Black hole thermodynamics from logotropic uids Salvatore Capozziello1 2 3Rocco DAgostino2 3yAlessio Lapponi4 5zand Orlando Luongo6 7 8x 1Dipartimento di Fisica E. Pacini Universit a di Napoli Federico II Via Cinthia 9 80126 Napoli Italy.

2025-05-01 0 0 542.93KB 13 页 10玖币

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Black hole thermodynamics from logotropic ﬂuids

Salvatore Capozziello,1, 2, 3, ∗Rocco D’Agostino,2, 3, †Alessio Lapponi,4, 5, ‡and Orlando Luongo6, 7, 8, §

1Dipartimento di Fisica “E. Pacini”, Universit`a di Napoli “Federico II”, Via Cinthia 9, 80126 Napoli, Italy.

2Scuola Superiore Meridionale, Largo San Marcellino 10, 80138 Napoli, Italy.

3Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Napoli, 80126 Napoli, Italy.

4Scuola Superiore Meridionale, Largo San Marcellino 10, 80138 Napoli, Italy

5Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Napoli, 80126 Napoli, Italy

6Divisione di Fisica, Universit`a di Camerino, Via Madonna delle Carceri 9, 62032 Camerino, Italy.

7Dipartimento di Matematica, Universit`a di Pisa, Largo B. Pontecorvo 5, 56127 Pisa, Italy.

8Institute of Experimental and Theoretical Physics,

Al-Farabi Kazakh National University, 050040 Almaty, Kazakhstan.

We show that the Einstein ﬁeld equations with a negative cosmological constant can admit black

hole solutions whose thermodynamics coincides with that of logotropic ﬂuids, recently investigated

to heal some cosmological and astrophysical issues. For this purpose, we adopt the Anton-Schmidt

equation of state, which represents a generalized version of logotropic ﬂuids. We thus propose a

general treatment to obtain an asymptotic anti-de Sitter metric, reproducing the thermodynamic

properties of both Anton-Schmidt and logotropic ﬂuids. Hence, we explore how to construct suitable

spacetime functions, invoking an event horizon and fulﬁlling the null, weak, strong and dominant

energy conditions. We further relax the strong energy condition to search for possible additional

solutions. Finally, we discuss the optical properties related to a speciﬁc class of metrics and show

how to construct an eﬀective refractive index depending on the spacetime functions and the ther-

modynamic quantities of the ﬂuid under study. We also explore possible departures with respect to

the case without the ﬂuid.

I. INTRODUCTION

Over the last years, studies on the black hole (BH)

entropy have revealed close connections between the

thermodynamic properties and the event horizon of a

BH [1–3]. Among all diﬀerent kinds of BHs provided

with diﬀerent geometries and thermodynamic features,

Schwarzschild BH represents the simplest case, where

part of the radiation is absorbed by the BH mass

[4,5]. Other interesting BH solutions include Reissner-

Norstr¨om BH, whose thermodynamics is similar to that

of regular BH [6–8], Hoˇrava-Lifshitz BH [9,10], charac-

terized by rich thermodynamic properties and so forth.

Moreover, asymptotic BH solutions to the de Sitter

space can be obtained from the Einstein equations with

a positive cosmological constant (Λ >0) [11,12]. In such

a case, it has been shown that the surface gravity of the

BH horizon would determine the temperature of particles

emitted from the BH [13]. The same, however, happens

even for the cosmological event horizon, so that a thermal

equilibrium may occur only if the two surfaces coincide

[14–16]. Further, BHs that are asymptotic to the anti-de

Sitter (AdS) space can be found as solutions to the Ein-

stein ﬁeld equations with Λ <0 [17]. As in the case of

the asymptotically ﬂat space, the entropy and tempera-

ture of AdS BH are equal to 1/4 of the event horizon area,

whereas, diﬀerently from the ﬂat space case, such objects

∗capozziello@na.infn.it

†rocco.dagostino@unina.it

‡alessio.lapponi-ssm@unina.it

§orlando.luongo@unicam.it

admit, at a given temperature, a stable equilibrium with

radiation, and a positive speciﬁc heat [18]. Throughout

recent years, the physics of asymptotically AdS BHs has

gained a renewed interest due to the AdS/CFT duality

[19–21]. In this context, particular attention was given

to the study of thermal ﬁeld theories living on the AdS

boundary and, from the bulk perspective, to the several

phase transitions that these types of BH exhibit.

Furthermore, in treating the cosmological constant as

the thermodynamic pressure, it has been shown that the

thermodynamics of a charged/rotating AdS BH exactly

coincides with that of Van der Waals’s ﬂuid [22–24]. Sub-

sequently, an asymptotically AdS solution to the Einstein

equations was obtained by matching the BH thermody-

namic parameters with those of a particular class of poly-

tropic gas [25]. Then, an additional AdS BH solution

was found in the thermodynamic framework of modiﬁed

Chaplygin gas [26,27].

Motivated by those ﬁndings, in the present study, we

focus on the class of logotropic models, whose thermody-

namic features permit to heal astrophysical issues related

to dark matter distribution in galaxies, and unify the

cosmological dark sector [28–31]. As a prominent result,

these models can be generalized to the well-known Anton-

Schmidt ﬂuid [32,33]. These scenarios have been recently

proposed in the cosmological context as a uniﬁed dark en-

ergy model [34–37]1. In this respect, the Anton-Schmidt

ﬂuid has been also studied in the Tolman-Oppenheimer-

Volkov formalism [45,46] to obtain analytical solutions

for a static and spherically symmetric BH [47]. In partic-

1For alternative approaches to dark energy, see also [38–44].

arXiv:2210.02431v3 [gr-qc] 13 Feb 2023

ular, from the relation between the Anton-Schmidt free

parameters and the BH mass, one can ﬁnd spacetime so-

lutions describing Schwarzschild-de Sitter BH and naked

singularities. Thus, it appears natural to investigate the

thermodynamic consequences to check whether the inclu-

sion of a logotropic and/or the Anton-Schmidt equation

of state (EoS) may lead to reasonable results in the BH

description.

In this paper, we search for a BH solution to the

Einstein ﬁeld equations, whose corresponding thermody-

namics coincides with that of logotropic models. Starting

from the Anton-Schmidt EoS, we propose a general treat-

ment to obtain an asymptotic Schwarzschild-AdS metric,

which reproduces the thermodynamic properties of the

involved ﬂuids, i.e. the pressure and the density. In par-

ticular, we motivate this choice since in a homogeneous

and isotropic universe, those quantities appear crucial in

order to write the energy-momentum tensor, as it will be

clariﬁed later in the text. Thus, to determine the most

suitable metric functions, we present a general method

involving any possible density term contribution. More-

over, in order to have a physical BH, we invoke the ex-

istence of an event horizon and investigate under which

circumstances the energy conditions may hold. We also

explore the possibility of violating the strong energy con-

dition, in order to ﬁnd additional physical properties. We

then discuss the physical consequence of this recipe in

view of the free constants emerging from the integra-

tion procedure. With the aim of distinguishing among

diﬀerent thermodynamic BHs, we consider the optical

properties of our solutions and show how to construct an

eﬀective refractive index, following the standard proce-

dure adopted for static and spherically symmetric space-

times. In particular, we show that the net dependence

of the refractive index on the underlying spacetime can

lead to diﬀerent outcomes. The refractive index increases

signiﬁcantly under the choice of particular constant val-

ues, whereas the asymptotic regime is investigated in

terms of density, showing the limit to Schwarzchild-AdS.

Hence, we explore possible departures with respect to the

case without the logotropic ﬂuid, corresponding to a pure

Schwarzschild-AdS case.

The paper is organized as follows. After this intro-

duction, in Sec. II we introduce the Anton-Schmidt EoS

and its limit to logotropic models. There we postulate

the metric ansatz for a static, spherically symmetric met-

ric that is consistent with an asymptotic AdS spacetime.

We thus analyze the thermodynamic properties of the

Anton-Schmidt BH in terms of its mass, temperature and

entropy. In Sec. III, we constrain BH solutions requiring

the presence of an event horizon and checking the validity

of the energy conditions. In particular, we discuss how

the violation of the strong energy condition may lead to a

metric solution containing a factor that can be associated

with a refractive index. Finally, in Sec. V, we summarize

our ﬁndings and draw the conclusions of our work. In

this study, we use Planck units c=~=G= 1.

II. LOGOTROPIC BLACK HOLES

Let us start by considering the Einstein ﬁeld equations

with the cosmological constant in the form

Gµν + Λgµν = 8πTµν ,(1)

where Gµν ≡Rµν −1

2Rgµν is the Einstein tensor, gµν

is the metric tensor, and Tµν is the stress-energy tensor

of the source ﬂuid. According to recent studies [48,49],

in the extended phase space one can interpret Λ as a

thermodynamic pressure, namely2

P=−Λ

8π=3

8πl2,(2)

where lis the AdS curvature constant. Our aim is to

construct an asymptotic AdS BH whose thermodynamics

matches that of the Anton-Schmidt ﬂuid with pressure

given by

P=Aρ

ρ∗−n

ln ρ

ρ∗,(3)

where the density ρis normalized to a reference density

ρ∗, while A > 0 and n6=−1 are constants.

This class of ﬂuid has been introduced in [33] for crys-

talline solids, where the Anton-Schmidt EoS gives the

empirical expression of crystalline solid’s pressure under

isotropic deformation. Afterwards, in the ﬁeld of cosmol-

ogy, see e.g. [34], it has been argued that, in analogy with

solid state physics, the pressure naturally changes its

sign, showing how the cosmic speed-up naturally emerges

as the universe volume changes under the action of cos-

mic expansion. To account for this mechanism, one can

assume the nparameter to depend upon the Gr¨uneisen

index, γG, i.e., n=n(γG), related to the speciﬁc heat

at constant volume and to the bulk modulus. This semi-

empirical relation provides a temperature dependence of

the free parameter nthat can be tested experimentally.

We here consider ﬁxing the index nto a constant, namely

without assuming the temperature dependence through

the Gr¨uneisen index. In fact, this allows one to inves-

tigate a given epoch of the universe dynamics that may

correspond to our time, where the temperature eﬀects are

negligible. Hence, we shall model our BH conﬁguration

through the pressure and the density only, which clearly

represent the main ingredients of the energy-momentum

tensor at late times. In so doing, our black hole con-

ﬁguration shows a cosmological constant contribution to

density and pressure that matches the Anton-Schmidt

ﬂuid given by Eq. (3).

Our strategy is to start from Eq. (3), which general-

izes the logotropic models with n= 0. In so doing, we re-

cover the logotropic thermodynamics as a limiting case of

2Alternatively, it is possible to work out the same recipe adopting

the conjugate variable of pressure, namely the volume [49].

the Anton-Schmidt ﬂuid. The logotropic thermodynam-

ics is of utmost importance, especially in the framework

of dark matter conﬁguration. Indeed, it is possible to

show that pressureless dark matter leads to cuspy den-

sity proﬁles, disfavoured by observations that, instead,

suggest a constant density core. If the dark matter halo

shows a polytropic EoS, that describes both dark matter

halos and the cosmological evolution, then the logotropic

solution appears as the most natural one. Again, we thus

require that our main thermodynamical properties to in-

vestigate are pressure and density as above reported. In

addition to what we discussed above, for the sake of com-

pleteness, generalized versions of logotropic models have

been also investigated [31,35,50] and criticized, see e.g.

[51], but lie beyond the purposes of this work.

Bearing in mind the above considerations, we shall con-

sider the static and spherically symmetric line element

ds2=f(r, ρ)dt2−dr2

f(r, ρ)−r2dΩ2,(4)

where

f(r, ρ) = −2M

r+r2

l2−h(r, ρ).(5)

Here, Mis the BH mass, and h(r, ρ) is an unknown

auxiliary function to determine. Clearly, in the case of

h(r, ρ)→0, the postulated metric represents an asymp-

totic AdS spacetime. In this scheme, the term h(r, ρ) rep-

resents a correction when it is considered diﬀerent from

zero. Using Eq. (2), we can rewrite Eq. (5) as

f(r, ρ) = −2M

r+8

3πr2P−h(r, ρ).(6)

In what follows, we seek a class of metrics such that the

BH thermodynamics predicted by the latter coincide ex-

actly with the Anton-Schmidt EoS, containing the lo-

gotropic models in the limit n= 0. The cosmological

pressure P, aﬀecting the BH thermodynamics, can be

therefore associated with the pressure of the ﬂuid P(ρ).

In this way, the function h(r, ρ) accounts for the metric

correction that occurs by considering an EoS diﬀerent

from the case of a pure cosmological constant3, namely

P=−ρ.

A. Black hole thermodynamics

To ﬁnd the most suitable form of h(r, ρ), we start from

the standard BH entropy in terms of the horizon radius

rhand area Aas [3,55,56]

S=A

4=πr2

h.(7)

3Although degenerating with a pure cosmological constant, the

case of dark ﬂuid [52–54] has not been explored here.

We can thus relate the BH thermodynamic properties

to the parameters of the Anton-Schmidt and logotropic

ﬂuids. In particular, the BH mass could be obtained from

the deﬁnition of horizon radius, namely f(rh, ρ) = 0:

M=4

3πr3

hP−rh

2h(rh, ρ).(8)

Since the EoS of the cosmological constant is ρ+P= 0,

the enthalpy associated with Λ is vanishing. For a BH

with volume V, the total energy within Vis E=M−P V ,

and then M=E+P V . Hence, it is natural to associate

the mass of the BH with its enthalpy H, such that M=

H(S, P ) [57,58].

One can thus use the standard thermodynamics rela-

tions to calculate the volume and temperature of the BH

by exploiting Eqs. (7) and (8):

V=∂H

∂P S

=4πr3

3−rh

∂h(rh, ρ)

∂ρ ∂P

∂ρ −1

,(9)

T=∂H

∂S P

= 2rhP−1

4πrh

∂(rh(r, ρ))

∂r rh

.(10)

From the ﬁrst law of thermodynamics, dE =T dS −

P dV , and assuming the following integrability condition

∂2S

∂T ∂V =∂2S

∂V ∂T ,(11)

one ﬁnds

S=ρ+P

TV . (12)

Therefore, plugging Eqs. (3), (7), (9) and (10) into

Eq. (12) and considering the solution for a generic r≥rh,

we obtain

8πr2(P−2ρ)Pρ+ 6(ρ+P)hρ−3(rh)0Pρ= 0 ,(13)

where the subscript ρand the prime denote the partial

derivatives with respect to the density and radial coor-

dinate, respectively. Starting from the theoretical setup

presented in [22], we seek a solution of Eq. (13) for a

generic rby implementing a general method that makes

use of combinations of linearly independent functions of

the density. In particular, a similar approach has been

employed in [25,26], but imposing a priori the functional

expressions for Ri(ρ). In our treatment, we relax this hy-

pothesis as illustrated in more detail in appendix A. We

thus write

h(r, ρ) = X

Xi(r)Ri(ρ),(14)

where the coeﬃcients Xidepend on the radial coordinate,

r. In this way, Eq. (13) takes the formal expression

ξj(r)Fj(ρ) = 0 ,(15)

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BlackholethermodynamicsfromlogotropicuidsSalvatoreCapozziello,1,2,3,RoccoD'Agostino,2,3,yAlessioLapponi,4,5,zandOrlandoLuongo6,7,8,x1DipartimentodiFisica\E.Pacini",UniversitadiNapoli\FedericoII",ViaCinthia9,80126Napoli,Italy.2ScuolaSuperioreMeridionale,LargoSanMarcellino10,80138Napoli,Italy.3Istit...

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Black hole thermodynamics from logotropic uids Salvatore Capozziello1 2 3Rocco DAgostino2 3yAlessio Lapponi4 5zand Orlando Luongo6 7 8x 1Dipartimento di Fisica E. Pacini Universit a di Napoli Federico II Via Cinthia 9 80126 Napoli Italy..pdf

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Black hole thermodynamics from logotropic uids Salvatore Capozziello1 2 3Rocco DAgostino2 3yAlessio Lapponi4 5zand Orlando Luongo6 7 8x 1Dipartimento di Fisica E. Pacini Universit a di Napoli Federico II Via Cinthia 9 80126 Napoli Italy.

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