Category Archive

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Seminar on

Mathematical General Relativity

Organizers: Sergiu Klainerman (Princeton)

Philippe G. LeFloch (Univ. Pierre et Marie Curie) Gabriele Veneziano (Collège de France)

With the financial support of  the Fondation des Sciences Mathématiques de Paris

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Monday March 7, 2011

Laboratoire Jacques-Louis Lions

Université Pierre et Marie Curie

Lecture rooms: 15/25  1-03 (morning) and 1-02 (afternoon)

11h   Jacques Smulevici (AEI, Potsdam)  Weakly regular T2 symmetric spacetimes

Abstract. Under weak regularity assumptions, only, we develop a fully geometric theory of vacuum Einstein spacetimes with T2–symmetry, establish the global well-posedness of the initial value problem for Einstein’s field equations, and investigate the global causal structure of the constructed spacetimes. Our weak regularity assumptions are the minimal ones allowing to give a meaning to the Einstein equations under the assumed symmetry and to solve the initial value problem. First of all, we introduce a frame adapted to the symmetry in which each Christoffel symbol can be checked to belong to Lp for some p. We identify certain cancellation properties taking place in the expression of the Riemann and Ricci curvatures, and this leads us to a reformulation of the initial value problem for the Einstein field equations when the initial data set has weak regularity. Second, we investigate the future development of a weakly regular initial data set. We check that the area R of the orbits of symmetry must grow to infinity in the future timelike directions, and we establish the existence of a global foliation by the level sets of the function R. Our weak regularity assumptions only require that R is Lipschitz continuous while the metric coefficients describing the initial geometry of the orbits of symmetry are in the Sobolev space H1 and the remaining coefficients have even weaker regularity. We develop here the compactness arguments required to cover the natural level of regularity associated with the energy of the system of partial differential equations determined from Einstein’s field equations. This is a joint work in collaboration with P.G. LeFloch (Paris).

14h   Erwann Delay (Univ. Avignon) Recollement des TT tenseurs

Abstract. La méthode de Corvino-Schoen permet de recoller localement et de façon lisse deux solutions des équations de contraintes. Nous verrons que l’on peut d’une façon générale recoller deux éléments du noyau de certains opérateurs à symbole surjectif et non injectif. Par exemple on peut recoller deux champs de vecteurs à divergence nulle ou deux TT-tenseurs. On en déduit par exemple que sur toute boule riemannienne ouverte, l’ensemble des TT-tenseurs lisses à support compact est de dimension infinie. On simplifie ainsi la construction de données initiales CMC.

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Seminar on

Mathematical General Relativity

Organizers: Sergiu Klainerman (Princeton)

Philippe G. LeFloch (Univ. Pierre et Marie Curie) Gabriele Veneziano (Collège de France)

With the financial support of  the Fondation des Sciences Mathématiques de Paris

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Monday February 14, 2011

Institut Henri Poincaré

Lecture room 01

 

11h – Qian Wang (IHES & Albert Einstein Institute, Potsdam) Breakdown criterion for Einstein Vacuum equations in CMC gauge

Abstract. I will report my recent work on the geometric criterion for the breakdown of Einstein vacuum space-times with the constant mean curvature (CMC) foliation. In this work, the criterion is formulated in terms of time-integrability of the sup- norms of the second fundamental form and derivatives of the lapse function associated to CMC foliation of the space-time. This result is obtained through deriving uniform lower bound on the null radius of injectivity of the light cones. I will sketch the main idea of the proof.

14h – Piotr Chrusciel (Vienna) On the general relativistic constraint equations

Abstract. I will present some new results with Rafe Mazzeo on the construction of solutions of the constraint equations with cylindrical ends.

 

 

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Seminar on

Mathematical General Relativity

Organizers: Sergiu Klainerman (Princeton)

Philippe G. LeFloch (Univ. Pierre et Marie Curie) Gabriele Veneziano (Collège de France)

With the financial support of  the Fondation des Sciences Mathématiques de Paris

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Monday January 31rst, 2011

Institut Henri Poincaré

Lecture room 314

 

11h   Lydia Bieri (Ann Arbor), Geometry of Spacetimes Solving the Einstein-Maxwell Equations in General Relativity and Gravitational Radiation

Abstract. A major goal of mathematical General Relativity (GR) and astrophysics is to precisely describe and finally observe gravitational radiation, one of the predictions of GR. In order to do so, one has to study the null asymptotical limits of the spacetimes for typical sources. Among the latter we find binary neutron stars and binary black hole mergers. In these processes typically mass and momenta are radiated away in form of gravitational waves. D. Christodoulou showed that every gravitational-wave burst has a nonlinear memory. In this talk, we discuss the null asymptotics for spacetimes solving the Einstein-Maxwell (EM) equations, compute the radiated energy and derive limits at null infinity and compare them with the Einstein vacuum (EV) case. The physical insights are based on geometric-analytic investigations of the solution spacetimes.

14h   Jérémie Szeftel (ENS, Paris) Around the bounded L2 curvature conjecture in general relativity

Abstract. We report on recent progress obtained on the construction and control of a parametrix to the homogeneous wave equation on a curved background, where the metric is a rough solution to the Einstein vacuum equations. Controlling such a parametrix as well as its error term when one only assumes L2 bounds on the curvature tensor of the metric is a major step towards the proof of the bounded L2 curvature conjecture. This is a joint work with Sergiu Klainerman and Igor Rodnianski.