Thermodynamic black hole with modified Chaplygin gas as a heat engine
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Thermodynamic black hole with modified Chaplygin gas as a heat engine Ujjal Debnatha Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711 103, India Received: 4 February 2020 / Accepted: 23 April 2020 © Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We assume the negative cosmological constant as a thermodynamic pressure and the anti-de Sitter (AdS) black hole thermodynamics with modified Chaplygin gas. We write the mass, volume, entropy and temperature of the black hole due to the thermodynamic system. We find a new solution of the Einstein’s field equations of AdS black hole with modified Chaplygin gas as a thermodynamic system. We examine the weak, strong and dominant energy conditions for the source fluid of black hole. We also show that the thermodynamic black hole with Chaplygin gas can be considered as a heat engine, and then we calculate work done and its efficiency by this system.
1 Introduction Thermodynamic properties of black hole have been studied during many years. In recent years, there is considerable interest in the physics of AdS black hole [1] due to AdS/CFT correspondence. Hawking et al. [2] have studied the thermodynamic properties of non-rotating uncharged Schwarzschild-AdS black hole. After that Chamblin et al. [3,4] have investigated the first-order phase transition in the non-rotating charged Reissner–Nordstrom–AdS black hole. When the charge and/or rotation of the AdS black hole are included, the behavior of the AdS black hole is qualitatively similar to the Van der Walls fluid [5,6]. The concepts of black holes from the viewpoint of chemistry, in terms of concepts such as Van der Waals fluids, reentrant phase transitions, and triple points have been studied in [7]. The Van dar Waals black hole has been determined by Rajagopal et al. [8]. Hendi et al. [9] have introduced a new ad hoc pressure which is related to the perfect fluid dark matter and found a first-order van der Waals like phase transition. The recent developments in thermodynamic phase transitions of black holes and their van der Waals-like behavior can be produced an interesting quantum interpretation of classical gravity [10]. Subsequently, the Van dar Waals black hole in d-dimensions has been described by Delsate et al. [11]. Also, the polytropic black hole has been formulated by Setare et al. [12]. Kubiznak et al. [13] have assumed the cosmological constant < 0 [14] in thermodynamic system, which represents the thermodynamic pressure [8,11,15,16] as
a e-mail: [email protected] (corresponding author)
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3 = 8π 8πl 2 and allows to vary in the first law of black hole thermodynamics [8,11] p=−
δ M = T δS + V δp + · · · with the black hole thermodynamics volume [8,11] ∂M , V = ∂ p S,...
(1)
(2)
(3)
where, M is the mass, S is the entropy and T is the temperature of the black hole. From this, an equation of state p = p(V, T ) can
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