Spherically symmetric de Sitter solution of black holes
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ORIGINAL PAPER
Spherically symmetric de Sitter solution of black holes M F Mourad1*, N H Hussein2, D A Eisa3 and T A S Ibrahim1 1
Mathematics Department, Faculty of Science, Minia University, Minia, Egypt 2
Mathematics Department, Faculty of Science, Assiut University, Egypt
3
Mathematics Department, Faculty of Science, Assiut University, New Valley, Egypt Received: 20 January 2020 / Accepted: 14 May 2020
Abstract: In this study, we obtain the solution of the spherically symmetric de Sitter solution of black holes using a general form of distribution functions which include Gaussian, Rayleigh and Maxwell–Boltzmann distributions as a special case. We investigate the properties of thermodynamic variables such as the Hawking temperature, the entropy, the mass and the heat capacity of black holes. Moreover, we show that the strong energy condition which includes the null energy condition is satisfied. Finally, we show the regularity of the solution by calculating the scalar curvature and invariant curvature in general distribution form. Keywords: Black hole; Spherically symmetric de Sitter solution; Thermodynamics; Distribution function; Hawking temperature
1. Introduction One of the greatest contributions of general relativity was the discovery that black holes (BHs) have thermodynamic (TD) properties. Moreover, for a quantum Schwarzschild black hole, the entropy was expressed in terms of the effective mass [1]. It is found that if temperature of black hole (BH) is larger than a critical value, anti-de Sitter (AdS) BH occurs, while a thermal gas solution happens when its temperature is less than this value. A BH has a natural temperature related to its surface gravity, and the entropy associated with its area is non-decreasing. Hawking and Page formulated the first-order phase transition which takes place between the BH solution and the thermal gas solution in the AdS [2]. It is really helpful for us to understand the holographic description of asymptotically AdS space-times [3]. In two and four dimensions, there are many solutions for a BH. The researchers believe that in three dimensions of space-time, these solutions do not exist. However, Deser et al. explained in refs. [4, 5] that it was possible to find a vacuum solution of Einstein’s gravity that could be considered a BH. The solution is
*Corresponding author, E-mail: [email protected]; [email protected]
characterized by a constant curvature, but the global topology is different from that of the 3D AdS solution. Thus, the causal structure of the solution is closer to Schwarzschild solution [6, 7]. Actually, in Gaussian distribution there exist a new inner horizon in the BH space-time, but anisotropic (i.e., pr = ph) smearing point of matter with energy momentum tensor (EMT) of anisotropic fluid. Tab = diag (q, pr, ph, pu), in a ‘‘self-consistent’’ way [8–11]. Park [12] investigated the de Sitter (dS) in three dimensions dS3 BHs for non-Gaussian smearing of hairs and gravastars for Gaussian distribution. In ref. [13], the deformed dS3 BHs have
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