Exact Differential and Corrected Area Law for Stationary Black Holes in Tunneling Method

TL;DR

This paper introduces a simple thermodynamic approach to derive the first law of black hole thermodynamics, explicitly calculates the Bekenstein-Hawking entropy for Kerr-Newman black holes, and extends to include quantum corrections and trace anomaly effects.

Contribution

It presents a new method to derive black hole entropy and temperature from basic thermodynamic principles, including quantum corrections and trace anomaly considerations.

Findings

Explicit calculation of Bekenstein-Hawking entropy for Kerr-Newman black holes

Derivation of corrected entropy considering quantum effects

Establishment of connection between correction coefficients and trace anomaly

Abstract

We give a new and conceptually simple approach to obtain the first law of black hole thermodynamics from a basic thermodynamical property that entropy (S) for any stationary black hole is a state function implying that dS must be an exact differential. Using this property we obtain some conditions which are analogous to Maxwell's relations in ordinary thermodynamics. From these conditions we are able to explicitly calculate the semiclassical Bekenstein-Hawking entropy, considering the most general metric represented by the Kerr-Newman spacetime. We extend our method to find the corrected entropy of stationary black holes in (3+1) dimensions. For that we first calculate the corrected Hawking temperature considering both scalar particle and fermion tunneling beyond the semiclassical approximation. Using this corrected Hawking temperature we compute the corrected entropy, based on properties of exact differentials. The connection of the coefficient of the leading (logarithmic) correction with the trace anomaly of the stress tensor is established . We explicitly calculate this coefficient for stationary black holes with various metrics, emphasising the role of Komar integrals.