux along the null cone and small data on an initial sphere of the cone. Furthermore, we make no global assumptions on the spacetime, as all our conditions deal only with the single null cone under consideration. This work is joint with S. Alexakis.

15h30  (Room 15-25-326)  Gustav Holzegel  (Princeton) Existence of dynamical vacuum black holes

Abstract. This is joint work with Mihalis Dafermos and Igor Rodnianski. We prove the existence of a large class of non-stationary vacuum black holes whose exterior geometry asymptotes in time to a fixed Schwarzschild or Kerr metric. The spacetimes are constructed by solving a backwards scattering problem for the vacuum Einstein equations with characteristic data prescribed on the horizon and at null infinity. The data admits the full functional degrees of freedom to specify data for the Einstein equations. An essential feature of the construction is that the solutions converge to stationarity exponentially fast with their decay rate intimately related to the surface gravity of the horizon and hence to the strength of the celebrated redshift effect which, in our backwards construction, is seen as a blueshift.

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Seminar on

Mathematical General Relativity

Organizers:

 Philippe G. LeFloch (Paris)

Ghani Zeghib (Lyon)

ANR Project

“Mathematical General Relativity. Analysis and geometry of spacetimes with low regularity”

Wed. November 14, 2012

Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris

Lecture room  15–16 309 (third level) 

Speakers

10h – Miguel Sánchez Caja (Granada) Recent interrelated progress in Lorentzian, Finslerian and Riemannian geometry

Abstract.    Recently, a correspondence between the conformal structure of a class of Lorentzian manifolds (stationary spacetimes) and the geometry of a class of Finsler manifolds (Randers spaces) has been developed. This correspondence is useful in both directions. On one hand, it allows a sharp description of geometric elements on stationary spacetimes in terms of Finsler geometry. On the other hand, the geometry of spacetimes suggests, both, new geometric elements and new results, for any Finsler manifold, including the Riemannian case. Here, three levels of this correspondence will be explained: (1) Causal structure of spacetimes: properties of Finslerian distances: 0903.3501.  (2) Visibility and gravitational lensing: convexity of Finsler hypersurfaces: 1112.3892, 0911.0360. (3) Causal boundaries: Cauchy, Gromov, and Busemann boundaries in Riemannian and Finslerian settings: arXiv:1011.1154.

11h30 Vladimir Matveev (Jena) Geodesic degree of mobility of Lorentzian metrics

Abstract. The degree of mobility of a metric can be defined as the dimension of the space of solutions of a certain linear PDE system of finite type whose coefficients depend on the metric, and, for a given metric, there are standard algorithms to determine it. The standard algorithms strongly depend on the metric and in most cases it is possible to find the maximal and sub-maximal values of the degree of mobility, only. I will show that the degree of mobility of a manifold is closely related to the space of parallel symmetric tensor fields on the cone over the manifold. In the case the metric is Einstein, it is essentially the tractor cone. I will use it to describe all possible values of the degree of mobility (on a simply connected manifold) for Riemannian and Lorentzian metrics. I will also consider the case when the metric is Einstein and, as a by-product, solve the classical Weyl-Petrov-Ehlers conjecture, and also show applications. Most these results are based on joint projects with  A. Fedorova and S. Rosemann.

14h30 Philippe LeFloch (Paris)  Injectivity radius and canonical foliations of Einstein spacetimes

Abstract. I will discuss recent results on the local geometry of spacetimes with low regularity, when no assumption on the derivatives of the curvature tensor is made, obtained in collaboration with Bing-Long. Chen. Specifically, I will establish that, under geometric bounds on the curvature and injectivity radius, only, there exist local foliations by CMC (constant mean curvature) hypersurfaces, as well as CMC–harmonic coordinates. Importantly, these coordinates are defined in geodesic balls whose radii depend on the assumed bounds, only, and the components of the Lorentzian metric have the best possible regularity.

14h45 Mehdi Belraouti (Avignon) Asymptotic behavior of level sets of a convex time function

15h00 Ghani Zeghib (Lyon) Actions on the circle and isometry groups of globally hyperbolic Lorentz surfaces (after D. Monclair)

Abstract. Let M be a globally hyperbolic spatially compact spacetime with dimension 1+1. A Cauchy surface in it is diffeomorphic to the circle and, more canonically, its family of lightlike geodesics is diffeomorphic to two copies of the circle and, under mild conditions, M embeds as an open set of the 2-torus.  The isometry group G of M acts naturally on these circles, so that G is a subgroup of Diff(S1). We will establish here that G tends to be included in PSL(2, R), the group of projective transformations of the circle S1, up to a global conjugacy by an element of the circle.

15h30 Eduardo Garcia-Rio (Santiago de Compostela) Quasi-Einstein and Ricci soliton Lorentzian metrics

Abstract. Quasi-Einstein metrics are natural generalizations of Einstein metrics and gradient Ricci solitons.  Moreover, they are closely related to the existence of warped product Einstein metrics. Such metrics are defined by an overdetermined equation involving the Ricci curvature and the Hessian of a potential function. I will present some results on the geometry of Lorentzian quasi-Einstein metrics by focusing primary on those which are locally conformally flat. In this setting the Ricci curvature determines the whole curvature tensor and thus the different possibilities depend on the geometry of the level sets of the potential function: warped product metrics and pp-waves appear in a natural way.

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Seminar on

Mathematical General Relativity

Organizers: 

 S. Klainerman (Princeton)

P.G. LeFloch (Paris)

A. Zeghib (Lyon)

Fondations des Sciences Mathématiques de Paris

ANR Project

“Mathematical General Relativity. Analysis and geometry of spacetimes with low regularity”

Thursday June 14, 2012

Laboratoire J-L Lions

Université Pierre et Marie Curie, Paris

Lecture room 15-25 104

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Jean-Philippe Nicolas (Brest) Perspectives in conformal scattering

Abstract. The origins of conformal scattering are to be found in a paper by Friedlander in 1980 “Radiation fields and hyperbolic scattering theory”, in which he realized that the Lax-Phillips theory was in fact providing an interpretation of scattering theory as the well-posedness of the characteristic Cauchy problem for the conformally rescaled wave-equation on null infinity. He clearly saw that the method provided an interesting geometrical short-cut to define a scattering operator with the advantage that all the analytical structure can be recovered a posteriori. The true power of the conformal approach to scattering theory lies in its complete indifference to time dependence. This talk will review the essential features of Lax-Phillips theory and its intimate link with conformal infinity via the Radon transform and the Whittaker formula, then describe the pinciples of conformal scattering with the crucial importance of the precise resolution of the Goursat problem; we will present some results (actual scattering constructions and studies of the Goursat problem) and explain the necessary steps of the extension of the method to black hole spacetimes, which is currently under development.

Jérémie Szeftel (Paris) The bounded L2 curvature conjecture in general relativity

Abstract.  In order to control locally a spacetime which satisfies the Einstein equations, what are the minimal assumptions one should make on its curvature tensor? The bounded L2 curvature conjecture roughly asserts that one should only need an L2 bound on the curvature tensor on a given space-like hypersurface. This conjecture has its roots in the remarkable developments of the last twenty years centered around the issue of optimal well-posedness for nonlinear wave equations. In this context, a corresponding conjecture for nonlinear wave equations cannot hold, unless the nonlinearity has a very special nonlinear structure. I will present the proof of this conjecture, which sheds light on the specific null structure of the Einstein equations. This is joint work with S. Klainerman and I. Rodnianski.

Thierry Barbot (Avignon) Particles in flat spacetimes in expansion

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Seminar on

Mathematical General Relativity

Organizers: 

 S. Klainerman (Princeton)

P.G. LeFloch (Paris)

A. Zeghib (Lyon)

Fondations des Sciences Mathématiques de Paris

ANR Project

“Mathematical General Relativity. Analysis and geometry of spacetimes with low regularity”

Thursday May 31, 2012

Laboratoire J-L Lions

Université Pierre et Marie Curie, Paris

Lecture room 15-25 104

2 pm Pieter Blue   (Edinburg)

Decay for the Maxwell field outside a Kerr black hole

Abstract.  This talk will repeat some of the material from last year on the same topic (January 12, 2011) and present some new results. The goal of this talk is to prove uniform energy bounds and Morawetz (integrated decay) estimates.  In the exterior of a Kerr black hole, one of the components of the Maxwell system satisfies a wave equation with a complex potential. Trapping and the complex potential interact to provide surprisingly difficult challenges. Pseudodifferential techniques can treat a model problem with both features. However, because of the structure of the original Maxwell system, a new idea suggests classical derivatives alone should be sufficient.

3:30 pm Jonathan Kommemi (Cambridge) 

Global structure of spherically symmetric spacetimes

Abstract. At the heart of the (weak and strong) cosmic censorship conjectures is a statement regarding singularity formation in general relativity. Even in spherical symmetry, cosmic censorship seems, at the moment, mathematically intractable. To give a framework in which to address these very difficult problems, we will introduce a notion of spherically symmetric ‘strongly tame’ Einstein-matter models, an example of which is given by Einstein-Maxwell-Klein-Gordon (self-gravitating charged scalar fields). We will demonstrate that for any ‘strongly tame’ model there is an a priori characterization of the spacetime boundary. In particular, for any ‘strongly tame’ Einstein-matter model, a ‘first singularity’ must emanate from a spacetime boundary to which the area-radius r extends continuously to zero.