Thermodynamics and phase transition of spherically symmetric black hole in de Sitter space from Rényi statistics


Schwarzschild black holes in a de Sitter background are studied in terms of their thermodynamics based on the Rényi statistics. This leads to thermodynamically stable black hole configurations for some certain range of black hole radii; namely, within this range the corresponding black holes have positive heat capacity. Moreover, for a certain background temperature there can exist at most three configurations of black hole, one among which is thermodynamically stable. These configurations are investigated in terms of their free energies, resulting in the moderate-sized stable black hole configuration being the most preferred configuration. Furthermore, a specific condition on the Rényi non-extensive parameter is required if a given hot spacetime were to evolve thermally into the moderate-sized stable black hole.

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  1. 1.

    Another way to fix the problem of incompatibility is to define an effective temperature as a conjugate variable to the Tsallis entropy (see, for example, [16]). Through this approach, a relation analogous to the standard zeroth law can be obtained.

  2. 2.

    In Rényi statistics, it is possible to have a thermodynamic system in which the two horizons are in thermal equilibrium. Such treatment on this possibility is described in 1.


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The authors acknowledge King Mongkut’s University of Technology Thonburi for funding Postdoctoral Fellowship to LT. PW is supported by the Thailand Research Fund (TRF) through grant no. MRG6180003. PW also would like to thank the Theoretical and Computational Physics Group (TCP) for hospitality during this work was in progress.

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Correspondence to Pitayuth Wongjun.



Zeroth law compatibility

In Sect. 2, the Rényi entropy is considered as thermodynamic entropy of the black hole system instead of the Tsallis entropy. The reason behind this is that the Tsallis entropy which obeys the following composition,

$$\begin{aligned} S_{12}=S_1+S_2+\lambda S_1 S_2, \end{aligned}$$

is not compatible with the zeroth law of thermodynamics which is crucial in defining temperature of a thermal system. In order to see this, let us consider a thermal isolated system with constant entropy, S. We can always consider this system as being composed of two weakly interacting subsystems, each of which has entropy of \(S_1\) and \(S_2\). Moreover, let us assume that the entropy composition rule is that of Gibbs–Boltzmann statistics as follows,

$$\begin{aligned} S=S_1+S_2. \end{aligned}$$

If the subsystems are allowed to exchange only thermal energy to one another, since the whole system is isolated, then according to the conservation of energy, the thermal energy gained by the subsystem 1 must be from the subsystem 2,

$$\begin{aligned} T_1 dS_1=-T_2 dS_2. \end{aligned}$$

If the system enter a thermal equilibrium, \(T_1=T_2\), then we obtain \(dS_1+dS_2=d\left( S_1+S_2\right) =dS=0\) which implies that there is no change in entropy which is reasonable for an isolate system. This statement is true only in the context of the Gibbs–Boltzmann entropy which obeys \(S=S_1+S_2\). On the other hand, when the Tsallis entropy is considered instead, the above analysis will imply a change in total entropy even when an isolated system is assumed, showing a contradiction between the Tsallis entropy and the zeroth law of thermodynamics. The more detailed investigation is reported in Ref. [14]. It is also worth mentioning that there is an approach which

On the thermal equilibrium between the two horizons

Although in the article the thermodynamic behaviour of the black hole event horizon is treated separately with those of the cosmic horizon, there is a possibility for the two horizons to be in thermal equilibrium in the language of Rényi statistics. The Gibbs–Boltzmann temperatures of both horizons can be found through their respective surface gravities, \(\kappa \) as follows,

$$\begin{aligned} T_{+}=\frac{\kappa _+}{2\pi },\quad T_c=\frac{\kappa _c}{2\pi }, \end{aligned}$$

where the subscripts \(+\) and c denote the quantity being evaluated at the black hole event horizon and at the cosmic horizon, respectively. Their corresponding Rényi temperatures are, after substituting the surface gravities,

$$\begin{aligned} T_{\text {R},+}&= \frac{1}{4 \pi r_{+}}\left( 1-\varLambda r_{+}^{2}\right) \left( 1+\lambda \pi r_{+}^{2}\right) , \end{aligned}$$
$$\begin{aligned} T_{\text {R},c}&= \frac{1}{4 \pi r_{c}}\left( \varLambda r_{c}^{2}-1\right) \left( 1+\lambda \pi r_{c}^{2}\right) . \end{aligned}$$

Note that the sign of \(T_{\text {R},c}\) is flipped because only the magnitude of the cosmic surface gravity is considered. In the situation where the two horizons are present, we must have \(9M^2\varLambda <1\) (where \(9M^2\varLambda =1\) implies an extremal case) and the black hole event horizon and the cosmic horizon can be expressed explicitly as follows,

$$\begin{aligned} r_+&=\frac{2}{\sqrt{\varLambda }}\cos \left[ \frac{\arctan \left( \sqrt{\frac{1}{9M^2\varLambda }-1}\right) +\pi }{3}\right] , \end{aligned}$$
$$\begin{aligned} r_c&=\frac{2}{\sqrt{\varLambda }}\cos \left[ \frac{\arctan \left( \sqrt{\frac{1}{9M^2\varLambda }-1}\right) -\pi }{3}\right] , \end{aligned}$$

which are functions of M and \(\varLambda \). Equating Eq. (26) with Eq. (27) allows us to find a suitable value of \(\lambda \) as a function of \(\varLambda \) and M so that the two horizons are in thermal equilibrium. Some of the possible configuration are given in Fig. 8 for specific values of \(\varLambda \).

Fig. 8

Possible \((\lambda ,M)\) configurations when thermal equilibrium between the black hole event horizon and the cosmic horizon is assumed are given for specific values of \(\varLambda \)

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Tannukij, L., Wongjun, P., Hirunsirisawat, E. et al. Thermodynamics and phase transition of spherically symmetric black hole in de Sitter space from Rényi statistics. Eur. Phys. J. Plus 135, 500 (2020).

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