From Lobachevsky-Beltrami-Fok space to Schwinger gravity and cosmology

The book is introduction to the Poincar´e model of the Lobachevsky geometry and beyond. This geometry is in relativity interpreted as the Lobachevsky-Beltrami-Fok velocity space of moving particles. It involves suprisingly the decay of the neutral π-meson and all decays of elementary particles. The generalization is performed and new angle of parallelism is derived. The nonlinear transformations between coordinate systems involves the maximal acceleration as an analogue of the maximal velocity in special relativity. The crucial consequence of it is the Thomas precession by acceleration. Quantum gravity is given in the Schwinger form and it enables to define the gravitational propagator with rative corrections. The application to the Binary is original. The power spectral formula of the Cherenkov radiation of gravitons at zero and finite temperature is original. The non-relativistic quantum energy shift of H-atom electrons due to Gibbons-Hawking thermal bath is presented. The space-time metric generated by its deformation is the new original idea of author.