The Mass Lowest Limit of a Black Hole: The Hydrodynamic Approach to Quantum Gravity

Abstract

In this work the quantum gravitational equations are derived by using the quantum hydrodynamic approach that allows to define the energy-impulse tensor density of the gravitational equation. The outputs of the work show that the quantum uncertainty principle opposes itself to the gravitational collapse so that an equilibrium condition becomes possible. In this case, when the maximum collapse is reached, all the mass is inside the gravitational radius of the black hole if it is larger than the Planck's one.  The quantum-gravitational equations of motion show that the quantum potential generates a repulsive force that opposes itself to the gravitational collapse. The eigenstates in a central symmetric black hole realize themselves when the repulsive force of the quantum potential becomes equal to the gravitational one. The work shows that, in the case of maximum collapse, the mass of the black hole is concentrated inside a sphere whose radius is two times its Compton length. The mass minimum is determined requiring that the gravitational radius is bigger than or at least equal to the radius of the state of maximum collapse.