application/xmlThe fate of black holes in an accelerating universePedro F. González-Dı́azCarmen L. SigüenzaPhantom energyBlack holesNegative temperaturePhysics Letters B 589 (2004) 78-82. doi:10.1016/j.physletb.2004.03.060journalPhysics Letters BCopyright © 2004 Elsevier B.V. All rights reserved.Elsevier B.V.0370-26935893-410 June 20042004-06-1078-82788210.1016/j.physletb.2004.03.060http://dx.doi.org/10.1016/j.physletb.2004.03.060doi:10.1016/j.physletb.2004.03.060http://vtw.elsevier.com/data/voc/oa/OpenAccessStatus#Full2014-01-01T00:14:32ZSCOAP3 - Sponsoring Consortium for Open Access Publishing in Particle Physicshttp://vtw.elsevier.com/data/voc/oa/SponsorType#FundingBodyhttp://creativecommons.org/licenses/by/3.0/JournalsS300.2PLB20828S0370-2693(04)00514-310.1016/j.physletb.2004.03.060Elsevier B.V.Astrophysics and CosmologyThe fate of black holes in an accelerating universePedro F.González-Dı́azp.gonzalezdiaz@imaff.cfmac.csic.esCarmen L.SigüenzaColina de los Chopos, Centro de Fı́sica “Miguel A. Catalán”, Instituto de Matemáticas y Fı́sica Fundamental, Consejo Superior de Investigaciones Cientı́ficas, Serrano 121, 28006 Madrid, SpainEditor: P.V. LandshoffAbstractBabichev, Dokuchaev and Eroshenko have recently found that accretion of phantom energy onto a black hole induces a gradual decrease of the black hole mass, and that masses of all possible black holes tend to zero as a result of this process as the universe approaches the big rip singularity. The implications of these results on the thermodynamics of black holes and phantom energy are explored in this Letter, showing that such results become consistent with the two first laws of thermodynamics only if phantom energy is characterized by a negative temperature depending on the scale factor and by a positive definite entropy which tends to become constant if phantom energy largely dominates the universe. It is also argued that the loss of quantum coherence of semiclassical black hole physics is no longer present if the universe is filled with phantom energy.PACS98.80.-k04.70.-sKeywordsPhantom energyBlack holesNegative temperatureOne of the most intriguing recent results in cosmology is the discovery by Babichev, Dokuchaev and Eroshenko [1] that all black holes in a universe filled with a fluid violating the dominant energy condition will steadily loss all of their mass to fully disappear all at once at the big rip, no matter their initial mass or the moment at which they were formed. In fact these authors found that as a result of dark energy accretion the mass M of a black hole in a universe filled with a quintessence scalar field with equation of state p=ωρ, varies at a rate given by [1](1)Ṁ=4πAM2(ρ+p), where A is a dimensionless positive constant and  ̇=d/dt. Using then the equation of state p=ωρ, Eq. (1) becomes (2)Ṁ=4πAM2(1+ω)ρ. Thus, since ρ>0, accretion of dark energy leads to a black hole mass increase if ω>−1 and a mass loss if ω<−1. The stability of the Schwarzschild–de Sitter universe is ensured by the fact that a positive cosmological constant corresponds to ω=−1 for which Ṁ=0. Now, for the flat geometry, it has been obtained [2] that the most general expression of the scale factor is given by (3)a(t)=a03(1+ω)/2+32(1+ω)t2/[3(1+ω)], in which a(0)=a0 is the initial value of the scale factor at the onset of the accelerating regime. For ω>−1 the scale factor steadily increases with t tending to infinity as t→∞. On the phantom energy regime ω<−1 [3] there is a super-accelerated expansion which reaches the big rip singularity as t approaches the finite value (4)t=t∗=23(|ω|−1)a03(|ω|−1)/2. On the other hand, the integration of the cosmic conservation law for energy, ρ̇+3ρ(1+ω)ȧ/a=0, leads then to an expression for the dark energy density (5)ρ=ρ0a−3(1+ω), with ρ0=const. Inserting Eqs. (3) and (5) into Eq. (2) and integrating we finally obtain the black hole mass time-evolution equation11The authors of Ref. [1] used for a phantom-energy universe a scale factor a=(1−t/τ)−2/[3(|ω|−1)], with τ the time at the big rip, obtaining the same conclusions as in the present Letter.(6)M(t)=Mi1−(1+ω)MiṀ0t̃tt̃+3(1+ω)t/2, where Mi is the initial mass of the black hole, t̃=a03(1+ω)/2 and Ṁ0=(4πAρ0)−1.For ω>−1, M monotonically increases with time t, tending to a maximum value (7)Mmax=Mi1−2Mi3M0̇t̃, as t→∞. If ω=−1, the black hole mass remains constant, i.e., the black hole does not accrete any energy from vacuum. Finally, and more importantly, such as it was pointed out by Babichev, Dokuchaev and Eroshenko [1], if ω<−1 M steadily decreases with time t and tends to vanish as t approaches the big rip time t∗. Near that singularity, (8)M→Ṁ0t̃t̃−32(|ω|−1)t|ω|−1, which does not depend on the initial mass of the black hole. According to Babichev, Dokuchaev and Eroshenko [1], that result means that all black holes in a universe filled with phantom energy will tend to be equal as the big rip is approached, and that phantom-energy accretion prevails over Hawking radiation, at least semiclassically.That result can be generalized by considering the case where a positive cosmological constant Λ=3λ is added to the dark energy fluid. Then the scale factor a(t) for t0=0 reads [2](9)a(t)=2πG3Cλ1/[3(1+ω)]e3(1+ω)λt/2−Ce−3(1+ω)λt/22/[3(1+ω)], where (10)C=λ+8πGa0−3(1+ω)/3−λλ+8πGa0−3(1+ω)/3−λ, with a0 the initial value of the scale factor. Since 0<C<1 we can also have a big rip singularity for ω<−1. That singularity takes now place at a time given by (11)t=t∗=−lnC3(|ω|−1)λ. Using solution (9) we can now obtain from Eqs. (2) and (5)(12)M(t)=Mi1−MiλṀ0e3(1+ω)λt−1e3(1+ω)λt−C, where now Ṁ0=(1−C)/(2Cρ0). For ω>−1, M again monotonically increases with time t towards a maximum Mmax=Mi/(1−Miλ/Ṁ0), which also occurs at t=∞, and remains constant when ω=−1. For ω<−1, Eq. (12) can be cast in the form (13)M(t)=Mi1+MiλṀ01−e−3(|ω|−1)λte−3(|ω|−1)λt−e−3(|ω|−1)λt∗. Once again for ω<−1 M decreases with time t, tending to vanish on the neighborhood of t=t∗. Thus, all behaviours obtained from solution (3) are matched when one uses solution (9), so confirming the result pointed out by Babichev, Dokuchaev and Eroshenko [1].At first sight, that result can, however, get into serious conflict with thermodynamics. On one hand, one could always assume the current existence of semiclassical black holes with very small masses and hence very large temperatures which induced Hawking radiation prevailing over accretion of the phantom fluid which should be characterized by quite lower positive temperatures [4]. On the other hand, one should also expect that if the above result holds, then the generalized entropy of the universe S+Sbh (where S and Sbh∝M2 are the entropy of the phantom fluid and of the black hole, respectively) decreased with time, so violating the generalized second law. In what follows we are going to show nevertheless that none of these two apparent difficulties actually holds because, contrary to a recent claim [4], the temperature of the phantom fluid is definite negative which would allow us to re-interpret results from a different, “quantum” standpoint.It has been in fact recently pointed out [4] that the entropy for a phantom energy fluid is always negative, and that therefore a cosmic regime with phantom energy should be ruled out. There exists, however, a rather general argument which is valid for all conceivable cosmic phantom-energy models that prevents that conclusion. In fact, if the first law of thermodynamics is assumed to hold, then in a general Friedmann–Robertson–Walker flat universe filled with dark energy satisfying the equation of state p=ωρ (with ω=const) the entropy per comoving volume stays constant [5], while the temperature of the universe is given by the general expression (14)T=κ(1+ω)a−3ω, where κ is a positive small constant. From Eqs. (5) and (14) one can readily derive the following generalized Stefan–Boltzmann law (15)ρ=ρ0Tκ(1+ω)(1+ω)/ω. Note that for ω=1/3, Eq. (15) consistently reduces to the usual law for radiation, and that for 0>ω>−1 the dark energy density decreases with the temperature. However, for the regime where ω<−1, in order to preserve ρ positive, we must necessarily take T<0, which is a condition that really directly stems from Eq. (14). It follows that ρ will always increase with |T| along the entire phantom regime. The energy of such a regime is bounded from above and allows therefore the occurrence of a negative temperature [6]. Since on the phantom regime ρ increases with the scale factor a(t) it also follows that |T| increases as the universe expands on that regime. Finally, a general expression for the dark energy entropy can also be obtained [4] which in the present case reads (16)S=C0T1+ω1/ωV, where C0 is a positive constant and V is the volume of the considered portion within the dark energy fluid. It follows that, contrary to the claim in Ref. [4], the entropy of a dark-energy universe is always positive, even on the phantom regime. Actually, by inserting Eq. (14) into Eq. (16) one in fact attains that S=const when we take V as the volume of the entire universe.It is nonetheless the temperature which becomes negative instead of entropy for ω<−1. Even though it is not very common in physics and therefore can be listed as just another more weird property of the phantom scenario, a negative temperature is not unphysical or meaningless [7]. Systems with negative temperature have already been observed in the laboratory and interpreted theoretically. In the case of phantom energy it means that the entropy of a phantom universe monotonically decreased if one would be able to add energy to that universe. Hence, a ω<−1 universe would always be “hotter” that any ω>−1 universe, and if two copies of the universe were taken, one with positive and other with negative temperature, and put them in thermal contact, then heat would always flow from the negative-temperature universe into the positive-energy universe. It could yet be argued that negative temperature is a quantum-statistical mechanics phenomenon and therefore cannot be invoked in the classical realm. However, a negative temperature given by Eq. (14) when ω<−1 [3] can still be heuristically interpreted along a way analogous to how, e.g., black hole temperature can be interpreted (and derived) without using any quantum-statistical mechanics arguments; that is by simply Wick rotating time, t→iτ, and checking that in the resulting Euclidean framework τ is periodic with a period which precisely is the inverse of the Hawking temperature [8]. Thus, the Euclideanized black hole turns out to be somehow “quantized”. Similarly in the present case, the phantom regime can be obtained by simply Wick rotating the classical scalar field [9], φ→iΦ, which can be generally seen to be equivalent to rotating time so that t→iτ, too. It is in this sense that the phantom fields are also “quantized” and that the emergence of a negative temperature in the phantom regime becomes consistent.In any event, since a system with negative temperature is “hotter” than any other systems having positive temperature, even if that temperature is +∞ [7], we see that the first of the two thermodynamical difficulties pointed out above becomes fully solved, at least in the case that ω<−1. On the other hand, if the universe is assumed to contain a black hole of mass M, the entropy of the dark energy fluid will be smaller than that given by (16) (when V∝a3), and can be expressed as (17)S=C0κ1/ω1−VbhV, where Vbh∝M3 is the volume occupied by the black hole and V is now the volume of the entire universe. Thus, for ω>−1 S decreases and for ω<−1 S increases, as dark energy is accreted onto the black hole. Moreover, after varying S with respect to Vbh and multiplying by T, one can obtain from Eq. (17)(18)δS∝±(1+ω)δET=−(ρ+p)δVbhT, where δE=∓ρδVbh and the upper sign corresponds to ω>−1 and the lower sign corresponds to ω<−1. On the other hand, if on the black-hole spacetime we take t∝M, we can derive from (1)(19)δSbh∝(1+ω)δETbh, where Sbh∝M2 and Tbh∝M−1 are the entropy and temperature of the black hole.Let us first analyze the total balance of entropy in the case of dark energy with ω>−1. Thus, from Eqs. (18) and (19) we see that the accretion of a given amount of dark energy onto the black hole leads to an increase of dark energy entropy which will exceed the corresponding decrease of black hole entropy, so preserving the second law, provided that T>Tbh, i.e., if Mbha−3ω⩾1, a condition which appears to be of general applicability along the entire late accelerating evolution of the universe. As to the case of phantom energy for ω<−1, we ought to recall that according to the Carnot equality the coupling between a negative-temperature system and a positive-temperature system leads to a Carnot engine of greater than 100 percent efficiency [10]. Thus, in the present case (20)F=1+Tbh|T|>1, so that the generalized entropy S+Sbh ought to inexorably decrease along phantom energy accretion, so violating the second law. In fact, Eqs. (18) and (19) tell us that in the phantom case the entropy S increases by an amount which is smaller than the decrease of Sbh provided the reasonable condition |T|>Tbh again holds. This violation of the second law was to be expected because, as it was mentioned earlier, negative temperatures are compatible with observationally checked “quantum” Carnot equalities violating the second law [10]. If all existing black holes will simultaneously disappear at the big rip leaving no Hawking radiation, then the information initially lost during formation of the black holes should be recovered. Actually, the so-called quantum coherence loss paradox, long championed by Hawking [11], according to which an initial pure state is transformed into a final mixed state during the whole process of black hole formation and subsequent complete evaporation, is here naturally solved in at least the phantom-energy regime in the sense that neither Hawking radiation nor black holes are left in the final state at the big rip+naked black hole singularity.AcknowledgementsThe authors thank Eugeny Babichev and Ademir Lima for useful explanations and enlightening correspondence. This work was supported by MCYT under Research Project No. BMF2002-03758.References[1]E.BabichevV.DokuchaevYu.Eroshenkogr-qc/0402089[2]P.F.González-Dı́azPhys. Rev. D682003084016[3]R.R.CaldwellPhys. Lett. 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