Theories of Space and Time Compatible with the Inertia Principle

TL;DR

This paper develops a formal framework for theories of space and time that align with the inertia principle, defining inertial frames, transformation relations, and presenting examples of such theories.

Contribution

It introduces a formal definition of inertial frames and transformation coefficients, establishing cocycle relations and a general rest frame map to construct compatible theories.

Findings

Defined inertial frames and their equivalence

Established cocycle relations among transformation coefficients

Presented four example theories of space and time

Abstract

A general formal definition of a theory of space and time compatible with the inertia principle is given. The formal definition of reference frame and inertial equivalence between reference frames are used to construct the class of inertial frames. Then, suitable cocycle relations among the coefficients of space-time transformations between inertial frames are established. The kinematical meaning of coefficients and their reciprocity properties are discussed in some detail. Finally, a rest frame map family is introduced as the most general constitutive assumption to obtain the coefficients and to define a theory of space and time. Four meaningful examples are then presented.