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F. Debbasch (LERMA-ERGA Université Paris 6) and M. Bustamante, C. Chevalier, Y. Ollivier Statistical Physics and relativistic gravity ( )

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1.Why extend conventional statistical physics to include the gravitational field? 2.Standard mean field theories of Newtonian gravity and Maxwell electromagnetism 3.A new mean field theory for relativistic gravitation 4.Classical black hole thermodynamics 5.Statistical ensembles of classical black holes 6.Conclusion

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To achieve a fully consistent treatment of non- quantal self-gravitating systems To properly take into account the finite resolutions with which many astrophysical and cosmological observations are carried out Because black hole thermodynamics suggests an intimate link between relativistic gravity, statistical physics, and quantum physics Because it is a first step towards a better understanding of quantum fluctuations of the space- time geometry Why extend conventional statistical physics to include the gravitational field?

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Gravitation is encoded in a potential (t, r) The potential is generated by the mass distribution: = - 4 G The trajectory of a point mass is modified by gravity: Standard mean field theories of Newtonian gravity and Maxwell electromagnetism 1. Newtonian gravity: Non statistical treatment drdr dt m = p dpdp dt = - grad = F

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Statistical ensemble of self-consistent potentials (t, r, ) and mass densities (t, r, ) For each , ( ) = - 4 G ( ) For each , The mean gravitational field is described by the potential Standard mean field theories of Newtonian gravity and Maxwell electromagnetism 2. Newtonian gravity: statistical treatment dpdp dt ()() = - grad ( ) = F ( ) (t, r) = where = mean value over

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Since all equations are linear: with and Standard mean field theories of Newtonian gravity and Maxwell electromagnetism 2. Newtonian gravity: statistical treatment dpdp dt ()() = - grad = F (t, r) = - 4 G (t, r) = < > for any test-mass located at point r at time t t, r

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Standard mean field theories of Newtonian gravity and Maxwell electromagnetism A ( ), F ( ) = ∂ A ( ) - ∂ A ( ), j ( ) and, for all : ∂ F ( ) = - 4 j 3. Maxwell electromagnetism: statistical treatment dp ds = q F ( ) u ()()

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Standard mean field theories of Newtonian gravity and Maxwell electromagnetism For all (t, r) = x, A (x) =, F (x) = j (x) = ∂ F = - 4 j 3. Maxwell electromagnetism: statistical treatment dp ds = = q F (x) u ()() < > x,u for any charged test particle situated at point x with velocity u

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A general relativistic space-time is equipped with two different structures, a metric g and a connection or covariant derivative operator Given a coordinate system, the connection is represented by a set of position-dependent numbers The `source’ of gravity is energy-momentum, represented by a stress-energy tensor field T A new mean field theory for relativistic gravitation

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Choose a fixed base manifold M Consider a statistical ensemble of metrics g( ), connections ( ) and stress-energy tensor fields T( ) defined on M Have these fields satisfy, for each , the equations of general relativity: A new mean field theory for relativistic gravitation

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( ) g ( ) = 0 (1) G ( ( ), g( )) = g ( ) g ( ) T ( ) (2) (1) ( ) is entirely determined by g( ) (Levi- Civita connection) The ( )’s are the Christoffel symbols of g( ); they depend non linearly on g( ) Equation (2) is also non linear in g( ) (at fixed T( ), which generally also depends on g( )) A new mean field theory for relativistic gravitation

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Problem: Define a mean metric, a mean connection and a mean stress-energy tensor that satisfy Einstein’s theory Theorem: In general, the connection represented by the coefficients is not compatible with any metric Consequence : The only natural and simple solution seems to be: g (x) = and = Levi-Civita connection of g Is this definition physically reasonable? A new mean field theory for relativistic gravitation

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The motion of test point masses is governed by the geodesics equation: Therefore dp d = 1 2 ∂ g ( ) v v with v = ()() dx d x, v x dp d x, v ()() 1 2 ∂ g v v < > = x A new mean field theory for relativistic gravitation

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Einstein equation then fixes the stress-energy tensor field of the mean space-time: G (, g) = g g T In general, T is different from The difference describes how the small-scale fluctuations of the gravitational field add up as an effective `source’ to the large-scale averaged field A new mean field theory for relativistic gravitation

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A few general remarks: The theory can accomodate other gauge fields, for example the Maxwell field The apparent large-scale values of the gauge charge densities depend on the `real’ small-scale values of these charges and on the small-scale fluctuations of the gravitational field The theory may have important cosmological consequences, but only if non linearity turns out to play an important role on cosmological scales A new mean field theory for relativistic gravitation

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A black hole is a space-time with a future event horizon. An event horizon is a null surface and, therefore, can only be crossed in one direction (`no escape out of the region inside the horizon’) by non quantum matter All stationary black hole solutions of the vacuum Einstein-Maxwell equations are known These solutions depend on three parameters only: their masses M, their charges q and their angular momenta J= j M. They describe a black hole if (j/j P ) 2 + (q/q P ) 2 < (M/m P ) 2 Classical black hole thermodynamics

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Quantum field theory in curved space-times has revealed that all stationary black holes exhibit thermodynamical properties, i.e. they have an entropy and (generically) a temperature One can write dM = TdS + Vdq + dJ, S(M, q, J), T(M, q, J), etc… For a Schwarzschild black hole: q = 0, J = 0, V = 0, = 0 and R H = 2 M (l P /m P ) T = (1/8 M) m P T P S = 4 M 2 /m P 2

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Quite generally, in Planck units 2 T (M, q, j) = /[2M(M + ) - q 2 ] and S(M, q, j) = (M + ) 2 with (M, q, j) = (M 2 - j 2 - q 2 ) 1/2 (with j = J/M) Extreme black holes correspond to = 0 They have no temperature but a non vanishing entropy Note that S is not extensive and T is not intensive: S( M, q, j) = 2 S(M, q, j) T( M, q, j) = T(M, q, j) 1 Classical black hole thermodynamics

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Classical black holes are thus to be considered as statistical ensembles The fact that their entropies and temperatures involve h (via m P ) suggests that the involved `microscopic’ degrees of freedom partake of quantum gravity Indeed, it is possible to derive the values of black hole entropies by considering them as microcanonical ensembles of strings/branes Classical black hole thermodynamics

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1. A toy model for finite precision observations of a Schwarzschild black hole Let R 4 = (t, r) be the base manifold and consider (t, r) as Kerr- Schild coordinates for the usual Schwarzschild space-time, associated to metric coefficients g (t, r) Consider the statistical ensemble of metrics g( ) defined by g (t, r, ) = g (t, r + ) where 3-ball of radius a and p( ) uniform Remark: Typical current observations of Sgr A* are associated to x = (a/M) (m P /l P ) ~ 400. Near future ones (10 years) should have x ~ a few units For all x < 2: Statistical ensembles of classical black holes

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1. A toy model for finite precision observations of a Schwarzschild black hole The mean space-time still describes a black hole The horizon radius of this black hole reads R H (M, x) = 2M (1+ x 2 /20) 1/2 The total energy of the mean space-time is still M but The stress-energy tensor field does not vanish outside the horizon and decreases as R -6 at infinity. It describes apparent matter of negative energy density. The radial pressure is negative and the angular pressures are identical and positive. All energy conditions are violated, yet The mean space-time is thermal Statistical ensembles of classical black holes

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1. A toy model for finite precision observations of a Schwarzschild black hole T(M, x) ~ (1/8 M) (1 + x 2 /20) for small x S(M, x) ~ 4 M 2 (1 + x 4 /800) The mean space-time is governed by a two-parameter thermodynamics. Natural choices are (T, R H ) and (T, A H ) S(T/ , R H ) = 2 S(T, R H ) but M(T/ , R H ) = M(T, R H ) dM = T dS + f dR H = TdS + dA H f ~ - x 2 /40, ~ - x 2 /(640 M) Thus, a standard averaging over the non quantum space-time degrees of freedom modifies the black hole thermodynamics Statistical ensembles of classical black holes

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2. Generating thermal space-times by averaging non thermal solutions of the classical Einstein equations Some ensembles of classical extreme black holes, analytically extended into complex space-times, can be interpreted as finite temperature real black holes The finite temperature apparently traces a `Zitter-Bewegung’ in at least some of the complex degrees of freedom The calculations permit, in theory, the identification of the system’s proper modes Statistical ensembles of classical black holes

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Conclusion The apparent properties of astrophysical objects depend on the finite resolutions with which they are observed Systematic biases in cosmology due to large-scale averagings? Possible links with the dark matter problem? What do the calculations performed on classical black holes tell us about quantum gravity? Link with superstring theory? Link with spinor and/or twistor formalism? Link with emergent gravity? `Genericity’ of black hole thermodynamics

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Conclusion Links with Galilean hydrodynamics, solid state physics/elasticity, superconductivity and optics The propagation of acoustic waves (phonons) in potential flows of Galilean barotropic fluids is governed by an acoustic Lorentzian metric Because flows can get supersonic, acoustic waves can get trapped in some regions of space called dumb holes A surface `gravity’ can be associated to the boundary of the dumb holes What is the hydrodynamical equivalent of the classical averaging procedure and of the black hole calculations presented here? Link with the general investigation of real and complex singularities in Euler flows?

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l P 2 = G h/2 c 3 m P = l P c 2 /G q P = l P c/G 1/2 (c.g.s. units)

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