The possibility to avoid physical realizability of cosmological singularity (singularity of Big Bang of the Universe) directly in the orthodoxal general theory of relativity (GR) and in its improved version – the relativistic gravithermodynamics (RGTD) is substantiated.

For the collective gravithermodynamic Gibbs microstates the connection between all thermodynamic potentials and parameters of matter have been found. This connection is realized with the help of four hidden wave functions that can take arbitrary values with certain probability.

The solution to the gravitational field equations of a flat galaxy has been found. It is shown that at the edge of the galaxy the excessively strong ordinary (unreduced) centrifugal pseudo-forces of inertia are compensated mainly by centripetal pseudo-forces of evolutionary self-contraction of matter in the background Euclidean space of expanding Universe, and not by the weak gravitational pseudo-forces at the edge of the galaxy.

The solution to the gravitational field equations of a flat galaxy has been found.. The strength of the dynamic gravitational field of spiral and other flat (or superthin) galaxies, according to their two-dimensional topology, is inversely proportional to the radial distance, not to its square.

According to the RGTD equations, the configuration of the dynamic gravitational field of a galaxy in a quasi-equilibrium state is standard (canonical in RGTD). Because it is not determined at all by the spatial distribution of the average mass density of its non-continuous matter.

The cardinal difference between relativistic gravithermodynamics (RGTD) and general relativity is that in RGTD the extranuclear thermodynamic characteristics of matter are used in the tensor of energy-momentum to describe only its quasi-equilibrium motion. For the description of the inertial motion in RGTD only the hypothetical intranuclear gravithermodynamic characteristics of matter are used.

According to the Relativistic Gravithermodynamics (RGTD) equations, the configuration of the dynamic gravitational field of a galaxy in a quasi-equilibrium state is standard (canonical in RGTD). That is because it is not determined at all by the spatial distribution of the average mass density of its non-continuous matter.

The cardinal difference between relativistic gravithermodynamics (RGTD) and general relativity (GR) is that in RGTD the extranuclear thermodynamic characteristics of matter are used in the tensor of energy-momentum to describe only its quasi-equilibrium motion.

The cardinal difference between relativistic gravithermodynamics (RGTD) and general relativity (GR) is that in RGTD the extranuclear thermodynamic characteristics of matter are used in the tensor of energy-momentum to describe only its quasi-equilibrium motion.

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.