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Thermodynamics of a Schwarzschild black hole observed with finite precision

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.

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Thermodynamics of a Schwarzschild black hole observed with finite precision

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  1. Thermodynamics of a Schwarzschild black hole observed with finite precision C. Chevalier1, F. Debbasch1, M. Bustamante2 and Y. Ollivier3 1ERGA-LERMA, 2LPS ENS Paris, 3UMA ENS Lyon, France Albert Einstein Century International Conference 2005 July 20th, 2005, UNESCO, Paris July 2005 Einstein Conference Paris

  2. 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 theeddy 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

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

  4. Mean gravitational field in GR (Debbasch ’03,’04, ‘05) • 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 : July 2005 Einstein Conference Paris

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

  6. All calculations carried out under the assumption a < 2M • Horizon :F(R) and 1/G(R) have a common zero : The mean space-time describes a BH with a bifurcate Killing horizon R = RH. RH vs. x ( 0 < x < 2 ) 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 βin imaginary time : • x = a/M << 1 T vs. x ( 0 < x < 2 ) July 2005 Einstein Conference Paris

  7. Apparent matter (Debbasch & Ollivier ’05) • Non vanishingmean 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. July 2005 Einstein Conference Paris

  8. 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.  For 0 < x = a/M < 2 : → Mass measured at infinity is not changed by the coarse-graining. Esurf,, Evol and E vs. x ( 0 < x < 2 ) July 2005 Einstein Conference Paris

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

  10. The coarse-graining changes repartition of energy in space-time but the total mass is not modified. Conclusions and perspectives • 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. • 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 ? July 2005 Einstein Conference Paris

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