A Simple Determination of the (LOGARITHMIC) Corrections of Black Hole Entropy "without Knowing the Details of Quantum Gravity"

TL;DR

This paper presents a simple, quasi-classical method to derive logarithmic corrections to black hole entropy without relying on detailed quantum gravity theories, aligning with results from more complex approaches.

Contribution

It introduces a straightforward, approximate approach to determine black hole entropy corrections that matches results from quantum gravity theories, without requiring their detailed knowledge.

Findings

Logarithmic correction coefficient is -0.5.

Method reproduces known quantum gravity results.

Black hole entropy correction derived from classical potential energy.

Abstract

In this work, starting by simple, approximate (quasi-classical) methods presented in our previous works, we suggest a simple determination of the (logarithmic) corrections of (Schwarzschild) black hole entropy "without knowing the details of quantum gravity"(Fursaev). Namely, in our previous works we demonstrated that all well-known important thermodynamical characteristics of the black hole (Bekenstein-Hawking entropy, Bekenstein entropy/surface quantization and Hawking temperature) can be effectively reproduced starting by simple supposition that black hole horizon circumference holds integer number of reduced Compton wave lengths corresponding to mass (energy) spectrum of a small quantum system. (Obviously it is conceptually analogous to Bohr quantization postulate interpreted by de Broglie relation in Old, Bohr-Sommerfeld, quantum theory.) Especially, black hole entropy can be presented as the quotient of the black hole mass and the minimal mass of small quantum system in ground mass (energy) state. Now, we suppose that black hole mass correction is simply equivalent to negative classical potential energy of the gravitational interaction between black hole and small quantum system in ground mass (energy) state. As it is not hard to see absolute value of the classical potential energy of gravitational interaction is identical to black hole temperature. All this, according to first thermodynamical law, implies that first order entropy correction holds form of the logarithm of the surface with coefficient -0.5. Our result, obtained practically quasi-classically, "without knowing the details of quantum gravity", is equivalent to result obtained by loop quantum gravity and other quantum gravity methods for macroscopic black holes.