# 85`eme > Rencontre entre Mathématiciens et

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85`eme > Rencontre entre Mathématiciens et

85ème <<RCP>> Rencontre entre Mathématiciens et Physiciens “Geometric and Probabilistic aspects of General Relativity” I.R.M.A. - Université de Strasbourg - 10,11,12 Juin 2010 Organizers : Jacques FRANCHI , Athanase PAPADOPOULOS Dates : from Thursday 10-6-2010 (9:00) to Saturday 12-6-2010 (11:30) Conferences : - Ismael BAILLEUL (Cambridge) : Lifetime of relativistic diffusions ; - Mauro CARFORA (Pavia) : Ricci flow and Einstein Equations ; - Christos CHARMOUSSIS (Paris Sud) : Higher order gravity theories and their geometric origin ; - Fabrice DEBBASCH (Paris 6) : Averaging classical relativistic gravitation ; - Nadav DRUKKER (Berlin) : Dualities of supersymmetric field theories, curves on Riemann surfaces and Dehn’s theorem ; - Helmut FRIEDRICH (Golm-Potsdam) : On the asymptotic structure of gravitational fields ; - Hubert GOENNER (Göttingen) : Probability and geometry : A poor man’s view ; - Mark HEINZLE (Vienna) : Probab. aspects of spacelike singularities in General Relativity ; - Jérôme MARTIN (IAP, Paris) : Inflation: theoretical aspects and observational status ; - Vladimir MATVEEV (Jena) : Can one reconstruct a metric by unparameterized geodesics ? - Niall Ó MURCHADHA (Cork) : Gravity on conformal superspace ; - Ramesh SHARMA (New Haven) : Weyl Curvature and Spatial Isotropy of Synchronous Spacetimes ; Abstracts : M. CARFORA : Ricci flow and Einstein Equations Ricci flow theory has a number of applications in general relativity, ranging from deformations of initial data sets for Einstein equations to black hole physics. In this talk I review, at an introductory level, some aspects of such an interaction with emphasis on open mathematical and physical problems. C. CHARMOUSSIS : Higher order gravity theories and their geometric origin We will review gravity theories stemming from Lovelock’s theorem. We will study their basic properties and discuss their geometric origin. We will then discuss Birkhoff’s theorem in this context and give the relavant black hole and soliton solutions. We will apply these backgrounds and to braneworld cosmology in order to find the modified Friedmann equations for a codimension 2 braneworld. F. DEBBASCH : Averaging classical relativistic gravitation General Relativity is a non linear theory of non quantum gravitation. Averaging a relativistic gravitational field is thus a highly non trivial operation. I will start by presenting the basics of the first general mean field theory of classical relativistic gravitation, proposed in 2004, and I will go on by adressing applications to black hole physics and cosmology. In particular, I will show that at least part of black hole thermodynamics can thus be understood in a completely novel manner and that the theory also sheds new light on the cosmological back reaction problem. H. GOENNER : Probability and geometry : A poor man’s view The concepts of probability and geometry meet in many areas of physics and mathematics ; their range extends from the statistical mechanics of black holes to stochastic analysis on path spaces. The focus of my review talk is on relativistic diffusion, Brownian motion and kinetic theory. While the emphasis is put on physical applications, I will try to also describe some of the conceptual and technical generalizations investigated by mathematicians. V. MATVEEV : Can one reconstruct a metric by unparameterized geodesics ? I will explain that certain astronomic observations allow to determine unparameterized geodesics of the space-time metric only. This naturally leads to the following mathematical problem explicitly asked by Weyl and Ehlers: how to determine the metric by unparameterized geodesics.I will show that generally the problem is not solvable (by showing examples of Lagrange and Levi-Civita of two different metrics with the same geodesics). The main mathematical theorem of my talk (I will give a rigorous proof) will be that 4D Einstein metrics of non constant curvature are geodesically rigid, in the sense that unparameterized geodesics determine such metrics uniquely. This result is joint with V. Kiosak. I will also explain that in the generic situation unparameterized geodesics determine the metric, and give a algorithm how to reconstruct a Ricci-flat 4D metric by unparameterized geodesics. In the rest of my talk I discuss metrics with other stress-energy tensor. N. Ó MURCHADHA : Gravity on Conformal Superspace The standard method of solving the Einstein constraints on a compact manifold involves specifying a base 3-metric, a TT (transverse-tracefree) metric relative to this metric, and a constant, which is the trace of the extrinsic curvature. These are the input into the Lichnerowicz-York (L -Y) equation, a nice elliptic equation, the solution of which is a conformal factor which is used to construct a CMC (constant mean curvature) solution to the Einstein constraints. This structure is conformally covariant, so the base 3-metric can be regarded as a representative of a conformal 3-geometry. A count of the degrees of freedom gives 2 per space point in the conformal 3-geometry, 2 per space point in the TT tensor (which can be regarded as a velocity of the conformal geometry), and one extra constant ( the value of the trace of K). A newly discovered symmetry of the L-Y equation allows us to remove this Trace K degree of freedom, thus returning the degrees of freedom to exactly the expected 2 + 2 per space point. Therefore, given a point and a velocity in conformal superspace, the Einstein equations generate a unique curve in conformal superspace. R. SHARMA : Weyl Curvature and Spatial Isotropy of Synchronous Spacetimes We consider synchronous space-time cosmological models (M,g) with space-like slices having pure trace extrinsic curvature and show that (i) the electric part of the Weyl tensor of g vanishes if and only if each slice is Einstein, and (ii) the full Weyl curvature of g vanishes if and only if each slice has constant curvature, i.e. (M,g) is spatially isotropic. We also indicate a relationship of the result (ii) with Penroses’s Weyl curvature hypothesis. PROGRAM Thursday 10-6-2010 09:00 : H. GOENNER : Probability and geometry : A poor man’s view 10:00 : Tea and coffee break 10:30 : M. CARFORA : Ricci flow and Einstein Equations 11:30 : H. FRIEDRICH : On the asymptotic structure of gravitational fields 12:30 : Free time for lunch 14:30 : M. HEINZLE : Probabilistic aspects of spacelike singularities in General Relativity 15:30 : F. DEBBASCH : Averaging classical relativistic gravitation 16:30 : Tea and coffee break 17:00 : I. BAILLEUL : Lifetime of relativistic diffusions 19:30 : Dinner at the Restaurant “le petit bois vert”, 2 quai de la Bruche, Strasbourg ; offered to the participants Friday 11-6-2010 09:00 : R. SHARMA : Weyl Curvature and Spatial Isotropy of Synchronous Spacetimes 10:00 : Tea and coffee break 10:30 : N. Ó MURCHADHA : Gravity on conformal superspace 11:30 : J. MARTIN : Inflation: theoretical aspects and observational status 12:30 : Free time for lunch 14:00 : N. DRUKKER : Dualities of supersymmetric field theories, curves on Riemann surfaces and Dehn’s theorem 16:00 : Departure for touristic boat tour around the old city (offered to the participants) 17:30 : Reception-appetizer at the Mairie (City Hall), place Broglie (offered to participants) Saturday 12-6-2010 09:00 : Coffee & tea 09:30 : C. CHARMOUSSIS : Higher order gravity theories and their geometric origin 10:30 : V. MATVEEV : Can one reconstruct a metric by unparameterized geodesics ? J. Franchi , A. Papadopoulos