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.

It is shown, that special and general relativities reflect the gauge of effect on matter of, correspondingly, motion and gravity. This doesn’t allow us to observe in intrinsic space and time of the matter any changes, appeared because of this effect. The solution of gravitational field equations that corresponds to astronomical objects, alternative to black holes, is found. The eternity of Universe existence both in the future and in the past is shown.

It is shown here that Lorentz transformations are caused by gauge effect of motion on matter (principle nonobservability of effect of motion on matter). This gauge effect of motion is caused by interdependence and mutual determination of propagation velocity of interaction between matter elementary particles and of rate of course of matter proper (standard) time. The Lorentz transformations are derived without any linearity assumptions and being based only on the presence of relativistic shrinkage of the length of moving body and on clock desynchronization at its slowest transfer along this body.

The possibility of getting a linear element (interval) of Schwarzschild frame of reference of spatial coordinates and time (FR) is shown, founded on the existence of fundamental (Newton absolute) space, which is formally independent on matter and is only a container for it [1]. In addition to it, the presence of evolutionary changeability and spatial inhomogeneity of properties of the physical vacuum (PV), filling all this absolutely rigid (nonexpanding) Euclidean (noncurved) infinite space, is assumed.

It is shown here, that relativistic shrinkage of the length of moving body is realizing strongly synchronously with the change of its velocity and is continuously self-sustaining without the effect of any forces. This shrinkage is caused by inertial isobaric self-contraction of body matter and by propagation of changes of the strengths of inertia forces field together with the front of body intrinsic time. A mechanism of kinetic energy filling (accumulation) of a body is considered and propagation of the phase waves of perturbation of gravitational field at supraluminal velocity is substantiated here.