# On Regular Black Holes at Finite Temperature

Authors: S. C. Ulhoa; E. P. Spaniol; R. Gomes; A. F. Santos; A. E. Santana
Journal: Advances in High Energy Physics
 (2020)

---

## Abstract

The Thermo Field Dynamics (TFD) formalism is used to investigate the regular black holes at finite temperature. Using the Teleparalelism Equivalent to General Relativity (TEGR), the gravitational Stefan-Boltzmann law and the gravitational Casimir effect at zero and finite temperature are calculated. In addition, the first law of thermodynamics is considered. Then, the gravitational entropy and the temperature of the event horizon of a class of regular black holes are determined.


---

## Body

## 1. Introduction

The existence of singularities in the theory of general relativity has been problematic since its inception, especially those linked to black holes. The so-called fundamental singularities do not allow the application of the laws of physics, which are a type of difficult giving rise to some proposals to address the problem. One of them is the well-known cosmic censorship that was proposed by Penrose in the last century [1]. Another more particular proposal came with the solution of regular black holes whose first solution was obtained by Bardeen [2]. It is interesting to note that solutions describing regular black holes have an event horizon and therefore share many features with singular black holes. So, if black holes are real objects, there is a good chance that the regular ones are the objects that will be experimentally perceived. In this way, the theoretical study of such solutions becomes very relevant. Particularly the analysis of the thermodynamics of regular black holes can reveal substantial experimental implications. For this, it is necessary to define how the temperature is introduced and how from it the gravitational entropy is obtained. In addition, other effects can be predicted by means of this termalization, such as the gravitational Stefan-Boltzmann law and Casimir effect. In this sense, we will work with Thermo Field Dynamics (TFD) because it has proved to be a very powerful tool to deal with the termalization of a given field [3, 4]. This approach requires doubling the Fock space which allows both time and temperature to be system variables. This is an advantage over Matsubara’s approach [5]. Thermo Field Dynamics also requires the characterization of the field under analysis by means of the corresponding Green function. Regarding the gravitational field, we will use an alternative description that is known as Teleparalelism Equivalent to General Relativity (TEGR). When the gravitational field is described in this alternative way, some unique predictions are revealed.Teleparalelism Equivalent to General Relativity is described in terms of tetrads which has many advantages over the metric formulation of gravitation as general relativity, since equivalence occurs only in relation to field equations. Among them, the most notorious is the existence of a gravitational energy that is obtained naturally through the Hamiltonian formulation of the TEGR [6]. Moreover, the propagator of graviton obtained in general relativity does not coincide with that predicted by the TEGR [7]. This opens up scope for the exploration of regular black holes via Thermo Field Dynamics. In this paper, we will use TFD to calculate the Stefan-Boltzmann law and the Casimir effect through the TEGR.This article is organized as follows. In Section2, Thermo Field Dynamics is introduced. In Section 3, the ideas of the Teleparallelism Equivalent to General Relativity are briefly recalled. In Section 4, the gravitational Stefan-Boltzmann and entropy are calculated for regular black holes. In addition, we also calculate for the same gravitational system, the Casimir effect, which is given at zero and finite temperature. Finally in the last section, we present our conclusions. In this article, we use the natural unities system, G=c=1. We denote the Lorentz symmetry by Latin indices, a=0,1,2,3, while diffeomorphisms are denoted by greek indices, μ=0,1,2,3.


## 2. Thermo Field Dynamics (TFD)

TFD is a thermal quantum field theory [3, 8–15]. This formalism is used when it is desirable to have explicit time dependence in addition to the temperature. TFD is based on two basic ingredients: (i) a doubling of the Hilbert space, S, of the original field system, giving rise to ST=S⊗S~, where S~ is the tilde (dual) space. This doubling is defined by the tilde conjugation rules. (ii) The Bogoliubov transformation which introduces thermal effects through a rotation between tilde (S~) and nontilde (S) operators. These ingredients allow to interpret the statistical average of an arbitrary operator A, as the expectation value in a thermal vacuum. The thermal vacuum, 0β, describes the thermal equilibrium of the system, where β=1/kBT, T is the temperature, and kB is the Boltzmann constant.By taking an arbitrary operatorA and A in the Hilbert space S and in tilde space S, respectively, the Bogoliubov transformation is
(1)AαξA~†α=UαAkξA~†k,where ξ=−1 for bosons and ξ=+1 for fermions. The Bogoliubov transformation, Uα, is defined as
(2)Uα=uα−wαξwαuα,with u2α+ξw2α=1. Here, the α parameter is the compactification parameter defined by α=α0,α1,⋯αD−1. The temperature effect is described by the choice α0≡β and α1,⋯αD−1=0, where β=1/kBT with kB being the Boltzmann constant.Any field in the TFD formalism can be written in terms of theα-parameter. As an example, consider the scalar field. Using the Bogoliubov transformation, the scalar field dependent of α-parameter becomes
(3)ϕx;α=UαϕxU−1α,ϕ~x;α=Uαϕ~xU−1α.Then, the propagator for the scalar field in terms ofα-parameter is written as
(4)G0ABx−x′;α=i0,0~τϕAx;αϕBx′;α0,0~=i∫d4k2π4e−ikx−x′G0ABk;α,where A and B represent the duplicate notation with AandB=1,2 and τ is the time ordering operator. Here
(5)G0ABk;α=U−1αG0ABkUα.It is important to note that the physical quantities are given by the nontilde variables. Using the Bogoliubov transformation in equation (5), the Green function becomes
(6)G011k;α=G0k;α=G0k+v2k;αG0k−G0∗k,with
(7)G0k=1k2−m2+iε,and v2k;α being the generalized Bogoliubov transformation [16] which is given as
(8)v2k;α=∑s=1d∑σs2s−1∑lσ1,⋯,lσs=1∞−ηs+∑r=1slσrexp−∑j=1sασjlσjkσj,where d is the number of compactified dimensions, η=1−1 for fermions (bosons), σs denotes the set of all combinations with s elements, and k is the 4-momentum.


## 3. Teleparalellism Equivalent to General Relativity (TEGR)

General relativity which is the standard approach for gravitation is based on the Riemann geometry in which the field variables are the components of the metric tensor. Such a formulation does not lead to gravitational conserved quantities, partly because of the inclusion of the local Lorentz symmetry and partly due to the difficulty to formally establish a reference frame. Those problems are solved by TEGR which is formulated in terms of the tetrad field in a Weitzenböck manifold. In the 1930 decade, Einstein tried to establish a unified field theory using the concept of distant teleparallelism [17] which led to the formulation of a New General Relativity by Hyashi and Shirafuji [18]. Then a Hamiltonian formulation was successfully established which yielded conserved quantities such as the gravitational energy-momentum tensor and angular momentum [6].Let us consider a Weitzenböck manifold endowed with the Cartan connection [19]
(9)Γμλν=eaμ∂λeaν,then the associated torsion tensor is
(10)Taλν=∂λeaν−∂νeaλ.The Cartan connection is curvature free. On the other hand, it is identically related to the Christoffel symbols0Γμλν, which exist in the realm of the Riemannian geometry, by
(11)Γμλν=Γ0μλν+Kμλν,where Kμλν is the contortion tensor which is defined by
(12)Kμλν=12Tλμν+Tνλμ+Tμλν,with Tμλν=eaμTaλν.In order to establish a Lagrangian density for TEGR, we firstly note that the scalar curvature constructed out of the Christoffel symbols is written in terms of the torsion tensor, due to the identity in equation (12), as
(13)eRe≡−e14TabcTabc+12TabcTbac−TaTa+2∂μeTμ.Then, getting rid of the total divergency which does not alters the field equations, we have(14)Leaμ=−κeΣabcTabc−LM,where κ=1/16π, LM is the Lagrangian density of matter fields, and Σabc is defined by
(15)Σabc=14Tabc+Tbac−Tcab+12ηacTb−ηabTc,with Ta=eaμTμ. If a derivative with respect to tetrad field is performed in the Lagrangian density, it yields
(16)∂νeΣaλν=14κeeaμtλμ+Tλμ, where
(17)tλμ=κ4ΣbcλTbcμ−gλμΣabcTabc,is the gravitational energy-momentum tensor [20]. The symmetry of Σaλν leads to
(18)∂λ∂νeΣaλν≡0.This allows one to define the total energy-momentum vector. It reads(19)Pa=∫Vd3xeeaμt0μ+T0μ,which may be written in the following form
(20)Pa=4κ∫Vd3x∂νeΣa0ν.It is important to point out that the energy-momentum verctor respects the Lorentz symmetry, and it is invariant under coordinate transformation.It is possible to use the Lagrangian density of TEGR above to establish a two-point Green function considering the tetrads as the observable fields on space-time. Thus, from(21)gμν=ημν+hμν,and expression (14), the graviton propagator is [7]
(22)ebλ,edγ=Δbdλγ=ηbdκqλqγ.Then, the Green function reads(23)G0x,x′=−iΔbdλγgλγηbd.Explicitly, it is(24)G0x,x′=−i64πq2,with q=x−x′, where x and x′ are four vectors. With the weak field approximation, the gravitational energy-momentum tensor tλμ becomes
(25)tλμx=κgμα∂γebλ∂γebα−gμγ∂αebλ∂γebα−gμα∂λebγ∂γebα−∂λebγ∂αebγ−2gλμ∂γebα∂γebα−∂αebγ.In order to avoid divergences, we adopt the usual procedure to write the energy-momentum tensor at different points in space-time and then taking the proper limit. Hence(26)tλμx=0tλμx0,=limxμ→x′μ4iκ−5gλμ∂′γ∂γ+2gμα∂′λ∂αG0x−x′,where ecλx,ebαx′=iηcbδαλG0x−x′. In this sense, it is possible to use the metric of a regular black hole to introduce thermal effects via TFD as explained in the last section. It worths to notice that in the weak field approximation, TEGR becomes a usual field which is very different from the metric formulation that cannot dissociate metric from space-time.


## 4. Gravitational Casimir Effect and Stefan-Boltzmann Law at Finite Temperature for Regular Black Holes

In this section, the framework of TFD is used to calculate the mean value of gravitational energy-momentum (26) which is obtained in the weak field approximation of TEGR. Thus, we have
(27)tλμABx;α=limx→x′4iκ−5gλμ∂′γ∂γ+2gμα∂′λ∂αG0ABx−x′;α.If we use the Casimir prescription(28)TλμABx;α=tλμABx;α−tλμABx,then
(29)TλμABx;α=limx→x′4iκ−5gλμ∂′γ∂γ+2gμα∂′λ∂αG¯0ABx−x′;α,where
(30)G¯0ABx−x′;α=G0ABx−x′;α−G0ABx−x′.It is possible to describe a class of regular black holes by the following line element [21]
(31)ds2=−frdt2+dr2fr+r2dθ2+sin2θdϕ2,with
(32)fr=1−2M0r1+r0/rqp/q,where M0 is the the mass of the regular black hole; in fact, it coincides with the ADM mass in the limit r⟶∞. The parameter r0 can be seen as a fundamental length of the regular black hole. Such a line element reproduces known solutions for a suitable choice of the parameters p and q which has to be integers. For instance, the Bardeen solution is achieved for p=3 and q=2, while the Hayward solution for p=q=3. Thus, it should be noted that the metric in equation (31) represents a class of solutions. Such regular black holes arose in order to deal with an open problem in gravitation, the existence of singularities. In fact, the Bardeen solution was the first class of regular black holes which can be understood as a magnetic monopole coupled to the Einstein equation [22]. It is important to point out that although the metric in equation (31) has no singularity at r=0, it does have an event horizon given by the solution of fRH=0. On the other hand, the stability of such solutions need to be investigated.


### 4.1. Gravitational Stefan-Boltzmann Law

In order to analyze the gravitational Stefan-Boltzmann law, we have to chooseα=β,0,0,0 in the TFD formalism. Then, the Bogoliubov transformation is
(33)v2β=∑j0=1∞e−βk0j0.Hence, we have to use the following Green function:(34)G¯011x−x′;β=2∑j0=1∞G011x−x′−iβj0n0,where n0=1,0,0,0. Thus, from (29), we can calculate the energy with AB=11 which is the physical component in the matrix obtained in TFD. It reads
(35)T0011x;α=εr,T=limx→x′∑j0=1∞4κi−31+4M0r1+r0/rqp/q∂0′∂0+5∂1′∂1+5r21+2M0r1+r0/rqp/q∂2′∂2+1sin2θ∂3′∂3G011x−x′−iβj0n0.Once the Riemann zeta function is defined byζ4=∑j0=1∞1/j04=π4/90, then the gravitational Stefan-Boltzmann energy is
(36)εr,T=32π4151+6M0r1+r0/rqp/qT4.Here, we have to notice that for the vanishing of the physical parameter of the regular black holeM0, the energy does not goes to zero. Therefore, we need to regularize such an expression by requiring Er,T=εr,T−32π4T4/15. That leads to
(37)Er,T=192π4M015r1+r0/rqp/qT4,which is the regularized energy.The first law of thermodynamics states(38)E+P=T∂P∂TV.Then, if we use the regularized energy above, such first-order differential equation has the solutionP=E/3. This is very interesting to note that it is equal to the photon state equation. In terms of the temperature, the pressure is explicitly
(39)Pr,T=192π4M045r1+r0/rqp/qT4.In order to calculate the gravitational entropy, we recall the definitionP=−∂F/∂V, where F is the free energy and S=−∂F/∂T. Therefore, from ∂P/∂TV=∂S/∂VT, the entropy is
(40)S=3072M0π545T3∫0Rr1+r0/rqp/qdr,which for the Bardeen and Hayward solutions may be represented graphically by Figure 1.Figure 1
Gravitational entropy. They axis is 90S/3072M0π5R2T3 and x=r0/R.We choose to analyze the dependency with respect tor0 for a constant R, instead of the temperature whose dependence is quite simple. Thus, the role of black hole geometry is better understood. It is interesting to note that there is a discontinuity for the point around x=0.5 for the Hayward solution. Such a feature reflects a natural limit for the r0 scale; after all, it is not expected that the geometric structure of the regular black hole coincides with the integration surface itself. The integration hyper-surface is a sphere of radius R. Thus, the gravitational entropy exists on an arbitrary portion of space which is a different approach from the usual black hole thermodynamics. Usually, a black hole has a fixed entropy written in terms of its event horizon area. Here, the following principles are assumed: (i) the entropy is a function of macroscopic parameters. If the “no hair” theorems are valid [23], then the entropy is a function of mass, angular momentum, and charge. (ii) The Penrose process leads to an arbitrary manipulation of these parameters provided that the event horizon area remains unchanged. That holds for regular black holes [24]. It implies that the entropy has to be a function of area. It should be noted that the metric of the regular black hole tends to the Schwarzschild metric for a position far from the event horizon. Therefore, the entropy tends to be proportional to the event horizon area; hence, it is reasonable to admit that SH=AH/4 for a regular black hole, where AH=4πRH2, with RH being the solution of
(41)2M0RH1+r0/RHqp/q=1.Therefore, the temperature of the event horizon of a class of regular black holes defined by (31) is
(42)TH=90R23072M0π4∫0Rrdr/1+r0/rqp/q1/3.Thus, for the Bardeen and Hayward solutions, the horizon temperature assumes the form as that in Figure2.Figure 2
The horizon temperature over90R2/3072M0π41/3 is shown as a function of x=r0/R.It should be noted that the same discontinuity of entropy in Hayward solution also appears in the temperature. This discontinuity can also be interpreted as an improbability of the Hayward solution to be an experimental reality. On the other hand, the marked difference between the solutions for smallr0 indicates an immediate choice when an experiment to measure the temperature of the event horizon of a regular black hole can be performed. Such an expression is a unique prediction of TFD applied to TEGR.


### 4.2. Gravitational Casimir Effect

If the Casimir effect description is desired, then the choiceα=0,i2d,0,0 has to be made which leads to the following Bogoliubov transformation
(43)v2d=∑l1=1∞e−i2dk1l1.If the Green function is given by(44)G¯011x−x′;d=2∑l1=1∞G011x−x′−2dl1r,then
(45)T0011d,r=εcd,r=limx→x′∑l1=1∞4κi−31+4M0r1+r0/rqp/q∂′0∂0+5∂′1∂1+5r21+2M0r1+r0/rqp/q∂′2∂2+1sin2θ∂′3∂3G011x−x′−2dl1n1,where n1=0,1,0,0. Hence, the energy associated to the gravitational Casimir effect for regular black holes is
(46)εcd,r=∑l1−4d4l141−2M0r1+r0/rqp/q+2M0r1+r0/rqp/q5dl1r,which for the approximation d≪r becomes
(47)εcd,r=−2π445d41−2M0r1+r0/rqp/q.It should be noted that the vacuum contribution is negative and has a dependency ofd−4. In order to take into account only the regular black hole contribution, a regularization procedure is necessary. Thus, subtracting the vacuum energy, we have
(48)Ecd,r=4π4M045d4r1+r0/rqp/q,where Ecd,r is the regularized Casimir energy. It is interesting to note that on the event horizon, it is exactly minus the vacuum energy.Similarly, the Casimir pressure is(49)T3311d,r=ρcd,r=limx→x′∑l1=1∞4κi1−2M0r1+r0/rqp/q51+2M0r1+r0/rqp/q∂′0∂0−31−2M0r1+r0/rqp/q∂′1∂1−5r2∂′2∂2+1sin2θ∂′3∂3G011x−x′−2dl1n1.This yields(50)ρcd,r=−4d4∑l11l143−18M0r1+r0/rqp/q−6dl1M0r21+r0/rqp/q,which, after the limit d/r≪1 is taken, reads
(51)ρcd,r=−2π415d41−6M0r1+r0/rqp/q.Again, in order to consider the nonvacuum contribution, a regularized pressure is necessary. Thus, the regularized gravitational Casimir pressure is(52)Pcd,r=12π4M015d4r1+r0/rqp/q.It should be noted that both the regularized Casimir energy and pressure are very small due to the weak field approximation. We would like to point out that in this formalism, the vacuum itself has some gravitational features which explain why a regularization is necessary.


### 4.3. Gravitational Casimir Effect at Finite Temperature

The choiceα=β,i2d,0,0 is suitable to describe the Casimir effect at finite temperature. As a consequence, following the TFD prescription, the Bogoliubov transformation is given by
(53)v2k0,k3;β,d=v2k0;β+v2k1;d+2v2k0;βv2k1;d,=∑j0=1∞e−βk0j0+∑l1=1∞e−i2dk1l1+2∑j0,l1=1∞e−βk0j0−i2dk1l1,where the first term takes into account temperature effects, the second term stands for the Casimir effect only, and the last term the interaction between both. The Green function is then
(54)G¯011x−x′;β,d=4∑j0,l1=1∞G011x−x′−iβj0n−2dl1r.As before, the gravitational Casimir energy is obtained from expression (29) which reads
(55)εcβ,d=limx→x′∑j0,l1=1∞4κi−31+4M0r1+r0/rqp/q∂′0∂0+5∂′1∂1+5r21+2M0r1+r0/rqp/q∂′2∂2+1sin2θ∂′3∂3G011x−x′−iβj0n−2dl1r,where εcβ,d=T0011β;d. It worths to obtain the regularized Casimir energy at finite temperature which is achieved by subtracting the vacuum energy from εcβ,d; explicitly, it is
(56)Ecβ,d=−64∑j0,l1=1∞14d2l121+2M0/r1+r0/rqp/q+j021−2M0/r1+r0/rqp/qβ234d2l121+12M0r1+r0/rqp/q2+2M0r1+r0/rqp/q1+5dl1r1+2M0r1+r0/rqp/q−j023−72M0r1+r0/rqp/q3+30dl1M0r21+r0/rqp/q+4M0r1+r0/rqp/q26+5dl1rβ2+64∑j0,l1=1∞4d2l12−3j02β24d2l12+j02β23,where Ecβ,d is the regularized expression. Similarly, the gravitational Casimir pressure, ρcβ,d, at finite temperature is given by
(57)ρcβ,d=limx→x′∑j0,l1=1∞4κi1−2M0r1+r0/rqp/q51+2M0r1+r0/rqp/q∂′0∂0−31−2M0r1+r0/rqp/q∂′1∂1−5r2∂′2∂2+1sin2θ∂′3∂3G011x−x′−iβj0n−2dl1r,where ρcβ,d=T1111β;d. As the regularized energy, the regularized Casimir pressure is
(58)Pcβ,d=−64∑j0,l1=1∞14d2l121+2M0/r1+r0/rqp/q+j021−2M0/r1+r0/rqp/qβ234d2l123+8M0r1+r0/rq2−6dl1M0r21+r0/rq1−2M0r1+r0/rq1−2M0r1+r0/rq2+j0224Mr2+6dl12Mr23+2Mr−11−2Mr2β2+64∑j0,l1=1∞4d2l12−j02β24d2l12+j02β23,where Pcβ,d=ρcβ,d+64∑j0,l1=1∞4d2l12−j02β2/4d2l12+j02β23. The regularized expressions take into account only the contributions of the regular black holes. They are small corrections to the vacuum quantities which have the known limits for β⟶∞. It should be noted that the gravitational Casimir effect is a very controversial idea due to the energy problem in general relativity. As a matter of fact, the lack of a gravitational energy-momentum tensor in this approach prevents one from analyzing such effect. On the other hand in the framework of TEGR, the gravitational Casimir effect can be explored.


## 4.1. Gravitational Stefan-Boltzmann Law

In order to analyze the gravitational Stefan-Boltzmann law, we have to chooseα=β,0,0,0 in the TFD formalism. Then, the Bogoliubov transformation is
(33)v2β=∑j0=1∞e−βk0j0.Hence, we have to use the following Green function:(34)G¯011x−x′;β=2∑j0=1∞G011x−x′−iβj0n0,where n0=1,0,0,0. Thus, from (29), we can calculate the energy with AB=11 which is the physical component in the matrix obtained in TFD. It reads
(35)T0011x;α=εr,T=limx→x′∑j0=1∞4κi−31+4M0r1+r0/rqp/q∂0′∂0+5∂1′∂1+5r21+2M0r1+r0/rqp/q∂2′∂2+1sin2θ∂3′∂3G011x−x′−iβj0n0.Once the Riemann zeta function is defined byζ4=∑j0=1∞1/j04=π4/90, then the gravitational Stefan-Boltzmann energy is
(36)εr,T=32π4151+6M0r1+r0/rqp/qT4.Here, we have to notice that for the vanishing of the physical parameter of the regular black holeM0, the energy does not goes to zero. Therefore, we need to regularize such an expression by requiring Er,T=εr,T−32π4T4/15. That leads to
(37)Er,T=192π4M015r1+r0/rqp/qT4,which is the regularized energy.The first law of thermodynamics states(38)E+P=T∂P∂TV.Then, if we use the regularized energy above, such first-order differential equation has the solutionP=E/3. This is very interesting to note that it is equal to the photon state equation. In terms of the temperature, the pressure is explicitly
(39)Pr,T=192π4M045r1+r0/rqp/qT4.In order to calculate the gravitational entropy, we recall the definitionP=−∂F/∂V, where F is the free energy and S=−∂F/∂T. Therefore, from ∂P/∂TV=∂S/∂VT, the entropy is
(40)S=3072M0π545T3∫0Rr1+r0/rqp/qdr,which for the Bardeen and Hayward solutions may be represented graphically by Figure 1.Figure 1
Gravitational entropy. They axis is 90S/3072M0π5R2T3 and x=r0/R.We choose to analyze the dependency with respect tor0 for a constant R, instead of the temperature whose dependence is quite simple. Thus, the role of black hole geometry is better understood. It is interesting to note that there is a discontinuity for the point around x=0.5 for the Hayward solution. Such a feature reflects a natural limit for the r0 scale; after all, it is not expected that the geometric structure of the regular black hole coincides with the integration surface itself. The integration hyper-surface is a sphere of radius R. Thus, the gravitational entropy exists on an arbitrary portion of space which is a different approach from the usual black hole thermodynamics. Usually, a black hole has a fixed entropy written in terms of its event horizon area. Here, the following principles are assumed: (i) the entropy is a function of macroscopic parameters. If the “no hair” theorems are valid [23], then the entropy is a function of mass, angular momentum, and charge. (ii) The Penrose process leads to an arbitrary manipulation of these parameters provided that the event horizon area remains unchanged. That holds for regular black holes [24]. It implies that the entropy has to be a function of area. It should be noted that the metric of the regular black hole tends to the Schwarzschild metric for a position far from the event horizon. Therefore, the entropy tends to be proportional to the event horizon area; hence, it is reasonable to admit that SH=AH/4 for a regular black hole, where AH=4πRH2, with RH being the solution of
(41)2M0RH1+r0/RHqp/q=1.Therefore, the temperature of the event horizon of a class of regular black holes defined by (31) is
(42)TH=90R23072M0π4∫0Rrdr/1+r0/rqp/q1/3.Thus, for the Bardeen and Hayward solutions, the horizon temperature assumes the form as that in Figure2.Figure 2
The horizon temperature over90R2/3072M0π41/3 is shown as a function of x=r0/R.It should be noted that the same discontinuity of entropy in Hayward solution also appears in the temperature. This discontinuity can also be interpreted as an improbability of the Hayward solution to be an experimental reality. On the other hand, the marked difference between the solutions for smallr0 indicates an immediate choice when an experiment to measure the temperature of the event horizon of a regular black hole can be performed. Such an expression is a unique prediction of TFD applied to TEGR.


## 4.2. Gravitational Casimir Effect

If the Casimir effect description is desired, then the choiceα=0,i2d,0,0 has to be made which leads to the following Bogoliubov transformation
(43)v2d=∑l1=1∞e−i2dk1l1.If the Green function is given by(44)G¯011x−x′;d=2∑l1=1∞G011x−x′−2dl1r,then
(45)T0011d,r=εcd,r=limx→x′∑l1=1∞4κi−31+4M0r1+r0/rqp/q∂′0∂0+5∂′1∂1+5r21+2M0r1+r0/rqp/q∂′2∂2+1sin2θ∂′3∂3G011x−x′−2dl1n1,where n1=0,1,0,0. Hence, the energy associated to the gravitational Casimir effect for regular black holes is
(46)εcd,r=∑l1−4d4l141−2M0r1+r0/rqp/q+2M0r1+r0/rqp/q5dl1r,which for the approximation d≪r becomes
(47)εcd,r=−2π445d41−2M0r1+r0/rqp/q.It should be noted that the vacuum contribution is negative and has a dependency ofd−4. In order to take into account only the regular black hole contribution, a regularization procedure is necessary. Thus, subtracting the vacuum energy, we have
(48)Ecd,r=4π4M045d4r1+r0/rqp/q,where Ecd,r is the regularized Casimir energy. It is interesting to note that on the event horizon, it is exactly minus the vacuum energy.Similarly, the Casimir pressure is(49)T3311d,r=ρcd,r=limx→x′∑l1=1∞4κi1−2M0r1+r0/rqp/q51+2M0r1+r0/rqp/q∂′0∂0−31−2M0r1+r0/rqp/q∂′1∂1−5r2∂′2∂2+1sin2θ∂′3∂3G011x−x′−2dl1n1.This yields(50)ρcd,r=−4d4∑l11l143−18M0r1+r0/rqp/q−6dl1M0r21+r0/rqp/q,which, after the limit d/r≪1 is taken, reads
(51)ρcd,r=−2π415d41−6M0r1+r0/rqp/q.Again, in order to consider the nonvacuum contribution, a regularized pressure is necessary. Thus, the regularized gravitational Casimir pressure is(52)Pcd,r=12π4M015d4r1+r0/rqp/q.It should be noted that both the regularized Casimir energy and pressure are very small due to the weak field approximation. We would like to point out that in this formalism, the vacuum itself has some gravitational features which explain why a regularization is necessary.


## 4.3. Gravitational Casimir Effect at Finite Temperature

The choiceα=β,i2d,0,0 is suitable to describe the Casimir effect at finite temperature. As a consequence, following the TFD prescription, the Bogoliubov transformation is given by
(53)v2k0,k3;β,d=v2k0;β+v2k1;d+2v2k0;βv2k1;d,=∑j0=1∞e−βk0j0+∑l1=1∞e−i2dk1l1+2∑j0,l1=1∞e−βk0j0−i2dk1l1,where the first term takes into account temperature effects, the second term stands for the Casimir effect only, and the last term the interaction between both. The Green function is then
(54)G¯011x−x′;β,d=4∑j0,l1=1∞G011x−x′−iβj0n−2dl1r.As before, the gravitational Casimir energy is obtained from expression (29) which reads
(55)εcβ,d=limx→x′∑j0,l1=1∞4κi−31+4M0r1+r0/rqp/q∂′0∂0+5∂′1∂1+5r21+2M0r1+r0/rqp/q∂′2∂2+1sin2θ∂′3∂3G011x−x′−iβj0n−2dl1r,where εcβ,d=T0011β;d. It worths to obtain the regularized Casimir energy at finite temperature which is achieved by subtracting the vacuum energy from εcβ,d; explicitly, it is
(56)Ecβ,d=−64∑j0,l1=1∞14d2l121+2M0/r1+r0/rqp/q+j021−2M0/r1+r0/rqp/qβ234d2l121+12M0r1+r0/rqp/q2+2M0r1+r0/rqp/q1+5dl1r1+2M0r1+r0/rqp/q−j023−72M0r1+r0/rqp/q3+30dl1M0r21+r0/rqp/q+4M0r1+r0/rqp/q26+5dl1rβ2+64∑j0,l1=1∞4d2l12−3j02β24d2l12+j02β23,where Ecβ,d is the regularized expression. Similarly, the gravitational Casimir pressure, ρcβ,d, at finite temperature is given by
(57)ρcβ,d=limx→x′∑j0,l1=1∞4κi1−2M0r1+r0/rqp/q51+2M0r1+r0/rqp/q∂′0∂0−31−2M0r1+r0/rqp/q∂′1∂1−5r2∂′2∂2+1sin2θ∂′3∂3G011x−x′−iβj0n−2dl1r,where ρcβ,d=T1111β;d. As the regularized energy, the regularized Casimir pressure is
(58)Pcβ,d=−64∑j0,l1=1∞14d2l121+2M0/r1+r0/rqp/q+j021−2M0/r1+r0/rqp/qβ234d2l123+8M0r1+r0/rq2−6dl1M0r21+r0/rq1−2M0r1+r0/rq1−2M0r1+r0/rq2+j0224Mr2+6dl12Mr23+2Mr−11−2Mr2β2+64∑j0,l1=1∞4d2l12−j02β24d2l12+j02β23,where Pcβ,d=ρcβ,d+64∑j0,l1=1∞4d2l12−j02β2/4d2l12+j02β23. The regularized expressions take into account only the contributions of the regular black holes. They are small corrections to the vacuum quantities which have the known limits for β⟶∞. It should be noted that the gravitational Casimir effect is a very controversial idea due to the energy problem in general relativity. As a matter of fact, the lack of a gravitational energy-momentum tensor in this approach prevents one from analyzing such effect. On the other hand in the framework of TEGR, the gravitational Casimir effect can be explored.


## 5. Conclusion

The regular black holes are studied at finite temperature. The temperature effects are introduced using the TFD formalism. TFD is a tool that allows to analyze temperature effects in addition to the time dependence. Using the Teleparalelism Equivalent to General Relativity, the gravitational thermodynamics to the regular black holes is investigated. This gravitational theory has a well defined energy-momentum tensor that allows to calculated the gravitational Stefan-Boltzmann law and Casimir effect associated to the regular black holes. A regularized gravitational Stefan-Boltzmann law for the regular black hole is obtained. Using the first law of thermodynamics, the gravitational pressure and the gravitational entropy are determined. The relation between gravitational energy and pressure is equal the relation that describes the photon. The gravitational entropy obtained here exists on an arbitrary portion of space. Then, it is a different approach from the usual black hole thermodynamics, since the usual black hole has a fixed entropy given in terms of its event horizon area. In addition, the temperature of the event horizon for regular black holes has been calculated. Furthermore, the gravitational Casimir energy and Casimir pressure at zero and finite temperature for this class of regular black holes are determined. It is interesting to note that such results can be experimentally verified; once confirmed, it suggests that the torsion tensor is the true quantity responsible by gravitation instead of curvature as the mainstream approach for the gravitational field.



---

*Source: 5712084-2020-06-10.xml*