Institut de Recherche Mathématique Avancée, UMR 7501

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10 juin 2010

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09h00

Helmut Goenner - Göttingen

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.

10h00

Tea and coffee break

10h30

Mauro Carfora - Pavia

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.

11h30

Helmut Friedrich - Golm-Potsdam

12h30

Free time for lunch

14h30

Mark Heinzle - Vienna

15h30

Fabrice Debbasch - Paris 6

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.

16h30

Tea and coffee break

17h00

Ismael Bailleul - Cambridge

19h30

Dinner at the Restaurant "Le petit bois vert", 2 quai de la Bruche, Strasbourg ; offered to the participants.

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11 juin 2010

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09h00

Ramesh Sharma - New Haven

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.

10h00

Tea and coffee break

10h30

Niall O’murchadha - Cork

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.

11h30

Jérôme Martin - IAP, Paris

12h30

Free time for lunch

14h00

Nadav Drukker - Berlin

16h00

Departure for touristic boat tour around the old city (offered to the participants)

17h30

Reception-appetizer at the Mairie (City Hall), place Broglie (offered to the participants).

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12 juin 2010

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09h00

Coffee and tea.

09h30

Christos Charmoussis - Paris Sud

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 relevant 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.

10h30

Vladimir Matveev - Jena

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 proofwill be that 4D Einstein metrics of non constant curvature are geodesically rigidin the sense that unparameterized geodesics determine such metrics uniquelyThis result is joint with VKiosakI will also explain that in the generic situation unparameterized geodesics determine the metricand 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.