Simultaneity as an Invariant Equivalence Relation

TL;DR

This paper analyzes simultaneity as an invariant equivalence relation in classical and relativistic physics, providing new proofs and clarifications of existing results, and discusses its implications for the debate on the conventionality of simultaneity.

Contribution

It offers a comprehensive examination of invariant equivalence relations in $ ext{R}^4$ and refines understanding of Malament's theorem and related results.

Findings

Provides alternative proofs and corrections of existing results.

Clarifies the role of simultaneity as an invariant equivalence relation.

Argues against the relevance of this view to the conventionality debate.

Abstract

This paper deals with the concept of simultaneity in classical and relativistic physics as construed in terms of group-invariant equivalence relations. A full examination of Newton, Galilei and Poincar\'e invariant equivalence relations in $\R^4$ is presented, which provides alternative proofs, additions and occasionally corrections of results in the literature, including Malament's theorem and some of its variants. It is argued that the interpretation of simultaneity as an invariant equivalence relation, although interesting for its own sake, does not cut in the debate concerning the conventionality of simultaneity in special relativity.