Category Archive

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Seminar on

Mathematical General Relativity

Organizers:

 Philippe G. LeFloch (Paris)

Jérémie Szeftel (Paris) 

Ghani Zeghib (Lyon)

ANR Project

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

May 28, 2014

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

Lecture room  1525-321

14h Erwann Delay (Avignon) A study of some curvature operators near the Euclidian metric

Abstract. We will show that some curvature operators of Ricci (or Einstein) type are locally invertible, in some weighted Sobolev spaces on Rn, near the euclidian metric. In the smooth case, we then deduce that the image of some Riemann-Christoffel type operators are smooth submanifolds in the neighborhood of the Euclidian metric.

15h30 Mahir Hadzic (London) Stability problem in the dust-Einstein system with a positive cosmological constant

Abstract. The dust-Einstein system models the evolution of a spacetime containing a pressureless fluid, i.e. dust. We will show nonlinear stability of the well-known Friedman-Lemaitre-Robertson-Walker (FLRW) family of solutions to the dust-Einstein system with positive cosmological constant. FLRW solutions represent initially a quiet fluid evolving in a spacetime undergoing accelerated expansion. We work in a harmonic-type coordinate system, inspired by prior works of Rodnianski and Speck on Euler-Einstein system, and Ringstrom’s work on the Einstein-scalar-field system. The main new mathematical difficulty is the additional loss of one degree of differentiability of the dust matter. To deal with this degeneracy, we commute the equations with a well-chosen differential operator and derive a family of elliptic estimates to complement the high-order energy estimates. This is joint work with Jared Speck.

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Seminar on

Mathematical General Relativity

Organizers:

 Philippe G. LeFloch (Paris)

Jérémie Szeftel (Paris) 

Ghani Zeghib (Lyon)

___________________________________________________________________________________________________________________________________________

ANR Project

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

February 12, 2014

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

Lecture room  1525-103

14h  Florian Beyer  (Dunedin)  Graceful exit from inflation for minimally coupled Bianchi A scalar field models

Abstract. We consider the dynamics of Bianchi A scalar field models which undergo inflation. The main question is under which conditions does inflation come to an end and is succeeded by a decelerated epoch. This so-called ‘graceful exit’ from inflation is an important ingredient in the standard model of cosmology, but is, at this stage, only understood for restricted classes of solutions. We present new results obtained by a combination of analytical and numerical techniques.

15h30  Cécile Huneau (ENS, Paris)  Vacuum constraint equations for asymptotically flat space-times with a translational Killing field

Abstract. In the presence of a space-like translational Killing field, vacuum Einstein equations in 3+1 dimensions reduces to 2+1 Einstein equations with a scalar field. Minkowski space-time is a trivial solution of vacuum Einstein equation with a translational Killing field. A natural question is therefore the nonlinear stability of Minkowski solution in this setting. A first step in addressing this problem is the study of the constraint equations. The main difficulty in that case is due to the delicate inversion of the Laplacian. In particular, we have to work in the non constant mean curvature setting, which enforces us to consider the intricate coupling of the Einstein constraint equations.

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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”

February 20, 2013

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

Lecture room  1525-103

14h  Florian Beyer  (Dunedin)  Asymptotics and conformal structures of solutions to Einstein’s field equations

Abstract. Roger Penrose’s idea that the essential information about the asymptotics of solutions of the Einstein’s field equations is contained in the conformal structure and the associated conformal boundary has led to astonishing successes. In his original work, he provided several examples which made the importance of his idea evident. However, the question whether general solutions of Einstein’s field equations are compatible with this proposal remained unanswered. Motived by this, Helmut Friedrich has initiated a research programme to tackle this problem based on his so-called conformal field equations. In this talk I report on the status of this work and some of Friedrich’s results, but also on joint work with  collaborators at the University of Otago.

15h30  Julien Cortier (IHES, Bures-sur-Yvette)  On the mass of asymptotically hyperbolic manifolds

Abstract. By analogy with the ADM mass of asymptotically Euclidean manifolds, a set of global charges can be defined for asymptotically hyperbolic manifolds. We will review their various definitions and , in particular, focus on the notion of “mass aspect” tensor, which gives rise to the  energy-momentum vector and arises  in the hyperbolic formulation of the positive mass theorem. We will compute these quantities for examples such that the Schwarzschild-anti de Sitter metrics, and we will present a family of counter-examples with “non-positive” mass when completeness is not assumed.