On the local integrability of almost-product structures defined by space-time metrics

TL;DR

This paper investigates the conditions under which space-time metrics can be locally expressed in a time-orthogonal form, linking the integrability of the associated Pfaff equation to the local structure of space-time solutions in general relativity.

Contribution

It analyzes the local integrability of almost-product structures in space-time metrics, connecting geometric properties to the physical solutions of Einstein's equations.

Findings

Many known exact solutions are completely integrable.

Some physically interesting solutions are not integrable.

Integrability depends on the Pfaff equation associated with the metric.

Abstract

The splitting of the tangent bundle of space-time into temporal and spatial sub-bundles defines an almost-product structure. In particular, any space-time metric can be locally expressed in time-orthogonal form, in such a way that whether or not that almost-product structure is locally generated by a coordinate chart is a matter of the integrability of the Pfaff equation that the temporal 1-form of that expression for the metric defines. When one applies that analysis to the known exact solutions to the Einstein field equations, one finds that many of the common ones are completely-integrable, although some of the physically-interesting ones are not.