Discussion of a Possible Corrected Black Hole Entropy

Received Nov 21, 2017; Revised Jan 31, 2018; Accepted Feb 12, 2018

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 .

1. Introduction

Black hole thermodynamics has been proposed for many years since the entropy and temperature were found by Bekenstein and Hawking [1, 2], even getting many interesting results, like the four laws of black hole thermodynamics. It established the connection between the gravity and thermodynamics.

The entropy is assumed to be proportional to its horizon area [1], and it is well-known that the so-called area formula of black hole entropy holds only in Einstein gravity. However, when some higher order curvature terms appear in some gravity theory, the area formula has to be modified [3]. A logarithmic term often occurs in the correction like the black hole entropy in loop quantum gravity (quantum geometry) [4–6] and thermal equilibrium fluctuation [7, 8]. The correction of entropy has been studied in Gauss-Bonnet gravity [9], Lovelock gravity [10], and $f(R)$ gravity [11]. In the apparent horizon of FRW universe, the entropy also has a correction [12].

Einstein’s equation can be derived from the thermodynamics [13]; on the other side, the thermodynamic route to the gravity field equation, which could get the first law of thermodynamics in Einstein gravity, was proposed by Padmanabhan [14–16]. It indicated a generic connection between thermodynamics of horizons and gravity, although it is not yet understood at a deeper level [17]. This technique has been used in Gauss-Bonnet gravity and Lanczos-Lovelock gravity [15]; the corrected entropy is the same in [18, 19], respectively.

The EiBI gravity was inspired by Bañados and Ferreira [20]. It is completely equivalent to the Einstein gravity in vacuum, but in the presence of matter it would show many interesting results, like an alternative theory of Big Bang singularity in early universe [21] and the mass inflation in EiBI black holes [22, 23]. However, there are few investigations for the thermodynamic properties of black hole in EiBI gravity, as it has a complicated spherically symmetric solution when the electromagnetic field is considered [24].

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