Abstract
Einstein’s equation could be interpreted as the first law of thermodynamics near the spherically symmetric horizon. Through recalling the Einstein gravity with a more general static spherical symmetric metric, we find that the entropy would have a correction in Einstein gravity. By using this method, we investigate the Eddington-inspired Born-Infeld (EiBI) gravity. Without matter field, we can also derive the first law in EiBI gravity. With an electromagnetic field, as the field equations have a more general spherically symmetric solution in EiBI gravity, we find that correction of the entropy could be generalized to EiBI gravity. Furthermore, we point out that the Einstein gravity and EiBI gravity might be equivalent on the event horizon. At last, under EiBI gravity with the electromagnetic field, a specific corrected entropy of black hole is given.
Highlights
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
We use this technique in Eddington-inspired Born-Infeld (EiBI) gravity and get the known first law; the results show a more general formula of entropy, which holds for AdS Schwarzschild black hole and AdS R-N black hole
Since it provided a convenient approach to study the black hole thermodynamics just from the field equation, we investigated the black hole thermodynamic in EiBI gravity
Summary
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. 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 [14,15,16] It indicated a generic connection between thermodynamics of horizons and gravity, it is not yet understood at a deeper level [17]. In this paper, inspired by Padmanabhan [14,15,16], we derive the first law of black hole thermodynamics with the commonly accepted thermodynamics quantities from Einstein’s equation We use this technique in EiBI gravity and get the known first law; the results show a more general formula of entropy, which holds for AdS Schwarzschild black hole and AdS R-N black hole.
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