New approach to phase transitions in black holes

0.41 MB·English

by  Rabin Banerjee

New approach to phase transitions in black holes
Rabin Banerjee∗ and Sujoy Kumar Modak†
S. N. Bose National Centre for Basic Sciences,
Block-JD, Sector-III, Salt Lake, Kolkata 700 098, India
Saurav Samanta‡
Narasinha Dutt College, 129, Belilious Road, Howrah-711101, India
We develop an analogy between fluids and black holes to study phase transitions in the latter.
Theentropy-temperaturegraphshowstheonsetofaphasetransitionwithoutanylatentheat. The
nature of this continuous (higher order) phase transition is examined in details. We find that the
secondorderderivativesofthefreeenergydivergeatthecriticaltemperature. Also,thetransitionis
smeared instead of sharp, so that the usual Ehrenfest’s scheme breaks down. A generalised version
of this scheme is formulated which is shown to be consistent with the phase transition curves.
1
1
0 Although black holes are well known thermody- analogous to the first law of thermodynamics,
2 namical systems, they are not well understood. In
fact there is no microscopic or statistical description dE =TdS−PdV. (2)
n
a behind their thermodynamical behaviour. Studying
J phase transition in black holes is very important as it The fact that the internal energy (E) of black holes
couldmanifesttheunderlyingmicro-structureofblack being represented by its mass (M), together with
2
1 holethermodynamics. Althoughtherearesomeinves- Hawking’s discovery that black holes have tempera-
tigations [1], these are more concerned with the geo- ture T = 2κπ, set the analogy between (1) and (2),
] metrical rather than the thermodynamical aspects of associating an entropy to black holes S = A. Several
c 4
phase transition. alternative approaches [3] have been developed which
q
- In this paper we systematically develop a new ap- reproduce identical results for black hole temperature
r proach to study phase transitions in black holes from and entropy.
g
a thermodynamical viewpoint. To do that we first We observe that while the first terms in the right
[
build an analogy between fluids and black holes that hand sides of (1) and (2) set a mathematical analogy

1 parallelstheanalogybetweenbetweenfluidsandmag- between black holes and ideal fluids, it is the second
v netic systems. From this analogy we then exploit the terms which actually establish the physical motiva-
8 known results in fluids to infer the corresponding sit- tion. It leads to the correspondence P ←→ −Ω and
1 uations in black holes. Following this approach we V ←→ J. One can realise the physical importance
3
find that the entropy-temperature graph (as well as of this analogy in the following manner. In ideal flu-
2
the Gibb’s energy-temperature graph) of the Kerr- ids,bothforisothermalandisentropicprocessesifone
.
1 AdS black hole shows the onset of phase transition increases the pressure its volume shrinks so that the
0 without any latent heat. Moreover this transition is density increases. Thus the P −V plot for fluids has
1 smeared round the critical temperature. The details a negative slope. On the other hand the relation be-
1 of this continuous (higher order) phase transition are tween the angular velocity and angular momentum of
v: examined by looking at various plots of entropy with Kerr-AdS black hole is given by [4, 5]
i specific heat, volume expansion and compressibility.
X All these plots manifest an infinite divergence at the S(cid:113) SΩ2 (cid:0)(π+S)2−S2Ω2(cid:1)2
r criticalpoint. Furthermore,thesmearednatureofthe (π+S)(π+S−SΩ2)
J = . (3)
a phase transition gets highlighted. 2π3/2(π+S)3
Duetothissmearedtransitionadirectapplication
We plot the Ω−J graph in figure-1 by taking vari-
of the Gibbsian approach (Ehrenfest’s scheme) to ex-
ous constant values for entropy. All these plots have
hibit and classify phase transitions fails. We thus de-
velop a generalized Ehrenfest scheme to account for
(cid:87)
this smearing. We show that such a generalization is
0.20
compatible with our graphical analysis.
The discovery that black holes are indeed thermo- 0.15
dynamicalsystemsisbuiltonamathematicalanalogy
0.10
betweenlawsofblackholemechanicsandlawsofther-
modynamics [2]. For chargeless, rotating black holes 0.05
the first law of black hole mechanics relates the in-
J
finitesimal change in black hole mass (M) with the 0.2 0.4 0.6 0.8
infinitesimal changes in its horizon area (A) and an-
FIG. 1: Angular velocity (Ω) vs. angular momentum (J)
gular momentum (J), given by
plots. Forblueline(lower)S =1;forpurpleline(middle)
S =3; and for grey line (upper) S =5.
κ
δM = δA+ΩδJ, (1)
8π positiveslopeswhichgiveaphysicalcredibilitytothe
correspondance,
whereκandΩaresurfacegravityandangularvelocity
at the event horizon, respectively. This is very much E ←→M; P ←→−Ω; V ←→J. (4)
2
S G Second, it is seen that the curvature of the two arms
2.0 (cid:230)(cid:230)(cid:230) 0.10 (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) in fig.2 (G−T) change sign. Since the curvature of
(cid:230)(cid:230) (cid:230)(cid:230)(cid:230) G is just the specific heat, we observe the occurence
11..05 (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) 00..0089 (cid:230)(cid:230) (cid:230) oacrftiittvhieceaqlpuphaaodsirneattn.rtanosfittihoenCcuΩrv−esSinptlohtespeopseitriavteedanbdyntehge-
(cid:230) 0.07 (cid:230)
(cid:230) (cid:230) Clearly a Gibbsian approach, which has its root
0.5 (cid:230)(cid:230) 0.06 (cid:230) (cid:230) in mean field theory, breaks down in order to analyse
(cid:230) (cid:230)
(cid:230) (cid:230) (cid:230) (cid:230) 0.05 this infinitely diverging, smeared phase transition. In
T T the following we make a modification to known Gibb-
0.25 0.30 0.35 0.40 0.25 0.30 0.35 0.40
sian approach (Ehrenfest’s scheme [7]), incorporating
asmearingeffect,toanalysethisnewphasetransition.
FIG.2: Entropy(S)andGibb’sfreeenergy(G)plotwith
We shall build our machinary considering a fluid sys-
respect to temperature (T) for fixed Ω=0.5
tem which undergoes a smeared phase transition and
finallyusing(4)weshallrecastrelevantquantitiesfor
black holes.
Nowweareinapositiontousetheknowntoolsof
NowtogeneralizeEhrenfest’sschemewestartfrom
thermodynamics to black holes. In order to follow a
the fact that in a smeared phase transition the values
Gibbsian approach to exhibit and classify black hole
of entropy or volume of two phases at the phase tran-
phase transitions it is customary to define the Gibb’s
sition point are not exactly same as it should be in a
free energy. Gibb’s free energy and temperature of
second order phase transition, instead there are small
thisblackholewascalculatedearlierin[4,5]andthese
smearing terms (s,v) in the following manner
are given by
S =S +s (8a)
2 1
G(S,Ω)=M −TS−ΩJ (5)
V =V +v (8b)
2 1
(cid:18) S(π+S)3 (cid:19)12
T(S,Ω)= × where the subscripts 1 and 2 denote the values in the
π+S−SΩ2
twophases. Ifwecharacterisethephasetransitionby
π2−2πS(Ω2−2)−3S2(Ω2−1) the temperature (T) and pressure (P) for which (8a)
. (6)
4π32S(π+S)2 a(cnhdar(a8cbt)erhisoeldd,baytTa(cid:48)d=iffeTre+ntdpTh,aPse(cid:48) =traPnsi+tiodnPp)otihnet
following equations will be true
TherelevantS−T andG−T plotsareshowninfig.2.
The temperature-entropy plot is continuous. Also, S +dS =S +dS +s(cid:48) (9a)
there exists a minimum temperature (T = 0.2574) 2 2 1 1
c V +dV =V +dV +v(cid:48) (9b)
after which the slope of this curve tends to change its 2 2 1 1
signveryslowly. Thesefeaturesarequalitativelysim-
From (8a) and (9a) we find
ilar to fluids (or magnetic systems) [6] revealing that
there is no first order phase transition involving la- dS =dS +(s−s(cid:48)) (10)
1 2
tent heat. Also, the same analogy indicates the onset
ofacontinuous(higherorder)phasetransitionwitha Taking S as a function of T and P we write the in-
criticaltemperatureT . Contrarytowhathappensin finitesimal change in entropy as dS = (cid:0)∂S(cid:1) dT +
c ∂T P
usual fluids, however, the transition appears smeared (cid:0)∂S(cid:1) dP. Using the Maxwell relation (cid:0)∂V(cid:1) =
instead of being sharp. Similar conclusions are also −∂(cid:0)P∂ST(cid:1) , this equation takes the form, ∂T P
drawn from the G−T plot. ∂P T
The situation becomes more transparent when, C
once again following the analogous treatment for flu- dS = PdT −χdP, (11)
T
ids, we plot the specific heat at constant angular ve-
where,
locity (C ) (analogue of C for fluids) with entropy.
Ω P (cid:18) (cid:19) (cid:18) (cid:19)
The expression of C is given by [5], ∂V ∂S
Ω χ = ; C =T (12)
∂T P ∂T
P P
C =2S(π+S)(π+S−SΩ2)×
Ω
are, respectively, the volume expansion index and the
π2−2πS(Ω2−2)−3S2(Ω2−1)
specific heat at constant pressure. Since (11) is inde-
(π+S)3(3S−π)−6S2(π+S)2Ω2+S3(4π+3S)Ω4 pendently true for two phases we write
(7)
C C
dS = P1dT −χ dP; dS = P2dT −χ dP.(13)
TheC −S plot(figure3)isaconcreteevidenceofon- 1 T 1 2 T 2
Ω
set of a phase transition. Now if we want to compare
Now substituting this in (10) we finally get the gen-
this plot with usual higher order phase transition we
eralized first Ehrenfest’s equation
find one important difference. This is the appearance
ofasmearedregiononeithersideofthecriticalpoint. (cid:18)∂P(cid:19) C −C s−s(cid:48)
= P2 P1 + . (14)
Let us just briefly discuss how the effects mani- ∂T T(χ −χ ) (χ −χ )(T(cid:48)−T)
fested in the curves of fig.2 are concretised in fig.3, S 2 1 2 1
which is quite akin to what happens for fluids, albeit In the absence of any smearing in entropy, the second
with some crucial difference. First, the smeared na- term on the right hand side is zero and one recovers
ture of the phase transition is prominently displayed. the first Ehrenfest’s equation in its usual form.
3
Next considering (8b) and (9b) we find C(cid:87) Χ
80
dV =dV +(v−v(cid:48)). (15) 60 20
1 2 Phase-2 Phase-2
40
10
Taking V as a function of T and P, we find, 20
(cid:18)∂V (cid:19) (cid:18)∂V (cid:19) (cid:45)20 0.5 1.0 1.5 2.0 S 0.5 1.0 1.5 2.0 S
dV = dT + dP (16)
∂T ∂P (cid:45)40 Phase-1 (cid:45)10 Phase-1
P T
(cid:45)60
= χdT +ξdP (17) (cid:45)20
where ξ is the compressibility FIG. 3: Semi-classical Specific heat (CΩ) and volume ex-
pansion index (χ) vs entropy (S) plot. The solid line
(cid:18) (cid:19) stands for Ω = 0.5 and the dashed line corresponds to
∂V
ξ = (18) its infintely small variation (Ω(cid:48) =Ω+5%×Ω).
∂P
T
Ξ
and χ has been defined in (12). Following the previ-
J
ous steps we find the generalized second Ehrenfest’s 0.14
equation 2 0.12
Phase-2
0.10
(cid:18)∂P(cid:19) χ −χ (v−v(cid:48)) 1 0.08
= 2 1 + . (19) 0.06
∂T ξ −ξ (ξ −ξ )(T(cid:48)−T) S
V 2 1 2 1 0.5 1.0 1.5 2.0 0.04
0.02
We can expect that in a smeared phase transition, (cid:45)1 Phase-1 S
0.5 1.0 1.5 2.0
(14) and (19) will be satisfied. In the following part
ofourpaperweshallinvestigatethisinthecontextof
black hole phase transition. FIG. 4: Moment of inertia index (ξ) and angular momen-
tum (J) vs entropy (S) plot. The solid line stands for
Using(4),(14)and(19)wenowproposethefollow-
Ω = 0.5 and the dashed line corresponds to its infintely
inggeneralizedEhrenfest’sequationfortheKerr-AdS small variation (Ω(cid:48) =Ω+5%×Ω).
black hole,
(cid:18)∂Ω(cid:19) C −C s−s(cid:48)
− = Ω2 Ω1 + (S =1.208). For these values of Ω and S the critical
∂T T(χ −χ ) (χ −χ )(T(cid:48)−T) c
S 2 1 2 1 temperature is found to be T =0.2574 from (6).
c
(20)
Now the left hand sides of (20), (21) evaluated
(cid:18)∂Ω(cid:19) χ −χ (j−j(cid:48)) by using (22c), (22d) at the critical point (S =
− = 2 1 + . (21) c
∂T J ξ2−ξ1 (ξ2−ξ1)(T(cid:48)−T) 112.2.80082,T7can=d 102..27597245.foTroΩcalc=ula0t.e5)thaerreigfhotuhnadndtosibdee
In order to check the validity of (20) and (21) we wefirstdrawtheCΩ−S, χ−S.ξ−S andJ−S plots
first recall the following equations from [5] (figures 3,4) using (7), (22a), (22b) and (3), respec-
tively. Let us first consider the smearing independent
χ= terms in (20), (21). The respective values of relevant
quantities are borrowed from their plots. We choose
6S2(π+S)3Ω−2S3(π+S)(2π+3S)Ω3
such points in two phases (say S for phase 1 and
1
(π+S)3(3S−π)−6S2(π+S)2Ω2+S3(4π+3S)Ω4 S for phase 2) by avoiding the smearing region as
2
(22a) in this part values widely change for two neighbour-
ing points. Our prescription of choosing a point is
that it should be very close to the critical point and
(cid:114) S(π+S)3 belongtoaregionwheretheslopeofthecurveisvan-
ξ = S F (22b) ishingly small. For Ω = 0.5 we take S = 0.653 and
π+S−SΩ2 1
S = 1.763. Thus we have a smeared region of width
(cid:113) 2
(cid:18)∂Ω(cid:19) 4π32(π+S−SΩ2)2 (πS+(Sπ+−SS)Ω32) s = S2−S1 = 1.11 (see (8a)). For this set of values
− = − the changes in C ,χ,ξ between the two phases are
∂T SΩ(S(π+S)(2π+3S)Ω2−3(π+S)3) Ω
S found to be ∆CΩ = 23.80 ,∆χ = 7.72, ∆ξ = 0.90.
(22c) The values thus found for the first terms of the right
(cid:115) hand side of (20) and (21) are 11.9817 and 8.578 re-
(cid:18)∂Ω(cid:19) π+S−SΩ2
spectively. We now observe that, since the l.h.s. of
− = H (22d)
∂T S(π+S)3 (20) equals 12.8027, the r.h.s. almost matches. This
J
does not hold for (21). This is reminiscent of recent
where F and H are defined as [9]. The physically al- studies [8] which show that if one neglects the smear-
lowed values for Ω are 0 ≤ Ω ≤ 1, else the Hawking ing effect for black hole phase transitions then only
temperature becomes negative [5]. We choose a par- the first Ehrenfest’s equation is satisfied and the sec-
ticular value around the central range (Ω = 0.5) to ondoneisviolated. Effectively,thereforethesmearing
examine how this generalized scheme works. For this termshaveanontrivialcontributiononlyfor(21). We
value of Ω the plots of C , χ and ξ (see figures 3 and now discuss this issue.
Ω
4) show a smeared divergence at the critical entropy In order to obtain the second (smearing depen-
4
1st generalized Ehrenfest’s reln. 2nd generalized Ehrenfest’s reln.
Ω l.h.s=−(cid:0)∂Ω(cid:1) X Y r.h.s=X +Y l.h.s=−(cid:0)∂Ω(cid:1) X Y r.h.s=X +Y
∂T J 1 1 1 1 ∂T S 2 2 2 2
0.1 72.1469 64.6853 0.0 64.6853 66.8405 6.232 39.50 45.732
0.2 35.6305 31.6156 Do 31.6156 35.5666 9.470 20.56 30.030
0.3 23.2151 20.5940 Do 20.5940 23.2601 10.329 12.36 22.689
0.4 16.8036 15.0467 Do 15.0467 16.7774 9.833 5.13 14.963
0.5 12.8027 11.9817 Do 11.9817 12.7985 8.578 2.924 11.502
0.6 9.9750 8.9837 Do 8.9837 9.9852 7.484 1.26 8.744
0.7 7.7547 7.1375 Do 7.1375 7.7544 5.944 0.596 6.540
0.8 5.8653 5.4616 Do 5.4616 5.8674 5.025 0.262 5.287
0.9 4.0034 3.3300 Do 3.3300 4.0030 2.970 0.188 3.158
TABLEI:SummaryofresultsforvalidityofgeneralizedEhrenfest’srelationsforKerr-AdSblackholeforvariousangular
velocities.
dent) terms we first make a small change (5%) in an- ernment of India, for financial help.
gular velocity and see the behaviour of all the plots.
Letusfirstconsider(20). Itisclearfromallplotsthat
the smearing width suffers no significant change, i.e.
s−s(cid:48) ≈ 0. We may neglect this small change as it is
furtherweakendbythedenominatorof(s−s(cid:48))in(20). ∗ Electronic address: [email protected]
Asaresulttherighthandsideofthefirstequational- † Electronic address: [email protected]
most remains unchanged. Regarding (21), however, ‡ Electronic address: [email protected]
this small change in Ω changes the smearing in J. It [1] S.W. Hawking and D.N. Page, Comm. Math. Phys.
is obtained by finding j(= J(S = 1.763,Ω = 0.5)− 87, 577 (1983); E. Witten, Adv. Theor. Math. Phys.
2
J(S = 0.653,Ω = 0.5)) and j(cid:48)(= J(S = 1.763,Ω = 2,505(1998);S.CarlipandS.Vaidya,Class.Quant.
1 2 Grav. 20, 3827 (2003); S. Stotyn, R. Mann, Phys.
0.525)−J(S =0.653,Ω=0.525)) from (3) and then
1 Lett. B 681 472 (2009).
taking their difference. This value of j−j(cid:48) in (21) is
[2] J. M. Bardeen, B. Carter and S. W. Hawking, Com-
comparablewithitsdenominator. Thusitisnolonger mun. Math. Phys. 31, 161 (1973).
negligible. The contribution from this extra term is [3] V.IyerandR.M.Wald,Phys.Rev.D50,846(1994)
found to be j−j(cid:48) = (0.091−0.086) = 2.924. ; G. ’t Hooft, Int. J. Mod. Phys. A 11, 4623 (1996)
(ξ2−ξ1)(T(cid:48)−T) 0.0019×0.9 ; M. K. Parikh and F. Wilczek, Phys. Rev. Lett. 85,
This value when added with the other term (8.578) 5042 (2000); S. Shankaranarayanan, K. Srinivasan
gives the right hand side of (21) as 11.502. This is in and T. Padmanabhan, Mod. Phys. Lett. A 16, 571
reasonable agreement with the l.h.s. value (12.7925). (2001) ; S. P. Robinson and F. Wilczek, Phys. Rev.
In order to get a full picture we consider various al- Lett. 95, 011303 (2005) ; R. Banerjee and S. Kulka-
lowed values of Ω and check the validity of (20) and rni, Phys. Rev. D 77, 024018 (2008); R. Banerjee, S.
Kulkarni,Phys.Lett.B659,827(2008);R.Banerjee
(21). The results are summarised in a tabular form
and S. K. Modak, JHEP 0905, 063 (2009).
(seeTable-I[10]). Itclearlyshowsthevalidityofthese
[4] M. M. Caldarelli, G. Cognola and D. Klemm, Class.
new equations for black hole phase transition. Quant. Grav. 17, 399 (2000)
We have discussed a new approach to study phase [5] R. Banerjee, S. K. Modak and S. Samanta,
transitions in black holes that is based on their anal- arXiv:1005.4832 [hep-th].
ogywithfluids. Severalfeaturesofthisanalogyplayed [6] H. E. Stanley, Introduction to phase transitions and
a crucial role. Unlike the liquid to vapour transition, critical phenomena, Oxford University Press, New
York (1987).
here it is not first order. Rather, according to our
[7] M. W. Zemansky and R. H. Dittman, Heat and
analysis this is a smeared continuous (higher order)
Thermodynamics, McGraw-Hill International Edi-
phase transition. Following a generalized Gibbsian tions, Sixth Edition (1981) (Chapter 14).
approachwehavederivedthegoverningequationsfor [8] R. Banerjee, S. K. Modak and S. Samanta, Eur.
such transitions. These equations were verified for Phys.J.C70,317(2010);R.Banerjee,S.Ghoshand
phase transition in Kerr-AdS black hole. Our calcu- D. Roychowdhury, Phys.Lett.B 696, 156 (2011).
lations are robust. We checked this by taking other [9] F = (π−3S)(π+S)3−2S(π+S)2(4π+3S)Ω2+S2(2π+3S)2Ω4
3
values of S and S close to those taken here. We 2π2[(π−3S)(π+S)4+6S2(π+S)3Ω2−S3(π+S)(4π+3S)Ω4]
feel that th1e approa2ch discussed here can be pushed H = 4π3/2S(π+S)2Ω((3(π+S)2−S(2π+3S)Ω2))
(π+S)3(3S−π)+2S(π+S)2(4π+3S)Ω2−S2(2π+3S)2Ω4
further, by exploiting the analogy with fluids, to [10] The new variables in table-II are defined as X =
1
gainadeeperinsightintoblackholethermodynamics. CΩ2−CΩ1 ,Y = s−s(cid:48) , X = χ2−χ1, Y =
Tc(χ2−χ1) 1 (Tc(cid:48)−Tc)(χ2−χ1) 2 ξ2−ξ1 2
j−j(cid:48)
One of the authors S. K. M thanks the Council of
(Tc(cid:48)−Tc)(ξ2−ξ1)
Scientific and Industrial Research (C. S. I. R), Gov-