glassy phase transition and stability in black holes
TRANSCRIPT
Eur. Phys. J. C (2010) 70: 317–328DOI 10.1140/epjc/s10052-010-1443-y
Regular Article - Theoretical Physics
Glassy phase transition and stability in black holes
Rabin Banerjee1,a, Sujoy Kumar Modak1,b, Saurav Samanta2,c
1S.N. Bose National Centre for Basic Sciences, JD Block, Sector III, Salt Lake, Kolkata 700098, India2Narasinha Dutt College, 129, Belilious Road, Howrah 711101, India
Received: 21 May 2010 / Revised: 25 August 2010 / Published online: 6 October 2010© Springer-Verlag / Società Italiana di Fisica 2010
Abstract Black hole thermodynamics, confined to thesemi-classical regime, cannot address the thermodynamicstability of a black hole in flat space. Here we show thatinclusion of a correction beyond the semi-classical approx-imation makes a black hole thermodynamically stable. Thisstability is reached through a phase transition. By usingEhrenfest’s scheme we further prove that this is a glassyphase transition with a Prigogine–Defay ratio close to 3.This value is well within the desired bound (2 to 5) for aglassy phase transition. Thus our analysis indicates a veryclose connection between the phase transition phenomenaof a black hole and glass forming systems. Finally, we dis-cuss the robustness of our results by considering differentnormalisations for the correction term.
1 Introduction
Black holes are the most striking predictions of Einstein’sgeneral theory of relativity of gravitation. Acceptance of thepossible identification of black holes as thermodynamicalsystems has unanimously evolved over the years.
The study of black hole thermodynamics is thereforequite important. However, if one analyses properties likethermodynamical stability, phase transition, black holeevaporation etc. the semi-classical results in the canonicalensemble are not much encouraging [1]. This is because thespecific heat is always negative and therefore in a canoni-cal ensemble it is not possible to attain a stable equilibriumwith the surroundings. However, results in a microcanonicalensemble indicate the possibility of stability [1, 2].
The aim of the present paper is to make a thoroughanalysis of the possibility of phase transition and stability
a e-mail: [email protected] e-mail: [email protected] e-mail: [email protected]
in black holes, within the framework of a canonical ensem-ble. As a specific example we consider the Kerr black hole,which provides a striking difference from the usual spheri-cally symmetric solutions of Einstein’s equation. The mainpoint is the appearance of a work term like Ω dJ , whereΩ and J are the angular velocity and angular momentum,respectively, in the first law of black hole thermodynam-ics. As shown later, this is the crucial term for classify-ing a phase transition. As we have already hinted in theprevious paragraph it is necessary to go beyond the stan-dard semi-classical approximation to discuss phase transi-tion. The results for various thermodynamic variables, forthe Kerr black hole, beyond the semi-classical approxima-tion were recently obtained by two of us [3]. In particu-lar, the entropy was shown to receive logarithmic correc-tions. Such corrections have some interesting applicationsin braneworld cosmology [4]. In this picture the FRW equa-tions get modified leading to a different early/late time cos-mology. Also, it has been suggested that [4] in the presenceof logarithmic corrections, small black holes can be stableeven in a flat background. Our calculations in this paper ac-tually quantify such suggestions by systematically analysingphase transition and aspects of stability. Although the loga-rithmic corrections to be used here were obtained in [3] theutility of such terms in the context of phase transition wereneither stated nor analysed in [3]. This is the new input inthis paper.
Using the corrected expressions of different thermody-namic entities derived in [3] we show that there exists aphase transition through which a black hole can switch overto a stable phase from an initial unstable phase. The conti-nuity of the corrected entropy for any value of the black holemass rules out the existence of a first order phase transition.However, we find that there is a discontinuity in specific heatfor a critical value of the mass. At this mass the black holetemperature is maximum. Then it sharply decreases finallyattaining the absolute zero for a vanishing mass. Thereforethis indicates a transition from an unstable phase to a stable
318 Eur. Phys. J. C (2010) 70: 317–328
phase. Having found this possibility of phase transition, wethen analyse the nature of this phase transition. We first de-rive two Ehrenfest like equations for the rotating Kerr blackhole. If this phase transition is second order in nature theEhrenfest equations must be satisfied. A numerical analysisreveals that the first Ehrenfest equation is satisfied for a longrange of angular velocity but the second Ehrenfest equationis violated. Hence a second order phase transition is alsoruled out.
The final possibility is the occurrence of a glassy phasetransition. It is found from experiments that in a liquid toglass phase transition, the specific heat (Cp), compressibil-ity (k) and thermal expansivity (α) i.e. the second orderderivatives of the Gibbs free energy, make a smeared jump.For this type of transition the Prigogine–Defay (PD) ratio[5] measures the deviation from the second Ehrenfest equa-tion. For a second order phase transition Π = 1 exactly butfor a glassy phase transition Π varies from 2 to 5.
In the present paper we have shown that the second deriv-atives of free energy suffer a smeared discontinuity at thephase transition point and the PD ratio as found by us forthe Kerr black hole is nearly 3. This enables us to concludethat the transition of the black hole from an unstable to astable phase is a glassy phase transition.
In this paper the coefficient of the correction term to thesemi-classical entropy is taken to be 1
90 which was obtainedin [3]. To check the robustness of our conclusion it is de-sirable to take other normalisations. However, here we arefaced with a lack of data. This is because, apart from ourwork [3], there is no other computation which explicitlygives the coefficient of the logarithmic term for the Kerrblack hole. We therefore choose arbitrarily some values forthis normalisation. Specifically we have taken this factor tobe 1
2 (which is also less than 1, like 190 ) and 3
2 (which isgreater than 1) as well as 1. In all cases we observe a glassyphase transition with the same PD ratio. This establishes thesoundness of our conclusion.
The novelty of this paper is not only confined to its resultsbut also to the approach. The issues related to the thermody-namic stability of a black hole have never been explored bygoing beyond the semi-classical regime. As we have shownit is essential to go beyond this approximation to study ther-modynamic stability through a phase transition. Of coursethere is a wide literature on the Hawking–Page [29] phasetransition which however occurs only in a black hole of AdSspace [30–37]. In this paper, on the other hand we discussphase transition and thermodynamic stability in an asymp-totically flat Kerr black hole.
The paper is organised in the following manner. In Sect. 2we study the Kerr black hole as a thermodynamical system,staying within the semi-classical regime. In the next sectionwe go beyond the semi-classical approximation and discovera phase transition for the Kerr black hole. Section 4 is de-voted to analyse the nature of this phase transition. Here
we show that it is a glassy phase transition which makesthe Kerr black hole thermodynamically stable. We give ourconclusions in Sect. 5. There are two appendices presentedat the end of the paper. In Appendix A we derive two Ehren-fest’s equations considering the Kerr black hole as a thermo-dynamical system and in Appendix B basic results are com-piled to prove the robustness of our conclusions for variousnormalisations of the correction (logarithmic) term.
2 Kerr black hole as a thermodynamical system;a semi-classical study
The Kerr black hole is an axisymmetric, non-static solu-tion of the Einstein equation having two Killing vectors. Ithas two conserved parameters mass (M) and angular mo-mentum (J ) corresponding to those vectors. The location ofouter/event (r+) and inner/Cauchy (r−) horizons are givenby
r± = M ±√
M2 − a2, (1)
where
a = J
M(a �= 0) (2)
is the angular momentum per unit mass. The extremal condi-tion (where the inner and the outer horizon coincide) for theKerr black hole is given by M2 = J 2
M2 . The angular velocityand area of the event horizon are defined as [6]
ΩH = a
r2+ + a2(3)
and
A = 4π(r2+ + a2) (4)
respectively. The semi-classical Hawking temperature andentropy of the rotating Kerr black hole are given by [6]
T = �κ
2π= �
2π
√M2 − a2
(r2+ + a2)(5)
and
S = A
4�= π(r2+ + a2)
�(6)
respectively. Note that for the extremal Kerr black holethe semi-classical Hawking temperature vanishes. To cal-culate the semi-classical specific heat it is desirable to firstmake the appropriate identifications. Comparison betweenthe work terms of the first law of thermodynamics and firstlaw of black hole thermodynamics,
dE = T dS − P dV, (7)
dM = T dS + ΩH dJ, (8)
Eur. Phys. J. C (2010) 70: 317–328 319
yields the following map
P → −ΩH , V → J. (9)
This is a well known analogy where one can treat −ΩH dJ
as the external work done on the black hole [6].The semi-classical specific heat for the Kerr black hole
can now be written, following its standard thermodynamicdefinition, as,
CΩH= T
(dS
dT
)
ΩH
=(
dM
dT
)
ΩH
− ΩH
(dJ
dT
)
ΩH
, (10)
where, in the second equality, (8) has been used. For conve-nience, instead of working with the variables (M,ΩH ,J ),we take (M,ΩH ,a) as the basic variables and express( dJdT
)ΩHas
(dJ
dT
)
ΩH
= a
(dM
dT
)
ΩH
+ M
(da
dT
)
ΩH
, (11)
where (2) has been used. From (10) and (11) we find
CΩH= (1 − aΩH )
(dM
dT
)
ΩH
− MΩH
(da
dT
)
ΩH
. (12)
To calculate CΩH, we first recast the semi-classical Hawking
temperature (5), by eliminating r+ using (3),
T = �ΩH
√M2 − a2
2πa= T (M,ΩH ,a). (13)
Likewise from (1) and (3), we find
a = 4M2ΩH
4M2Ω2H + 1
. (14)
Exploiting this, one of the variables (M,a) can be replacedin favour of the others and thus (13) can be written in eitherof the following ways,1
T = T (M,ΩH ) = 1
8πM
(1 − 4M2Ω2
H
), (15)
T = T (ΩH ,a) = (1 − 2aΩH )
4π
√ΩH
a(1 − aΩH ). (16)
Now using these two equations together with (14) we findthe semi-classical specific heat (12) as
CΩH= CΩH
(M,ΩH ) = −8πM2(1 − 4M2Ω2H )
(1 + 4M2Ω2H )3
. (17)
1From now on by a,T ,M and ΩH we mean a√�, T√
�, M√
�and
√�ΩH
respectively. With this convention, � no longer appears explicitly inany of the equations.
In the above equation we took M and ΩH as the two inde-pendent variables for the purpose of graphical analysis. Weshall stick to this convention for other cases also. Using (1)and (14), the semi-classical entropy (6) can be expressed interms of M,ΩH as
S = 2πM
(M +
√M2(1 − 4M2Ω2
H )2
(1 + 4M2Ω2H )2
). (18)
For the non-extremal region (T > 0), it follows from (15)that (1 − 4M2Ω2
H ) > 0 and therefore one has the follow-ing expression for the semi-classical entropy for Kerr blackhole,
S = S(M,ΩH ) = 4πM2
(1 + 4M2Ω2H )
. (19)
Following the thermodynamical definition Gibbs free en-ergy, for the Kerr black hole we define
G = M − ΩH J − T S. (20)
For Kerr black hole it is found to be
G = M
1 + 4M2Ω2H
− 1
4
(1 − 4M2Ω2
H
)(M +
√(M − 4M3Ω2
H )2
(1 + 4M2Ω2H )2
). (21)
In Fig. 1 we plot the semi-classical Hawking tempera-ture (15), entropy (18), specific heat (17) and Gibb’s freeenergy (20) with respect to M , keeping the other parameterΩH fixed at a particular value ΩH = 103. The temperaturediverges as the mass of the black hole tends to zero. The
point at which T = 0 is the extremal case (M2 = J 2
M2 ) wherethe Hawking temperature vanishes. From (15) this yieldsM2 = 1
4Ω2H
= 25 × 10−8. Hence M = 0.0005, which is also
reflected in the T –M graph (Fig. 1(a)). Beyond this point,
the temperature acquires a negative value (M2 ≤ J 2
M2 ) andthe curve does not have any physical significance. The en-tire non-extremal region where T is positive, corresponding
to M2 ≥ J 2
M2 , is relevant. In this non-extremal region spe-cific heat is negative (Fig. 1(c)) for all values of the mass andthus renders the system unstable. The semi-classical entropy(Fig. 1(b)) and Gibbs free energy (Fig. 1(d)) are continuousfor the entire non-extremal region. The absence of discon-tinuity in S and CΩH
suggests that there is no phase tran-sition. Thus at the semi-classical level, the system remainsunstable all throughout. Later we shall repeat this analysisby considering a correction to the semi-classical Hawkingtemperature and entropy of the Kerr black hole which willreveal some dramatic changes.
320 Eur. Phys. J. C (2010) 70: 317–328
Fig. 1 Semi-classical Hawking temperature (T ), entropy (S), specific heat (CΩH) and Gibbs free energy (G) vs. mass (M) plot: In all curves
ΩH = 103
3 Kerr black hole as a thermodynamical system;beyond semi-classical approximation
Now we want to investigate the situation when a correctionto the semi-classical approximation is taken into account. Ina recent work involving two of us [3] the black hole temper-ature and entropy for Kerr black hole were found to be
T̃ = T
(1 +
√�
180πMr++ O
(�
2))−1
, (22)
and
S̃ = A
4�+ 1
90log
A
4�+ O(�), (23)
where the usual semi-classical expressions for A and T weregiven in (4) and (5) respectively. We remark that the WKBtype methods adopted in [3] were also successfully em-
ployed [7–13] in other black hole geometries reproducingwell known results found by other approaches [14–23].
With these structures, the first law of black hole thermo-dynamics for the Kerr black hole is now given by [3]
dM = T̃ dS̃ + ΩH dJ. (24)
In the literature there exist several approaches which findthe logarithmic correction to the semi-classical Bekenstein–Hawking entropy for various black holes [14–23]. However,the coefficient of the correction term is not identical in theseapproaches. Also as far as we are aware, there exists no otherpaper apart from [3] which fixes the coefficient of the loga-rithmic correction to the semi-classical entropy for the Kerrblack hole.
In order to calculate the specific heat we follow themethod discussed in the previous section and write T̃ as a
Eur. Phys. J. C (2010) 70: 317–328 321
function of (M,ΩH ) and (a,ΩH ) in the following way:
T̃ = T̃ (M,ΩH ) = T (M,ΩH )
(1 + 1 + 4M2Ω2
H
360πM2
)−1
, (25)
T̃ = T̃ (a,ΩH ) = T (a,ΩH )
(1 + ΩH
90πa
)−1
, (26)
where T is just the semi-classical expression defined in (15)and (16). Using (14), the corrected entropy (23) can be writ-ten as a function of M,ΩH in the following way
S̃ = S + 1
90logS, (27)
where the semi-classical entropy S is defined in (18). Fol-lowing the semi-classical analysis, the specific heat is de-fined as
C̃ΩH= T̃
(dS̃
dT̃
)
ΩH
= (1 − aΩH )
(dM
dT̃
)
ΩH
− MΩH
(da
dT̃
)
ΩH
. (28)
It is now straightforward to derive the corrected specific heatfor the Kerr black hole by using (25), (26) and (28). This fi-nally leads to
C̃ΩH= − (1 + 4M2Ω2
H + 360πM2)2(1 − 4M2Ω2H )
45(1 + 4M2Ω2H )2(16M4Ω4
H + 1440πM4Ω2H + 16M2Ω2
H + 360πM2 − 1). (29)
The Gibbs free energy, in the presence of quantum correc-tion, is found to be
G̃ = M
1 + 4M2Ω2H
+M(4M2Ω2
H − 1)(180πM(M +√
(M−4M3Ω2H )2
(1+4M2Ω2H )2 )) + log(2Mπ(M +
√(M−4M3Ω2
H )2
(1+4M2Ω2H )2 ))
2 + 8M2(Ω2H + 90π)
. (30)
Making use of equations (25), (27) and (29), we plot T̃ , S̃,C̃ΩH
and G̃ with mass M for the previously fixed value ofΩH = 103 (Fig. 2). Let us discuss the physical significanceof the various plots one by one.
Figure 2(a) shows that the corrected Hawking temper-ature is not diverging when the mass of the black holegoes to zero, rather it vanishes. This behaviour was ab-sent in the semi-classical case. The extremality conditionis still satisfied for M = 0.0005, since at this point thecorrected temperature vanishes. There is a critical mass(Mcrit = 0.0002425) for which the temperature is maximumgiven by Tmax = Tcrit = 0.006756. From Fig. 2(c) one cansee that specific heat CΩH
now suffers a discontinuity atMcrit. At this point C̃ΩH
becomes positive from the ini-tial negative value. This point can be elaborated in the fol-lowing manner. During the black hole evaporation as themass decreases the temperature of the black hole initially in-creases. This is shown in the region of phase 1 of Fig. 2(c).At M = Mcrit a phase transition is taking place where thespecific heat switches to a positive value. Consequently theblack hole cools down during further evaporation and finallyat zero mass the temperature vanishes. This behaviour isfound in phase 2 region of Fig. 2(c). The corrected entropy
(S̃) in Fig. 2(b) and corrected free energy (G̃) in Fig. 2(d)are continuous for the entire region.
Note that we cannot write the corrected temperature (22)
as T̃ = T (1−√
�
180πMr+ + O(�2)) since at the phase transition
point,√
�
180πMr+ ∼ 105. As this is considerably greater thanone, a naive simplification of (22) by a series expansion isnot possible.
So far it seems reasonable from the plots that there isa phase transition which eventually leads to the thermody-namic stability of the Kerr black hole. In the next section weshall make a systematic study to identify the type of phasetransition.
4 Nature of the black hole phase transition
In the previous section we found a signature of phase tran-sition with the corrected versions of Hawking temperatureand entropy of Kerr black hole. This phase transition is nota first order phase transition as there is no discontinuity inthe corrected entropy for the entire physical region. Instead,the discontinuity in specific heat suggests that there may bea second order phase transition. If that is the case then one
322 Eur. Phys. J. C (2010) 70: 317–328
Fig. 2 Corrected Hawking temperature (T̃ ), entropy (S̃), specific heat (C̃ΩH) and Gibbs free energy (G̃) vs. mass (M) plot: In all curves ΩH = 103
naturally expects that there must be some analog of Ehren-fest’s equations in black hole thermodynamics which mustbe satisfied. Till now Ehrenfest’s relations were not stud-ied in the context of black hole thermodynamics. Hence weshall systematically derive (see Appendix A) the Ehrenfestequations and then check the validity of these relations af-terwards. These (first and second) Ehrenfest’s equations aregiven by (A.18) and (A.24) respectively. It is noteworthythat, since in a second order phase transition both S and J
remain constant, left hand sides of these two equations havethe same value. In the next subsection we shall check thevalidity of these two equations in the context of black holethermodynamics.
Let us now convince ourselves that there is indeed adiscontinuity present in the behaviour of corrected specific
heat CΩH(A.14), analog of volume expansion coefficient
α (A.13) and compressibility kT (A.23). As far as CΩHis
concerned, its values at two different phases are given byCΩH1
= 0 and CΩH2= 0.0222, so that CΩH2
− CΩH1=
0.0222 (Fig. 2(c)). For others, the expressions for α and kT
in terms of M and ΩH are derived and appropriate graphsare plotted.
Comparing (A.13) and (11) we find
Jα = a
(∂M
∂T
)
ΩH
+ M
(∂a
∂T
)
ΩH
. (31)
Using the expressions of the corrected Hawking temperaturefrom (25) and (26) and substituting a from (14) this equa-tion is simplified to
Jα = − 4M2ΩH (1 + 4M2(90π + Ω2H ))2(4M2Ω2
H + 3)
45(1 + 4M2Ω2H )2[16M4(90πΩ2
H + Ω4H ) + 8M2(2Ω2
H + 45π) − 1] . (32)
Eur. Phys. J. C (2010) 70: 317–328 323
Fig. 3 Jα, JkT vs. mass (M) plot for ΩH = 103
The plot of Jα with the mass (M) for our chosen constantangular velocity ΩH = 103 of Kerr black hole is shown inFig. 3(a). There is a discontinuity present in this plot ex-actly at the same mass value where the specific heat also suf-fered a discontinuity. The values of Jα in different phasesare given by Jα1 = −0.000022 and Jα2 = 0.
To express kT in terms of M and ΩH we proceed asfollows. Using (2) we re-express (A.23) in the followingway,
JkT =(
∂J
∂ΩH
)
T̃
=(
∂(Ma)
∂ΩH
)
T̃
= M
(∂a
∂ΩH
)
T̃
+ a
(∂M
∂ΩH
)
T̃
. (33)
Using the property of partial differentiation,
(∂a
∂ΩH
)
T̃
(∂ΩH
∂T̃
)
a
(∂T̃
∂a
)
ΩH
= −1;(
∂M
∂ΩH
)
T̃
(∂ΩH
∂T̃
)
M
(∂T̃
∂M
)
ΩH
= −1
(34)
we obtain the following two equalities
(∂a
∂ΩH
)
T̃
= − ( ∂T̃∂ΩH
)a
( ∂T̃∂a
)ΩH
;(
∂M
∂ΩH
)
T̃
= − ( ∂T̃∂ΩH
)M
( ∂T̃∂M
)ΩH
.
(35)
These two equations can easily be simplified by using (25)and (26). Substituting their values in (33) and using (14) toeliminate a, we finally obtain,
JkT = −4[M3 + 4M5(7Ω2H − 90π) + 16M7(540πΩ2
H + 7Ω4H ) + 64M9(270πΩ4
H + Ω6H )]
(1 + 4M2Ω2H )2[16M4(Ω4
H + 90πΩ2H ) + 8M2(2Ω2
H + 45π) − 1] . (36)
This is plotted in Fig. 3(b) for constant ΩH = 103. Thecurve shows a discontinuity at the same critical mass value(Mcrit = 0.0002425). This plot gives JkT1 = −4.1 × 10−10
and JkT2 = 0.The above analysis strongly suggests the occurrence
of phase transition. All the relevant physical parameters(CΩH
, Jα and JkT ) appearing in the Ehrenfest equa-tions show discontinuity at the same critical value ofmass. To conclusively argue in favour of a phase tran-sition, we explicitly check the validity of the Ehrenfestequations. In the next subsection we shall address this is-sue.
4.1 Validity of Ehrenfest’s equations
Let us consider the first Ehrenfest equation (A.18). Usingthe chain rule of partial differentiation we have2
(∂T̃
∂ΩH
)
S̃
=(
∂T̃
∂ΩH
)
a
+(
∂T̃
∂a
)
ΩH
(∂a
∂ΩH
)
S̃
. (37)
2Under the change of variables u = u(x, y) and v = v(x, y) for a func-
tion f = f (x, y), (∂f∂x
)y = (∂f∂u
)v(∂u∂x
)y + (∂f∂v
)u( ∂v∂x
)y . In the special
case when u = x, (∂f∂x
)y = (∂f∂u
)v + (∂f∂v
)u( ∂v∂x
)y .
324 Eur. Phys. J. C (2010) 70: 317–328
The derivatives of the corrected temperature (r.h.s.) are com-puted from (26). To calculate ( ∂a
∂ΩH)S̃
, we first derive an ex-pression for the corrected entropy. On using (3) and (27) thisis written as
S̃ = S̃(a,ΩH ) = πa
ΩH
+ 1
90log
πa
ΩH
. (38)
Exploiting the rule of partial differentiation, one obtains
(∂a
∂ΩH
)
S̃
= − ( ∂S̃∂ΩH
)a
( ∂S̃∂a
)ΩH
. (39)
From (38) and (39) we find ( ∂a∂ΩH
)S̃
= aΩH
. Substituting thisin (37) and using (26) yields,
(∂ΩH
∂T̃
)
S̃
= 2(90πa + ΩH )(1 − aΩH )2
45aΩH (2aΩH − 3)
√ΩH
a(1 − aΩH ).
(40)
This is the left hand side of the first Ehrenfest equation.Eliminating a in terms of M,ΩH by using (14), we findthe cherished expression,
−(
∂ΩH
∂T̃
)
S̃
= (1 + 4M2Ω2H )(1 + 4M2(Ω2
H + 90π))
180M3ΩH (16M4Ω4H + 16M2Ω2
H + 3).
(41)
For ΩH = 103 and M = Mcrit = 0.0002425, we get thevalue of the left hand side of the first Ehrenfest equation(A.18) that follows directly from (41),
−(
∂ΩH
∂T̃
)
S̃
= 1.487 × 105. (42)
The right hand side of (A.18) is now calculated by usingresults dictated earlier from different graphs (i.e. Figs. 2(a),2(c), 3(a)). From Fig. 2(a) we find, at M = Mcrit the criticaltemperature T̃ = Tcrit = 0.006756. Using these values in theright hand side of (A.18), we get
C̃ΩH2− C̃ΩH1
T̃crit(Jα2 − Jα1)= (0.222 − 0)
0.006756 × 0.000022
= 1.494 × 105. (43)
The remarkable agreement between (42) and (43) clearlyshows the validity of the first Ehrenfest equation for the Kerrblack hole.
Let us now take the second Ehrenfest equation (A.24).The left hand side of (A.24) can be calculated in the follow-ing manner. First we simplify (14) to find (1 − aΩH ) =
J
4M3ΩH. Substituting this in (26) and replacing a from
(2) we write the angular momentum in the followingway:
J = J (M,ΩH , T̃ ) = MΩH (45M − T̃ )
90(πT̃ + MΩ2H )
. (44)
For constant value of J (i.e. dJ = 0) we have(
∂J
∂M
)
ΩH ,T̃
dM +(
∂J
∂ΩH
)
M,T̃
dΩH
+(
∂J
∂T̃
)
M,ΩH
dT̃ = 0. (45)
Since T̃ , M and ΩH are connected by (25), we take onlythe variables M and ΩH as independent. Then dT̃ can bewritten as
dT̃ =(
∂T̃
∂M
)
ΩH
dM +(
∂T̃
∂ΩH
)
M
dΩH . (46)
Now multiplying (45) by ( ∂T̃∂M
)ΩHand (46) by ( ∂J
∂M)ΩH ,T̃
and subtracting one from the other we arrive at the re-sult
P
(∂ΩH
∂T̃
)
J
=( ∂J
∂T̃)M,ΩH
( ∂T̃∂M
)ΩH+ ( ∂J
∂M)T̃ ,ΩH
( ∂J∂M
)ΩH ,T̃
( ∂T̃∂ΩH
)M − ( ∂J∂ΩH
)M,T̃
( ∂T̃∂M
)ΩH
.
(47)
Derivatives of J and T̃ , obtained from (44) and (25) re-spectively, are substituted in (47) to finally yield the lefthand side of the second Ehrenfest equation (A.24) as
−(
∂ΩH
∂T̃
)
J
= ΩH (4M2Ω2H + 3)(1 + 4M2Ω2
H + 360M2Ω2H )2
45M[1 − 4M2(90π − 7Ω2H ) + 16M4(540πΩ2
H + 7Ω4H ) + 64M6(270πΩ4
H + Ω6H )] . (48)
For ΩH = 103 and M = Mcrit = 0.0002425 we find directlyfrom (48),
−(
∂ΩH
∂T
)
J
= 1.4875 × 105. (49)
To calculate the right hand side of (A.24) we employ thegraphical results, as stated earlier. This yields,
(Jα2 − Jα1)
(J kT2 − JkT1)= (2.2 × 10−5)
(4.1 × 10−10)= 5.37 × 104. (50)
Eur. Phys. J. C (2010) 70: 317–328 325
The disagreement between (49) and (50) spoils the pos-sibility of a second order phase transition in black hole ther-modynamics. In the following section we shall argue thatthis phase transition is very special and usually found inglass forming liquids when they are supercooled to formglasses.
4.2 Glassy phase transition and Prigogine–Defay ratiofor Kerr black hole
When some liquids are supercooled they solidify withoutcrystallisation and form glasses, most common of which isthe silicate glass. Though the underlying properties of ma-terials which favour a glassy phase transition have not beencompletely understood [24, 25], it is known that the rate ofcooling is an important factor. Metals and alloys which werepreviously known as non-glass forming substances are nowfound to form glasses under very fast cooling (105 K/s)[26].This is almost 107 times faster than the cooling rate to formnormal glass. In spite of the fact that in the glassy phase tran-sition the specific heat, expansion coefficient and isothermalcompressibility suffer discontinuity at the critical tempera-ture, one cannot identify this as second order phase transi-tion for the following reason. The critical temperature andthe width of the transformation region for the glassy transi-tion depend on the cooling rate [5]. As a consequence thesecond Ehrenfest equation (A.24) is violated for this type ofphase change.
For a normal second order phase transition, first orderderivatives of Gibb’s free energy are continuous at the phasetransition point and hence one can equate the left hand sidesof two Ehrenfest’s equations to get the result Π = 1, whereΠ , the Prigogine–Defay ratio [5], is defined by
Π = �Cp�k
T V (�α)2(51)
In the case of black hole phase transition, from (42) and (49)we see that the left hand sides of (A.18) and (A.24) are alsoidentical. Since in the previous subsection we discussed theviolation of the second Ehrenfest equation, we rectify it byputting a correction term (Π ) in the right hand side of (A.24)to get
−(
∂ΩH
∂T
)
J
= Πα2 − α1
kT2 − kT1
. (52)
Equating (52) and (42) we get the Prigogine–Defay ratio(which is the black hole analogue of (51))
Π = �CΩH�kT
T J (�α)2. (53)
For the Kerr black hole an explicit calculation gives Π =0.0222×4.1×10−10
0.006756×0.0000222 = 2.78. This is strikingly well within the
known limits 2 ≤ Π ≤ 5 for glassy phase transitions in liq-uids and polymers [27, 28].
Our studies further reveal that for values of ΩH (≥102),the critical point (Mcrit and T̃crit) is shifted but the value ofPD ratio remains same. For small values of ΩH (<10), thechange in JkT between two phases is extremely small andit is very difficult to check the validity of Ehrenfest’s equa-tions. When ΩH = 10 both of the Ehrenfest equations areviolated and only further studies can reveal the peculiarityof this point. We thus infer that the glassy phase transitionoccurs for ΩH � 10.
One may wonder whether the results are dependent onthe normalisation of the logarithmic term appearing in (23).While it is difficult to answer this question in complete gen-erality, nevertheless, we provide some arguments that estab-lish the robustness of our conclusions. The first point to noteis that, contrary to other black holes (like Schwarzschild andReissner–Nordstrom etc) where different approaches yielddifferent normalisations [14–23], explicit results for the Kerrmetric are only given in [3]. However that does not preventus from taking other normalisations and testing the infer-ences. Specifically, if the normalisation of the logarithmicterm in (23) is taken as 1
2 (instead of 190 ) we again get a
PD ratio close to 3, signifying a similar type of glassy phasetransition. Likewise, if we consider other normalisations like32 (which, contrary to the earlier 1
90 or 12 , is greater than one)
or 1, our basic result remains unaffected. For details, see Ap-pendix B. On the other hand, a negative normalisation of thelogarithmic correction term modifies the corrected Hawkingtemperature in such a way that it is found to be negative inthe entire non-extremal region which is clearly unphysical.So our conclusion is strictly valid for a positive normalisa-tion of the correction term.
5 Conclusions
In spite of considerable research in black hole physics, is-sues related to thermodynamical stability are often over-looked. This is because in most of the cases black holesare studied within the semi-classical regime where they arealways thermodynamically unstable. This implies that thepossibility of phase transition and stability can be studiedby going beyond the semi-classical approximation. Also, itis necessary to have the work term (PdV-type) in the law ofblack hole thermodynamics to properly identify and classifya phase transition. Both these criteria are satisfied by consid-ering the corrected expressions for the thermodynamic vari-ables in the Kerr black hole. These expressions were takenfrom an earlier work done by two of us [3]. This reference,however, did not consider at all the issues related to phasetransition.
326 Eur. Phys. J. C (2010) 70: 317–328
Here we showed that inclusion of correction beyond thesemi-classical approximation makes the Kerr black hole sta-ble via a phase transition. We discovered that this phasetransition cannot be classified either as first or second order.This type of phase transition has a close analogy with liquidto glass transition and is known as a glassy phase transi-tion. In particular, the Prigogine–Defay ratio found for theKerr black hole fits within the bound obtained from experi-mental results for a glassy phase transition. Furthermore, weshowed the robustness of our result by choosing three morearbitrary positive coefficients of the correction (logarithmic)term to the semi-classical entropy for the Kerr black hole. Inall these cases a glassy phase transition occurred with almostthe same PD ratio. This ratio, close to 3, was well within thebounds (2 to 5) quoted in the literature [27, 28] for glassyphase transitions in liquids and polymers. We feel that thiswork may open up a new field of research where the deepconnection between black hole physics and thermodynam-ics (statistical mechanics) can be further analysed.
Acknowledgements One of the authors (S.K.M) thanks the Councilof Scientific and Industrial Research (C.S.I.R), Government of India,for financial support.
Appendix A: Derivation of Ehrenfest’s equationsfor Kerr black hole
Using the first law of black hole thermodynamics (8) andthe definition of Gibb’s function (20), the differential formof Gibbs free energy is written as
dG = −J dΩH − S dT . (A.1)
From this equation, the black hole entropy and the angularmomentum can be written as the derivatives of the Gibbsfree energy as
S = −(
∂G
∂T
)
ΩH
, (A.2)
J = −(
∂G
∂ΩH
)
T
. (A.3)
Since, by definition, the Gibbs free energy (20) is a statefunction, dG is an exact differential and hence we get thefollowing Maxwell relation from (A.1)
(∂J
∂T
)
ΩH
=(
∂S
∂ΩH
)
T
. (A.4)
In a second order phase transition, Gibbs free energy and itsfirst order derivatives, i.e. entropy and angular momentum,
are all continuous. So at a phase transition point, charac-terised by some temperature T and an angular velocity ΩH ,
G1 = G2, (A.5)
S1 = S2, (A.6)
J1 = J2, (A.7)
where the subscripts 1 and 2 denote the values of the differ-ent physical quantities in the two phases.
Let us now consider the equality (A.6). If the temperatureand angular velocity are increased infinitesimally to T +dT
and ΩH + dΩH , then
S1 + dS1 = S2 + dS2. (A.8)
From (A.6) and (A.8) we find
dS1 = dS2. (A.9)
Taking S as a function of T and ΩH
S = S(T ,ΩH ) (A.10)
we write the infinitesimal change in entropy as
dS =(
∂S
∂T
)
ΩH
dT +(
∂S
∂ΩH
)
T
dΩH . (A.11)
Using (A.4), the above equation takes the form
dS = CΩH
TdT + Jα dΩH , (A.12)
where
α = 1
J
(∂J
∂T
)
ΩH
(A.13)
is the coefficient of change in angular momentum and
CΩH= T
(∂S
∂T
)
ΩH
. (A.14)
Since J is same in both phases,
dS1 = CΩH1
TdT + Jα1 dΩH , (A.15)
dS2 = CΩH2
TdT + Jα2 dΩH . (A.16)
Now use of the condition (A.9) requires the equality of(A.15) and (A.16),
CΩH1
TdT + Jα1 dΩH = CΩH2
TdT + Jα2 dΩH . (A.17)
Eur. Phys. J. C (2010) 70: 317–328 327
This gives the first Ehrenfest equation,
−(
∂ΩH
∂T
)
S
= CΩH2− CΩH1
T J (α2 − α1). (A.18)
We now take the constancy of angular momentum (A.7)which, under an infinitesimal change of temperature and an-gular velocity, gives
dJ1 = dJ2. (A.19)
Taking the angular momentum as a function of T and ΩH ,
J = J (T ,ΩH ) (A.20)
we get the differential relation
dJ =(
∂J
∂T
)
ΩH
dT +(
∂J
∂ΩH
)
T
dΩH (A.21)
= Jα dT − JkT dΩH , (A.22)
where
kT = 1
J
(∂J
∂ΩH
)
T
(A.23)
is the analog of compressibility. Since dJ is same for thetwo phases, we get the second Ehrenfest equation by mim-icking the previous steps used to derive (A.18):
−(
∂ΩH
∂T
)
J
= α2 − α1
kT2 − kT1
. (A.24)
The two Ehrenfest equations (A.18) and (A.24), which mustbe satisfied for a second order phase transition, play thesame role as the Clapeyron equation for the first order phasetransition.
Appendix B: Summary of results for distinctnormalisations of the logarithmic termin (23)
Here we summarise our findings for different normalisationsof the logarithmic term in (23). For instance, if the normal-isation is taken as 1
2 , the corrected entropy and temperatureare given by
S̃ = A
4�+ 1
2log
A
4�+ O(�), (B.1)
and
T̃ = T
(1 +
√�
4πMr++ O
(�
2))−1
. (B.2)
After carrying out the analysis with these corrected expres-sions one would find the following results for two Ehren-fest’s equations, which are the analogues of (42) and (43),obtained with the earlier normalisation ( 1
90 ),
−(
∂ΩH
∂T
)
S
= 6.6934 × 106,
(B.3)CΩH2
− CΩH1
T J (α2 − α1)= 7.0109 × 106.
and
−(
∂ΩH
∂T
)
J
= 6.68077 × 106,
(B.4)α2 − α1
kT2 − kT1
= 2.2500 × 106.
The above pair of equations (B.4) are the analogues of (49)and (50). This clearly shows that while the first Ehrenfestequation is closely satisfied the second one is violated insuch a way that the PD ratio (53) following from (B.4) be-
comes Π = 6.68077×106
2.2500×106 = 2.96.
Similarly, taking the normalisation as 32 of the logarith-
mic term in (23), we get
S̃ = A
4�+ 3
2log
A
4�+ O(�), (B.5)
T̃ = T
(1 + 3
√�
4πMr++ O
(�
2))−1
. (B.6)
Correspondingly, the results for the two Ehrenfest equationsare given by
−(
∂ΩH
∂T
)
S
= 2.00802 × 107,
(B.7)CΩH2
− CΩH1
T J (α2 − α1)= 2.14084 × 107,
and
−(
∂ΩH
∂T
)
J
= 2.00423 × 107,
(B.8)α2 − α1
kT2 − kT1
= 6.82923 × 106.
These results finally give a PD ratio Π = 2.93 for the abovechoice of the normalisation.
Finally, if we set the normalisation as unity, the correctedtemperature and entropy are given by
S̃ = A
4�+ log
A
4�+ O(�), (B.9)
T̃ = T
(1 +
√�
2πMr++ O
(�
2))−1
, (B.10)
328 Eur. Phys. J. C (2010) 70: 317–328
leading to the Ehrenfest equations
−(
∂ΩH
∂T
)
S
= 1.33868 × 107,
(B.11)CΩH2
− CΩH1
T J (α2 − α1)= 1.36735 × 107 (B.12)
and
−(
∂ΩH
∂T
)
J
= 1.33615 × 107,
(B.13)α2 − α1
kT2 − kT1
= 4.63415 × 106.
Once again one finds a PD ratio Π = 2.88, which closelymatches other cases.
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