Entropy of extremal black holes from entropy of quasiblack holes Jos´e P. S. Lemosa, Oleg B. Zaslavskiib aCentro Multidisciplinar de Astrof´ısica – CENTRA, Departamento de F´ısica, InstitutoSuperior T´ecnico - IST, Universidade T´ecnica de Lisboa - UTL, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal, & Institute of Theoretical Physics - ITP, Freie Universita¨t Berlin, Arnimallee 14 D-14195 Berlin, Germany. bAstronomical Institute of Kharkov V.N. Karazin National University, 35 Sumskaya St., Kharkov, 61022, Ukraine. 1 1 Abstract 0 2 Theentropyofextremalblackholes(BHs)isobtainedusingacontinuityargumentfromextremalquasiblack n holes (QBHs). It is shown that there exists a smooth limiting transition in which (i) the system boundary a approachestheextremalReissner-Nordstro¨m(RN)horizon,(ii)thetemperatureatinfinitytendstozeroand J quantumbackreactionremainsboundedonthehorizon,and(iii)thefirstlawofthermodynamicsissatisfied. 5 The conclusion is that the entropy S of extremal QBHs and of extremal BHs can take any non-negative ] value, only in particular cases it coincides with S = A/4. The choice S = 0 with non-zero temperature at c infinity is rejected as physically unsatisfactory. q - r Keywords: quasiblack holes, black holes, extremal horizon, entropy, thermodynamics g [ 2 1. Introduction Whatis the QBHapproach? AQBHis asystem v whose boundary approaches the would-be horizon 8 as nearly as one likes, and yet the system does not Theissueofblackhole(BH)entropyisoneofthe 6 collapse;a horizonis almostformedbut neverdoes 7 most intriguing in BH physics. For non-extremal 2 BHs the entropy S is given in terms of the horizon [8]. The approach consists in finding the limiting 1. area A by the Bekenstein-Hawking formula S = A properties of the system when the boundary tends 4 to its quasihorizon [9]. Properties of such systems 1 [1],apuzzlenotyetresolvedinfundamentalmicro- 0 level terms. Surprisingly, the issue becomes even arethencomparedwithpureBHproperties. Ithas 1 beenfoundthat,thoughworkedoutthroughtotally more intriguing in what concerns extremalBHs, as : different methods, QBHs and pure BHs share for v there are two mutually inconsistent results. There i is the prescription S = 0 obtained from the fact outside observers the same properties, such as the X mass formula and many others [9], although the that for extremal BHs the period of the Euclidean r interiorof both systems is totally different, interior a time is not fixed in a classical calculation of the action [2], and there is the usual S = A value ob- madeofmatterforQBHs,vacuuminteriorforpure 4 BHs. tainedfromstringtheory[3]. Therehavebeensome interesting proposals to further understand the is- Inthework[10]importantdevelopmentsonQBH sue, see [4] for a thermodynamical treatment and properties were advanced. The entropy of non- [5, 6] for a semiclassical approach, but the situa- extremal QBH systems was found thermodynam- tion remains contradictory up to now, see [7] for ically and shown to be equal to the BH entropy the latest comments. Here, we suggest a resolution S = A. This was achieved by using on one hand ofthisproblemonthebasisofpurethermodynamic 4 QBH procedures, and on the other hand the for- arguments. In doing so, we exploit the quasiblack malism for gravitating systems, such as BHs, of hole (QBH) approach. BrownandYork[11]forthedefinitionofquasilocal energy and other quasilocal thermodynamic quan- Email addresses: [email protected] (Jos´eP.S. tities. Now, QBHs can be obtained from a quite Lemos),[email protected] (OlegB.Zaslavskii) Preprint submitted toElsevier January 6, 2011 generic class of systems, but a simple realizationof We show that our consistent thermodynamic them is provided through thin chargedshells when treatment rejects definitely the choice S = 0 but these are brought to their own horizon radius. In does not give an unambiguous universal result for the works [12] these systems were thermodynami- S. The entropy depends on the properties of the cally studied but the analysis fell short of letting workingmaterialand,moreover,onthemannerthe the shell approach the horizon. In [4] a specially temperature approaches the zero value. In partic- arrangedshell system was imaginedin order to an- ular S = A is not singled out beforehand for the 4 alyzeitsbehaviorwhenloweringquasistaticallythe extremal BH entropy. shell into its own horizon, with the shell being im- mersedin an appropriatequantum vacuum. It was found that S = A, as well. Our QBH approachfor 2. Basic formulas 4 non-extremal systems [10] is generic and thus has none of the drawbacks of specialization to simple The study of extremal QBHs has one advantage thin shell systems. over the study of non-extremal ones. While non- Since many properties, in particular the entropy extremalQBHs show a sortof singular behavior at S of non-extremal QBHs, can be found and match the quasihorizon, such as a singular stress-energy the corresponding quantities of pure BHs, we con- tensor, extremal QBHs are nonsingular well be- tinue our pursue and want to shed light on the en- haved systems throughout [9]. In order to make tropy of extremal BHs by studying the entropy of the problem tractable we stick to spherically sym- QBHs, more specifically, extremal QBHs. One can metric systems. thenconsiderasequenceofstarsmadeofsomesort Consider a spherical symmetric compact body of usualmatter, each member of the sequence with with boundary at r = R such that r < R defines lesser radius say, in which the last member of the the inner region, r > R the outer one, and its to- sequenceis anextremalQBH.Byusing starsmade tal charge q is equal to its ADM mass m. The of matter one is enabled to consider more prosaic genericspace-timeline elementinthe usualcoordi- systems, i.e., systems that do not possess a hori- nates (t,r,θ,φ) is then zon, and study them through the usual textbook formalism of thermodynamics. Only for the last dr2 membersofthesequenceofstarsonetakesthelimit ds2 = V exp(2ψ)dt2+ +r2 dθ2+sin2θdφ2 , − V tothetransitiontotheQBHstate. Atthisverylast (cid:0) ((cid:1)1) stage of the sequence, in order to have a well de- where V and ψ are functions of r. In general one finedthermodynamicsystem,onehastouseresults also needs an expression for the electric potential from quantum field theory in curved background φ(r). For r R the space-time is described by [13]. The sequential procedure is of immense im- ≥ the extremal Reissner-Nordstro¨m (RN) metric, in portance,asduetothecontinuityofthecalculation which case ψ(r) = 0, and by the Coulomb electric process, the QBH approach enables to evade diffi- potential, culties connected with the often invoked potential discontinuity between non-extremal and extremal r 2 r + + V(r)= 1 , φ(r)= +constant, (2) BHs. Using the QBH, we follow the procedure de- (cid:16) − r (cid:17) r veloped in [10] (see also [4]), i.e., we calculate S of where r = m = q, r being the gravitational ra- the material system when the would-be horizon is + + dius of the body, i.e., the radius of the would-be approached. Since for an external observer a QBH horizon, and the constant can be chosen in con- and a BH cannot be distinguished [9], one expects venient terms. R r always holds here and at that the entropy of such a system without a true ≥ + R=r a QBH forms. horizon tends to the entropy of the corresponding + BH. In this sense our calculations not only give an Thewholesystem,compactbodyplusspacetime, answerfortheQBHentropybutelucidatethevalue is assumed to be in thermodynamic equilibrium at of the entropy of a BH to which the exterior of a somenon-zerotemperature. Theentropyofthesys- QBH tends due to the continuity of the process in temiscalculatedfromintegratingthefirstlawwhen the latter case. itundergoesareversibleprocess. Ingeneral,thein- tegrationrequiresknowledgeofthematterequation 2 ofstate,butweshowthat,whenonedealswithsys- according to our general results on pressure in [9]. tems on the thresholdof forming an extremal hori- When matter is absent in the inner region, as in a zon, deep conclusions can be drawn without that thin shell, this condition is exact. When there is knowledge. The first law of thermodynamics for matter, one can write quite generally p matter(r)= r our system can be written as [11] b(r+,R) 1 r+ , valid near R = r and with the 4πR2 − R + function(cid:0) b(r+,R(cid:1)) model-dependent. Note that we TdS =dE+λdA ϕde. (3) − do not need to impose that pr(R) = 0 for R > r+, the surface can move due to thermal motion or We now go through TdS, dE, λdA, and ϕde − somethingelse,orevenbe acoldstarinwhichcase carefully. p (R) = 0. The point is that if the body is suffi- r The local temperature T on the boundary R is ciently compressed it follows that prmatter(r+) = 0 relatedtothetemperatureT atinfinitybytheTol- [9]. Thus, finally, 0 man formula 1 b(r ,R) + λ= . (6) T T 0 0 8π R T = = . (4) V(R) 1 r+ − R On the other hand, the area A is defined as A = p Since S is the entropy of the system, dS is the 4πR2, so that change of the entropy upon changing the other dA=8πRdR. (7) quantities. The quasilocal energy E is given by [11], E = The electric potential ϕ represents the difference R 1 V(R) . It is seen from this and (1) that in electrostatic potential between a reference point for(cid:16)ou−rpextrema(cid:17)l system E = r+ does not depend with potential φ0 and the boundary R with poten- on R. Thus tial φ(R) = q/R, blue-shifted from infinity to R dE =dr . (5) throughthefactor1/√V,whereV isthetimecom- + ponent of the static metric. Thus, The gravitational pressure λ is found from the φ φ(R) 0 inner region. For the inner region r R the ϕ= − . (8) ≤ V(R) metric is given as in Eq. (1). Then the grav- p itational pressure λ at the boundary at r = For de, since at the quasihorizon limit e = r , one + R equals to [11] 8πλ = 1 V(R) 1 + has R(cid:16)p − (cid:17) de=dr (9) 1 1 dV(r) + V(r) dψ(r) , where r = + (cid:18)2√V(r) dr p dr (cid:19)r=R− We are now ready to analyze the entropy of quasi- R− means that the derivatives should be taken black holes. from the inner region. It follows from the tt and rr Einstein equations for the inner region and the boundary condition ψ(R) = 0 (mandatory for a 3. Entropy of quasiblack holes and entropy smooth matching with the outer region) that ψ = of extremal black holes 4π rdr¯r¯(pr(r¯)+ρ(r¯)). Here p is the radialpressure R V(r¯) r andRρ is the energy density, both include contribu- First,letusconsiderthesimplestcase: acharged tionsfromthematterandtheelectromagneticfield, shell with a flat space-time inside and an extremal i.e., p = pmatter +p em and ρ = ρmatter +ρem. r r r RN metric outside. Then, b = 0 since there is no It also follows from the tt Einstein equation that matter inside. Also, as the potential is everywhere V(r) = 1 2m(r), with m(r) = 4π rdr¯r¯2ρ(r¯). − r 0 constant inside, one has ϕ = 0. Then, we obtain Then, for our concrete system, usingR(2) and the the first law in the form, equationfor λ, one finds after some manipulations, 8πλR = 4π prmatterR2/(1−r+/R). Now, to make TdS =dr+. (10) progress we have to understand the system at the threshold of being a QBH. We have to take into It is instructive to recall that in the case of un- account that on the quasihorizon pmatter(r ) = 0 charged shells, as treated in [12], the integrability r + 3 condition of the first law yields T = T (r ), so one obtains from (13) that for QBHs S = 0 and 0 0 + T is not a function of R in such a case. Now our from (11) that T is positive and finite, not equal 0 0 caseisanextremalchargedshellratherthananun- to zero, T >0. This latter particular case of QBH 0 chargedone. Inthiscasetheintegrabilitycondition behavior is equivalent to the prescription given in for Eq. (10) is T = T(r ), i.e., the local tempera- [2] for extremal pure BHs. + ture is a function of r alone. On the other hand + We now argue that for extremal pure BHs the the temperature at infinity has thus the form, prescription T = 0 of [2] is unsatisfactory. As 0 r 6 + pointed in [13], the prescription that T = 0 is T =T(r ) 1 , (11) 0 0 + 6 (cid:16) − R(cid:17) an arbitrary finite quantity, is inconsistent with quantum backreaction. Indeed, the correspond- It contains a dependence on R, but, as usual, it ing quantum stress-energy tensor is of the form does not depend on r. With these remarks we can Tquantν = T4fν + hν where hν is a term finite now integrate Eq. (10) and obtain µ µ µ µ everywhere. Near the horizon the first term of r+ 1 Tquantν diverges as the local temperature T di- µ S =S(r )= dr¯ , (12) + Z + T(r¯ ) verges due to the redshift factor. This unstabilizes 0 + the system and is physically inappropriate at the wherethe constantofintegrationensuresthatS semi-classical level [13]. Actually, if this were true, → 0 when the system shrinks to nothing. To be sure, the temperature of the quantum fields and that of Eq. (12) is valid for any R r+. the BH itself would not coincide, making thermal ≥ equilibrium impossible. T = 0 and S = 0 can- 0 Second,weconsideramoregeneralconfiguration, 6 not be a solution. One is left with vanishing T (T 0 with the inside havingsome type or anotherof dis- finite) and S >0 undetermined for extremal BHs. tribution of matter other than vacuum. Clearly, one has to assume that the integrability conditions Our QBH approach gives consistency to this so- forthe systemarevalid, otherwisethere is no ther- lutionofthethermodynamicextremalBHproblem. modynamic system. Then, since S is a total dif- Indeed,theresultprovidedbyEqs.(11)and(13)is ferential one can integrate along any path. Choose free of difficulties. As the local temperature T(r ) + the path R=r+(1+δ) with δ constant and small, remains finite when R r+, the quantum stress- so that dS = (something)dr+. Then one can in- energy tensor Tquantνµ o→n the quasihorizonremains tegrate this equation to obtain S. Taking then at finite or even negligible. Moreover, the first law once the limit R r+, we obtain instead of (12) of thermodynamics is also satisfied with the choice → the following equation, (13). Thus, thermal equilibrium is kept in the sys- tem, the temperature tends to the Hawking value r+ D(r¯ ) S =S(r )= dr¯ + , (13) with a suitable rate, given by (11), and S > 0 is + + Z0 T(r¯+) somehow undetermined. Can we nevertheless say somethingmoredefiniteabouttheformofthefunc- where, tion S(r )? Eq. (13) tell us that the situation D(r )=1+b ϕ , (14) + + + + − is model-dependent, it depends on D(r )/T(r ), + + b+ = b(r+,R = r+) and ϕ+ = ϕ+(r+,R = r+). which depends on the properties of the particu- In general, we only require 1 + b+ ϕ+ > 0 to lar system under study. For instance, only for − ensure the positivity of the entropy. Note that if special cases, when the quantity D(r )/T(r ) is + + the density of matter inside vanishes at r = R, we given by D(r )/T(r ) = 2πr , can we obtain the + + + return to the thin shell situation, since b+ 0, Bekenstein-Hawkingvalue A whereAistheareaof → 4 ϕ+ 0, and so D(r+)=1. the quasihorizon surface. In addition, for a given → model,changingthe parameterT(r ),say,onecan + Thus, we can state the following. For QBHs, obtainanydesirablevaluefor S,withS >0, S =0 for any finite generic T(r ), one obtains a well- + being ruled out. defined positive entropy, S > 0, from (13), as well as a vanishing temperature at infinity, T0 0, In deriving that the entropy of extremal BHs is → from (11). In addition one can consider the case in model-dependent we are not alone. We were pre- which T(r+) is not finite, T(r+) as T(r+) = ceded by the results of [4]. In [4] particular thin T0/ 1− rR+ |R→r+. In this pa→rti∞cular instance (cid:0) (cid:0) (cid:1)(cid:1) 4 shells as working material were analyzed, and the thermodynamic picture can change drastically. In Gibbs-Duhem relation (which for self-gravitating particular, the fact that we cannot simply take the systems is, in general, not valid) was used, to sup- limit T 0 but, instead, shouldconsider different 0 → port the conclusion that extremal BH entropy is waysofitsapproachingtozerodependingonT(r ) + model-dependent. Ourapproachismuchmoregen- asinEq.(11),makesthisissuemuchmoreintricate eral, makes no use of thin shells neither of the than expected. Gibbs-Duhem relation. Moreover, in deriving that Westressedthe keyroleplayedbytheQBHcon- themannerinwhichthetemperatureT approaches 0 ceptandhaveshownhowtosubstantiatethechoice zero is not well fixed, as T(r ) is a free quantity, + fortheextremalBHentropyfromathermodynamic we are also not alone. We were preceded by the stand. The result is not universal, with S = 0, results of [5] and [6]. Indeed, remarkably, on a to- T = 0 being ruled out. We used continuity argu- tally different setting and actually in a work which 0 6 mentsandsoonequestionweshouldaskiswhether raised for the first time problems connected to ex- the limiting configuration in the QBH setup yields tremalBHsalone,itwasshownin[5]thatattheex- anentropyS thatcanbeconsidertheentropyofan tremal state fluctuations on the temperature grow extremal BH. Our approach stems from taking the unbound. Our work shows the appearance of un- horizon limit of matter configurations with time- usualfeaturesinthethermaldescriptionevenwith- likeboundaries,whereasBHs havefromthe starta out considering such fluctuations. This problem is lightlike horizon. Can we trust that by continuity a quite separate non-trivial issue needing further fromthe QBHapproachwegetthe correctentropy consideration. In addition, [6] has concluded that of an extremal BH? Non-extremal QBHs yield to the notion of zero temperature is ill-defined for ex- continuity arguments [10], but there one knew the tremal BHs, whereas we defined it but in a rather result beforehand. However,now the entropy of an delicate way (see Eq. (11)), so it changes when we extremal BH is unknown, so there is no gauge to go through the referred sequence of configurations. compare with. Thus, the situation is more tricky, and though we do not possess a rigorous proof, we can add arguments in its favor. When we change 4. Conclusions m and q, approaching the extremal RN BH metric from a non-extremal one, jumps in S are not ex- cluded. However,these jumps should be connected We have obtained the expression for the entropy with jumps in the temperature. If we take the pre- in Eq. (13) (see also Eq. (12)). This expression is scriptionof[2],T changesfromT 0forthenear- valid for any R > r . We have been interested in 0 0 + ≈ extremal configuration to finite T . In contrast, in the quasiblack hole limit R r in the course of 0 + → ourQBHapproachT 0smoothlywithnosource which the temperature at infinity T obeys T 0 0 0 0 → → of discontinuity. Moreover, using the standard ap- according to Eq. (11). In this regard, we want to proachforthe entropyofanextremalBH,therere- emphasize the difference between the system un- mains the difficulty ofits calculationanddefinition der discussion and traditional thermodynamics. In within thermodynamics. If one takes the prescrip- the latter, the state is characterized by its ther- tion of [2], T is finite and arbitrary, but backre- modynamic parameters with no memory on how 0 action destroys the horizon. If, instead, one puts their values were achieved. Therefore, in the limit T to zero in accordancewith its Hawking value, it when T 0 and R r one could naively ex- 0 0 + → → is not quite clear how to obtain an entropy by dif- pect to obtain some unambiguous quantity for S ferentiating the system’s free energy with respect corresponding to R =r , T =0. Instead, our ap- + 0 to a fixed zero temperature. On the other hand, proach implies either that the entropy of extremal the QBH approach evades these problems since a QBHS, and by inference of extremal BHs, is not horizon is absent and at each stage it has a well- a full-fledged unambiguous quantity, in the sense defined small non-zero T . It seems appropriate to that any desirable value of S can be achieved by 0 considerthe limiting entropyofthe sequenceofthe tuning T(r ) say, or that T(r ) is unique and can + + QBH configurations precisely as a definition of ex- befoundonfundamentalgroundsinasemiclassical tremal BH entropy, analogously to the operational theory. 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