A framework for structure-preserving submersions and some theorems in relativistic flows

TL;DR

This paper introduces a general framework for structure-preserving submersions, enhancing the understanding of geometric flows and deriving new theorems in relativistic dissipationless and shear-free flows across various spacetime dimensions.

Contribution

It develops a unified, algorithmic approach to structure-preserving submersions and applies it to extend classical theorems in relativistic flow theory to higher dimensions.

Findings

Derived degrees of freedom for Riemannian and Weyl submersions

Generalized the Herglotz-Noether theorem to conformally flat spacetimes in all dimensions

Provided a partial proof of Ellis conjecture across dimensions

Abstract

In this paper we first propose a framework of structure-preserving submersions, which generalises the concept of a Riemannian submersion, and dualises the concept of subgeometry, or "structure-preserving immersions". The emphasis of our approach is on making precise the free variables and the degree of freedom in a given system, thus making the messy calculations in such problems more bearable and, more importantly, algorithmic. In particular, we derive the degrees of freedom of Riemannian submersions and of Weyl submersions. Then we apply our framework to the study of relativistic dissipationless flow and shear-free flows, obtaining generalisations of the classical Herglotz-Noether theorem to conformally flat spacetime in all dimensions and a partial result of Ellis conjecture to all dimensions.