Fields generated by a moving relativistic point mass and mathematical correction to Feynman's law

TL;DR

This paper develops a mathematical framework for fields generated by moving relativistic point masses, correcting Feynman's law, and explores solutions to n-body problems under relativistic forces, linking electromagnetic and gravitational theories.

Contribution

The paper introduces fundamental fields derived from Feynman's formula, proves their relation to Maxwell equations, and establishes existence and uniqueness results for relativistic n-body problems.

Findings

Fields coincide with Lienard-Wiechert potentials and satisfy Maxwell equations.

The Bogdan-Feynman Theorem unifies key aspects of relativistic point mass fields.

Existence and uniqueness of solutions for relativistic n-body systems are proven.

Abstract

Analysis of the original Feynman's formula for a moving point charge leads to the notion of a retarded time, which has to be treated as a field. The Lorentzian frame, the trajectory, and the retarded time field uniquely determine a system of fields over the frame, which the author calls the fundamental fields. By means of these fields one can represent and establish relations between the wave, Lorentz gauge, and Maxwell equations, and to prove that electromagnetic field generated by Lienard-Wiechert potentials, and the fields represented by the amended Feynman's formula coincide, and both fields satisfy Maxwell equations and homogenous wave equations. One of the main results is contained in Bogdan-Feynman Theorem for a moving point mass combining all important aspects of the development into a single mathematical theorem. This theorem leads, for instance, to a field generated by a point mass, that in its rest frame represents Newton's gravity field, if neglecting the constant of proportionality. The rest of the paper is devoted to the study of n-body problems for various general relativistic force fields and to prove the existence and uniqueness of solutions. Such problems lead to nonanticipating differential equations, solutions of which depend not only on the initial data, but also on the initial trajectory of the entire system of bodies. To analyze such equations author develops a theory of nonanticipating uniformly Lipschitzian operators on trajectories of such systems.