Static Fundamental Solutions of Einstein Equations and Superposition Principle in Relativistic Gravityv

Abstract

We show that Einstein equations are compatible with the presence of massive point particle idealization and find the corresponding two parameter family of solutions. They are complete defined by the bare mechanical mass $M>0$ and the Keplerian mass $m>0$ ($m < M$) of the point source of gravity. The global analytical properties of these solutions in the complex plane define a unique preferable radial variable of the one particle problem. These new solutions are fundamental solutions of the quasi-linear Einstein equations. We introduce and discuss a novel nonlinear superposition principle for solutions of Einstein equations and discover the basic role of the relativistic analog of the Newton gravitational potential. For the relativistic potential we introduce a simple quasi-linear superposition principle as a new physical requirement for the initial conditions for Einstein equations, thus justifying the instant gravistatic case for N particle system. This superposition principle allows us to sketch a new theory of the gravitational mass defect. In it a specific Mach-like principle for the Keplerian mass $m$ is valid, i.e. it depends on the mass distribution in the universe, in contrast to the bare mass $M$, which remains a true constant. Several basic examples both of discrete and of continuous mass distributions are considered.