Thermodynamics of black holes in finite boxes

arXiv:gr-qc/0302079v1 19 Feb 2003 Abstract We analyze the thermodynamical behavior of black holes in closed finite boxes. First the black hole mass evolution is analyzed in an initially empty box. Using the conservation of the energy and the Hawking evaporation flux, we deduce a minimal volume above which one black hole can loss all of its mass to the box, a result which agrees with the previous analysis made by Page. We then obtain analogous results using a box initially containing radiation, allowed to be absorbed by the black hole. The equilibrium times and masses are evaluated and their behavior discussed to highlight some interesting features arising. These results are generalized to N black holes + thermal radiation. Using physically simple arguments, we prove that these black holes achieve the same equilibrium masses (even that the initial masses were different). The entropy of the system is used to obtain the dependence of the equilibrium mass on the box volume, number of black holes and the initial radiation. The equilibrium mass is shown to be proportional to a positive power law of the effective volume (contrary to naive expectations), a result explained in terms of the detailed features of the system. The effect of the reflection of the radiation on the box walls which comes back into the black hole is explicitly considered. All these results (some of them counter-intuitive) may be useful to formulate alternative problems in thermodynamic courses for graduate and advanced undergraduate students. A handful of them are suggested in the Appendix. 1 Thermodynamics of black holes in finite boxes P.S.Custódio and J.E.Horvath Instituto de Astronomia, Geofı́sica e Ciencias Atmosféricas Rua do Matão, 1226, 05508-900 São Paulo SP, Brazil Email: [email protected] December 14, 2010 1 Introduction In a seminal work of the 70ś, Hawking [1], showed that black holes are capable of emitting radiation, and evaluated the quantum process by which these black holes lose mass due to vacuum polarization (induced by the gravitational energy of the black hole). This important work and follow-up contributions, together with Bekenstein’s arguments [2] about the entropy of black holes, prompted a new discipline of black hole research: thermodynamics of black holes. Today, the study of black hole thermodynamics is very active and several arguments indicate that it must embrace the partial (or perhaps complete) marriage of quantum mechanics, thermodynamics and gravitation. The original Hawking analysis showed that the particles created display a thermal spectrum f (E, T ) = eβE1±1 with β = 1/kB T , and a temperature determined by the black hole mass M given by Tbh h̄c3 M⊙ K = ∼ 10−7 8πkB GM M   (1) see [3] for details. Since this spectrum is thermal, the luminosity of the black hole can be calculated using the Stefan-Boltzmann law, using L ∝ 2 T 4 rg , where the gravitational radius rg ∝ M squared plays the role of the emitting area. Using these definitions, L(M) ∝ M −2 is readily obtained. 2 In addition to ”purely academic” cases (e.g. black holes inside ideal boxes), these emission (absorption) relations are in principle useful for investigating the properties of astrophysical and/or primordial black holes (hearafter PBHs) and their cosmological consequences. For example, some shorttimescale gamma-ray burst events have been possibly explained by evaporating PBHs in a terminal (explosive) phase. In fact, the derived relations between period, mass and spectra are consistent with the PBH model, see [4]. Even in the simplest cases (the black holes interacting with ambient radiation) it is interesting to understand what determines the equilibrium features of the system and how is it achieved. We shall perform an analysis restricted to finite volumes with the aim of illustrating some novel features of the thermodynamics of black holes using gedanken experiments. These excercises serve also as a prelude to the more complicated situation, in which evaporation of PBHs must be studied in an expanding universe. However, and quite independently of these further advanced considerations, ”boxed” black holes display interesting features which need careful clarification. 2 Classical absorption in an initially cold box Let us consider the following gedanken experiment: we start with a black hole and put it within a finite box (with linear size L). In the beginning, the box has no radiation, i.e. its temperature is zero. Then, since the black hole is hotter than the environment, it will emit thermal radiation, and thus this closed box will be increasingly filled with this radiation shortly afterwards. The evolution of the mass of the black hole is described by the well-known expression which balances the energy loses to the Hawking luminosity ∝ M −2 dM C =− 2 (2) dt M with an initial condition of the box Trad (t = ti ) = 0. The constant C depends on the degrees of freedom allowed to be radiated by the black hole [1] and is set to C ∼ 1026 g 3 s−1 throughout this work, and the initial mass is Mi . We assume that the box linear size is much larger than the black hole gravitational radius L ≫ rg to avoid a complicated non-linear feedback between the geometry and the radiation field, which would lead us far beyond 3 the purely thermodynamical approach. Since the box is closed, the total energy is conserved. Therefore it is easy determine a relation for the black hole mass and the thermal radiation contained within the box from ̺rad (Trad )Vbox + M(t) = Mi (3) where t > ti (we use natural units to set the speed of light c equal to 1). The radiation density is given by ̺rad (T ) = a∗ T 4 with a∗ ∼ 8 × 10−36 gcm−3 K −4 . After some time, the box temperature is Mi − M(t) Trad (t) = a∗ Vbox  1/4 (4) At an even later time t > ti , eq.(2) should not simply contain the Hawking term, but an absorption term is also needed. The simplest form of term is generally constructed as the product of the incoming flux times the gravitational cross-section of the radiation falling into the hole, B̺rad (T )M 2 , with 2 (or M27π B = 27πG 4 in natural units, see Ref. [5] for details and discussion). c3 pl Therefore the complete differential equation for the mass is given by C dM = − 2 + B̺rad (T )M 2 (5) dt M From this equation, it is clear that thermodynamical equilibrium between the black hole plus and the ambient radiation will be given by Ṁ = 0, and thus the mass of the black hole in equilibrium is Mbh = Mc (teq ) = D Trad (teq ) (6) where D = (C/a∗ B)1/4 ∼ 2 × 1026 gK. Solving the set of equations above we track the black hole mass evolution as it approaches the equilibrium (and determine the conditions for this equilibrium). If we substitute the eq.(4) into eq.(5) we obtain C Mi − M dM M2 =− 2 +B dt M Vbox   (7) The evolution of the mass of eq.(7) will be given by solving the integral 4 Z M (t) Mi dMM 2 = t −κ1 + κ2 M 4 + κ3 M 5 (8) and the constants κi are κ1 = C, κ2 = BMi /Vbox and κ3 = −B/Vbox . In the limit Vbox >> rg 3 the solution of eq.(7) approaches M(t) ∼ Mi [1 − (t/tevap )]1/3 M3 with tevap = 3Ci , as expected. The numerical solution of the integral eq.(7) is shown in Fig.1 for different box sizes but the same initial black hole mass Mi . In order to obtain the conditions for the equilibrium we solve the algebraic equation Trad (teq ) = THaw (Meq ), rewritten as  Mi − M(teq ) a∗ Vbox 1/4 = C M(teq ) (9) Inserting the respective constants above, eq.(9) becomes [1 − µ(teq )]µ4 (teq ) = D(Vbox /cm3 ) × (M⊙ /Mi )5 (10) (teq ) and D ∼ 4.2 × 10−97 . where µ(teq ) = MM i The interpretation of the solution is quite evident, when the volume of the box is such that the right side of eq.(9) is larger than one, the black hole evaporates completely in the absence of additional quantum corrections at the Planck scale. But if D(Vbox /cm3 ) × (M⊙ /Mi )5 ∼ 1, the size of the box is small enough to stop the evaporation, and the system black hole+radiation achieves thermodynamical equilibrium before its mass vanishes completely. The linear size of the box below which the black hole achieves its equilibrium is given by Lc (Mi ) ∼ 1.3 × 1032 (Mi /M⊙ )5/3 cm (11) The time for reaching equilibrium teq is determined by the initial mass and the size of the box only, i.e. teq = teq (Mi , Vbox ). For the same initial mass, smaller boxes will contain black holes with larger final masses at equilibrium (see Fig. 1). Our analysis agrees with the previous work by Page [7]. From now on, we define the critical volume by Vcrit (M) = Lc (M)3 using eq.(10), with its clear physical interpretation given above. In the next sections we revisit the derivation of eq.(10), extend it to more complicated cases and consider causality features. 5 Figure 1: The approach to equilibrium of black holes in finite boxes. The curves represent qualitatively the temporal behavior of the mass of the black hole for two different values of the box size Vbox (initially devoid of radiation); with the initial value of the mass Mi held fixed. 3 Classical absorption in a closed box with initial radiation Let us now evaluate the behavior of the black hole mass when introduced in a box with a non-zero initial radiation content. We expect the black hole to achieve thermodynamical equilibrium earlier than in the case without initial radiation, and with a smaller equilibrium mass (considering boxes with the same size). In this case, is easy show that Z M (t) Mi dMM 2 =t −κ1 + κ4 M 4 (Ei − M) (12) where Ei = Mi + a∗ Ti 4 Vbox and κ4 = B/Vbox . The plot of M(t) as a function of time is actually similar to the former case. As before, we impose the equilibrium condition in to obtain the relation between the black hole mass and the box volume. Following the same steps as before, we obtain µ∗ (teq )4 [F (Ti , Mi ) − µ∗ (teq )] = D(Vbox /cm3 ) × (M⊙ /Mi )5 (13) Ei , and the teq is different from the previous case. where F (Ti , Mi ) = M i Actually the black hole achieves thermodynamical equilibrium sooner, as expected. 4 N black holes plus radiation in a closed box The generalization to N black holes seems quite straightforward, although it will become clear that the initial states and other details must be carefully defined to achieve consistent results. First, we shall consider just two black holes immersed in a closed box (with constant volume) in the following initial 6 situation: one black hole has initial mass M1 and the other black hole is more massive than the first, M2 > M1 . Initially the box has no radiation, and therefore these objects begin to evaporate immediately. Moreover, we shall consider that Vbox ∼ Vcrit (M1 ). Then at the initial time ti we have  dM2 dt  =− C M22 (14)  dM1 dt  =− C M12 (15) ti ti Since M1 < M2 one may consider |(Ṁ1 )evap | ≫ |(Ṁ2 )evap | initially. Afterwards, when the box starts to be filled with the emitted radiation, some energy can be absorbed by the black holes. Therefore, their evolution will be given by C dM2 = − 2 + B̺rad (T )M2 2 dt M2 (16) C dM1 = − 2 + B̺rad (T )M1 2 dt M1 (17) M1 (t) + M2 (t) + Vbox ̺rad (T ) = M1 (ti ) + M2 (ti ) (18) where the last condition displays the conserved energy of the box. Using this constraint, we can describe this system by the following set of equations C B dM2 =− + M2 (t)2 [Ei − M1 (t) − M2 (t) 2 dt M2 (t) Vbox (19) dM1 C B =− + M1 (t)2 [Ei − M1 (t) − M2 (t)] 2 dt M1 (t) Vbox (20) combining eqs.(17) and (18) and using the conservation of energy yields −C  1 1 + 2 + B̺rad [M12 + M22 ] + Vbox ̺˙ rad = 0 2 M1 M2  (21) where Ei = M1 + M2 is the initial energy. Note that the set of coupled equations does not include the effect of black hole motion due to their mutual gravity. Motion is likely to affect the 7 emission/absorption properties of the black holes in the relativistic regime [8], [9]. Thus, the analysis is strictly valid whenever vbh ≪ c. Let us take a look at the global behavior of the solutions. First, it is easy to show that the equilibrium between two black holes is possible. If we impose that the thermodynamical equilibrium will be achieved, then asymptotically dM1 dM2 = =0 (22) dt dt C From these equations B Vbox = M1 4 [Ei − M1 − M2 ] immediately follows, and the same is true for the other black hole. Therefore, it follows that M1,eq = M2,eq (23) The same results of these equilibrium masses could have been obtained from the expression of the entropy. In fact, more complicated cases can be worked out either using the above approach or using that the entropy must be an extreme for a system in equilibrium. For instance, let us consider N black holes enclosed in the box. The total entropy for this system is given by Stotal (t) = 1X 3 Ai (t) + g∗ Trad (t)Vbox 4 i (24) Since the horizon area for the i-th black hole is Ai = 4πrg,i2 we have Stotal (t) = 4π X 3 Mi (t)2 + g∗ Trad (t)Vbox 4 Mpl i In equilibrium Trad = Tbh1 = Tbh2 = ... = Tbhi = 2 Mpl , 8πMeq (25) and therefore Smax (N, Vbox ) = β∗ Nµeq 2 + l∗ /µeq 3 (Vbox /cm3 ) (26) where β∗ = 2.5 × 1040 , l∗ = 8 × 1033 α∗ and µeq = (Meq /1015 g) the equilibrium mass scaled to a convenient reference value.  Extremizing the entropy dSmax dMeq = 0 and solving for Meq yields Meq (N, Vbox ) ∼ 5.4 × 1013 α∗ 1/5 N −1/5 (Vbox /cm3 ) 1/5 g with α∗ = 26.7g∗ . After substituting back into Smax we find 8 (27) Figure 2: Graphical solution of the non-linear equation for the equilibrium mass (eq.30) as a function of time for a fixed value of Vbox and a different number of black holes N. It may be said that the equilibrium mass ”feels” the presence of other black holes. Smax (N, Vbox ) ∼ 1.2 × 1038 α∗ 2/5 N 3/5 (Vbox /cm3 )2/5 (28) Since N = nVbox by definition, we have Smax ∼ 1.2 × 1038 n3/5 (Vbox /cm3 ). Therefore, the thermodynamic approach leads us to conclude that the equilibrium black hole mass is bigger for bigger boxes. This result is quite counterintuitive, but it has a reasonable explanation given below. To confirm the dependence first, we resort to another method. Let us rewrite the equation for the mass of the i-th black hole as dMi C B =− + Mi (t)2 [Ei − NMi (t)] 2 dt Mi (t) Vbox (29) where we prepare the system with N black holes with the same initial mass (this particular condition is not over-restrictive and simplifies the algebra). From Ṁi = 0 we obtain µeq 4 − K 1 µeq 5 = (Vbox /cm3 ) µi Nµi (30) where K ∼ 3.5 × 10−6 . This algebraic equation has non-trivial solutions which can be found numerically. For the sake of definiteness we adopt N = 1, µi = 10, and a small Vbox = 103 cm3 , which yields µeq ∼ 0.137. If we enlarge the box, say Vbox = 104 cm3 instead, then µeq ∼ 0.244, and so on. If we fix the size of the box instead, say Vbox = 2.8 × 105 cm3 , and change the number of black holes, the equilibrium mass will reflect the effect of the presence of other black holes accordingly. For example, taking µi = 10 and N = 1 we have µeq ∼ 1.027 but for N = 103 we obtain µeq ∼ 0.178 (Fig. 2). These results agree with the previous entropy analysis, and confirm that the coupled equations correctly describe the evolution of the system. It is quite interesting to propose the construction of a plot showing the behavior of the solutions and a discussion about the reasons for that behavior. 9 As a further related work it is proposed to show that if we consider the existence of an initial thermal radiation with energy Ei , the effective volume becomes Vef f (N, Mi ) = Vbox N + Ei /Mi (31) which holds for N black holes with the same initial masses. This effective volume must be used instead of (Vbox /N) in eq.(26) and Meq (N, Vbox ) becomes 1/5 Vbox ∝ [ N +E ] . i /Mi The puzzling growth of the equilibrium mass in these systems must be further explained in more detail to improve the understanding. To proceed, let us consider two boxes with volumes Vbox,2 and Vbox,1 where the second box is larger than the first. Let us immerse two black holes with the same initial masses in both boxes. Therefore, the energy content is forced to be the same. Then, in equilibrium we will have (dropping some inessential numerical factors) 4 4 Ebox = 2Meq,1 + Trad,1 Vbox,1 = 2Meq,2 + Trad,2 Vbox,2 (32) Since the equilibrium mass is given by Meq ∝ [Vbox ]1/5 for both boxes, then we have 4 4 Vbox,1 1/5 − Vbox,21/5 = Trad,2 Vbox,2 − Trad,1 Vbox,1 (33) both terms are negative due to the assumption Vbox,2 > Vbox,1 then we obtain from the r.h.s. of this equation  Trad,2 Trad,1 4 Vbox,1 < Vbox,2   <1 (34) then Trad,2 < Trad,1 . Therefore, the bigger box will have smaller radiation 1 temperature. Since in equilibrium Trad,2 = Tbh,2 ∝ Meq,2 then the mass of the second black hole must be bigger than the first, in agreement with our previous analysis. Previous work by Page [7] (specifically his eq.(10)) has explored this problem, which has been solved here using the conservation of energy only, with the absorption terms explicitly present. 10 5 Causality considerations The above considerations implicitly assumed that the black hole absorption starts instantaneously as the former is created inside the box. One might wonder whether a black hole immersed within an initially empty box is actually able to absorb ambient radiation immediately; since the Hawking radiation emitted by it will be available to absorption only after ∼ L/c as required by causality. In order to gain some insight, we use the fact that if the box is large enough, when the radiation comes back into the black hole, the object has evaporated completely. Then, a critical condition is obtained imposing this time interval to be bigger than the timescale for evaporation. Thus L M3 > c 3C (35) 29 Therefore, it follows that Lcrit (M) ∼ 10 cm  M 1015 g 3 . Then, any box (without initial radiation) containing a black hole with mass M whose linear size L is bigger than the critical size Lcrit above would let the black hole to evaporate completely, thus precluding equilibrium. Note that the functional dependence and numerical value are very different from the naive approach used previously where we have considered that the absorption term turns on instantaneously (that is, as soon we immersed the black hole in the box). If we boldly compare Lcrit with the size of the Hubble horizon today ∼ 1027 cm, we deduce that a PBHs with masses smaller than ∼ 2 × 1014 g can evaporate completely. However, it is clear that in a realistic case other sources of energy must be considered and our comment is intended just for pedagogical purposes. The result of eq.(35) can be interpreted as follows. If we put one hot black hole (at ti ) with initial mass Mi inside a closed box with volume Vbox , it will evolve according to Ṁ = −C/M 2 for a time ti < t < Lbox . But from t > c Lbox /c on, the black hole has to absorb radiation from the box (originated by itself!). Then the absorption term ∝ T 4 Vbox contributes to avoid its complete evaporation. If the box is slightly larger (but smaller than the critical volume defined by eq.(10)) the onset of absorption is delayed, then this black hole will be hotter than the previous case. This black hole is then filling the box with higher temperature radiation, and after some time it will be absorbing 11 radiation at higher temperature, hence the absorption term ∝ T 4 M 2 will be larger. This term drives the black hole into the larger equilibrium mass, as given by eq.(27). It should be remembered that a maximal number for black holes enclosed in a box with volume Vbox must exist, a naive estimate of Nmax can be obtained from the close-packing condition Nmax rg3 (Meq ) ∼ Vbox . 6 Conclusions In this work we have analyzed the behavior of black holes+radiation systems, considering some gedanken experiments within finite boxes (initially with and without radiation) (see [10] for a clear pioneering discussion of these issues). These gedanken experiments are important to understand the evolution of black holes in more complex situations, since it is hoped that we can always approximate the thermodynamical behavior of the universe with that of a finite box. In these simple cases the energy available for absorption by the black holes is finite at all times and there is no need to consider other sources. Strictly speaking, and with an eye on the actual cosmological formation and evolution of black holes, it may be argued that the problem is oversimplified, since actual primordial black hole masses must be formed with a substantial fraction of the horizon mass [6], and therefore finite size effects should be considered from the scratch. The actual situation is much more complicated but not impossible to tackle. The real difficulty, which can be viewed as an advanced exercise dealing with the (generalized) second law of thermodynamics not related to laboratory systems, is that (unlike laboratory gases) the black hole ”gas” will have varying masses and therefore a complicated kinetic equation must be used to describe their evolution. This provides a concrete example, possibly realized in nature, of a variable-mass gas needing a deep understanding of statistical mechanics principles for its very formulation. The simplest approach taken here was enough to show that one black hole plus radiation can achieve thermodynamical equilibrium if the box volume is smaller than a critical volume (in agreement with a previous treatment made by Page [7]). Some surprises arise in more general cases where we have two (or more) black holes (plus radiation) within these boxes. Even in their simplest versions the cases of black holes in boxes are very instructive to analyze and may serve as an good issues for a graduate course 12 of thermodynamics/statistical mechanics (see [11] for a discussion of related formal aspects of black hole thermodynamics) as an alternative to traditional problems dealing with the approach to equilibrium. They pose several questions of deep physical meaning, which are also strongly entangled and sometimes puzzling to interpret. References [1] S.W.Hawking, Particle creation from black holes, Comm. Math. Phys.43, 199- 220 (1975). [2] J.Bekenstein, Black holes and entropyPhys. Rev.D7, 2333-2346 (1973) ; ibid Black Hole Thermodynamics, Phys.Today 33, 24-31 (1980). [3] V.Frolov and I.Novikov, Physics of Black Holes, (Kluwer Academic Publishers, Dordretch 1995). [4] A.Belyanin et al, Gamma Ray Bursts from the final stage of primordial black hole evaporation, MNRAS283, 626-634(1996); D. Cline and W.Hong, Very short gamma ray bursts and primordial black hole evaporation, Astropart.Phys5, 175-182 (1996). [5] P.S.Custódio and J.E. Horvath, Bounds on the cosmological abundance of primordial black holes from diffuse sky brightness: single mass spectra Phys.Rev.D65, 024023 1-8 (2002). [6] B.Carr, The Primordial Black Hole mass spectrum, Astrophys.J201, 119 (1975). [7] D.N. Page, Black hole thermodynamics, mass inflation and evaporation, in Black Hole Physics : Proccedings of the NATO Advanced Study Institute on Black Hole Physics (Eds. V. De Sabbata and Zhenjiu Zhang), 185- 196 (Kluwer Academic Publishers, Dordretch 1992). [8] A.S. Majumdar, P. Das Gupta and R. P.Saxena, Baryogenesis from black hole evaporationInt. Jour. Mod. Phys. D4, 517-529 (1995). 13 [9] P.S. Custodio and J.E. Horvath, Evolution of a primordial black hole population, Phys. Rev.D58, 023504 1-7 (1998) ; ibid Dynamics of black hole motionPhys. Rev. D60, 083002 1-9 (1999). [10] S. W. Hawking, Black holes and thermodynamics, Phys. Rev. D13, 191197 (1976). [11] B.R. Parker and R.J. McLeod, Black hole thermodynamics in an undergraduate thermodynamics course, Am. J. Phys.48, 1066-1070 (1980). 7 Appendix A complementary set of simple problems intended to reinforce the conceptual aspects of black hole evaporation. Except for the problem 4, their mathematical complexity is quite straightforward. 1) Using eq.(2), prove that the timescale for evaporation for one black hole is proportional to M 3 . Verify that for one black hole with the solar mass (M ∼ 2 × 1033 g); tevap (M) ∼ 1065 years. Evaluate this for a initial mass ∼ 1015 g and compare with the age of the universe. 2) Verify the expression eq.(6) for the critical mass. Evaluate this number today, considering that Trad (t0 ) ∼ 3K. 3) Derive eq.(11). Use for the linear size of the box the present size of the universe. Evaluate the corresponding mass at equilibrium. 4) Deduce the set of eqs.(19)-(21) and interpret the solutions graphically. 5) Deduce the Meq (N, Vbox ) for the cases without initial radiation and with a radiation filling the box. Interpret these results and compare with the cosmological situation plugging representative numbers for the latter. 6) Discuss how all these results may be modified if the box expands following a known temporal dependence. 14 M (t) M equilibrium i Vbox(2) V box(1) t t eq (M i , V box ) M(t) Mi N=1 V box IL[HG N = 1000 teq t