Godbillon-Vey Invariants of Non-Lorentzian Spacetimes and Aristotelian Hydrodynamics

TL;DR

This paper explores the geometric properties of non-Lorentzian spacetimes using the Godbillon-Vey class, linking it to fluid dynamics and revealing new conservation laws and obstructions to steady flows.

Contribution

It introduces a novel geometric framework connecting Godbillon-Vey invariants with Aristotelian structures and hydrodynamics, providing new insights into fluid flow obstructions and conservation laws.

Findings

Godbillon-Vey class measures local spin of spatial leaves.

Obstruction to integrability of Aristotelian structures identified.

Conditions for obstructions to steady fluid flow established.

Abstract

We study the geometry of foliated non-Lorentzian spacetimes in terms of the Godbillon-Vey class of the foliation. We relate the intrinsic torsion of a foliated Aristotelian manifold to its Godbillon-Vey class, and interpret it as a measure of the local spin of the spatial leaves in the time direction. With this characterisation, the Godbillon-Vey class is an obstruction to integrability of the $\mathsf{G}$-structure defining the Aristotelian spacetime. We use these notions to formulate a new geometric approach to hydrodynamics of fluid flows by endowing them with Aristotelian structures. We establish conditions under which the Godbillon-Vey class represents an obstruction to steady flow of the fluid and prove new conservation laws.