Thermodynamics of phantom black holes in Einstein-Maxwell-dilaton theory

Manuel E. Rodrigues* and Zui A. A. Oporto†

Universidade Federal do Espırito Santo, Centro de Ciencias Exatas - Departamento de Fısica Av.Fernando Ferrari s/n - Campus de Goiabeiras, CEP29075-910 - Vitoria/ES, Brazil

(Received 19 February 2012; published 10 May 2012)

A thermodynamic analysis of the black hole solutions coming from the Einstein-Maxwell-dilaton

theory in 4D is done. By considering the canonical and grand-canonical ensemble, we apply standard

method as well as a recent method known as geometrothermodynamics. We are particularly interested in

the characteristics of the so called phantom black hole solutions. We will analyze the thermodynamics of

these solutions, the points of phase transition and their extremal limit. The thermodynamic stability is also

analyzed. We obtain a mismatch between the results of the geometrothermodynamics method when

compared with the ones obtained by the specific heat, revealing a weakness of the method, as well as

possible limitations of its applicability to very pathological thermodynamic systems. We also found that

normal and phantom solutions are locally and globally unstable, except for certain values of the coupled

constant of the Einstein-Maxwell-dilaton action. We also show that the anti-Reissner-Nordstrom solution

does not possess extremal limit nor phase transition points, contrary to the Reissner-Nordstrom case.

DOI: 10.1103/PhysRevD.85.104022 PACS numbers: 04.70.�s, 04.20.Jb, 04.70.Dy

I. INTRODUCTION

Since the discovery made by Hawking [1] of the ther-modynamic properties of the black holes related to thequantum phenomena, there has been much interest instudying the properties of various kinds of solutions ob-tained from general relativity and their modifications. Inclose analogy to the usual thermodynamics, the black holethermodynamics is based upon four basic principles, thezero law and the other three laws, which are all analogousto the usual ones [2,3]. Therefore, a new black hole solu-tion can be physically interpreted also by the analysis oftheir thermodynamic properties. Furthermore, we can ana-lyze the thermodynamic stability of a new solution throughits properties.

There are several methods to study the thermodynamicsproperties and stability of a black hole. A basic reference onthis is the work of Davies [4]. We can mention other morerecent methods, such as geometrothermodynamics [5] andHamiltonian thermodynamics [6]. For the present work, thefirst two methods will developed, always in parallel so wecan compare their results. We want to stress that the maingoal of this paper is to analyze the thermodynamics prop-erties and stability of the solutions known as phantom blackholes, specifically those coming from the Einstein-(anti)Maxwell-(anti)dilaton (EMD) theory. These solutionscome from the minimal coupling of the Einstein-Hilbertaction with a scalar field that could be either dilatonic orphantom, which at the same time is coupled with aMaxwellfield that can be a spin-1 normal or phantom field. Thephantom term furnishes the contribution of negative energydensity, which justifies the nomenclature.

In order to develop the analysis of the thermodynamicproperties of this class of black holes, let us briefly illus-trate the interest i studying phantom solutions in black holephysics. The programs of evolution of our universe, spe-cially the ones for the spectrum of anisotropies of thecosmic background radiation on the one hand and for therelation magnitude versus redshift of the tye Ia supernovaeon the other, have pointed out today an accelerated expan-sion of the universe, dominated by an exotic fluid ofnegative pressure. Furthermore, there is evidence suggest-ing this exotic fluid could be of phantom nature [7]. Hence,several classes of black holes (also wormholes, see [8])have been found that have phantom characteristics. Animportant class of phantom solutions is the Gibbons andRasheed’s of the EMD theory [9]. Several other general-izations were obtained, such as the higher-dimensionalblack holes by Gao and Zhang [10] and the higher-dimensional black branes by Grojean et al. [11]. Theanalysis of the algebra produced by a metric with 2 timesin higher dimensions, which provides phantom fields in4D, was developed by Hull [12] and for sigma models byClement et al. [13]. In this work, we will study somesolutions coming from the EMD theory, which werestudied in detail in [14].There are some methods of analysis in black hole ther-

modynamics theory dubbed as geometrical, because theymake use of differential geometry to determine thermody-namic properties such as: points at which black holesbecome extremal or they pass through a phase transitionand thermodynamic stability of the system. One of the firstmethods was proposed by Rao [15], subsequently devel-oped by other authors [16]. Later, the works of Weinhold[17] and Ruppeiner [18] were frequently used for the studyof the black hole thermodynamics. The method we willexplore in this work is known as geometrothermodynamics

*esialg@gmail.com†azurnasirpal@gmail.com

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(GTD); however, the results obtained with this method willalways be compared with the ones obtained by the usualnongeometric methods. The GTD had been shown to beequivalent,and, in some situations, even superior in manyaspects when compared with the usual nongeometric ap-proaches. The GTD has been widely used in the literatureto study the most diverse classes of black holes [19,20].The results obtained by the GTD methods reconcile someinconsistencies between the Weihold and Ruppeiner meth-ods; for example, we can mention some cases in which ablack hole suffers a phase transition according to onemethod but not by other one. Yet, there is still a goodconcordance when different thermodynamic potentialsare chosen, like, for example, the mass and entropy repre-sentations, this in virtue of the invariance of the formalismby Legendre transformations. Finally, we have mentionedthat the results are independent of the particular thermody-namic ensemble considered.

However, we also have to point out that the GTDmethodcan contain some inconsistencies when compared with themore usual analysis done by the specific heat. Recently ithas shown that for the cases of Reissner-Nordstrom-AdSand (phantom case) anti-Reissner-Nordstrom-AdS blackholes [21], the GTD method does not reproduce the resultsobtained by the specific heat method. We will arrive to thesame conclusion here in the case of phantom black holeswithin the EMD theory.

This paper is organized as follows: In Sec. II is asummary of the static and spherically symmetric solutionfor the EMD theory in 4D1 and the derivation of thermo-dynamics variables is presented. In Sec. III, the GTDmethod is reviewed in some detail. In Sec. IV, the GTDmethod is applied to the classes of asymptotically flat blackholes of the EMD theory in Sec. IVA and the anti-Reissner-Nordstrom case in Sec. IVB In Sec. V the con-clusions and perspectives of the work are presented.

II. PHANTOM BLACK HOLE SOLUTIONS ANDTHE FIRST LAW OF THERMODYNAMICS

In this section we present the class of solutions comingfrom Einstein-Maxwell-dilaton theory, their relevant pa-rameters, the formulation of the first law of thermodynam-ics, as well as the fundamental ingredients necessaries for adetailed analysis of the thermodynamics properties of thesesolutions.

We begin by defining the Einstein-Maxwell-dilaton ac-tion as

S ¼Z

dx4ffiffiffiffiffiffiffi�g

p ½R� 2�1g��r�’r�’

þ �2e2�’F��F���; (1)

where the first term corresponds to the usual Einstein-Hilbert action, the second one is the kinetic term of thescalar field (dilatonic or phantom), and the third one is thecoupling term between the scalar and Maxwell fields, withreal valued coupling constant �. The coupling constant �1

takes the values �1 ¼ þ1 (dilaton) or �1 ¼ �1 (antidila-ton). The parameter �2 can be �2 ¼ þ1 (Maxwell) or�2 ¼ �1 (anti-Maxwell). Depending on whether the con-tribution of the energy density is positive or negative,which is determined by �1;2, the kinetic term of the scalar

field and the coupling term with the Maxwell field can benormal (ordinary) or phantom.Now, we use some well-established results about the

class of black hole solutions in EMD theory. Accordingto [14], Eq. 2.19 in Sec. 2, let us choose the solutions of!ðuÞ and JðuÞ as the function sinhðuÞ, where the horizon isnondegenerate and u0 > 0, and let us perform a reparamet-rization of the radial coordinate as

u ¼ 1

ðrþ � r�Þ ln�fþf�

�; f� ¼ 1� r�

r; (2)

with

r� ¼ � 2a

1� e�2au0ðrþ � r� ¼ 2aÞ; (3)

then we obtain the solution

dS2 ¼ fþf��dt2 � f�1þ f��� dr2 � r2f1��� d�2; (4)

F ¼ � q

r2dr ^ dt; e�2�’ ¼ f1��� ; (5)

where ’0 ¼ 0, � ¼ ��=�þ (for �1 ¼ 1,�1<�< 1, and�1 ¼ �1, � 2 ð�1;�1Þ [ ð1;þ1Þ), 0< r� < rþ for�2�þ > 0, and finally r� < 0< rþ for �2�þ < 0. Thisis the exact solution of a spherically symmetric black hole,asymptotically flat, electrically charged and static, withinternal horizon ‘‘r�’’

2 and event horizon ‘‘rþ’’, which isrelated to the physical parameters, mass and charge of theblack hole, through the relations

M ¼ rþ þ �r�2

; (6)

q ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi1þ �

2

s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2rþr�

p: (7)

Now we are interested in the geometrical analysis rep-resenting semiclassical gravitational effects of the blackhole solutions mentioned before. By semiclassical, wemean to quantize the called matter fields, while leavingclassical the background gravitational field. Therefore wewill work with the semiclassic thermodynamics of blackholes, studied first by Hawking [1], and further developedby many other authors [22].

1From here it will be implied that the black hole solutionsdiscussed here are static and spherically symmetric solutioncoming from the EMD theory in 4D.

2A detailed discussion of the causal structure for the phantomcase can be found in [14].

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There are several techniques to derive the Hawkingtemperature law. For example. we can mention theBogoliubov coefficients [23] and the energy-momentumtensor methods [4,22], by the Euclidianization of themetric [24], the transmission and reflection coefficients[25,26], the analysis of the anomaly term [27], and bythe black hole superficial gravity [28]. Since all thesemethods have been proved to be equivalents [29], thenwe opt, without loss of generality, to calculate theHawking temperature by the superficial gravity method.

The superficial gravity of a black hole is given by theexpression [30]

� ¼�

g0002

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�g00g11p

�r¼rþ

; (8)

where rþ is the radius of the event horizon. The relation-ship between the Hawking temperature of the black holeand its superficial gravity is given by the formula [1,28]

T ¼ �

2�: (9)

Therefore, for the black hole defined by (4), the super-ficial gravity (8) takes the form

� ¼ ðrþ � r�Þ�2r1þ�

þ; (10)

and the corresponding Hawking temperature (9) would be

T ¼ ðrþ � r�Þ�4�r1þ�

þ: (11)

We define the area of the horizon of the black hole as

A ¼Z 2�

0

Z �

0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffig22g33

pd�d

��������r¼rþ¼ 4�r2f1���

��������r¼rþ

¼ 4�r1þ�þ ðrþ � r�Þ1��: (12)

Then the entropy of the black hole can be defined as [2]

S ¼ 1

4A ¼ �r1þ�

þ ðrþ � r�Þ1��: (13)

On the other hand, working out (5), we obtain the electricpotential at the event horizon, which reads

A0 ¼Z r

þ1F10ðr0Þdr0

��������r¼rþ¼ q

rþ: (14)

Using Eqs. (6), (7), and (13), we can write the differen-tial forms of the mass, charge and entropy

dM ¼ 1

2ðdrþ þ �dr�Þ;

dq ¼ �2

�1þ �

4q

�ðr�drþ þ rþdr�Þ;

dS ¼ �r1þ�þ

ðrþ � r�Þ���2� ð1þ �Þ r�

rþ

�drþ � ð1� �Þdr�

�:

(15)

These relations, together with (11) and (14), satisfy the firstlaw of black hole thermodynamics [2]

dM ¼ TdSþ �2A0dq: (16)

Note that when �2 ¼ �1, the first law is generalized forthe Einstein-anti-Maxwell-dilaton sector, where the secondterm, which is related to the work, suffers a change ofsignal as a consequence of a negative energy contributionto the system. The exact formula of Eq. (16), originallyknown as the Smarr formula [31], can be integrated,resulting in

M ¼ 2TSþ �2A0q: (17)

In this case we also have a change of signal of the workterm for the phantom case.We can write the horizons in terms of the mass and

charge parameters of the black hole as follows:

rþ ¼ MþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 � 2�2�q

2

ð1þ �Þ

s; (18)

r� ¼ 1

�

�M�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 � 2�2�q

2

ð1þ �Þ

s �: (19)

It is important to note that the possible extremal case,i.e., rþ ¼ r�, has to be analyzed carefully, because accord-ing to the causal structure described in [14], for the ordi-nary EMD theory r� is a curvature singularity, and for thephantom case in which �2�þ < 0 (r� < 0) is a true singu-larity. Hence, in the last case we have a causal structureidentical to the Schwarzschild solution with just one hori-zon. In some phantom cases where � takes discrete (inte-ger) values, we have two horizons that can be crossedwhere the radial coordinate r for the internal horizon isno longer r�, but r ¼ 0, with r� being a singularity wherethe geodesics ends. When this is so, it is not simple todetermine if the extremal limit exists, because rþ ! 0represents a regime which has not yet been analyzed inthe literature. The analysis by the geometrothermodynam-ics method could provide a new insight to understand thispathological solution, but we will see that, just as the usualmethod using the specific heat, some subtleties are stillunavoidable. Maybe this subtleties can be understood onlywithin the context of more fundamental quantum analysis.When the normal EMD theory is considered, the ex-

tremal limit still provides a structure quite similar to whatis called a naked singularity; classified as a lightlike nakedsingularity, which can be reached only after an infinitelapse of time. A semiclassical analysis of such a structure,i.e., a nonasymptotically flat black hole, has been studied in[25], but the analysis does not contain a conclusive result inrelation to the thermodynamic properties. So once again itseems like some fundamental quantum theory of gravityshould be considered at this point to depth into thisquestion.

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We have briefly discussed here the causal structure andthe extremal limit of this class of solutions; now we areinterested in their thermodynamic properties, in which wewill see present subtleties precisely for the particular casesconsidered in this work.

Taking the Eqs. (13), (14), (18), and (19), we can rewritethe temperature, entropy and the electric potential as func-tions of the mass and electric charge as follows:

TðM;qÞ ¼ 1

4���

�Mþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 � 2�2�q

2

ð1þ �Þ

s �1þ� � (20)

��ð�� 1ÞMþ ð�þ 1Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 � 2�2�q

2

ð1þ �Þ

s ��1��;

(21)

SðM;qÞ ¼ ����1

�Mþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 � 2�2�q

2

ð1þ �Þ

s �1þ�

��ð�� 1ÞMþ ð�þ 1Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 � 2�2�q

2

ð1þ �Þ

s �1��

;

(22)

A0ðM;qÞ ¼ �2

�q

�M�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 � 2�2�q

2

ð1þ �Þ

s �: (23)

Later on, we will use the entropy as thermodynamicpotential. If we want to consider the mass, which is equiva-lent to the energy, as thermodynamic potential we need toinvert (22) in order to write the mass in terms of the entropyand the electric charge. This is by no means evident fromthe general case (4). Thus, if we want to proceed with theanalysis in terms of at least two different thermodynamicpotentials, i.e., entropy and mass, it is convenient to spe-cialize to the case � ¼ 0, � ¼ 1 (with �1 ¼ �2 ¼ 1),which is known as the Reissner-Nordstrom, or(��1 ¼ �2 ¼ �1) which would be the anti-Reissner-Nordstrom. For these cases, the mass, electric charge,entropy, and electric potential are given by

M ¼ 1

2ðrþ þ r�Þ; q ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

�2rþr�p

;

T ¼ rþ � r�4�r2þ

; S ¼ �r2þ; A0 ¼ q

rþ; (24)

and satisfy the first law of black hole thermodynamics (16)and the Smarr formula ([31]).

Finally, from (24) we have

rþ ¼ MþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 � �2q

2q

; r� ¼ M�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 � �2q

2q

;

(25)

which gives us the temperature, entropy, and electricpotential, which explicitly read

T ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 � �2q

2p

2�ðMþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 � �2q

2p Þ2

; (26)

S ¼ �ðMþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 � �2q

2q

Þ2;A0 ¼ �ðM�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 � �2q

2q

Þ=q: (27)

We have introduced all the necessary prerequisites tostart with the study of the thermodynamic properties bymeans of the framework of geometrothermodynamics. Inthe next section we will introduce this method and furtherapply it to our particular cases of interest.

III. THE GEOMETROTHERMODYNAMICSMETHOD

Geometrothermodynamics (GTD) makes use of differ-ential geometry as a tool to represent the thermodynamicsof physical systems. Let us consider the (2nþ 1)-dimensional space T, which coordinates are representedby the thermodynamic potential �, the extensive variableEa, and the intensive variables Ia, where a ¼ 1; . . . ; n. Ifthe space T has a nondegenerate metric GABðZCÞ, whereZC ¼ f�; Ea; Iag, and the so called Gibbs 1-form � ¼d�� abI

adEb, with ab the delta Kronecker; then thestructure (T;�; G) is said to be a contact Riemannianmanifold if � ^ ðd�Þn � 0 is satisfied [32]. The space Tis known as the thermodynamic phase space.We can define a n-dimensional subspace E � T, with

extensive coordinates Ea, by the map ’:E ! T, with � ��ðEaÞ, such that

’�ð�Þ � 0 )8<:d� ¼ abI

adEb;@�@Ea ¼ abI

b:(28)

We call the space E the thermodynamic space of theequilibrium states; the first Eq. (28) is called the ‘‘first lawof thermodynamics,’’ and the second relation would bereferred as the ‘‘condition of thermodynamic equilibrium.’’We impose as a necessary condition the ‘‘second law ofthermodynamics’’

� @2�

@Ea@Eb 0; (29)

where the signal (�) depends on the chosen thermodynamicpotential; for example, in the case of the mass we have (þ),and for the case of entropy we have (�). This is known asconvexity of the thermodynamic potential condition. Thethermodynamic potential is defined so that it satisfies thehomogeneity condition �ð�EaÞ ¼ ���ðEaÞ. By differen-tiation with respect of �, using the second equation of (28)and taking � ¼ 1, we have that

��ðEaÞ ¼ abIaEb: (30)

By differentiation of this last equation, and using the firstequation of (28), we have

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ð1� �ÞabIadEb þ abE

adIb ¼ 0: (31)

For � ¼ 1, we get the Euler identity (30) and the Gibbs-Duhem formula (31). The pullback ’�:T�ðTÞ T�ðTÞ !T�ðEÞ T�ðEÞ,3 induces a metric on E, such that ’�ðGÞ¼g.

Hernando Quevedo has developed and improved a pos-sible metric G for the GTD, which, for the case of blackholes, can be written as [5]

dL2 ¼ GABdZAdZB ¼ �2 þ ðabE

aIbÞð�abdEadIbÞ;

(32)

where �ab ¼ f�1; 1; . . . ; 1g. The case �ab¼f�1;1;...;1gholds for second-order phase transition. The metric on Einduced by the pullback ’� is

dl2 ¼ gabdEadEb ¼ @ZA

@Ea

@ZB

@EbGABdE

adEb

¼�Ec @�

@Ec

���ad

di @2�

@EiEb

�dEadEb: (33)

Since the thermodynamics system has to be independentof the particular choice of thermodynamic potentials, andalso invariant under Legendre transformation, we have thatthe metrics (32) and (33) should be invariant underLegendre transformations of the form

� ¼ �� abEaIb; Ea ¼ �Ia; Ib ¼ Eb: (34)

The n-dimensional space E, with metric gab, contains theinformation about the thermodynamic interaction, phasetransitions, and fluctuations or stability of the thermody-namic system. With the help of metric (33), we can derivethe scalar curvature R, which gives us information abouttwo aspects of the theory: when there are thermodynamicinteraction and when there are phase transitions; it alsotells us at which points of the thermodynamic equilibriumspace these transitions take place.

As we mentioned in the introduction, this methodpresents some important results. The first one is that theuse of the Weinhold and Ruppeiner metrics providesresults that are in contradiction to each other, even incontradiction to themselves when, for example, differentthermodynamic potentials are used to describe the samesystem. We can cite the case of the Reissner-Nordstrom(RN) black hole, for which the use of the Ruppeiner metricin the entropy representation provides a flat space E, hencewithout phase transition [33]. However, when the repre-sentation is given by the internal energy of the black hole,the same method points out a nonzero scalar curvature witha singularity, i.e., with a phase transition in the curvedspace E [34]. This contradictory result is resolved in theGTD by the metric (32), which is invariant under Legendretransforms (34), reconciling the results of RN black hole

thermodynamics regardless of the choice of thermody-namic potential [35]. The second result is the agreementwith the usual analysis of the thermodynamic system bymeans of the specific heat of the black hole [20]. A thirdresult is that the description of the thermodynamic systemdepends, in many cases, on the ensemble’s choice [36,37],because this choice leads to different specific heats. Thisproblem is solved by the description of GTD, so it results ina consistent description of the thermodynamic systemdescribed by different ensembles [38].We saw that the metric of the thermodynamic equilib-

rium space E can be obtained by the pullback of the metricdefined on the contact Riemannian space. By the definitionof the line element (33) in E space, we can define thedistribution of probabilities to get a physical state withextensive variable Ea within the interval Ea þ dEa

PðEaÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidet½gab�

pð2�Þðn=2Þ exp

�1

2gabdE

adEb

�; (35)

where PðEaÞ satisfiesZ Yn

a¼1

dEaPðEaÞ ¼ 1: (36)

It can be shown that by taking the derivative of (35) withrespect of V�1, where V is the volume of the system, weobtain the expression for the second fluctuations (in thethermodynamic limit V ! 1) [39]

h�Ea�Ebi ¼ �gab; (37)

where �Ea ¼ Ea � Eað0Þ, and gab is the inverse of gab. A

more realistic analysis requires the constants to beadjusted. If the fluctuations are small and real valued,then we say the system is stable.Another criterion used to determine the stability is

through the following geometric objects

pð1Þ1 ¼ g11 > 0; pð2Þ

1 ¼ g22 > 0; . . . ; pðnÞ1 ¼ gnn > 0;

(38)

pð1Þ2 ¼

�������� g11 g12g12 g22

��������>0; pð2Þ2 ¼

�������� g22 g23g23 g33

��������>0;

(39)

pð1Þ3 ¼

�������������g11 g12 g13

g12 g22 g23

g13 g23 g33

�������������>0; pn ¼ det½gab�> 0:

(40)

The positive (negative) signal of pn, which would dependon the choice of the thermodynamic potential, determinesthe local (un)stability of the thermodynamic system; morespecifically if we have

3Where T�ðTÞ and T�ðEÞ represent the cotangent spaces of Tand E, respectively.

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pi > 0; i ¼ 1; . . . ; n; (41)

then we can affirm the system is globally stable.Since we are not considering rotating black holes, an-

other way to determine the global (un)stability is throughthe Helmholtz free energy. In terms of black hole thermo-dynamics variables, the Helmholtz free energy is no otherthing than the Legendre transformation of mass (energy)MðS; qÞ:

FðT; qÞ ¼ MðS; qÞ � TS: (42)

When we have

FðEaÞ< 0; 8EaðEa 2 IðEaÞÞ; (43)

where IðEaÞ is an interval and Ea are the extensive varia-bles, then the thermodynamic system is said to be globallystable. In the case of usual thermodynamics the Helmholtzfree energy is given by FðT; VÞ ¼ U� TS.

We can also define the Gibbs potential as

GðT; A0Þ ¼ MðS; qÞ � TS� �2A0q; (44)

in this case the global stability is determined by

GðEaÞ< 0; 8EaðEa 2 IðEaÞÞ: (45)

We introduced the signal �2 in (44) to compensate for thecontribution of the work term. Later we will make use ofthe Gibbs potential to determine the global stability of thethermodynamic system. For a system shown stable in thisensemble, we must have

@2G

@T2;

@2G

@A20

;@2G

@T@A0

� 0: (46)

In the next section we will apply these methods todetermine the thermodynamic properties of the blackhole solutions coming from the EMD theory.

IV. THERMODYNAMICS OF PHANTOMBLACK HOLES

A. Application of the geometrothermodynamicsmethod

1. Einstein-(anti)Maxwell-dilaton solutions

To begin with, let us define the thermodynamic variablesof the system. For the Einstein-Maxwell-dilaton blackholes, we will have always a solution with two physicalparameters, the mass M and the electric charge q. Othervariables (parameters) such as entropy S, temperature Tand electric potential A0, can be defined as implicit func-tions of the parameters mentioned before. The contactRiemannian manifold T is, in this case, 5-dimensionaland the space E of thermodynamic equilibrium states is a2-dimensional manifold.

The thermodynamic description is the entropy represen-tation, SðM;qÞ, which is identified as the thermodynamicpotential �, according to what was defined in the previous

section. The extensive variables are the mass M and theelectric charge q, which are represented by the coordinatesEa. The intensive variables are the temperature T and theelectric potential A0, represented by the coordinates Ia.We have then a coordinate system for the thermody-

namic phase space T as being ZA ¼ fSðM;qÞ;M; q; T; A0g,together with the Gibbs 1-form given by4

�S ¼ dS� 1

TdMþ �2A0

Tdq; (47)

such that ’�ð�SÞ ¼ 0 is satisfied, which is no other thingthan first law of black hole thermodynamics dM ¼ TdSþ�2A0dq (�2 ¼ �1).For a second-order phase transition, the line element

(32) of the space T is

dL2 ¼�dS� 1

TdMþ �2A0

Tdq

�2 þ

�M

T� �2A0

Tq

�

���d

�1

T

�dMþ d

���2A0

T

�dq

�: (48)

The first and second laws, and the equations of the equi-librium state as are given by

d� ¼ abIadEb ! dS ¼ 1

TdM� �2A0

Tdq; (49)

@2S

@M2;

@2S

@M@q;

@2S

@q2� 0; (50)

@�

@Ea¼ abI

b ! @S

@M¼ 1

T;

@S

@q¼ ��2A0

T: (51)

Now we have to specify a solution, where we considerfirst the general case (4). The line element (33) of theequilibrium space, taking into account (22), would be

dl2 ¼�M

@S

@Mþ q

@S

@q

��� @2S

@M2dM2 þ @2S

@q2dq2

�; (52)

¼ gMMdM2 þ gqqdq

2; (53)

4This expression comes from the first law of thermodynamics(16), which was inverted in order to isolate the entropy.

MANUEL E. RODRIGUES AND ZUI A. A. OPORTO PHYSICAL REVIEW D 85, 104022 (2012)

104022-6

gMM ¼ 16�2�2�

�Mþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 � 2�2�q

2

ð1þ�Þq �

1þ2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 � 2�2�q

2

ð1þ�Þq

��M2 � �2q

2 þM

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 � 2�2�q

2

ð1þ �Þ

s �

��ð�� 1ÞMþ ð�þ 1Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 � 2�2�q

2

ð1þ �Þ

s ��1�2�

��ð1þ �ÞM� ð1þ 3�Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 � 2�2�q

2

ð1þ �Þ

s ��; (54)

gqq ¼ �32�2�2�1þ2�

�Mþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 � 2�2�q

2

ð1þ�Þq ��1þ2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 � 2�2�q

2

ð1þ�Þq

��M2 � �2q

2 þM

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 � 2�2�q

2

ð1þ �Þ

s �� (55)

��ð�� 1ÞMþ ð�þ 1Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 � 2�2�q

2

ð1þ �Þ

s ��1�2�

��M3 þ ðM2 � �2q

2ÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 � 2�2�q

2

ð1þ �Þ

s �: (56)

We notice that the phantom contribution can switch thesignature of themetric of the space E. To calculate the scalarcurvature associated with the metric (53), we can make useof a particular formula valid for 2-dimensional spaces

RðM;qÞ ¼ � 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijdet½g�jp�@q

�@qgMM � @MgMqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijdet½g�jp

�

þ @M

�@Mgqq � @qgMqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijdet½g�jp

��� det½HS�

2ðdet½g�Þ2 ; (57)

HS ¼gMM gMq gqq

@MgMM @MgMq @Mgqq

@qgMM @qgMq @qgqq

0BB@

1CCA: (58)

Replacing (53) into (57), we have

RðM;qÞ ¼ NðM;qÞDðM;qÞ ; (59)

where

NðM;qÞ¼S2MMðFFqSqqqþ2SqqðF2q�FFqqÞÞ

þFSqqð�SqqFMSMMMþFðS2MMq�SMMMSqqqÞÞþSMMð�2S2qqF

2MþFSqqð�FqSMMq

þFMSqqMþ2SqqFMMÞþF2ðSMMqSqqq

�S2qqM�2SqqðSMMMM�SqqMMÞÞÞ; (60)

with @i . . . @jS ¼ Si...j, @i . . . @jF ¼ Fi...j, and

DðM;qÞ ¼ 2F3S2MMS2qq; (61)

where

F ¼ MSM þ qSq; (62)

and S as given before by (22).With the help of mathematical software, we find

the zeros of the scalar curvature RðM;qÞ [see (59)]

are the mass values M1 ¼ �qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2�2�=ð1þ �Þp

, M2 ¼�q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2ð1þ �Þ=2p

. We also have another zero for the scalar

curvature for the particular case �2 ¼ �1 at M3 ¼q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ �Þ=2ð1þ 2�Þp. This tell us that in general the

scalar curvature is nonzero—which implies the existenceof thermodynamic interaction for this class of black holes[40]—and is zero only when the mass takes the valuesM1,M2 or M3. These values characterize when the black holebecomes extremal. Let us note that only when we set rþ ¼r� in (18) and (19), we obtain the value M2 for the mass.Probably this is related to the fact we pointed out before,that in the normal and phantom cases, r� is a true singu-larity of the curvature, therefore, the extremal limit in thisregime is not viable. The two values M1 andM2, for �2 ¼� ¼ 1 (RN), are M1;2 ¼ �q, in total agreement with [4].

We also want to note that there exists an extremal limitat M3, when �2 ¼ �1. The point at which the scalarRðM;qÞ diverge is given by the mass M4 ¼ �qð1þ 3�Þ=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2�2ð1þ �Þð1þ 2�Þp

. Then, when �2 ¼ � ¼ 1, M4 ¼�2q=

ffiffiffi3

p, also in agreement with the previous work of

Davies [4]. However, for the anti-RN solution M4 is notreal, which means that there is no phase transition for thephantom when work with the entropy representation.To corroborate our analysis, let us calculate the specific

using the well-known formula

Cq ¼�@M

@T

�q¼

�@M

@S

�q

�@2M

@S2

�q

¼ ��@S

@M

�2

q

�@2S

@M2

�q: (63)

Doing this, we obtain

Cq ¼ 4���

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 � 2�2�q

2

ð1þ �Þ

s �Mþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 � 2�2�q

2

ð1þ �Þ

s �1þ�

�

�ð�� 1ÞMþ ð1þ �Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 � 2�2�q

2

ð1þ�Þq �

1��

ð1þ �ÞM� ð1þ 3�ÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 � 2�2�q

2

ð1þ�Þq : (64)

The zeros of (64) give us the information about thepoints at which the black hole becomes extremal; theseare precisely M1 and M2 found by the previous analysis,thus corroborating our analysis. In the same way, we getthat the phase transition point at which Cq diverges is M4.

Remarkable is the fact that M3 is not a zero of the specific

THERMODYNAMICS OF PHANTOM BLACK HOLES IN . . . PHYSICAL REVIEW D 85, 104022 (2012)

104022-7

heat, hence, it should be some spurious zero that results fromafailure of applicability of the analysis for this class of patho-logical solutions. We will see below that not only a new(nonphysical) extremal case is revealed by the GTDmethod,but also a new critical point is revealed when we choose themass of the black hole as the thermodynamic potential.

2. (anti)Reissner-Nordstrom solution

To study in detail the thermodynamics of the EMDsystem, let us consider the particular case � ¼ 1, thiscorresponds to the (anti)Reissner-Nordstrom [(anti)RN]solution. For this case, we have that the entropy is

S ¼ �ðMþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 � �2q

2q

Þ2; (65)

that, when replaced the into (53), furnishes the following:

dl2 ¼�4�2 ðM2��2q2þM

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2��2q

2p Þ

ðM2��2q2Þ2

�fðMþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2��2q

2q

Þ3ð�Mþ2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2��2q

2q

ÞdM2

þ�2ðMþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2��2q

2q

Þ½ðM2��2q2Þ3=2þM3�dq2g:

(66)

The scalar curvature derived from this metric is given by(59), where

NðM;qÞ ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 � �2q

2q

ðMþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 � �2q

2q

Þ� ½1024M14 � 3840�2q

2M12 þ 5504q4M10

� 3408�2q6M8 þ 460q8M6 þ 377�2q

10M4

� 122q12M2 þ 5�2q14 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 � �2q

2q

� ð1024m13 � 3328�2q2M11 þ 3968q4M9

� 1776�2q6M7 � 100q8M5 þ 251�2q

10M3

� 35q12M�; (67)

DðM;qÞ ¼ 4�2½64M10 � 208�2q2M8 þ 248q4M6

� 137�2q6M4 þ 34q8M2 � 2�2q

10

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 � �2q

2q

� ð64M9 � 176�2q2M7

þ 168q4M5 � 71�2q6M3 þ 11q8MÞ�2: (68)

The zeros of the numerator, NðM;qÞ, in (67), are M5 ¼�q

ffiffiffiffiffiffi�2

pand M6 ¼ �iq

ffiffiffiffiffiffiffiffiffiffiffi�2=3

p, which depend on the

choice of �2. The RN case, when �2 ¼ 1, has the zeroM5 ¼ �q (where we did rþ ¼ r�) that corresponds to theextremal RN black hole. The anti-RN case, �2 ¼ �1, has

the zero M6 ¼ �q=ffiffiffi3

p, but this result reveals a weakness

of this method, because it points out the presence of anextremal anti-RN black hole, which we know does notexist. The analysis of the causal structure performed in[14] shows that the anti-RN black hole has a causal struc-

ture identical to the Schwarzschild black hole, hence, thereis no existing extremal limit for this case. We do not haveany explanation for such irregularity of the thermodynamicsystem when described by the GTD method. We believethat the extremal limit for nontrivial black holes, just likethe anti-RN case, reveals pathologies which are not welldescribed by the GTD.The zero of the denominator DðM;qÞ, in (68), is given

by M7 ¼ �2qffiffiffiffiffiffiffiffiffiffiffi�2=3

p, which is real only for �2 ¼ 1. It

occurs then that the RN black hole has a second-orderphase transition point at M7, in good correspondencewith Davies [4]. On the other hand, the anti-RN blackhole, �2 ¼ �1, does not have any phase transition point;consequently, there is no extremal analogue, nor phasetransition for the anti-RN case.As before, in order to check our results, let as calculate

the specific heat of these model. The specific heat (63),calculated in combination with (65), is

Cq ¼ 2�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 � �2q

2q ðMþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

M2 � �2q2

p Þ2ðM� 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2 � �2q

2p Þ

: (69)

From (69) we have that the black hole is extremal (Cq ¼ 0)

when M ¼ M5 ¼ �qffiffiffiffiffiffi�2

p, and it has a point of phase

transition (Cq ! 1) at M ¼ M7 ¼ �2qffiffiffiffiffiffiffiffiffiffiffi�2=3

p. These re-

sults are in total agreement with the previous results wefound by the GTDmethod, but wewant to stress that beforewe obtained an extra extremal anti-RN black hole solution.This reinforces our previous statement about the weaknessof the GTD method for some pathological situations.As was mentioned before, we will now consider the

mass representation analysis. We start by inverting (65)to write the mass in terms of the entropy and electriccharge, this is

MðS; qÞ ¼ Sþ �2�q2

2ffiffiffiffiffiffiffi�S

p : (70)

The Gibbs 1-form reads in this case

�M ¼ dM� TdS� �2A0dq: (71)

Then, using (32) we obtain

dL2 ¼ ðdM� TdS� �2A0dqÞ2 þ ðTS� �2A0qÞ� ½�dTdSþ dð��2A0Þdq�: (72)

The pullback ’� induces a metric on the space E,

dl2 ¼�S@M

@Sþ q

@M

@q

���@2M

@S2dS2 þ @2M

@q2dq2

�(73)

¼ ðSþ 3�2�q2Þ

4S

�ðS� 3�2�q2Þ

8�S2dS2 þ �2dq

2

�: (74)

Finally, with this metric we can compute the scalar ofcurvature,

MANUEL E. RODRIGUES AND ZUI A. A. OPORTO PHYSICAL REVIEW D 85, 104022 (2012)

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RðS; qÞ ¼ �2

288�2q2S2ðS� �2�q2Þ

ðS� 3�2�q2Þ2ðSþ 3�2�q

2Þ3 : (75)

We can see that when the thermodynamic potential is themass of the RN black hole, the interpretation by thismethod of the extremal limit (rþ ¼ r�) is clear: the zerosof the numerator of RðS; qÞ in (75) exist only for the valuesS1 ¼ �q2, which clearly represent the extremal RN blackhole. When the anti-RN (phantom) case, �2 ¼ �1,is considered, we see that there is no extremallimit, as it should be. This last conclusion solves in somesense the unphysical prediction we get in the entropyrepresentation, by changing to the mass representation.But, in contrast, a new problem arises, we get a new pointof phase transition for the anti-RN case at S2 ¼ 3�2�q

2,with rþ ¼ 3�2r�. This indicates a breach of the invarianceof the theory for describing the system regardless of choiceof thermodynamic potential. Once again,this result con-firms our claim about the failure of the GTD method whenapplied to pathological solutions, which despite having asimple causal structure, apparently present nontrivial solu-tions. For the RN solution we have a extremal limit at S1and a second-order phase transition point at S2, in agree-ment with [4].

Finally, let us compute the specific heat. Using (63)together with (70), we get

Cq ¼ �2S

�S� �2�q

2

S� 3�2�q2

�: (76)

From this formula the correct interpretation can be read:for the RN black hole we have an extremal limit when S ¼S1 ¼ �q2, and a phase transition point at S ¼ S2 ¼ 3�q2.On the other hand, for the anti-RN case, we conclude thereis no extremal limit, nor point of phase transition.

B. Local and global stability

It is usual to study local stability of a thermodynamicsystem by means of the specific heat. Alternatively, withinthe GTD, we can study the metric components of thethermodynamic equilibrium space E or even more, theHessians of the entropy and mass. To determine the globalstability it can be achieved by the analysis of all thecomponents of the metric as well as their correspondingdeterminants; but also by means of the Helmholtz freeenergy or by the Gibbs potential. Here we will continuewith the study of local and global stability of EMD blackholes solutions.

Let us start calculating the Hessian of the entropy for thegeneral case (22), which is defined as

HS ¼@2S@M2

@2S@M@q

@2S@M@q

@2S@q2

0@

1A: (77)

Using (22) and considering �2 ¼ 1 (including � ¼ 1) weget local instability in view of the fact that SMMðM;qÞ isalways positive, and SqqðM;qÞ and SMqðM;qÞ are positive

as long as q > 0. In the same manner, when �2 ¼ �1,SMMðM;qÞ and SqqðM;qÞ are always positive and

SMqðM;qÞ is positive provided that q > 0; therefore we

have that the solutions are locally unstable.Analogously, we can calculate the Hessian matrix of the

mass for the particular choice � ¼ 1. Using (70) we foundthat

HM ¼@2M@S2

@2M@S@q

@2M@S@q

@2M@q2

0@

1A (78)

¼3�2�q

2�S

8ffiffiffi�

pS5=2

�ffiffiffi�

p�2q

2S3=2

�ffiffiffi�

p�2q

2S3=2�2

ffiffiffi�

pS1=2

0B@

1CA; (79)

The Hessian of the mass also leads us to local instabilities.More precisely, when �2 ¼ 1, then MSS and MSq can take

positive or negative values. Similarly, when �2 ¼ �1,MSq

can take positive or negative values, whereasMSS andMqq

are always negative.Once again, let us consider the analysis by the study of

the specific heat (64), which in terms of rþ and r� can bewritten as

Cq ¼ �2�r1þ�þ

ðrþ � �r�Þðrþ � r�Þ1��

½rþ � ð1þ 2�Þr�� : (80)

From this expression, we can establish, for example,local stability (Cq > 0) for the event horizon interval

�r� < rþ < ð1þ 2�Þr�, 0< �< 1 (EMD with �1 ¼1). The specific heat, with � ¼ �2 ¼ 1 or � ¼ ��2 ¼ 1,corresponds to the RN or anti-RN cases, respectively. Theanti-RN case with � ¼ 1 is locally unstable because r� <0 implies Cq < 0. The only case where the phantom solu-

tions are locally stable will be when �r� < rþ < ð1þ2�Þr� for � <�1.As we mentioned before, the components of the metric

of the thermodynamic equilibrium space, E, give us alsoinformation about local stability. According to (52), thecomponents of gMM and gqq, written in terms of rþ and r�,read

gMM ¼ �16�2r2ð1þ�Þþ

½rþ � ð1þ 2�Þr��ðrþ � �r�Þðrþ � r�Þ2�

; (81)

gqq ¼ �8�2�2r1þ2�

þ½r2þ þ ð�� 1Þr� þ 2�ð�þ 1

2Þr2��ðrþ � �r�Þðrþ � r�Þ2�

:

(82)

These components always have opposite signals for eachof the two cases, normal and phantom, including the case� ¼ 1. Therefore, by simple inspection of the componentsof the metric on E, we assert that the system is locallyunstable; additionally, it tells us that there is global insta-bility, just as was mentioned in Sec. III.

THERMODYNAMICS OF PHANTOM BLACK HOLES IN . . . PHYSICAL REVIEW D 85, 104022 (2012)

104022-9

For the next step, let us study the global stability of theclass of solutions of interested. For the general case we canwrite

M ¼ q

2A0

�1þ 2�2�A

20

ð1þ �Þ�; TS ¼ q

4A0

�1� 2�2A

20

ð1þ �Þ�;

(83)

rþ ¼ 1

4�T

�1� 2�2A

20

ð1þ �Þ�: (84)

The analysis starts with the grand-canonical ensemble.Using (83) and (84), we calculate the Gibbs potential(44)

GðT; A0Þ ¼ 1

16�T

�1� 2�2A

20

ð1þ �Þ�1þ�

: (85)

We know that T > 0, so we have to analyze the termbetween brackets. Consider first the normal case with�2 ¼ 1; when ð1þ �Þ is an odd integer, then the

system is globally stable only for � > 0 and A0 2ð�1;� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ �Þ=2p Þ [ ð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ �Þ=2p

;þ1Þ. The critical

electric potential values are A0ðcÞ ¼ � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ �Þp(

ffiffiffi2

p �1)

(by setting rþ ¼ 0 in (84)). Now let us consider the phan-tom solutions with �2 ¼ �1; in this case the system is

globally unstable, except for � <�1 and A0 2ð�1;� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij1þ �j=2p Þ [ ð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij1þ �j=2p

;þ1Þ. A straightfor-ward calculation of the derivatives in (46) shows that

@2G

@T2¼ 1

8�T3

�1� 2�2A

20

ð1þ �Þ�1þ�

; (86)

@2G

@A20

¼ ��2

ð1� 2�2A20Þ

4�T

�1� 2�2A

20

ð1þ �Þ���1

; (87)

@2G

@T@A0

¼ �2A0

4�T2

�1� 2�2A

20

ð1þ �Þ��: (88)

From here we can see that (88) always breaks thestability criteria, because A0 can be positive or negative(remember that q is real). Then it follows that the system isalways locally unstable.

The corresponding analysis by the canonical ensembleformalism is slightly different. We start by writing theHelmholtz free energy (42)

F ¼ 1

16�T

�1� 2�2A

20

ð1þ �Þ���1þ 2�2A

20

�1þ 2�

1þ �

��: (89)

The analysis of this formula is the following: the system is

globally stable for two cases. The first one is when A0 2ð�1;� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ �Þ=2p Þ [ ð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ �Þ=2p

;þ1Þ for � > 0 odd

integer and �2 ¼ 1. The second case is when A02ð�1;� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ�Þ=2ð1þ2�Þp Þ [ ð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ�Þ=2ð1þ2�Þp

;þ1Þfor � even integer and �2 ¼ �1 or A0 2

ð�1;� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ �Þ=2p Þ [ ð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ �Þ=2p;þ1Þ for � 2 <.

When we use the specific heat formula (80), we found

second-order phase transition points given by A0ð1Þ ¼� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

�2ð1þ �Þ=2ð1þ 2�Þpand A0ð2Þ ¼ � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

�2ð1þ �Þ=2p(even for � > 1). The point A0ð3Þ ¼ � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

�2ð1þ �Þ=ð2�Þp,

as well as A0ð2Þ with � < 1, represent the extremal case

where rþ ! 0 (see [14] for details).Finally, we can calculate the minimum temperature for

this system using the formula @T�1=@rþ ¼ 0 [41], so wehave

T0 ¼ 1

4�rþ

��

1þ �

��: (90)

From the specific heat, Eq. (80), we identify a phasetransition point at rþ ¼ 3r�, which gives us the criticaltemperature

Tc ¼ 1

4�rþ

�2�

1þ 2�

��; (91)

combining this with (90) we get the relation Tc ¼T0½2ð1þ �Þ=ð1þ 2�Þ��.

V. CONCLUSIONS

The thermodynamic properties of the class of solutionsknown as phantom black holes have not been studied indetail yet. The physical stability of these solutions can bedetermined also by studying their thermodynamic proper-ties. The main objective of the present work was to fill thisgap by establishing a detailed analysis of this kind ofsolutions.The zeroth, second, and third laws of thermodynamics

remain unmodified by these solutions. However, the firstlaw had to be generalized to take into account the contri-bution of the work done on (or by) the system with the‘‘wrong’’ sign (when compared to the usual case). Thedifferential form of the generalized first law was written in(16) whereas the exact expression was written in ([31]).The use of geometrothermodynamics as a tool of analy-

sis has proven to be, once more, equivalent to the mostusual methods, but with an important exception that wasthe phantom case with �2 ¼ �1. More specifically, wesaw for the phantom solutions that when we choose theentropy as the thermodynamic potential, the method intro-duces a new value of the mass parameter which could beinterpreted as a (nonphysical) extremal black hole limit. Ifwe choose the mass as the thermodynamic potential, thenwe get a new critical point for the system. The new massparameter and the new critical point have to be consideredas spurious zeros of the numerator and denominator of thescalar curvature of the space E. This can be understood bythe fact that EMD solutions switch to the phantom sector(i.e., with�2 ¼ �1) by the symmetry transformation q2 !�q2, as was shown in [13]. In the case of the entropyrepresentation, with the choice � ¼ 1, that kind of

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symmetry leads to the appearance of a new real valued zeroof the scalar curvature which originally was purely imagi-nary. In the case of the mass representation, the samesymmetry transformation entails the appearance of a newcritical point, i.e., a divergence of the scalar curvature.When we carried out the study of the thermodynamicproperties by means of specific heat, we have not foundnew spurious critical points nor new divergence points ofthe scalar curvature, just as was expected. This resultrevealed the fragility of the GTD method when appliedto pathological solutions.

In regard to the stability analysis, we found that the onlypossible local stability would correspond to the followingcases: �r� < rþ < ð1þ 2�Þr�, for �1 ¼ �2 ¼ 1 and 0<�< 1 or for �1 ¼ �2 ¼ �1 and � <�1. All the othersolutions, normal or phantom, have been shown to belocally unstable. The global stability can be establishedfor some particular situations that restrict the values theelectric potential A0 and the parameter � can take.

The perspective of the present work is to study in detailthe subtleties that arise from the phantom solutions, withthe hope to strengthen some weak aspects of the promising

novel method which proved to be the geometrothermody-namics. TThus, we expect some physical limitations couldappear for the application of this method. This has alreadybeen shown to be true in the cases of Reissner-Nordstrom-AdS and anti-RN-AdS black holes [21]. We propose toelucidate this issue with more details in a forthcomingwork.Finally, with respect to the global stability of one portion

of the class of phantom solutions, we expect this to be anindication for the stability of the space-time of these classof solutions. That shall also be a topic to be studied in asubsequent work.

ACKNOWLEDGMENTS

We are grateful to Gabriela Conde Saavedra for the helpin the elaboration of the manuscript. M. E. Rodriguesthanks UFES for the hospitality during the developmentof this work and Z. Oporto thanks Centro Latino-Americano de Fısica (CLAF) and the Conselho Nacionalde Desenvolvimento Cientıfico e Tecnologico (CNPq-Brazil) for Grant No. 141579/2008-0.

[1] S. Hawking, Commun. Math. Phys. 43, 199 (1975).[2] J.M. Bardeen, B. Carter, and S.W. Hawking, Commun.

Math. Phys. 31, 161 (1973).[3] C. O. Lousto, Nucl. Phys. B410, 155 (1993); B449, 433

(1995).[4] P. C.W. Davies, Proc. R. Soc. A 353, 499 (1977).[5] H. Quevedo, A. Sanchez, S. Taj, and A. Vazquez, Gen.

Relativ. Gravit. 43, 1153 (2011).[6] Jorma Louko and Stephen N. Winters-Hilt, Phys. Rev. D

54, 2647 (1996).[7] S. Hannestad, Int. J. Mod. Phys. A 21, 1938 (2006); J.

Dunkley et al., Astrophys. J. Suppl. Ser. 180, 306 (2009).[8] K. A. Bronnikov, M. S. Chernakova, J. C. Fabris, N. Pinto-

Neto and M. E. Rodrigues, Int. J. Mod. Phys. D 17, 25(2008).

[9] G.W. Gibbons and D.A. Rasheed, Nucl. Phys. B476, 515(1996).

[10] C. J. Gao and S. N. Zhang, arXiv:hep-th/0604114.[11] C. Grojean, F. Quevedo, G. Tasinato, and I. Zavala, J. High

Energy Phys. 08 (2001) 005.[12] C.M. Hull, J. High Energy Phys. 07, (1998) 021.[13] Mustapha Azreg-Ainou, Gerard Clement, Julio C. Fabris

and Manuel E. Rodrigues, Phys. Rev. D 83, 124001(2011).

[14] Gerard Clement, Julio C. Fabris and Manuel E. Rodrigues,Phys. Rev. D 79, 064021 (2009).

[15] C. R. Rao, Bull. Calcutta Math. Soc. 37, 81 (1945).[16] S. Amari, Differential-Geometrical methods in Statistics

(Springer-Verlag, Berlin, 1985).[17] F. Weinhold, J. Chem. Phys. 63, 2479 (1975); 65, 559

(1976).

[18] G. Ruppeiner, Phys. Rev. A 20, 1608 (1979); Rev. Mod.Phys. 67, 605 (1995); 68, 313 (1996).

[19] Danny Birmingham and Susan Mokhtari, Phys. Lett. B697, 80 (2011); Hernando Quevedo and Maria N.Quevedo, arXiv:1111.5056v1; Hernando Quevedo andAlberto Sanchez, Phys. Rev. D 79, 024012 (2009);Alexis Larranaga and Alejandro Cardenas,arXiv:1108.2205v1; Alexis Larranaga and Sindy Mojica,Braz. J. Phys. 41, 154 (2011); Hernando Quevedo andAlberto Sanchez, Phys. Rev. D 79, 087504 (2009); LaszloA. Gergely, Narit Pidokrajt, and Sergei Winitzki, Eur.Phys. J. C 71, 1569 (2011); Hernando Quevedo, J. Math.Phys. (N.Y.) 48, 013506 (2007); Hernando Quevedo,Alberto Sanchez and Alejandro Vazquez,arXiv:0811.0222v1; Hernando Quevedo and AlejandroVazquez, AIP Conf. Proc. 977, 165 (2008); M. Akbar,H. Quevedo, K. Saifullah, A. Sanchez, and S. Taj, Phys.Rev. D 83, 084031 (2011); H. Quevedo, A. Sanchez, andA. Vazquez, arXiv:0805.4819v5; Alejandro Vazquez,Hernando Quevedo and Alberto Sanchez, J. Geom.Phys. 60, 1942 (2010); J. L. Alvarez, H. Quevedo, andA. Sanchez, Phys. Rev. D 77, 084004 (2008).

[20] Hernando Quevedo, Alberto Sanchez, Safia Taj andAlejandro Vazquez, J. Phys. A 45, 055211 (2012); SafiaTaj and Hernando Quevedo, arXiv:1104.3195v1; W.Janke, D. A. Johnston, and R. Kenna, J. Phys. A 43,425206 (2010); H. Quevedo, A. Sanchez, S. Taj, and A.Vazquez, Gen. Relativ. Gravit. 43, 1153 (2011); PengChen, Int. J. Mod. Phys. A 26, 3091 (2011).

[21] Deborah F. Jardim, Manuel E. Rodrigues and StephaneJ.M. Houndjo, arXiv:1202.2830v2.

THERMODYNAMICS OF PHANTOM BLACK HOLES IN . . . PHYSICAL REVIEW D 85, 104022 (2012)

104022-11

[22] N. D. Birrell and P. C.W. Davies, Quantum Fields inCurved Space (Cambridge University Press, Cambridge,England, 1982).

[23] L. H. Ford, in Particles and Fields: Proceedings of theIXth Jorge Andre Swieca Summer School, Brazil, 16-28February 1997, edited by J. C. Barata, Sergio F. Novaes,and Adolfo P. C. Malbouisson (World Scientific,Singapore, 1998), pp. 345-388, arXiv:gr-qc/9707062v1.

[24] G.W. Gibbons and S. Hawking, Phys. Rev. D 15, 2752(1977).

[25] G. Clement, J. C. Fabris, and G. T. Marques, Phys. Lett. B651, 54 (2007).

[26] Panagiota Kanti and John March-Russell, Phys. Rev. D 66,024023 (2002); Wontae Kim and John J. Oh, J. KoreanPhys. Soc. 52, 986 (2008); Kazuo Ghoroku and Arne L.Larsen, Phys. Lett. B 328, 28 (1994).

[27] S. P. Robinson and F. Wilczek, Phys. Rev. Lett. 95, 011303(2005).

[28] T. Jacobson and G. Kang, Classical Quantum Gravity 10,L201 (1993).

[29] Glauber Tadaiesky Marques and Manuel E. Rodrigues,Eur. Phys. J. C 72, 1891 (2012).

[30] R.M. Wald, Genaral Relativity (Univ. of Chicago Press,Chicago, 1984).

[31] L. Smarr, Phys. Rev. Lett. 30, 71 (1973).[32] R. Hermann, Geometry, Physics and Systems (Marcel

Dekker, New York, 1973); G. Hernandez and E.A.Lacomba, Diff. Geom. and Appl. 8, 205 (1998).

[33] Jan Aman, Ingemar Bengtsson and Narit Pidokrajt, Gen.Relativ. Gravit. 35, 1733 (2003).

[34] Jianyong Shen, Rong-Gen Cai, Bin Wang and Ru-KengSu, Int. J. Mod. Phys. A 22, 11 (2007).

[35] Hernando Quevedo, Gen. Relativ. Gravit. 40, 971 (2008).[36] X. N. Wu, Phys. Rev. D 62, 124023 (2000).[37] Y. S. Myung, Y.W. Kim, and Y. J. Park, Phys. Lett. B 663,

342 (2008).[38] Hernando Quevedo and Alberto Sanchez, J. High Energy

Phys. 09 (2008) 034.[39] George Ruppeiner, Phys. Rev. D 75, 024037 (2007); Rev.

Mod. Phys. 67, 605 (1995); 68, 313 (1996).[40] H. Quevedo, A. Sanchez, S. Taj, and A. Vazquez,

arXiv:1011.0122v1.[41] R. Banerjee, S. Kumar Modak and D. Roychowdhury,

arXiv:1106.3877v2.

MANUEL E. RODRIGUES AND ZUI A. A. OPORTO PHYSICAL REVIEW D 85, 104022 (2012)

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