A modern view of the classical Herglotz-Noether theorem

TL;DR

This paper provides a new, dimension-independent proof of the classical Herglotz-Noether theorem, showing that certain rigid flows in Minkowski spacetime are generated by isometries, within a broader framework of structure-preserving submersions.

Contribution

It introduces a novel, general proof of the Herglotz-Noether theorem applicable in all dimensions and illustrates a framework for structure-preserving submersions.

Findings

All rotational shear-free and expansion-free flows are generated by Killing vector fields.

The proof is valid for all spacetime dimensions.

Framework for structure-preserving submersions is outlined.

Abstract

In this paper we give a new proof, valid for all dimensions, of the classical Herglotz-Noether theorem that all rotational shear-free and expansion-free flows (rotational Born-rigid flows) in Minkowski spacetime are generated by Killing vector fields (isometric flows). This is aimed as an illustration of a general framework for working with problems that can be described as a structure-preserving submersion, which we will describe in a subsequent paper.