A Fermi Model of Quantum Black Hole

Abstract

We propose a quantum model of the Schwarzschild black hole as a quantum mechanics of a system of fermionic degrees of freedom. The system has a constant density of states and a Fermi energy that is inversely proportional to the size of the system. Assuming equivalence principle, we show that the degeneracy pressure of the Fermi degrees of freedom is able to withstand the collapse of gravity if the radius of the system is given precisely by the horizon radius of the Schwarzschild black hole. In our model, the fermionic degrees of freedom at each energy level can be entangled in certain different ways, giving rise to a multitude of degenerate ground states of the system. The counting of these microstates reproduces precisely the Bekenstein-Hawking entropy. This simple Fermi model is universal and works also for the Reissner-Nordstr\"om charged black hole as well as black hole with a cosmological constant. From the properties of the Fermi variables, we propose that quantum gravity is characterized by a principle of {\it maximal capacity of states} where there can be no more than $V /l_P^3$ quantum states in any volume $V$. It implies a loss of spatial locality below the Planck length and suggests that any singularity predicted by general relativity is resolved and replaced by a quantum space in quantum gravity. In our model, a black hole spacetime is equipped with an uniform distribution of energy levels. This is another reason why black hole can be considered a simple harmonic oscillator of quantum gravity.