Mathematical general relativity, compressible fluids, and more

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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.

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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 24, 2012

Laboratoire J-L Lions

Université Pierre et Marie Curie, Paris

Lecture room 15-25 104

11:00 am A. Shadi Tahvildar Zadeh (Rutgers) Zero-gravity limit of Kerr-Newman spacetimes and their electromagnetic fields

Abstract.  We discuss the limit of vanishing G (Newton’s constant of universal gravitation) of the Kerr–Newman electrovacuum spacetimes. We investigate the topologically nontrivial spacetime emerging in this limit and show that it consists of two copies of flat Minkowski spacetime glued at a timelike cylinder. The electromagnetic fields of the Kerr–Newman spacetimes converge to nontrivial solutions of Maxwell’s equations on this background spacetime. We show how to obtain these fields by solving Maxwell’s equations with singular sources supported only on a circle in a spacelike slice of the manifold. These sources do not suffer from any of the pathologies that plague the alternate sources found in previous attempts to interpret the Kerr–Newman fields on the topologically simple Minkowski spacetime.

2:00 pm James Isenberg (Eugene)  AVTD behavior in smooth solutions of Einstein’s equations

Abstract.  One of the more useful approaches to studying the Strong Cosmic Censorship conjecture in a family of solutions of Einstein’s equations is to first verify that generic solutions in that family exhibit AVTD (asymptotically velocity term dominated) behavior near their singular regions. It has been proven (by Ringstrom) that AVTD behavior occurs in generic Gowdy spacetimes, and it has also been shown that it occurs in at least some vacuum spacetimes with T2 isometry, and in some with U(1) isometry. These T2 and U(1) results have been proven using Fuchsian techniques, and have the unfortunate feature that, like many Fuchsian-based results, they require that the spacetimes be analytic. In work done with Florian Beyer, Philippe LeFloch, and Ellery Ames, we show that the analyticity condition can be removed, at least for the T2 case. To prove this result, we have developed a variant of the Fuchsian technique which does not require analyticity. It is very likely that this variant can be applied to U(1) symmetric vacuum spacetimes as well as to those with T2 symmetry.

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Seminar on

Mathematical General Relativity

Organizers:

 Philippe G. LeFloch (Univ. Pierre et Marie Curie)

Ghani Zeghib (Ecole Normale Supérieure, Lyon)

ANR Project

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

Thursday April 5, 2012

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

Lecture room 15-25 101 (first level)

10:00 am Lars Andersson (Potsdam) Hidden symmetries and conserved charges

Abstract.  A proof of decay estimates for test fields with non-zero spin, e.g. Maxwell and linearized gravity, on the Kerr background is an important step towards understanding the black hole stability problem. Fields with non-zero spin on Kerr admit non-radiating modes which must be eliminated in order to prove decay. In this talk I will discuss the relation between conserved charges and hidden symmetries for linearized gravity on Minkowski space and vacuum spaces of Petrov type D, and outline the application of these ideas in proving estimates for the higher spin fields on the Kerr background.

11:30 am  François Béguin (Orsay) On the BKL conjecture for vacuum spatially homogeneous models

Abstract. About forty years ago, Belinski, Kalatnikov, and Lifschitz proposed a rather complete description (based on heuristic arguments) of the asymptotic behavior of spatially homogeneous spacetimes close to their initial singularity. In particular, a “generic” vacuum spatially homogeneous spacetime is expected to exhibit a “chaotic” oscillatory behavior closed to its initial singularity. It was only in 2010 that the first rigorous mathematical results going in this direction were established. I will review here what is known (from a rigorous mathematical viewpoint) on the behavior of vacuum spatially homogeneous spacetimes close to their initial singularity. I will also try to explain what one can (or cannot) hope to prove about this asymptotic behavior and will sketch the proofs.

2:00 pm Simone Calogero (Granada) Dynamics of spatially homogeneous cosmological models

Abstract.  Understanding the asymptotic dynamics of the Universe is one of the main goals of theoretical cosmology. In the context of spatially homogeneous (SH) cosmological models, where the Einstein equations of general relativity reduce to a system of ordinary differential equations, our knowledge on this problem has increased substantially over the years, thanks to the application of methods from the theory of finite dimensional dynamical systems. After reviewing the results known for the dynamics of perfect fluid cosmological models, this talk will focus on the more complicated case of a space-time filled with anisotropic matter. In the latter case the qualitative global behavior of SH solutions depends strongly on the asymptotic values of the principal pressures in the limit toward the singularity. Such behavior has been completely characterized for locally rotationally symmetric (LRS) solutions in the class A of Bianchi models. It has been found that there exist matter models, compatible with the standard energy conditions, for which Bianchi type IX LRS solutions are singularity-free for an initial data set with positive measure, and others for which the approach toward the singularity of generic solutions is oscillatory. This talk is based on a series of works in collaboration with Mark Heinzle.

3:30 pm Ghani Zeghib (Lyon) Projective transformation groups

Abstract.  To pseudo-Riemannian (in particular Riemannian or Lorentzian) metric one naturally associates its Levi-Civita connection. It is also natural to ask whether, conversely, the connection determines the metric, that is, (essentially) whether two metrics having the same geodesics coincide?  More strongly, two metrics are said to be projectively equivalent if their geodesics coincides as geometric non-parametrized curves. One then asks when projectively equivalent metrics coincide? The answer to this classical problem is `no’, but only for very special metrics.  This confirms a real interest for this problem since it allows the emergence of special structures. We will deal here with a parallel (i.e. weaker) rigidity question: classify metrics admitting an essential projective transformation, that is, diffeomorphism preserving, non-parametrized geodesics without being an isometry?