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.
Highlights
Gravity is the only fundamental interaction which is not quantized at present
One of the sufficient conditions is that the interaction potential has a negative lower bound at a finite length scale, Rbound . (Note that, at least for laboratory systems, there seems to be no example which violates this sufficient condition but retains the Boltzmann formula.) in Section 3, we show the necessary condition for the existence of thermal equilibrium states of quantum system, for the case that the interaction among many particles is a sum of two-particle interactions and multi(≥3)-particle interactions do not exist
If we assume that the criterion of phase transition in ordinary thermodynamics is applicable to black hole, it is concluded from the behavior of FBH shown in Figure 3 that an unstable equilibrium state is transformed to a stable one under the environment of constant temperature, because FBH (2M < rw < 3M ) < FBH (3M, rw )
Summary
Gravity is the only fundamental interaction which is not quantized at present. By combining classical physics (general relativity) of black holes and quantum field theory in black hole spacetime, it is theoretically very reasonable to regard the stationary black hole as a thermal equilibrium state of gravity whose temperature is determined by the thermal spectrum of Hawking radiation [1,2,3,4,5,6,7,8,9,10]. Since no counter-example to the sufficient conditions seems to be found at least in laboratory systems, it seems to be empirically reasonable that the sufficient conditions for the validity of Boltzmann formula hold in quantum gravity In this case, the interaction potential among microstates of underlying quantum gravity is bounded below at a finite length scale Rbound , unlike the Newtonian gravity. When the underlying quantum gravity satisfies the necessary condition for the existence of thermal equilibrium states and the sufficient conditions for the validity of Boltzmann formula, the two-body interaction should be repulsive within the length scale Rbound. This Rbound may be the Planck length, at which the quantum gravitational effect appears significantly. A reader who knows the topic can skip the corresponding review section
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