Abstract
We have searched for a universal property of quantum gravity, where "universal" means to be independent of any existing model of quantum gravity (such as the super string theory, loop quantum gravity, causal dynamical triangulation, and so on). To do so, we have investigated carefully the basis of black hole thermodynamics. The uniqueness of Bekenstein-Hawking entropy lets us extract a reasonable universal property of quantum gravity from the conditions justifying Boltzmann formula. The universal property indicates a repulsive gravity at Planck length scale, otherwise stationary black holes can not be regarded as thermal equilibrium states of 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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