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

Department of Physics, Daido University, 10-3 Takiharu Minami-ku, Nagoya 457-8530, Japan

Received: 22 June 2011 / Revised: 28 August 2011 / Accepted: 31 August 2011 / Published: 5 September 2011

(This article belongs to the Special Issue Black Hole Thermodynamics)

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# Abstract

This paper consists of three parts. In the first part, 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 part, 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 part, 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.*Keywords:*black hole entropy; black hole thermodynamics; quantum gravity; axiomatic thermodynamics; Boltzmann formula; thermodynamic limit of quantum mechanics

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