Generalized second law and universal relations of cosmological black hole
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Arabian Journal of Mathematics
Ashfaque H. Bokhari · M. Akbar · Tayeb Brahimi
Generalized second law and universal relations of cosmological black hole
Received: 30 December 2018 / Accepted: 26 June 2019 © The Author(s) 2019
Abstract The objective of this paper is to investigate the validity conditions for the generalized second law of thermodynamics, and the universal relations for multi-horizon dynamical spacetime. It is found that there are three horizons of McVittie universe termed as event horizon, cosmological apparent horizon, and virtual horizon. The mass-dependent and mass-independent area product relations are formulated in terms of areas of the dynamical event horizon, cosmological horizon and virtual horizon. It is noted that whereas the area sum relation is mass independent, the area product relation is explicitly mass dependent. Moreover, we have also analyzed and listed explicit mass-independent and mass-dependent relations. Mathematics Subject Classification
83F05
1 Introduction In recent times, considerable interest has been shown in exploring universal relations constructed from the area product and sum relations of multi-horizon stationary black holes in super gravity, Einstein gravity and other modified theories of gravity [1–6]. In most cases, whereas the area product and sum relations are mass independent, they depend on various black hole parameters including charge, angular momentum, and cosmological constant. However, in some cases such as asymptotically non-flat spacetime, these relations fail to execute mass-independent relations [7–10]. Visser [11] found that the mass-independent area product relations fail in Schwarzschild-de Sitter black hole even if the contributions of the virtual horizon is included. Several authors [12–14] argued that the contribution of virtual horizon area should be added to build up mass-independent area-product and sum relations. However, in some cases, such contributions of virtual horizon could not sustain significantly [11]. It is known that a class of black holes admits more than one horizons, such as black hole event horizon (outer horizon), cauchy horizon (inner horizon), and virtual horizon (un-physical horizon). It has been argued [15–18] that like event horizon, other horizons are also associated with the notion of temperature and entropy. An extension of these notions has also been made for dynamical apparent horizon of FRW universe [19–28]. Despite the multiple horizons of stationary black holes, there are dynamical spacetime metrics containing dynamic multiple horizons. The universal relations for dynamical A. H. Bokhari Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia E-mail: [email protected] M. Akbar (B) · T. Brahimi Effat University, Jeddah 21478, Saudi Arabia E-mail: [email protected] T. Brahimi E-mail: [email protected]
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Arab. J. Math.
multi-horizons geometries need to be investigated for more physical interest. In this context, we co
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