Representation of fields associated with any moving point mass by means of fundamental fields corresponding to its trajectory in the frame of Einstein's special theory of relativity

TL;DR

This paper introduces fundamental fields derived from a moving point mass's trajectory in special relativity, showing that any field in the event set can be expressed through these fields, forming a differentiable manifold.

Contribution

It demonstrates that all fields on the event set can be represented as functions of three fundamental fields, establishing a manifold structure diffeomorphic to the event set.

Findings

Fundamental fields form a differentiable manifold.

Any field on the event set can be expressed via fundamental fields.

The manifold is diffeomorphic to the set of events.

Abstract

Assume that in a Lorentzian frame is given a relativistically admissible trajectory of a point mass. An event in such a frame can be described by four coordinates, first three representing the position and the last one the time of the event. Let G denote the set of all events that do not lie on the trajectory. The trajectory uniquely determines on the set G a system of fields called by the author the fundamental fields. The most important are the following three: (1) The retarded time field, representing the time a wave should be emitted from the trajectory to arrive at some point of the set of events G; (2) The delayed time field, representing the difference between the actual time of the event and the retarded time; (3) The unit vector field representing the direction in which the wave should be emitted. In the paper arXiv:0909.5240 the author used the fundamental fields to prove, that the fields of the amended Feynman's Law satisfy the homogeneous system of Maxwell equations, and to obtain explicit formulas for Feynman fields in terms of the fundamental fields. In this note the author proves that any field on the set G of events can be represented as a function of the three fields mentioned above. The joint range of these three fields represents a differentiable manifold M diffeomorphic with the set G of the events. The manifold consists of the Cartesian product of the space R of reals, the space of positive real numbers, and the unit sphere in 3 dimensional Euclidean space.