Quantum Gravity and the Distance-Velocity Profile of a Black Hole

Abstract

Newton's law of universal gravitation implies that the circular velocity of a low-mass body at a distance r from the center is given by the relation v²=GM/r, while the velocity v remains unbounded. According to Einstein's theory of relativity, the relative velocity of motion is limited by a maximum speed, which is the speed of light. Furthermore, a massive object cannot reach the speed of light. In General Relativity, the circular velocity from a gravitational center is also determined by the same expression used to describe the gravitational field according to Newton.

By using this fundamental relationship in combination with the relativistic limit of maximum velocity to derive the magnitude of binding energy—and consequently the binding mass of a bound state of two massive objects—we obtain two equations. According to the first, the value of the gravitational coupling parameter K can be determined as a function of the distance from the center; according to the second, the velocity of circular motion at that distance can be established.

The derived equation for the coupling parameter, which we shall call the "vacuum equation," can be expressed as the sum of three equations. The first is the fermion equation, the second is the boson equation, and the third is the gravitation equation. Solving these equations individually, as well as their partial sums for the parameter K=2, leads to determining the volume ratios of selected regular polytopes in four-dimensional space (with properties described by L. Schläfli, which are in agreement with the mass ratios of selected elementary particles. The concept of regular polyhedra in three-dimensional space in connection with planetary orbits was previously used by J. Kepler.

For the parameter K∈(0,1), the solution of the gravitation equation leads to determining the mass of the resulting so-called black hole after the collapse of two massive objects. The first set of solutions to the equation for the gravitational coupling parameter determines the velocity profile based on circular velocity at a distance from the gravitational center both outside and inside the black hole. This gravitational parameter is established by solving the gravitation equation, which depends on a parameter that distinguishes massive objects based on their composition—whether they are fermionic, bosonic, or a mixture of these basic components. The Planck black hole, which lies on the boundary between the masses of elementary particles and massive objects such as black holes, can be considered the limiting threshold.

The second set of solutions for the parameter K > 2 determines the mass of a particle equivalent to the anti-gravitational effect expected in dark energy. Simultaneously, this group of solutions allows for the determination of conditions for the quantum mechanical evaporation of black holes based on particleanti particle creation with rest mass. This tri-particle creation involves a dark energy particle, a dark matter particle, and a baryonic particle, whose masses are in the expected ratio corresponding to their observed distribution in the universe.