Two-body problem. Part II: Einstein's original relativistic model
Keywords:
Gravitational force. Two-body problem. Perihelion precession of Mercury. General Theory of Relativity.Abstract
In this paper, we present the second part of a long work on the classical two-body problem for an interaction represented by a force that acts along the straight-line joining them, a work that has begun, in its part I, with an analysis of the orbital motion found with Newton's gravitational force (1687), including the complete and detailed mathematical resolution, with the exact analytic solution, in full agreement with Kepler's law. Here in this second part, we present the unpublished translation to Portuguese of the original paper by Einstein (in German), published in November 1915, in which he applies the General Theory of Relativity (GTR) to find an approximate solution for the gravitational two-body-problem, finding elliptic orbits with a rate of perihelion's precession in very good agreement to the value observed for planet Mercury ($43$'' by century), but, as he himself admits at the end of the paper, it is not in agreement with the values observed for the other planets of the solar system. That will allow us, at the third part of this work (to be published in this same journal), to compare the relativistic model with the exact solution furnished by Relational Mechanics (1989). We are intending, with this long work, to show that, though Einstein's GTR and Relational Mechanics yield the same rate of perihelion's precession for Mercury, the equations of motion are not identical. We shall also show that the advance of perihelion predicted by Relational Mechanics takes place with respect to the fixed stars, just as it has been observed and measured by astronomers, which does not occur in relativistic models.
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