Observers and Splitting Structures in Relativistic Electrodynamics

TL;DR

This paper develops a coordinate-free, relativistic splitting framework to translate electromagnetic fields from four-dimensional space-time to three-dimensional observer space, providing new insights and classifications without relying on traditional frames.

Contribution

It introduces a minimal, coordinate-free mathematical structure for relativistic splitting, offering a novel classification of observers and applications to classical paradoxes.

Findings

Provides a concise classification of splitting structures

Enables analysis of the Ehrenfest paradox

Facilitates understanding of Schiff's question in GR

Abstract

We introduce a relativistic splitting structure as a means to map fields and equations of electromagnetism from curved four-dimensional space-time to three-dimensional observer's space. We focus on a minimal set of mathematical structures that are directly motivated by the language of the physical theory. Space-time, world-lines, time translation, space platforms, and time synchronization all find their mathematical counterparts. The splitting structure is defined without recourse to coordinates or frames. This is noteworthy since, in much of the prevalent literature, observers are identified with adapted coordinates and frames. Among the benefits of the approach is a concise and insightful classification of splitting structures that is juxtaposed to a classification of observers. The application of the framework to the Ehrenfest paradox and Schiff's "Question in General Relativity" further illustrates the advantages of the framework, enabling a compact, yet profound analysis of the problems at hand.