Entropy for the Interior of a Schwarzschild Black Hole Assuming the Mass Is Increasing with Time
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I, PARTICLES, FIELDS, GRAVITATION, AND ASTROPHYSICS
Entropy for the Interior of a Schwarzschild Black Hole Assuming the Mass Is Increasing with Time Sandip Duttaa,* and Ritabrata Biswasa,** a
Department of Mathematics, The University of Burdwan, Golapbag Academic Complex, Purba Burdwan, West Bengal, 713104 India *e-mail: [email protected] **e-mail: [email protected] Received October 25, 2019; revised October 25, 2019; accepted November 15, 2019
Abstract—Black hole thermodynamics is the area of study that seeks to reconcile the laws of thermodynamics with the existence of black hole event horizons. Here we calculate the entropy corresponding to the interior of a Schwarzschild black hole for massless modes, assuming the mass of the black hole increasing with time. We find that the entropy is proportional to the Bekenstein–Hawking expression. Also, we can see that the evaluated entropy satisfies the second law of thermodynamics. Using the thermodynamic law we get a relation between temperature and inverse temperature. The special relativistic corrections to thermodynamic quantities are considered. The change in thermodynamic properties are analyzed when the velocity of the considered system is comparable to the speed of light. The effect of presence of scalar charge is incorporated. DOI: 10.1134/S1063776120040056
1. INTRODUCTION There exist many theories regarding the compact objects like Black holes. Even the present day gravitational wave detections [1] of black hole mergers can be treated as strong evidence of their existence. For the first time, theoretically Hawking has shown that black holes can evaporate and leave thermal radiation [2, 3]. This concept attracts people and different aspects of black hole thermodynamics [4] has been discussed. Many thermodynamic quantities were calculated. The study was analogous to the previous studies of classical thermodynamics. Entropy for a black hole was treated to be a quantity which is proportional to its ever increasing area of event horizon. Bekenstein–Hawking’s entropy form [5] is usually expressed as
SBH = A , 4 G where A is the area of the black hole’s event horizon. The second law of thermodynamics states that the entropy increases with time. Another natural notion of entropy is the Von Neumann entropy given by S[ρ] = −tr(ρlnρ), where ρ is the density matrix of a quantum system. In classical level, we know that black hole mechanics follows the laws similar to the ordinary laws of thermodynamics [6]. A well known formula for the entropy of black hole in terms of Noether charge was later executed in the 90’s [7]. Yet, a question has arrived as unanswered in the area of classical thermodynamics of
black holes: what is the source of the entropy at the classical level? More generally, what are the classical microstates that corresponds to the entropy microstates? So, we found a long-standing issue in literature by this question, which has recently renewed: do extremal black holes have zero or non-zero entropy [8]? Recently, a new path of def
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