**Authors:** Miroslav Pardy

We derive the Poincar´e model of the Lobachevsky geometry from the Fermat principle. The Lobachevsky geometry is interpreted as the Lobachevsky-Beltrami-Fok velocity space geometry of moving particles. The relation of this geometry to the decay of the neutral π-meson is considered. The generalization of the Lobachevsky geometry is performed and the new angle of parallelism is derived. Then, we determine nonlinear transformations between coordinate systems which are mutually in a constant symmetrical accelerated motion. The maximal acceleration limit follows from the kinematic origin. Maximal acceleration is an analogue of the maximal velocity in special relativity. We derive the dependence of mass, length, time, Doppler effect, on acceleration as an analogue phenomena in special theory of relativity. We apply the derived nonlinear Lorentz group to the so called Thomas precession. The total quantum energy loss of binary is caused by the production of gravitons emitted by the rotation motion of binary. We have calculated it in the framework of the Schwinger theory of gravity for the situation where the gravitational propagator involves radiative corrections. We also derive the finite-temperature gravitational Cherenkov radiation involving radiative corrections. The graviton action in vacuum is generalized for the medium with the constant gravitational index of refraction. From this generalized action the power spectral formula of the Cherenkov radiation of gravitons is derived in the framework of the Schwinger theory at zero and nonzero temperature. The next text deals with non-relativistic quantum energy shift of H-atom electrons due to Gibbons-Hawking thermal bath. The seventh chapter deals with gravity as the deformation of the space time and it involves the light deflection by the screw dislocation. In conclusion, we consider the scientific and technological meaning and the perspectives of the results derived. Some parts of the complex are published in the reputable journals. 1

**Comments:** 89 Pages. The original ideas published in reputable journals

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