Causal monotonicity, omniscient foliations and the shape of space

TL;DR

This paper explores methods to define the shape of space in a spacetime using edgeless spacelike submanifolds and foliations by timelike curves, establishing conditions for their equivalence and meaningfulness.

Contribution

It introduces conditions under which spacelike submanifolds and foliations yield consistent and meaningful notions of space in a spacetime, linking local and homotopy properties.

Findings

Conditions for edgeless spacelike submanifolds to define space

Criteria for foliations to produce a valid shape of space

Relationship between local spacelike behavior and global space topology

Abstract

What is the shape of space in a spacetime? One way of addressing this issue is to consider edgeless spacelike submanifolds of the spacetime. An alternative is to foliate the spacetime by timelike curves and consider the quotient obtained by identifying points on the same timelike curve. In this article we investigate each of these notions and obtain conditions such that it yields a meaningful shape of space. We also consider the relationship between these two notions and find conditions for the quotient space to be diffeomorphic to any edgeless spacelike hypersurface. In particular, we find conditions in which merely local behavior (being spacelike) combined with the correct behavior on the homotopy level guarantees that a putative shape of space really is precisely that.