Entropy in the interior of a black hole and thermodynamics
aa r X i v : . [ g r- q c ] O c t Entropy in the interior of a black hole and thermodynamics
Baocheng Zhang ∗ School of Mathematics and Physics,China University of Geosciences, Wuhan 430074, China
Abstract
Based on a recent proposal for the volume inside a black hole, we calculate the entropy associatedwith this volume and show that such entropy is proportional to the surface area of the black hole.Together with the consideration of black hole radiation, we find that the thermodynamics associatedwith the entropy is highly possible to be caused by the vacuum polarization near the horizon. ∗ Electronic address: [email protected] . INTRODUCTION How big is a black hole? This is not an easy question, since the definition of the volumeof the space inside a black hole depends on how the spacetime is sliced into space and time,unlike the surface area of the black hole that is the same for all observers. Intuitively,a nice description for the volume should be slicing invariant, which was made firstly byParikh [1] and also discussed by others [2–6]. Recently, a different method was suggested byChristodoulou and Rovelli [7] based on a simple observation that the interior of Schwarzschildblack holes is not static, which leaded to another sensible description for the volume as thelargest volume bounded by the event horizon of a black hole. For a Schwarzschild blackhole, they showed that at late time the volume took such an expression, V CR ∼ √ πM υ (1)where M is the mass of the black hole and v is the advanced time. In the paper, we willcall the volume as CR volume for brevity. Instantly, this result was extended to otherbackgrounds of spacetime [8–10].Relative to the specific forms for the volume inside a black hole, ones concern more aboutits significance, that is, why do we want to investigate the volume? This was regarded tobe relevant to the black hole information loss paradox in past studies. So it is unavoidableto involve the black hole radiation [11] and the corresponding thermodynamics [12, 13] insuch studies, but up to now they only were involved with the qualitative discussions. In thispaper, we will try to make some quantitative calculations to estimate the entropy associatedwith the CR volume.As well-known, a black hole can emit the thermal radiation that makes the black hole havea lifetime ∼ M and thus the CR volume inside the black hole has an extraordinary formthat is proportional to M . A natural question arises: for such a large volume, how manyfield modes can be included in it. Furthermore, whether these modes are relevant to theinterpretation of Bekenstein-Hawking entropy statistically. When the radiation happens, thebackground geometry will be altered according to the semiclassical Einstein equation, which,in equilibrium, can be described with the first law of black hole thermodynamics. Thus onemight ask: whether the maximal volume is also included in the background geometry. If thevolume changes, how is the first law changed? In this paper, we will work on these problems.2he structure of the paper is as follows. Firstly, we will revisit the definition of thevolume inside a black hole by Christodoulou and Rovelli, and present explicitly the choiceof the maximal hypersurfaces [14] by maximal slicing [15, 16] in the second section. Then,we calculate the entropy in the volume using the standard statistical method in the thirdsection. In the forth section, we discuss the first law of black hole thermodynamics, inparticular for the newly obtained entropy, which is related to the vacuum polarization [17]near the horizon of the black hole. Finally, we will give a conclusion in the five section. II. BLACK HOLE VOLUME AND MAXIMAL SLICING
Start with the geometry of a collapsed object as in Ref. [7], which can be described withthe Eddington-Finkelstein coordinates, ds = − f ( r ) dv + 2 dvdr + r dθ + r sin θdφ (2)where f ( r ) = 1 − Mr and the advanced time v = t + R drf ( r ) = t + r + 2 M ln | r − M | . Inparticular, we have taken the units G = c = ~ = k B = 1. The advantage of the coordinatesover the static one is that there is no coordinate singularity at the event horizon. Thus it canbe analytically continued to all r >
0, which is required for the description of the geometryof the collapsed matter.In order to calculate the volume, a proper hypersurface has to be chosen. Withan transformation v → v ( T, λ ) , r → r ( T, λ ), the coordinates (2) becomes ds = (cid:16) − f (cid:0) ∂v∂T (cid:1) + 2 ∂v∂T ∂r∂T (cid:17) dT + (cid:16) − f (cid:0) ∂v∂λ (cid:1) + 2 ∂v∂λ ∂r∂λ (cid:17) dλ + r dθ + r sin θdφ where we haveassumed the cross term vanishes by taking the transformation properly. In particular, if thecondition − f (cid:0) ∂v∂T (cid:1) +2 ∂v∂T ∂r∂T = − dv = − √− f dT + dλ, dr = √− f dT which also removes the cross term simultaneously) is enforced, one will find thatthe hypersurface Σ: T =constant, is that chosen in Ref. [7] where the spherically symmetrichypersurface is taken as the direct product of a 2-sphere and an arbitrary curve parameter-ized by λ in v - r plane. In particular, it is noted that the hypersurface T =constant is ableto be gotten by r =constant according to the transformation dr = √− f dT .The hypersurface Σ, as in Ref. [7], is coordinatized by λ, θ, φ , and the line element of theinduced metric on it can be expressed as ds = (cid:0) − f ( r ) ˙ v + 2 ˙ v ˙ r (cid:1) dλ + r dθ + r sin θdφ (3)3here the dot represents the partial derivative with regard to the parameter λ , and − f ( r ) ˙ v +2 ˙ v ˙ r > V Σ = 4 π Z dλ p r ( − f ( r ) ˙ v + 2 ˙ v ˙ r ) . (4)with a proper choice of the curves. An investigation for geodesics in an auxiliary manifoldgave the maximization condition by choosing the curves [7], r = 32 M, (5)which, together with Eq. (4), gives the CR volume expressed in Eq. (1).On the other hand, the maximization can also be calculated in mathematical relativity[18] where a method called as maximal slicing can lead to hypersurfaces of maximal volumewhich have vanishing mean extrinsic curvature, K = 0, where K is the trace of the extrinsiccurvature of the hypersurface. Here we show the vanishing K is equivalent to the condition(5). According to Ref. [16], we use the coordinates (2) and take the spacelike hypersurfacesby r =constant, since the time and space are regarded as being interchanged across thehorizon of a Schwarzschild black hole. Take n as the future pointing timelike unit normalto the hypersurfaces Σ r , n = p − f (cid:18) ∂∂r + 1 f ∂∂v (cid:19) . It is easy to confirm that g µν n µ n υ = − n and the trace of the extrinsiccurvature tensor K µν , K = − ▽ · n where ▽ is the covariant derivative with respect to thespacetime metric. With the given metric (2), we have K = 12 √− f (cid:18) ∂f∂r + 4 f r (cid:19) = 0 , which gives the equation r = M , consistent with the condition (5). III. ENTROPY IN THE VOLUME
Since the CR volume is obviously different from the normal volume that might still existinside the black hole, it is significant to investigate how many modes of quantum fields canbe included in CR volume. In the paper, we involves only the scalar field. According to thestandard quantum statistical method [19], the number of quantum states in some volume4as to be counted for one certain phase-space which can be labeled here by λ, θ, φ, p λ , p θ , p φ .From the uncertainty relation of quantum mechanics, ∆ x i ∆ p i ∼ π , one quantum state cor-responds to a “cell” of volume (2 π ) in the phase-space. Therefore, the number of quantumstates is given by dλdθdφdp λ dp θ dp φ (2 π ~ ) . (6)In order to calculate the integral, we consider a massless scalar field Φ in the spacetimewith the coordinates, ds = − dT + (cid:0) − f ( r ) ˙ v + 2 ˙ v ˙ r (cid:1) dλ + r dθ + r sin θdφ . (7)It is noted that this metric is equivalent to such an form: ds = − dT + H ( T ) dλ + r ( T ) dθ + r ( T ) sin θdφ , which means that it is not static for the defined time T in the interior ofthe black hole. For this reason, sometimes the interior of the black hole is interpreted as acosmological model (see also Ref. [20]), and it evolves towards the singularity of the blackhole. It is not entirely clear what should be done for the statistics in a dynamic background,but fortunately, our calculation is made at v >> M and r = M . For the maximal slicing,the slices accumulate on a limiting hypersurface r = M when t is large enough (that isguaranteed by v >> M ) [15]. That is to say, near the maximal hypersurface, the proper timebetween two neighbouring hypersurfaces tends to zero as t increases, so nearly no evolutionhappened there. Thus our statistical calculation is not affected by the non-static characterof the metric, since it is calculated on approximately T =constant which is the hypersurfacethat leads to the CR volume. Therefore, in what follows we will use the common method inthe curved spacetime to discuss the motion of scalar field in the interior of the black hole.Using the WKB approximation, the field Φ can be written as Φ =exp[ − iET ] exp[ iI ( λ, θ, φ )], and then substituting it into the Klein-Gordon equationin curved spacetime, √− g ∂ µ ( √− gg µν ∂ ν Φ) = 0, we obtain E − − f ( r ) ˙ v + 2 ˙ v ˙ r p λ − r p θ − r sin θ p φ = 0 , (8)where p λ = ∂I∂λ , p θ = ∂I∂θ , p φ = ∂I∂φ . Thus, following the Eq. (6), the number of quantum states5ith the energy less than E is obtained as g ( E ) = 1(2 π ) Z dλdθdφdp λ dp θ dp φ = 1(2 π ) Z p − f ( r ) ˙ v + 2 ˙ v ˙ rdλdθdφ Z r E − r p θ − r sin θ p φ dp θ dp φ = 1(2 π ) Z p − f ( r ) ˙ v + 2 ˙ v ˙ rdλdθdφ (cid:18) π E r sin θ (cid:19) = E π (cid:20) π Z dλ p r ( − f ( r ) ˙ v + 2 ˙ v ˙ r ) (cid:21) = E π V CR , (9)where the relation p λ = p − f ( r ) ˙ v + 2 ˙ v ˙ r q E − r p θ − r sin θ p φ is used in the second line,the integral formula R R q − x a − y b dxdy = π ab is used in the third line, and in thefinal line we have used the condition (5). As expected, the number of quantum states isproportional to the CR volume, but this is still different from a normal situation, becausethe CR volume is the result of the curved spacetime.Temporarily ignoring the exotic feature of CR volume, we can continue to calculate thefree energy at some inverse temperature β , F ( β ) = 1 β Z dg ( E ) ln (cid:0) − e − βE (cid:1) = − Z g ( E ) dEe βE − − V CR π Z E dEe βE − − π V CR β . (10)Furthermore, the entropy is obtained as S CR = β ∂F∂β = π V CR β , (11)which looks like the entropy in the normal volume.Now we turn to the CR volume again, but under the consideration with Hawking radia-tion. Since Hawking radiation is thermal, the loss mass rate of a Schwarzschild black holecan be given by the Stefan–Boltzmann law, dMdv = − γM , γ > , (12)6here γ is a constant whose value does not influence the discussion in the paper. In Ref.[10], it is discussed that the large volume is remained until the final stage of black holeevaporation. Here we are concerned about the time that the radiation can last. Thus, for ablack hole with the mass M , we have v ∼ γM , (13)which also satisfies the requirement in Ref. [7], v >> M . Then using the Eqs. (1) and (11),and considering the temperature for a Schwarzschild black hole, β = T − = 8 πM , one findthe entropy S CR ∼ (cid:0) √ γ (cid:1) M (45 × ) = (cid:0) √ γ (cid:1) (90 × ) π A, (14)where A = 16 πM is the surface area of the Schwarzschild black hole. This is a surprisingand intriguing result that the entropy from quantum theory in the CR volume is proportionalto the surface area of the black hole horizon that bounds the volume. Note that the resultis dependent on the validity of the relation (12), which was shown [21] to hold so long asthe mass of a Schwarzschild black hole is greater than the Planck mass. Moreover, theexact description of final stage of black hole evaporation has not existed, but some specificconsiderations such as General Uncertainty Relation (i.e. see the review [22]) implied thatat the final stage, the loss mass rate becomes dMdv ∼ M [23], which is inconsistent withthe requirement for the calculation of the CR volume, beyond our problem obviously. Onthe other hand, although the entropy S CR is proportional to the surface area, a roughestimation finds that the parameter before the area is much smaller than . Then a naturalproblem: what is the entropy S CR ? Whether it will be included in the first law of black holethermodynamics. In what follows, we will discuss it. IV. FIRST LAW AND VACUUM POLARIZATION
Since the derivation of CR volume is in the case of v >> M , which means the black holehas formed by the collapse of the matter and is static for the external observers. Thus, dueto the Hawking radiation, the first law of black hole thermodynamics reads as dM = T dS H , (15)where the entropy S H = A . This relation was obtained [11, 12] by the analogy of blackhole physics to thermodynamics which will reach an equilibrium state after the relaxation7rocesses are completed. The change of the entropy S H is equivalent to the change of thesurface area, so the entropy S CR in the CR volume will be changed. Although our earlieranalysis shows that the entropy S CR is proportional to the surface area, its original form inEq. (11) is closely related to the CR volume. So in equilibrium, the thermal process requiresa term relevant to the change of CR volume. According to the general thermodynamics,such term should be written as P dV CR where P is the pressure. Thus, we expect that thefirst law can be written as dM = T dS − P dV CR , (16)where S = S H + S CR . In order to remain the validity of the first law, a relation is required, T dS CR ∼ P dV CR . Now, the vital problem is that from where and how much the pressure is,after the collapsed matter had been concentrated into the singularity (it is also noted thatsome studies pointed out that the nonlocal effect will prevent the black hole from collapsinginto a physical singularity, i.e. see Ref. [24]).In the original calculation of Hawking [11], the concept of particles are used in the asymp-totically flat region far from the black hole where they can be unambiguously defined. Whilethe particle flux carries away the positive energy, an accompanying flux of negative energygoes into the hole across the horizon, which can only be understood by the zero point fluc-tuations of local energy density in the quantum theory. This phenomena, called also asvacuum polarization, will play an important role in the neighborhood of the black hole.The vacuum polarization is usually considered in the semi-classical Einstein gravity, inwhich the fluctuations of gravitational field are small and so is the expectation value of theenergy-momentum tensor of the relevant quantized fields in the chosen vacuum state. Bysolving the modified Einstein equation, G µν = 8 π h T µν i , the quantum pressure at the horizoncaused by the vacuum polarization is given by [25–28], P = 1(90 × ) π M . (17)It is noted that the CR volume is calculated for the hypersurface r = M , while P is at thehorizon. But it is not difficult to clarify this, since the CR volume is just the volume forthe black hole, and the boundary for the volume is still the horizon. This point was alsoseen in Ref. [7], by the parameter λ that take the range covering the region from r = 0 to r = 2 M . In particular, some recent interesting studies considered the pressure as stemming8rom the cosmological constant, i.e. see Refs. [29–31], but they are different from ours sincethe cosmological constant is not involved here.The term P dV CR is calculated as P dV CR = 5 (cid:0) √ γ (cid:1) (90 × ) π dM, (18)which is on the level of 10 − for the parameter before dM , and is very small, as expected.The term T dS CR is also easy to be estimated as T dS CR = 4 (cid:0) √ γ (cid:1) (90 × ) π dM ∼ − dM. (19)It is seen that the term T dS CR is not equal exactly to the term P dV CR , but fundamentallythey can be canceled on the level of 10 − . The reason is that the values of the pressure P and the temperature T is not so exact in the present consideration. Actually, a directevidence that the thermodynamics in the CR volume is caused by vacuum polarization isthat the term P dV CR is not dependent on the black hole parameter M , as obtained for theterm T dS CR .Finally, it is noted that if taking such a term V CR dP , as made in Refs. [29–31], it canbalance exactly the change of thermodynamics caused by T dS CR , but no evidence showsthat why the pressure here should be changed, so the exact cancelation might be occasional.In fact, vacuum polarization causes the quantum correction to the usual black hole thermo-dynamics, so the corresponding thermodynamic variables would be corrected, which is thereason why the terms P dV CR and T dS CR cancel each other out approximately. V. CONCLUSION
In the paper, we have investigated the relation between the derivation of the volumeof the space inside a Schwarzschild black hole defined by Christodoulou and Rovelli andthe maximal slicing, and found explicitly that the CR volume was just obtained for thehypersurface whose mean extrinsic curvature is zero. We have also calculated the entropyin the CR volume through counting the number of quantum states in the volume with astandard statistical method. Differently from the normal situation, the entropy associatedwith the CR volume is proportional to the surface area of the black hole, but the parameter ismuch smaller than that required for the Bekenstein-Hawking entropy. The small parameter9as also been interpreted in the paper, from the perspective of black hole thermodynamics,for which a suggestive result is given that the thermodynamics associated with the entropyin the CR volume is caused by the vacuum polarization near the horizon, since the matterhas collapsed into the singularity when we investigate this phenomena. Thus, our resultverifies further the relation between black hole physics and quantum theory again.
VI. ACKNOWLEDGE
The author would like to thank Prof. C. Rovelli for reading this paper and his positivecomments. This work is supported by Grant No. 11374330 of the National Natural ScienceFoundation of China and by Open Research Fund Program of the State Key Laboratory ofLow-Dimensional Quantum Physics and by the Fundamental Research Funds for the CentralUniversities, China University of Geosciences (Wuhan)(CUG150630). [1] M. K. Parikh, Phys. Rev. D 73, 124021 (2006).[2] D. Grumiller, arXiv: 0509077 [gr-qc].[3] B. S. DiNunno and R. A. Matzner, Gen. Relativ. Gravi. 42, 63 (2009).[4] W. Ballik and K. Lake, arXiv: 1005.1116 [gr-qc].[5] M. Cvetiˇc, G. W. Gibbons, D. Kubizn´ak, and C. N. Pope, Phys. Rev. D 84, 024037 (2011).[6] W. Ballik and K. Lake, Phys. Rev. D 88, 104038 (2013).[7] M. Christodoulou and C. Rovelli, Phys. Rev. D 91, 064046 (2015).[8] I. Bengtsson, E. Jakobsson, arXiv:1502.01907 [gr-qc].[9] Y. C. Ong, JCAP 04, 003 (2015).[10] Y. C. Ong, arXiv:1503.08245 [gr-qc].[11] S. W. Hawking, Nature , 30 (1974); S. W. Hawking, Commun. Math. Phys. , 199 (1975).[12] J. D. Bekenstein, Phys. Rev. D , 2333 (1973).[13] J. M. Bardeen, B. Carter, and S. W. Hawking, Commun. Math. Phys. 31, 161 (1973).[14] J. E. Marsden and F. J. Tipler, Phys. Rep. 66 109 (1980).[15] F. Estabrook, H. Wahlquist, S. Christensen, B. DeWitt, L. Smarr, and E. Tsiang, Phys. Rev.D 7, 2814 (1973).
16] I. Cordero-Carri´on, J. M. Ib´a˜nez, and J. A. Morales-Lladosa, J. Math. Phys. 52, 112501 (2011).[17] V. P. Frolov and I. D. Novikov,
Black Hole Physics: Basic Concepts and New Developments (Kluwer Academic Publishers, Dordrecht, Netherlands, 1998).[18] E. Gourgoulhon, arXiv: qr-qc/0703035[19] B. Cowan,
Topics in Statistical Mechanics (Royal Holloway, Imperial College Press, London,UK, 2005).[20] S. M. Carroll, M. C. Johnson, and L. Randall, J. High Energy Phys. 11 (2009) 094.[21] S. Massar, Phys. Rev. D 52, 5857 (1995).[22] S. Hossenfelder, Living Rev. Relativity 16, 2 (2013).[23] L. Xiang, Phys. Lett. B 540, 9 (2002).[24] A. Saini and D. Stojkovic, Phys. Rev. D 89 044003 (2014).[25] W. G. Unruh, Phys. Rev. D 14, 870 (1976).[26] P. Candelas, Phys. Rev. D 21, 2185 (1980).[27] D. N. Page, Phys. Rev. D 25, 1499 (1982).[28] T. Elster, Phys. Lett. A 94, 205 (1983).[29] D. Kastora, S. Rayb, and J. Traschena, Classical Quantum Gravity 26, 195011 (2009).[30] B. P. Dolan, Classical Quantum Gravity 28, 235017 (2011).[31] D. Kubiznak and R. B. Mann, arXiv: 1404.2126.(Royal Holloway, Imperial College Press, London,UK, 2005).[20] S. M. Carroll, M. C. Johnson, and L. Randall, J. High Energy Phys. 11 (2009) 094.[21] S. Massar, Phys. Rev. D 52, 5857 (1995).[22] S. Hossenfelder, Living Rev. Relativity 16, 2 (2013).[23] L. Xiang, Phys. Lett. B 540, 9 (2002).[24] A. Saini and D. Stojkovic, Phys. Rev. D 89 044003 (2014).[25] W. G. Unruh, Phys. Rev. D 14, 870 (1976).[26] P. Candelas, Phys. Rev. D 21, 2185 (1980).[27] D. N. Page, Phys. Rev. D 25, 1499 (1982).[28] T. Elster, Phys. Lett. A 94, 205 (1983).[29] D. Kastora, S. Rayb, and J. Traschena, Classical Quantum Gravity 26, 195011 (2009).[30] B. P. Dolan, Classical Quantum Gravity 28, 235017 (2011).[31] D. Kubiznak and R. B. Mann, arXiv: 1404.2126.