Newtonian limit of Einsteinian gravity: from dynamics of Solar system to dynamics of stars in spiral galaxies

Abstract

Attempts to merge Einsteinian gravity with Newtonian run into inconsistencies because in Newton's gravity time is absolute and the speed of gravity is infinite. Such an assumption was in a focus of attention of scientists in 19th century interested in finding out if the speed of gravity is infinite. By analogy with electrodynamics, some retarded potentials replacing Newtonian were utilized. By using one of such potentials Gerber correctly calculated the perihelion shift for Mercury in 1902. Subsequent attempts at calculation of bending of light using Gerber-style calculations were not successful. Recently Gin\'e (Chaos, Solitons and Fractals 42, 1893 (2009)) reobtained both the perihelion shift and the bending of light using retarded potential. His equations however are not those obtained by Einstein and his results coincide with those by Einstein only at the level of leading order terms of infinite series expansions. The obtained differential equations of motion are of delay-type. When applied to two-body dynamics, such equations lead to orbital quantization. In this work the Einsteinian approach is used to reproduce this quantization. Numerous arguments justifying the superiority of Einsteinian approach, including uses of the Bertrand spacetimes for description of motion of stars around galactic centers are provided. The developed formalism is tested by calculating the number of allowed stable orbits for planets and those for regular satellites of heavy planets resulting in reasonable agreement with observational data. The paper also discusses possible quantum mechanical nature of rings of heavy planets as well as of rotation curves of stars in spiral galaxies.