July 2005 Einstein Conference Paris Thermodynamics of a Schwarzschild black hole observed with finite precision C. Chevalier 1, F. Debbasch 1, M. Bustamante 2 and Y. Ollivier 3 1 ERGA-LERMA, 2 LPS ENS Paris, 3 UMA ENS Lyon, France Albert Einstein Century International Conference 2005 July 20th, 2005, UNESCO, Paris

July 2005 Einstein Conference Paris Why a mean field theory for GR ? Observations and experiments are necessarily performed with finite precision measurements → access to mean fields only. Mean field theories produce unexpected effects in the case of non-linear theories, such as the eddy viscosity in hydrodynamics (Frisch & Dubrulle 1991). Construction of a general mean field theory for relativistic gravitation ( Debbasch ’03/’04/’05) → the separation between matter and gravitational field is scale-dependent. Main idea taken up by Kolb et al. (’05), for perturbative cosmological applications. Presentation of a non-perturbative astrophysical application : characterization of the effects of finite precision measurements on the study of a Schwarzschild BH.

July 2005 Einstein Conference Paris A mean field theory for GR (Debbasch ‘03,’04, ‘05) Non –linearity of Einstein’s equations  How are the mean physical quantities related to the averaged ones ?  Which equations do they verify ? Σ = Statistical ensemble of space-times labelled by the statistical variable ω Mean Space-time

July 2005 Einstein Conference Paris Mean gravitational field in GR  Mean metric : Study of the average motion of test point particles in space-time → collection of geodesics belonging to the various space- times of the statistical ensemble averages into a geodesic of the mean metric defined by :  Mean connection :  Mean stress-energy tensor : (Debbasch ’03,’04, ‘05)

July 2005 Einstein Conference Paris Finite resolution ↔ « average over r » → technically : averaging the ω - dependent metric : at fixed r over with the probability measure on Ω Schwarzschild metric in Kerr-Schild coord. : A Schwarzschild BH observed with finite precision Final expression of the mean metric, valid for r > a : (Debbasch & Ollivier ’05) → Mean metric :  description of the Shwarzschild BH as observed with a finite precision measurement of order a in the spatial Kerr-Schild coordinates.

July 2005 Einstein Conference Paris Horizon and temperature of the coarse-grained ST  Temperature : Analytic extension of the mean ST → Euclidean mean space-time.  Euclidean mean ST is naturally periodic of period β in imaginary time : x = a/M << 1 T vs. x ( 0 < x < 2 ) The mean space-time describes a BH with a bifurcate Killing horizon R = R H. R H vs. x ( 0 < x < 2 )  Horizon : F(R) and 1/G(R) have a common zero : All calculations carried out under the assumption a < 2M

July 2005 Einstein Conference Paris Non vanishing mean stress energy tensor :  The coarse-grained BH is not a vacuum solution of Einstein’s equation.  It appears as surrounded by an effective matter of negative energy density. Apparent matter (Debbasch & Ollivier ’05)

July 2005 Einstein Conference Paris  For 0 < x = a/M < 2 : → Mass measured at infinity is not changed by the coarse-graining. Mass of the mean ST Energy of the mean ST.  Komar formula : Gravitational mass affecting a distant object orbiting the BH in the Newtonian limit. E surf,, E vol and E vs. x ( 0 < x < 2 )

July 2005 Einstein Conference Paris Entropy S(R H, T) : When x = a/M << 1 : Calorific Capacity and Entropy of the mean ST 2 independent parameters : M and a → ( x, M ), ( R H, T ) or ( x, T ) = couples of thermodynamical variables. Calorific Capacity at constant R H : S vs. x ( 0 < x < 2 )

July 2005 Einstein Conference Paris Summary : - Construction of a mean field theory for GR. - Consequences of observing a vacuum Schw. BH with finite precision : → BH surrounded by matter. → Mass M remains the same. Conclusions and perspectives Mean field theory and black holes : – Application of the mean field formalism to more general BH : Kerr BH, Reissner-Nordström BH or Kerr-Newmann. → Possible interpretations of astrophysical BH observations. Mean field theory and cosmology : – A mean field theory for cosmology ? – Possible contribution to dark energy ? The coarse-graining changes repartition of energy in space-time but the total mass is not modified.