Space-time distributions

TL;DR

This paper explores the structure of space-time foliations compatible with gravitational fields, focusing on their geometric properties, causality conditions, and implications for decomposing space-time into spacelike surfaces.

Contribution

It provides a detailed analysis of space-time foliations, causal structures, and their relation to gravitational fields within a 4-manifold framework.

Findings

Space-time foliation determines a fibration with 3D spacelike leaves.

Stable causality corresponds to a foliation generated by an exact timelike 1-form.

Decomposition allows representing space-time as a union of spacelike 3-surfaces.

Abstract

The space-time foliation Sigma compatible with the gravitational field g on a 4-manifold M determines a fibration pi of M, pi : M -> N is a surjective submersion over the 1-dimensional leaves space N. M is then written as a disjoint union of the leaves of Sigma, which are 3-dimensional spacelike surfaces on M. The decomposition, TM=Sigma + T^0 M, also implies that we can define a lift of the curves on N to curves (non-spacelike) on M. The stable causality condition M coincides with Sigma being a causal space-time distribution, generated by an exact timelike 1-form omega^0 = dt where t is some real function on M. In this case M is written as a disjoint union of a family of spacelike 3-surfaces of constant t, which cover D^+(S) of a initial 3-surface S of M.