F. Debbasch (LERMA-ERGA Université Paris 6) and M. Bustamante, C. Chevalier, Y. Ollivier Statistical Physics and relativistic gravity (2003-2005)

F. Debbasch(LERMA-ERGA

Université Paris 6)

and

M. Bustamante, C. Chevalier, Y. Ollivier

Statistical Physics and relativistic gravity (2003-2005)

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

• 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 physicsto include the gravitational field?

€

• 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

drdt

m = p

dpdt = - grad = F

• 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



€

2. Newtonian gravity: statistical treatment

dp

dt() = - grad () = F ()

(t, r) = < (t, r, ) > where <…> = mean value over

Since all equations are linear:

with

and



€

2. Newtonian gravity: statistical treatment

dp

dt() = - grad = F (t, r)

= - 4 G

(t, r) = < (t, r, ) >

< >

for any test-mass located at point r at time t

t, r t, r



• A(), F () = ∂A() - ∂A(), j()

and, for all :

• ∂F () = - 4 j

•

3. Maxwell electromagnetism: statistical treatment

dp

ds= q F() u()



• For all (t, r) = x,

A (x) = < A(x, ) > , F (x) = < F (x, ) > j(x) = < j (x, ) >• ∂F = - 4 j

•

3. Maxwell electromagnetism: statistical treatment

dp

ds= < q F(x, ) u > = q F(x) u ()< >

x,u

for any charged test particle situated at point x with velocity u

• 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

• 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:

€

∇


• () 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())€

∇

€

∇

€

∇


• 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) = < g (x, ) > and = Levi-Civita connection of g• Is this definition physically reasonable?

€

∇


• The motion of test point masses is governed by the geodesics equation:

• Therefore

dp

d= 1

2∂g() vv with v = ()

dx

dx, v x

dp

d x, v

()12

∂g vv< > = x


• 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 < T() > • 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 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 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/jP)2 + (q/qP)2 < (M/mP)2

Classical black hole thermodynamics


•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

RH = 2 M (lP/mP) T = (1/8M) mPTP

S = 4M2/mP2

• Quite generally, in Planck units 2 T (M, q, j) = /[2M(M + ) - q2] and S(M, q, j) = (M + )2

with (M, q, j) = (M2 - j2 - q2)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 holes are thus to be considered as statistical ensembles

• The fact that their entropies and temperatures involve h (via mP) 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


1. A toy model for finite precision observations of a Schwarzschildblack hole

• Let R4 = (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() uniformRemark: Typical current observations of Sgr A* are associated to

x = (a/M) (mP/lP) ~ 400. Near future ones (10 years) should have x ~ a few units

For all x < 2:

€

∈

Statistical ensembles of classical black holes

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

RH(M, x) = 2M (1+ x2/20)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


1. A toy model for finite precision observations of a Schwarzschild black hole

• T(M, x) ~ (1/8M) (1 + x2/20) for small x• S(M, x) ~ 4M2 (1 + x4/800) • The mean space-time is governed by a two-parameter

thermodynamics. Natural choices are (T, RH) and (T, AH)• S(T/, RH) = 2 S(T, RH) but M(T/, RH) = M(T, RH)• dM = T dS + f dRH = TdS + dAH

f ~ - x2/40, ~ - x2/(640 M)

Thus, a standard averaging over the non quantum space-time degrees of freedom modifies the black hole thermodynamics


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


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

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?

lP2 = G h/2 c3

mP = lPc2/G

qP = lP c/G1/2 (c.g.s. units)

Documents

F. Debbasch (LERMA-ERGA Université Paris 6) and M. Bustamante, C. Chevalier, Y. Ollivier Statistical Physics and relativistic gravity (2003-2005)