Advances in High Energy Physics

Volume 2017 (2017), Article ID 1095217, 14 pages

https://doi.org/10.1155/2017/1095217

## Thermodynamics of the Schwarzschild-AdS Black Hole with a Minimal Length

^{1}School of Physics, Nankai University, Tianjin 300071, China^{2}Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut), Mühlenberg 1, 14476 Potsdam, Germany

Correspondence should be addressed to Yan-Gang Miao; nc.ude.iaknan@gyoaim

Received 10 October 2016; Accepted 14 December 2016; Published 24 January 2017

Academic Editor: George Siopsis

Copyright © 2017 Yan-Gang Miao and Yu-Mei Wu. 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. The publication of this article was funded by SCOAP^{3}.

#### Abstract

Using the mass-smeared scheme of black holes, we study the thermodynamics of black holes. Two interesting models are considered. One is the self-regular Schwarzschild-AdS black hole whose mass density is given by the analogue to probability densities of quantum hydrogen atoms. The other model is the same black hole but whose mass density is chosen to be a rational fractional function of radial coordinates. Both mass densities are in fact analytic expressions of the -function. We analyze the phase structures of the two models by investigating the heat capacity at constant pressure and the Gibbs free energy in an isothermal-isobaric ensemble. Both models fail to decay into the pure thermal radiation even with the positive Gibbs free energy due to the existence of a minimal length. Furthermore, we extend our analysis to a general mass-smeared form that is also associated with the -function and indicate the similar thermodynamic properties for various possible mass-smeared forms based on the -function.

#### 1. Introduction

Nonrenormalizability is a well-known puzzle when gravity is combined with quantum theory. Among many attempts to solve the problem, the idea of a nonperturbative quantum gravity theory is attractive. Recently, Dvali and collaborators propelled [1, 2] this idea by putting forward the “UV self-complete quantum gravity” in which the production of micro black holes is assumed to play a leading role in scattering at the Planck energy scale. According to the Heisenberg uncertainty relation, an incident photon needs to have higher energy in order to locate a target particle more accurately. When the photon energy is comparable to that of the particle, it is likely to create a new particle. If the energy of the incident photon gets further higher, say the Planck energy, so huge energy confined in a small scale may create a black hole, the so-called the hoop conjecture [3]. After the formation of micro black holes, the higher the incident photon energy is, the bigger the black hole is, which makes the probe meaningless. As a result, the existence of a minimum length may be deduced by the horizon radius of an extreme black hole.

Regular black holes, that is, the black holes without singularity at the origin, whose research could be traced back to Bardeen’s brilliant work [4], appeared in [5, 6] on the discussions of mass definition, casual structure, and other related topics. In 2005, for the purpose of constructing the noncommutative geometry inspired Schwarzschild black hole, Nicolini, Smailagic, and Spallucci suggested [7] that the energy-momentum tensor should be modified in the right-hand side of Einstein’s equations, such that it describes a kind of anisotropic fluid rather than the perfect fluid, where the original point-like source depicted by the -function should be replaced by a mass-smeared distribution, while no changes should be made in the left-hand side. By replacing the -function form of the point-like source by the Gaussian form^{1} and then solving the modified Einstein equations, they obtained a self-regular Schwarzschild metric and established a new relation between the mass and the horizon radius (i.e., the existence of a minimum radius and its corresponding minimum mass). They also discussed the relevant thermodynamic properties and deduced the vanishing temperature in the extreme configuration and thus eliminated the unfavorable divergency of the Hawking radiation. Besides, there is a correction to the entropy in the near-extreme configuration. Since then, a lot of related researches have been carried out, such as those extending to high dimensions [8, 9], introducing the AdS background [10], quantizing the mass of black holes [11], and generalizing to other types of black holes [12, 13].

In this paper, we at first introduce the AdS spacetime background to the self-regular Schwarzschild black hole and then investigate equations of state and phase transitions of the self-regular Schwarzschild-AdS black hole with two specific mass densities, where one mass density is assumed [14] by an analogue to a quantum hydrogen atom and the other to a collapsed shell. Although the mass densities of the two models are different, both of them can be regarded as a smeared point-like particle and then be depicted by an analytic expression of the -function. Next, we extend our discussions to a general mass-smeared form which can be written as an unfixed analytic expression of the -function and explain the similarity in thermodynamic properties for different mass distributions previously studied in [10, 15, 16].

The paper is organized as follows. In Section 2, we derive the relations between masses and horizons of the two models and give the limits of vacuum pressure under the consideration of the hoop conjecture. Then, we analyze the thermodynamic properties of the two models in detail in Section 3. We turn to our analysis in Section 4 for the self-regular Schwarzschild-AdS black hole with a general mass distribution. Finally, we give a brief summary in Section 5.

We adopt the Planck units in this article: .

#### 2. Two Specific Models

##### 2.1. Mass Density Based on Analogy between Black Holes and Quantum Hydrogen Atoms

Based on our recent work in which we made an analogy between black holes and quantum hydrogen atoms [14], we choose the probability density of the ground state of hydrogen atoms as the mass density of black holes:where is the total mass of black holes and with the length dimension will be seen to be related to a minimal length through the extreme configuration discussed below. We plot the mass density in Figure 1 and obtain , which means that the origin is no longer singular. Now the mass distribution of black holes takes the formwhich, divided by , can be understood as a kind of step functions with continuity (see Figure 1(b)). It is easy to see that such a function goes to the step function under the limit . Correspondingly, as done in [7, 10], we work out the self-regular Schwarzschild-AdS metric through solving the modified Einstein equations associated with the above mass distribution:where represents the curvature radius of the AdS spacetime and has the relation with the vacuum pressure as follows:One can easily find that the self-regular metric goes back to the ordinary Schwarzschild-AdS one if and that on the other hand it goes to the de Sitter form if approaches zero:where the equivalent cosmological constant can be understood as^{2} . Thus the singularity is avoided since the positive cosmological constant represents the negative (outward) vacuum pressure that prevents the black hole being less than a certain scale from collapsing. From , one can have the relation between the total mass and the horizon radius :which is shown in Figure 2 under different vacuum pressure.