Seminar on Mathematical General Relativity — June 14, 2012

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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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11:00 am 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.

2:00 pm Jérémie Szeftel (Paris)

Abstract.  TBA

3:30 pm Thierry Barbot (Avignon) Particles in flat spacetimes in expansion

Seminar on Mathematical General Relativity — May 31, 2012

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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 TBA

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.

Seminar on Mathematical General Relativity — May 24, 2012

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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.