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
from Thursday 10-6-2010 (9:00) to Saturday 12-6-2010 (11:30)
- 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
- 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
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.
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
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)
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