Universal Property of Quantum Gravity implied by Uniqueness Theorem of Bekenstein-Hawking Entropy

TL;DR

This paper proves the uniqueness of Bekenstein-Hawking entropy for black holes, explores quantum mechanics properties that justify the Boltzmann formula, and suggests quantum gravity may be inherently repulsive at the Planck scale, independent of specific models.

Contribution

It establishes the universal property of black hole entropy and proposes fundamental features of quantum gravity based on thermodynamics and quantum mechanics.

Findings

Bekenstein-Hawking entropy is uniquely determined by thermodynamics.

Quantum mechanics properties justify the Boltzmann entropy formula.

Quantum gravity potential may be bounded and repulsive at Planck length.

Abstract

This paper consists of three steps. In the first, we prove that the Bekenstein-Hawking entropy is the unique expression of black hole entropy. Our proof is constructed in the framework of thermodynamics without any statistical discussion. In the second, intrinsic properties of quantum mechanics are shown, which justify the Boltzmann formula to yield a unique entropy in statistical mechanics. These properties clarify three conditions, one of which is necessary and others are sufficient for the validity of Boltzmann formula. In the third, by combining the above results, we find a reasonable suggestion from the sufficient conditions that the potential of gravitational interaction among microstates of underlying quantum gravity may not diverge to negative infinity (such as Newtonian gravity) but is bounded below at a finite length scale. In addition to that, from the necessary condition, the interaction has to be repulsive within the finite length scale. The length scale should be Planck size. Thus, quantum gravity may become repulsive at Planck length. Also, a relation of these suggestions with action integral of gravity at semi-classical level is given. These suggestions about quantum gravity are universal in the sense that they are independent of any existing model of quantum gravity.