Advances in High Energy Physics

Volume 2019, Article ID 2539217, 15 pages

https://doi.org/10.1155/2019/2539217

## Reissner-Nordström Black Holes Statistical Ensembles and First-Order Thermodynamic Phase Transition

Faculty of Physics, Semnan University, 35131-19111 Semnan, Iran

Correspondence should be addressed to Hossein Ghaffarnejad; ri.ca.nanmes@dajenrafahgh

Received 2 November 2018; Accepted 28 February 2019; Published 10 April 2019

Academic Editor: Kazuharu Bamba

Copyright © 2019 Hossein Ghaffarnejad and Mohammad Farsam. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We apply Debbasch proposal to obtain mean metric of coarse graining (statistical ensemble) of quantum perturbed Reissner-Nordstöm black hole Then we seek its thermodynamic phase transition behavior. Our calculations predict first-order phase transition which can take Bose-Einstein’s condensation behavior.

#### 1. Introduction

Every observation in any arbitrary system is necessarily finite which deals with a finite number of measured quantities with a finite precision. A given system is therefore generally susceptible of different, equally valid descriptions and building the bridges between those different descriptions is the task of statistical physics (see introduction in [1] for more discussion). Nonlinearity property of Einstein’s metric equations causes their averaging to be nontrivial. Various possible ways of averaging the geometry of space time have already been proposed by [2–8], but none of them seems fully satisfactory (see section 7 in [1] for full discussion). Debbasch used an alternative way to averaging Einstein’s metric equation in [1]. To do so, he chose a general framework where the mean metric still obeys the equations of general theory of relativity. In his approach averaging and/or coarse graining a gravitational field changes the matter content of space time called “apparent matter” which in cosmological context is related to the dark energy (see [9–12]). So general relativity mean field theory can propose a physical meaning for unknown cosmological dark energy/matter via the “apparent matter”. In the Debbasch approach, statistical ensemble of metric is ensembles of histories and not ensembles of states. This is different basically with ordinary statistical mechanics of classical and/or quantum particles. From the latter point of view, it has been known for a long time that black holes in asymptotically flat space times do not admit stable equilibrium states in the canonical ensemble (see introduction in [13]). But from the former point of view the Debbasch gives in [1] general proposal to obtain a mean field theory for the general theory of relativity. In his model members of the ensembles will be labeled by the symbol where is an arbitrary probability space [14]. To each , there are corresponding metric tensor , compatible connections , and the Einstein metric equation (see [1] and section 2 in [10]). All members of the ensemble correspond to the same macroscopic history of the space time manifold, in particular to a given same mean metric and corresponding mean connection . As application of his model Debbasch and coworkers considered statistical ensemble of Schwarzschild black holes as nonvacuum solutions of mean Einstein metric equation by using Kerr-Schild coordinates They calculated nonvanishing temperature of mean metric where single Schwarzschild black hole is well known which has nonvanishing temperature as a vacuum solution of the Einstein equation. They discussed their results with special emphasis on their connections with the context of astrophysical observations [12]. Extreme RNBH with has vanishing temperature (see next section) and regular Kerr-Schild coordinates are not applicable to obtain mean metric similar to the Schwarzschild one because the coarse graining space time turns out not to be a black hole [9]. Hence Chevalier and Debbasch used analytic continuation of the Kerr-Schild coordinates as to obtain mean metric of extreme classical black hole in [11]. According to the Debbasch approach we are free to choose types of coarse graining and/or ensemble space to obtain mean metric of the space times ensemble under consideration. We should point that topology of ensemble space times must be similar to topology of their mean metric (see [9]) which restrict us to choose an analytic continuation of Kerr-Schild coordinates for extreme RNBH. In short, with Debbasch proposal the averaging process does not change topology between ensemble of the curved space times and the corresponding mean space time. Precisely, the averaging process modifies the horizon radius and changes the energy-momentum tensor of space time but not total energy or mass of the black holes ensemble. Really the averaging process just redistributes without any change in the total mass which means that the total energy of the black holes dose not change by the coarse graining proposal.

Similar to study of thermodynamic behavior of single RNBH [15], we seek thermodynamic aspect of mean metric of nonextreme RNBHs ensemble in this work, by applying the Debbasch approach to evaluate the mean and/or coarse graining metric. Organization of the paper is as follows.

In Section 2, we calculate mean metric of ensemble of RNBHs. In Section 3 we obtain locations of mean metric horizons. In Section 4 we calculate interior and exterior horizons entropy, temperature, heat capacity, Gibbs free energy, and pressure of RNBHs mean metric. In Section 5 we calculate interior and exterior horizons luminosity and corresponding mass loss equation of quantum perturbed RN mean metric. Section 6 denotes concluding remark and discussion.

#### 2. RNBHs Ensemble and Mean Metric

Exterior metric tensor of a single charged, nonrotating, spherically symmetric body is given byThis is metric solution of Einstein-Maxwell equation and is called RNBH in which and are corresponding ADM mass and electric charge defined in units where Equating for arbitrary spherically symmetric hypersurface , one can obtain apparent (exterior) horizon radius as and Cauchy (interior) horizon radius as which appear only for One can obtain mass independent relation between and as With particular choice (called extreme and/or Lukewarm RNBH) these horizons coincide as Clearly the RNBH metric solution (1) leads to Schwarzschild one by setting for which we will have and . Temperature of a single RNBH can be obtained for interior and exterior horizons as [1] which reduce to a zero value for extreme (Lukewarm) RNBH because of They show positive (negative) temperature for exterior (interior) horizons. Negative temperatures of systems have physical meaning and happen under particular conditions. More authors studied conditions where the physical systems are taken to have negative temperatures. See [16] for temperatures of interior and exterior horizons of Kerr-Newman black hole. One can see [17–19] for negative temperature of nongravitational systems. In the nature, materials are obtained which have interesting properties like negative refraction index and reversibility of the Doppler’s effect, and so the phase and group velocity (velocity of energy propagation) have opposite singes. In these systems temperature will have negative values (see [17] and references therein). Such systems are called dual system (left-handed) of direct counterpart (right-handed conventional materials). Absolute temperature is usually bounded to be positive but its violation is shown in [18] by Braun et al. They showed, under special conditions, however negative temperatures where high energy states are more occupied than low energy states. Such states have been demonstrated in localized systems with finite, discrete spectra. They used the Bose-Hubbard Hamiltonian and obtained attractively interacting ensemble of ultra-cold bosons at negative temperature which are stable against collapse for arbitrary atom number. Furman et al. studied in [19] behavior of quantum discord of dipole-dipole interacting spins in an external magnetic field in the whole temperature range They obtained that negative temperatures, which are introduced to describe inversions in the population in a finite level system, provide more favorable conditions for emergence of quantum correlations including entanglement. At negative temperature the correlations become more intense and discord exists between remove spins being in separated states. According to the documentation and looking to diagrams of the present work, one can be convinced that a quantum perturbed mean metric of coarse graining RNBHs will be exhibited finally with a first-order phase transition and Bose-Einstein condensation state microscopically. According to the Debbasch approach [1] ensemble of the nonextreme RNBHs is collection of coarse graining RNBHs indexed by a 3-dimensional real parameter where is the three balls of radius as follows:The metric solution (1) is convenient to be rewritten with Kerr-Schild coordinates by transformingas follows (see [10–12]):whereand is the Euclidean norm of the vector . It should be pointed that all metric solutions of Einstein’s field equation will have simple form by using Kerr-Schild coordinates. They are decomposed into the well-known flat Minkowski background metric and null vector fields as where and is a scalar function (see [20] and references therein). Now, we must choose a probability measure. Hence we follow the assumption presented in [11] and choose uniform probability measure in which is probability density of this measure with respect to Lebesgue measure as with Applying the Kerr-Schild radial coordinate (in case of extreme RNBH where we must use analytic continuation of the Kerr-Schild coordinates as (see discussion given in the introduction)), we extend single RNBH metric (4) to obtain metric of coarse graining and/or statistical ensemble of RNBHs as follows:where and Using perturbation series expansion method and averaging the metric (6) against we obtain mean metric of (6) such that (see [21] for details of calculations)where , ,andIt is simple to show that the mean metric (7) reduces to a single RNBH metric (4) by setting . We can rewrite the mean metric (7) in the static frame by defining the Schwarzschild coordinates. To do so, we first choose a suitable local frame with coordinates asandwhereIn the latter case the mean metric (7) readswhere we definedandWe now seek location of mean metric horizons.

#### 3. Horizons Location for Mean Metric

One can obtain event horizon location of the mean metric (15) by solving and location of apparent (interior and exterior) horizons by solving null condition which leads to the equation such thatThe above equation has not exactly analytic solution for but for small we can use perturbation series expansion to evaluate the event horizon location. To do so we first define for which the horizon equation (18) can be written as The latter equation has a real solution as for We know that for a single RN black hole and so the condition reads for which horizon of the ensemble of statistical RN black holes is not destructed by raising if we want to apply perturbation series expansion method to obtain asymptotically behavior of the event horizon solution versus the parameters Thus we must obtain perturbation series expansion form of the event horizon but for as follows. Insertingand solving (18) as order by order, we obtainwhere and denote apparent exterior and Cauchy (interior) horizon radiuses of the mean metric (7), respectively. Inserting (9) and (19), one can obtain perturbation series expansion of (12) which up to terms in order of becomeswhere we definedArea equation of apparent horizon hypersurface of the spherically symmetric static mean metric (15) is defined by which up to terms in order of readswhere we definedAccording to Bekenstein-Hawking entropy theorem we have the result that given by (23) will be entropy function of exterior (interior) horizon of the mean metric (15). Black holes containing multiple horizons have several corresponding temperatures. Such a black hole will be in-equilibrium thermally throughout the space time where the temperature has a gradient between the horizons. Thermal equilibrium is possible only if horizon radiuses and so the corresponding temperatures become equal (see, for instance, [22, 23]). The latter situations happen for an extreme RNBH where and so We now calculate thermodynamic characteristics of interior and exterior horizons of the nonextreme mean metric of RNBHs statistical ensemble.

#### 4. Mean Metric Thermodynamics

In the next section we will consider massless, chargeless quantum scalar field effects on luminosity of the quantum perturbed coarse graining RNBHs where its electric charge becomes invariant quantity. Hence it is useful to define dimensionless black hole mass and ensemble factor in what follows. In the latter case exterior horizon entropy of mean metric (15) can be obtained up to terms in order of as follows:and its interior horizon entropy becomeswhere andDiagrams of entropies (25) and (26) are plotted versus in Figure 4. They show that for a single RNBH () in limits but for an ensemble of RNBHs for which we use , they reach infinity In fact for physical systems the entropy itself must be positive function but its variations may reach some negative values. Hence we define difference between interior horizon entropy and exterior horizon entropy asand total entropy such as follows:Diagrams of and are plotted in Figure 3. Fortunately these diagrams show that, for a single RNBH where , we will have by decreasing and but for ensemble of RNBHs with we have while . Hence and should be considered as physical entropies of coarse graining RNBHs. Decrease of entropy causes some negative temperatures (see Figure 2) in thermodynamic systems containing bounded energy levels. In the latter case there is a critical temperature for which the system exhibits a phase transition reaching Bose-Einstein condensation state microscopically. In thermodynamics, increase of entropy means an increase of disorder or randomness in natural systems. It measures heat transfer of the system for which heat flows naturally from a warmer to a cooler substance. Decrease of entropy means an increase of orderliness or organization of microstates of a system. To do so the substance of a system must loose heat in the transfer process. Individual systems can experience negative entropy, but overall, natural processes in the universe trend toward positive entropy. Negative entropy was first introduced for living things by Ervin Schrödinger in 1944 as the reverse concept of entropy, to describe the order that can emerge from chaos [24]. The heat generated by computations in the information theory is other applications for negative entropy concept (see [25–28] for more discussions). However we consider and to be physical entropies of RNBHs statistical ensemble containing two horizons which is in accord with positivity condition of the Bekenstein-Hawking entropy theorem. Our coarse graining RNBHs can be considered as a two-level thermodynamical system with upper bound finite energy because it has two dual (interior and exterior) horizons. We now calculate exterior (interior) horizon temperature of the RNBHs mean metric (15) as follows:Their diagrams are plotted against in Figure 2 for . For we see that has some negative (positive) values and their sign is changed when . We also plotted diagram for versus in Figure 2. They show that for reaching zero value at for While , for after that to obtain a finite positive maximum value. This maximum has smaller value for with respect to situations where we choose In ordinary statistical physics, negative temperatures are taken into account when the system has upper bound (maximum finite) energy for which entropy is continuously increasing but the energy and temperature decrease and vice versa. In the latter case the system reaches Bose-Einstein condensation state microscopically. Energy upper bound of our system is its total mass for which we have Regarding quantum matter effects on mean metric we will show in Section 5 that mass of mean metric decreases finally as (see Figure 1). Bose-Einstein condensation state needs a phase transition which happens when sign of heat capacity is changed. Hence we now calculate interior and exterior horizon of mean metric heat capacity which up to terms in order of , at constant electric charge and ensemble radius , becomeTheir diagrams are plotted against in Figure 5. They show that sign of is changed at for but sign of is changed at for We plot also diagrams of versus in Figure 5. They show a changing of sign for when and but not for In case we see for but its absolute value exhibits a minimum value. When we see which decreases monotonically to negative infinite value for Changing of sign of exterior horizon heat capacity means that a phase transition happens when the quantum perturbed RNBHs ensemble reaches its stable state with minimum mass To determine order kind of this phase transition we should study behavior of the corresponding Gibbs free energy as follows.