The quantum equation of gravitational field have been found, the solutions of which set the spatial distribution of gravitational radius of matter in its every new gravithermodynamic (GTD) state with the polynomial function with the next more high degree.

Only conditional identity of inertial mass of moving matter to its gravitational mass only by gravity-quantum clock, which is located in the point, from which the matter started its inertial motion, and due to the usage of corrected value of gravitational constant in its pseudo-centric intrinsic frame of reference of spatial coordinates and time (FR), is justified.

The general solution of the equations of the gravitational field of the galaxy with an additional variable parameter n is found. The additional variable parameter n determines in GR the distribution of the average mass density mainly in the friable galactic nucleus

The equivalence of only inert free energy of matter (and not of the total internal energy) to gravitational and inert masses is justified. It was proved that total energy of matter of inertially moving body is equal in all frames of references of spatial coordinates and time that are moving relatively to matter. And that is precisely why there is a conform Lorentz-invariance of thermodynamic potentials and parameters in examined modification of transformations of the special theory of relativity

The most perfect modification of the special and general theories of relativity is considered. It is shown that equations of the gravitational field of General Relativity (GR) should be considered as equations of spatially inhomogeneous gravithermodynamic state of only utterly cooled down matter. This matter can only be the hypothetical substances such as ideal gas, ideal liquid and the matter of absolutely solid body. The real matter will be inevitably cooling down for infinite time and never will reach the state that is described by the equations of gravitational field of the GR.

The general solution of the equations of the gravitational field of the galaxy with an additional variable parameter n is found. The additional variable parameter n determines in GR the distribution of the average mass density mainly in the friable galactic nucleus. The velocity of the orbital motion of stars is close to Kepler only for n>225.

It is shown that equations of the gravitational field of General Relativity should be considered just as equations of spatially inhomogeneous thermodynamic state of utterly cooled down matter. This matter can only be the hypothetical substances such as ideal gas, ideal liquid and the matter of absolutely solid body. The real matter will be inevitably cooling down for infinite time and never will reach the state that is described by the equations of gravitational field. Gravitational field itself is the field of spatial inhomogeneity of thermodynamic state of dense matter of compact astronomical objects, as well as of strongly rarefied gas-dust matter of space vacuum.