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INVESTIGACION´ REVISTA MEXICANA DE FISICA´ 52 (6) 515–521 DICIEMBRE 2006

Some statistical mechanical properties of black holes

X. Hernandez, C.S. Lopez-Monsalvo, and S. Mendoza Instituto de Astronom´ıa, Universidad Nacional Autonoma´ de Mexico,´ Apartado Postal 70-264, Distrito Federal 04510, Mexico,´ e-mail: [email protected], [email protected], [email protected] R.A. Sussman Instituto de Ciencias Nucleares, Universidad Nacional Autonoma´ de Mexico,´ Apartado Postal 70-543, Distrito Federal 04510, Mexico,´ e-mail: [email protected] Recibido el 3 de noviembre de 06; aceptado el 17 de noviembre de 06

We show that if the total internal energy of a is constructed as the sum of N all having a fixed wavelength chosen to scale 2 with the Schwarzschild radius as λ = αRs, then N will scale with Rs. A statistical mechanical calculation of the configuration proposed 2 yields α = 4π / ln(2) and a total entropy of the system S = kB N ln(2), in agreement with the Bekenstein entropy of a black hole. It is shown that the critical temperature for Bose-Einstein condensation for relativistic particles of λ = αRs is always well below the Hawking temperature of a black hole, in support of the proposed internal configuration. We then examine our results from the point of view of recent loop quantum ideas and find that a natural consistency of both approaches appears. We show that the Jeans criterion for gravitational instability can be generalised to the special and general relativistic regimes and holds for any type of -energy distribution.

Keywords: of black holes; classical black holes; quantum aspects of black holes; evaporation; thermodynamics.

En este art´ıculo estudiamos la relacion´ entre la energ´ıa y la entrop´ıa de un gas de fotones tipo cuerpo negro, contenido dentro de un recinto adiabatico´ de radio R, cuando es comprimido hacia un regimen´ auto–gravitacional. Mostramos que este regimen´ coincide aproximadamente con el regimen´ de un agujero negro para el sistema, i.e., R ∼ Rs, donde Rs representa al radio de Schwarzschild del sistema. La entrop´ıa del sistema resulta estar siempre por debajo de la cota Holografica,´ incluso cuando R → Rs. Una posible configuracion´ cuantica´ para el gas de fotones a R → Rs se sugiere, la cual satisface todas las condiciones de agujero negro para la energ´ıa, entrop´ıa y temperatura. Finalmente, examinamos nuestros resultados desde el punto de vista de algunas ideas recientes de Loop .

Descriptores: F´ısica de agujeros negros; agujeros negros classicos; aspectos cuanticos´ de agujeros negros; evaporacion;´ termodinamica.´

PACS: 04.70.-s; 04.70.Bw; 04.70.Dy

1. Introduction Notice that dS/dM = (c2T )−1, fixing the internal en- ergy of the black hole as U = Mc2. Equation (1) has been In general, black holes are defined uniquely by their mass, an- about in speculative form since the 18th century, and was gular momentum and charge (cf. [1]). In this paper we shall given a firm theoretical footing within the framework of gen- deal exclusively with Schwarzschild black holes, where the eral relativity during the 1930s. Equations (2)-(3) are the and charge are both zero. These black result of the vigorous development in black hole thermody- holes of mass M are understood as systems defined uniquely namics of the 1960s and 1970s by various authors, notably by the condition R ≤ Rs and have the following properties: Bekenstein and Hawking (cf. [2] and references therein). 2GM It has been suggested (e.g. [3-5] for a review) that for any R = , (1) s c2 physical system, Eq. (2) should hold always, with ≤ replac- S A ing the equality, which in turn should hold only in the black = BH , (2) k 4A hole regime. This is termed the , from B P the fact that the information content of an object would be hc limited not by its 3D volume, but by its 2D bounding sur- TBH = 2 . (3) 8π kBRs face. The interesting connection implied between quantum Equation (1) defines the Schwarzschild radius in terms of the mechanics through AP and gravity through the particle hori- mass. Relation (2) establishes the entropy S of the system zon, has raised the hope that the validity of the holographic 2 principle would yield important clues regarding quantum the- as 1/4 of the horizon area ABH = 4πRs, in units of the 3 Planck area AP = ~G/c . Equation (3) states that the black ories of gravity. In this sense, even a heuristic study as to the hole radiates as a black body of the given temperature TBH . possible origin of this principle should prove valuable. This emission process implies a loss of energy for the system, We study the behaviour of a classical black body photon which results in an evaporation rate for the black hole given gas as it is compressed into a black hole, and propose a sim- by: dM/dt ∝ (~c4/G2)M −2. ple model for such a system using only photons confined to 516 X. HERNANDEZ, C.S. LOPEZ-MONSALVO, S. MENDOZA, AND R.A. SUSSMAN the Schwarzschild radius at their lowest possible momentum where C0 is a numerical factor of order unity. Two interesting level. Two parameters determine the model, with a restric- conclusions are immediately evident from this last equation. tion only on the product of both of them. A formal statisti- Firstly, it is obvious that for any black body photon gas hav- cal mechanical calculation is given through which these pa- ing Rs > Rp, a Schwarzschild radius larger than the Plank rameters are determined, leading to agreement with all black length, the holographic principle will be valid throughout the hole structural properties of Eqs. (1)-(3). The study of a self– contraction process, as the horizon area will always be greater gravitating (a geon) was first introduced by than ABH . Second, that the classical equations for the diluted Wheeler [6] in 1955. Considerable development in this area photon gas, Eqs. (4) and (5), cannot be valid in the black hole has taken place since then related to different physical prop- regime, since the required relationship for the entropy of such erties of stars (see e.g. [7,8]) and their stability. an object is Eq. (2) and the exponent of Eq. (8) is 4/3 and Given recent proposals of a physical model for a black not 1. The inconsistency of Eq. (8) with Eq. (2) indicates that hole interior within the framework of loop quantum grav- physical processes which the system being modelled surely ity [9], we analyse our model in this context. Taking the experiences, such as pair creation at high temperatures and view that the Bekenstein entropy has a statistical mechani- quantum effects related to the dimensions of the system be- cal origin in terms of counting states on the surface defined ing comparable to the typical de Broglie wavelengths of the by the Schwarzschild radius of a black hole, canonical quan- photons, are not been taken into account by the classical de- tum gravity has yielded scenarios in which this entropy can scription of the photon gas through Eqs. (4) and (5). So far, be derived from first principles. We find no inconsistencies we have assumed that the photon gas was being compressed with the quantum gravity approach, which in fact allows us by some external agency; however, if it is to form a self grav- to explicitly and independently re-evaluate the parameters in- itating object, an equilibrium configuration should exist, and troduced in the quantum simple model, obtaining the same possibly a collapse beyond this. This point can be estimated results. by evaluating the√ Jeans length RJ of the problem for a speed of sound vs = c/ 3, c 2. Classical Limit R = . (9) J (3Gρ)1/2 Most of the material in this section can be found elsewhere, In this context, the mass density ρ is equivalent to e.g. [10]; it is reproduced here for context. For a photon gas E /V . Notice that Eq. (9) is derived directly from Ein- having a black body spectrum the following well known re- EM stein’s equations, where the pressure term is directly dE/dV ; lations define the total electromagnetic energy E and en- EM hence no assumption of particle interactions is being made tropy S in terms of the volume and temperature: (see the appendix for details on this). From Eq. (9) we see 4 −2 π2 (k T )4 4πR3 that since ρ scales with T , RJ will scale with T , which E = B , (4) EM 15 (~c)3 3 is interesting given that under adiabatic conditions the radius of the system will scale with T −1. This means that gravita- 2 3 3 S 4π (kBT ) 4πR tional instability will occur, i.e. R > RJ above a certain crit- = 3 , (5) kB 45 (~c) 3 ical temperature, below a certain critical equilibrium radius Rc. The situation becomes increasingly unstable in going to- for a spherical region of radius R. wards smaller radii and larger temperatures. In general, for a If we think of an ideal adiabatic wall enclosing this spher- fluid of mass-energy M, radius R and sound speed vs, ical region, we immediately obtain the well known scaling of −1 µ ¶1/2 T ∝ R , and we can eliminate (kBT ) from Eq. (4) in favour 4πR3 R = v , (10) of S and R to obtain J s 3GM µ ¶4/3 ~c S which expressing M in terms of the Schwarzschild radius E = C , (6) R kB through Eq. (1) reads µ ¶1/2 µ ¶1/2 where C is a numerical constant of order unity. If we think 8π Rvs R RJ = . of the contraction as proceeding into the black hole regime, 3 c Rs we can think of the radius as reaching Rs, which in this case would yield If we take the critical condition R = RJ , the previous rela- 2GE tion gives EM µ ¶ µ ¶2 Rs = 4 . (7) R 3 c c J = . (11) To obtain Eq. (7) we have used Eq. (1), replacing M by Rs 8π vs 2 EEM /c . Substitution of EEM from Eq. (6) into Eq. (7) Equation (11) shows that, since for all non–relativistic systems v ¿ c, we should expect R À R . Indeed, for leads to µ ¶ s J s S 4/3 A most astrophysical applications, the Jeans radius of a system = C0 BH , (8) kB Ap is many orders of magnitude larger than the Schwarzschild

Rev. Mex. F´ıs. 52 (6) (2006) 515–521 SOME STATISTICAL MECHANICAL PROPERTIES OF PHOTON BLACK HOLES 517 radius. However, in going to a relativistic fluid, the condi- where the summation is over quantum states, µ is the chem- tion vs ' c will apply, leading to RJ ' Rs. In other words, ical potential, and ²k is the energy of the k − th state. Since the self-gravitating regime will appear only close to the black the total number of components of the system is given by: hole regime. For the adiabatic photon gas we have studied, µ ¶ √ X 1 taking vs = c/ 3 and Eqs. (4), (5), and (9), in correspon- N = , (15) e[²k−µ]/kB T − 1 dence with Eq. (11) we obtain also RJ ' Rs. This last k result shows that the self-gravitating regime for the photon gas we have studied does not appear until one is very close and given that in the system proposed all photons have the to the black hole regime, at scales where the analysis leading same energy, we can write to Eq. (8) already showed that the structure equations for the N gas (4) and (5) are no longer valid. In any case, the analy- N = . (16) e[²−µ]/kB T − 1 sis following Eq. (9) together with Eq. (11), strongly sug- gests that any self-gravitating photon gas will be very close Note that each photon is assumed to be in a distinct detailed to catastrophic collapse and black hole formation. quantum level, and hence the analogy with a condensate is The above results are in fact valid in the regime where not complete. Now, the self-gravity of the radiation field is important, as shown by [10]; instability sets in for R < 2Rs, but equilibrium max- [² − µ]/kBT = ln(2), (17) imum entropy configuration exists above this radius, which however also show the scaling of Eq. (8), [c.f. their results which when substituted back into Eq. (14) gives: following their Eq. (41)]. Ω = −kBTN ln(2). (18)

3. Quantum limit This last result now yields the entropy for the system through S = −∂Ω/∂(kBT )|V,N as S = N ln(2), provid- The and transition between the classi- ing a justification for the assumption of S/kB = βN made cal regime of Sec. 2 and a black hole will not be treated ex- above, in terms of simple statistical physics arguments. We plicitly. Advances in that direction can be found for example hence obtain β = ln(2) and α = 4π2/ ln(2). Note that the in Refs. 10 to 12. previous results are of general validity for ; the case of Being subject to the extreme gravitational regime of R → photons is obtained with µ = 0. Rs, it is reasonable to expect that the photons will be highly We also note that given the restriction of a fixed total in- limited in momentum space. At this point we introduce as a ternal energy, according to the hypothesis that this is to be hypothesis that all photons will have a wavelength λ = αRs, the sum of the energies of N photons, the maximum entropy with α a numerical constant. We can evaluate the internal state will be the one with the most photons. In that case, all energy of the system as: photons are at their lowest possible energy λ = αRs. Hence, the configuration proposed is also a maximum entropy state Nhc EEM = , (12) and is suggestive of a micro–physical origin for black hole αRs entropy. where N is the total number of photons. Establishing a corre- It has been argued [13] that the shortest scale that can en- ter into any physical theory is the . Although spondence between EEM and the internal energy of a black 2 so far we have been working under the assumption of macro- hole as EEM = Mc , and using Eq. (1) to express M in scopic black holes, we can extrapolate to the very small scales terms of Rs, the above expression yields: as follows. In the context of the ideas presented here a natu- α ABH ral lower limit for the Schwarzschild radius of order the Plank N = 2 . (13) 16π Ap length appears by setting N = 1 in Eq. (13), a single photon black hole. The energy e associated to this single photon is If we think of a correspondence between the total en- e ≈ ~c/(Gh/c3)1/2 ≈ 1028 eV. tropy of the system and the total photon number given by Note that using Eq. (1) to substitute M for Rs in Eq. (13) S/kB = βN, with β a proportionality constant expected to gives the following quantisation for the mass MN of a black be of order unity, we find by comparison with Eq. (2) that the hole in units of Planck mass: configuration we propose will satisfy all required black hole ³ ´ 2 M π 1/2 properties if the condition αβ = 4π is satisfied. N = N 1/2, (19) We can compute β directly by calculating the entropy of mp α the proposed system from first principles, through the ther- 1/2 modynamic potential Ω given by: where mP = (~c/G) , a quantisation suggested already by Bekenstein’s entropy equation (2). For sufficiently small X ³ ´ [µ−²k]/kB T black hole , this equation suggests a discrete spectrum Ω = kBT ln 1 − e , (14) k associated with the transitions N = 1 → 0, N = 2 → 1,

Rev. Mex. F´ıs. 52 (6) (2006) 515–521 518 X. HERNANDEZ, C.S. LOPEZ-MONSALVO, S. MENDOZA, AND R.A. SUSSMAN etc. The change in mass ∆MN corresponding to a ∆N = 1 transition is given by " #   µ ¶2 1/2  π mP ∆MN = MN 1 + − 1 . (20)  α MN 

In the limit of a macroscopic black hole, where MN À mp, the above equation implies that ∆M π m N = P . (21) mP 2α MN 2 It is interesting to note that c ∆MN approximately corre- sponds to kBTBH for a black hole mass MN . If we now identify the time scale ∆t for the mass loss ∆MN to take place with the limit of the Heisenberg uncer- 2 tainty principle, we can set up ∆t ∼ ~/c ∆MN . Under the above considerations, the mass evaporation rate for a black hole is 4 ∆MN ~c 1 ∝ 2 2 , (22) ∆t G MN which is within a numerical constant of the standard evapo- ration rate for a black hole [2], seen here as the macroscopic FIGURE 1. The solid line shows the black hole temperature, in limit of an intrinsically quantum process. units of this quantity for a black hole having a Schwarzschild radius If the structure of the photon configuration we are de- equal to the Planck length, as a function of the Schwarzschild ra- scribing is in any way related to quantum phenomena akin dius, in units of the Planck length. The dotted line gives the critical to Bose-Einstein condensation, we should expect the temper- temperature for Bose-Einstein condensation of photons in a black hole, Eq. (26), in the same units as the solid curve, as a function of ature to lie well below the critical temperature Tc for Bose- Einstein condensation. This, for relativistic particles, can be the same quantity. calculated in an analogous way to well-known Bose-Einstein critical temperatures for non-relativistic particles (e.g. [14]), −1/3 −1 As Tc scales with Rs and TBH scales with Rs , it is by integrating the expression clear that for all black holes larger than a certain limit, the 2 gV p dp condition TBH ¿ Tc will be satisfied. A comparison of both dN = £ ¤, (23) 2π2~3 e(²−µ)/kB T temperatures is shown in Fig. 1, as a function of Rs, from which we see that Tc is already over an order of magnitude for µ = 0, g = 2 (photons) and in this case ² = pc, giving greater than TBH at Rs = Rp. Any realistic black hole will Z∞ be at a temperature much lower than the critical temperature V (T k )3 z2dz N = c B , for Bose-Einstein condensation for photons, showing the in- π2~3c3 ez − 1 ternal consistency of the model. 0 where z = ²/(kBT ). Evaluation of the above integral yields The physics described so far is clearly highly idealised, 3 2.202 and so, using V = (4π/3)Rs, leads to however in a core collapse process within a massive star, as µ ¶ for the central region R → Rs, the typical speeds and vs of 8.808 R3(T k )3 N = s c B . (24) the constituent particles must necessarily tend to c, with de 3π (~c)3 Broglie wavelengths not greater than the Schwarzschild ra- If we now use the expression for N in Eq. (13), and writ- dius. ing T in units of T for a black hole of radius equal to the c BH At this point, quantum effects similar to Bose-Einstein Planck length R , which correspond to a temperature T , p BHp condensation could take place, packing all (or most) pho- we get µ ¶3 µ ¶ tons into the lowest energy state. In this sense, typical wave- T 6π3 αR c = p , (25) lengths of the order of the Schwarzschild radius would be T 1.101 R BHp s expected, as longer wavelengths would be prohibited by the which for α = 4π2/ ln(2), as determined through the statis- containment imposed by gravity, and shorter wavelengths tical mechanical calculation shown above, yields, would imply an expansion in momentum space. In this sense, µ ¶ µ ¶ the identification we have maintained of the constituent par- T R 1/3 c = 21.3 p . (26) ticles as photons is shown to be largely arbitrary, and any rel- TBHp Rs ativistic bosons will yield essentially identical conclusions.

Rev. Mex. F´ıs. 52 (6) (2006) 515–521 SOME STATISTICAL MECHANICAL PROPERTIES OF PHOTON BLACK HOLES 519

4. approach The number of links N` associated with the particles we are dealing with in this article must be proportional to the We now explore the ideas of the previous sections within the number of particles N. The simplest possible configuration framework of loop gravity. In particular, we shall see that this is the one in which the proportionality factor is equal to unity, allows us an independent re-evaluation of the constants α and and so β established in the last section, with the aid of some general N` = N. (34) results from loop quantum gravity. The loop quantisation of 3 + 1 is de- Using this relation and Eqs. (13) and (33), we can evaluate α scribed in terms of a set of network states which span the to obtain 4π2 on which the theory is based. These spin net- α = . (35) work states are labelled by closed abstract graphs with spins ln 2 assigned to each link and intertwining operators assigned to The parameter β previously defined through the rela- each vertex. tion S = βN can now be re-derived independently through A recent result that follows from the theory is that if a sur- Eq. (30) to yield: face Σ is intersected by a link ` of a spin network carrying i N` j β = ln(2j + 1)j=1/2 = ln 2. (36) the label i, it acquires an area [15, 16] N p AΣ(ji) = 8πApγ ji(ji + 1), (27) From (35) and (36) we can see that the product αβ = 4π2, as required by the considerations on Sec. 3. It is where γ is the . interesting that the model proposed in the previous sections Let us now consider for our purposes that the horizon Σ is is seen not to be in conflict with a loop gravity approach, intersected by a large number N` of links. Each intersection which indeed allows us to re-evaluate the scaling parameters with Σ represents a puncture. In the limit of large N`, one introduced earlier independently, and in accordance with the can say that each puncture is equipped with an internal space expectations of the physics discussed above. Hj (the space of all flat U(1) connections on the punctured sphere) of dimension [17] 5. Conclusions dim H = 2j + 1. (28) j A photon gas contained within an adiabatic enclosure will Each puncture of an edge with spin j increases the di- satisfy the holographic principle, at least until just before mension of the boundary Hilbert space by a factor of 2j + 1. reaching the black hole regime, which approximately coin- Under these considerations, it follows that the entropy can be cides with the self-gravitating condition and where the clas- calculated by sical description is no longer valid. Ã ! A configuration where all photons have the same energy Y within R = RS can be constructed to satisfy all black hole S(jp) = ln dim Hjp . (29) conditions. p This configuration gives rise to a discrete evaporation Statistically, the most important contribution comes from spectrum for RS close to the Planck length, and in the macro- scopic limit permits a re-derivation of the standard black hole those configurations in which the lowest possible spin jmin dominates, so we can write the entropy (29) as evaporation rate. Our results are consistent with the loop quantum gravity

S(jmin) = N` ln(2jmin + 1), (30) scheme, satisfying the constraints required to be in agreement with the Bekenstein-Hawking entropy for a black hole. where N` is given by A comparison of both regimes studied is highly sugges- tive of a heuristic proof of the holographic principle, as any ABH N` = . (31) real system would require an increase in entropy (at fixed en- AΣ(jmin) ergy and volume) to be turned into the photon gas we have Due to the fact that the assumed gauge group of loop studied. quantum gravity is SU(2), then it follows that jmin = 1/2, and so the Immirzi parameter is given by [16] Acknowledgements ln 2 γ = √ , (32) X. Hernandez acknowledges the support of grant UNAM π 3 DGAPA (IN117803-3), CONACyT (42809/A-1) and CONA- CyT (42748). C. Lopez-Monsalvo is grateful for economic and (31) becomes support from UNAM DGAPA (IN119203). S. Mendoza 1 ABH gratefully acknowledges financial support from CONACyT N` = . (33) 4 ln 2 Ap (41443) and UNAM DGAPA (IN119203). We thank the

Rev. Mex. F´ıs. 52 (6) (2006) 515–521 520 X. HERNANDEZ, C.S. LOPEZ-MONSALVO, S. MENDOZA, AND R.A. SUSSMAN anonymous referee for his comments and suggestions, which with constant index κ. This means that the sound velocity 2 2 improved the final version of the paper. vs is given by the relation vs = c (κ ¡− 1), and¢ so, the left 2 2 hand side of Eq. (A.2) can be written as vs /c de/dr. Seen A Relativistic Jeans criterion for gravitational in this way, the left hand side of Eq. (A.2) no longer repre- sents gradients of pressure which are in balance with self– instability gravitational forces related to the plasma. Indeed, the balance In order to show that the Jeans gravitational instability limit with the gravitational forces produced by the plasma is now is valid in the general relativistic regime, let us proceed as related to the gradients of its proper internal energy density e follows. The condition of hydrostatic equilibrium in general by relativistic fluid mechanics is obtained by using the fact that de the field is static. This means that one can describe the prob- (κ − 1) dr lem in a frame of reference in which the fluid is at rest, with ½ ¾ ke GM(r) 4π all hydrodynamical quantities independent of time. This also = − ³ ´ + Gr3ρ . (A.5) implies that the mixed space and time components of the met- 2GM(r) c2 c2 r r − c2 ric tensor are null. Under these assumptions, the equation of hydrostatic equilibrium is then given by [18] Let us now take the absolute value on both sides of 1 ∂p 1 ∂ Eq. (A.4), to the order of magnitude de/dr ≈ e/r and = − log g , (A.1) 3 w ∂r 2 ∂r 00 M(r) ≈ (4/3) π r ρ. A gravitational instability occurs when the absolute value of the left hand side of Eq. (A.2) where g is the time component of the metric tensor, 00 [or equivalently Eq. (A.5)] is less than the absolute value of w=e+p is the enthalpy per unit volume, e the internal en- its right hand side. Using all the above statements, it fol- ergy density and p the pressure. Oppenheimer & Volkoff [19] lows that this instability occurs when the radial coordinate r showed that Eq. (A.1) can lead to the form [20] ½ ¾ is such that dp (e + p) GM(r) 4π s = − ³ ´ + Gr3ρ , (A.2) dr 2GM(r) c2 c2 3 vs r r − 2 r & √ := Λ . (A.6) c 8π (3κ − 1) Gρ J where the mass-energy M(r) within a radius r is given by Zr The quantity ΛJ on the right hand side of Eq. (A.6) is of the same order of magnitude as the standard Jeans length M(r) = 4π ρ r2 dr, (A.3) used in non-relativistic fluid dynamics. In other words, the 0 criterion (A.6) means that the Jeans criterion for gravitational and ρ(r) := e/c2 is the mass–energy density of the fluid. collapse is also valid in the relativistic regime as well. Note that for the case of relativistic and non–relativistic dust For the particular case studied in this article, the constant particles, M(r) represents the mass of particles within ra- κ = 4/3 for a photon¡ gas and√ the¢ Jeans√ criterion can be ap- dius r. For the case of a photon gas, M(r) is the mass corre- plied to it, giving r & 1/2 2π vs/ Gρ . sponding to the internal energy. We see than the Jeans criterion can be generalised be- We now assume that the plasma obeys a Bondi-Wheeler yond an equilibrium between gas pressure and rest–mass equation of state self–gravity, to a very general equilibrium between energy– p = (κ − 1) e, (A.4) momentum flux and total self–gravity.

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Rev. Mex. F´ıs. 52 (6) (2006) 515–521