From Singularities of Fields to Equations of Particles Motion

Abstract

Due to the well known Einstein’s proposition, the non-linearity of the GR field equations allows to derive, from the singularities of the field, the geodesic principle i.e., the equations of motion of massive pointwise particles. In this paper, we illustrate how this construction can be realized explicitly in a simple case of a non-linear scalar field model. For a field singular at one point (a timelike curve in 4D description), we derive the inertial motion law. For a field with two singularities (two disjoint timelike curves), we obtain, in the lowest approximation, the second law of non-relativistic dynamics together with the proper expression of Newton’s law of attraction. The ordinary used method in such type of derivation is the integration over a tube near the singular line. Instead, we are working with the singular terms themselves. We compare the terms of the field equation that have the same order of divergence. The dynamical equation is derived as the relation between the coefficients of two leading singular terms—the agent of inertia and the agent of interaction.