Slicing, Threading & Parametric Manifolds

TL;DR

This paper unifies the slicing and threading decompositions of spacetime in general relativity, providing a comprehensive framework for metric and connection decomposition in both approaches, including recent developments on parametric manifolds.

Contribution

It introduces a unified treatment of slicing and threading decompositions, extending the understanding of spacetime foliations in general relativity.

Findings

Unified framework for slicing and threading decompositions

Recovery of recent results on parametric manifolds

Enhanced understanding of spacetime foliations

Abstract

We present a unified treatment of the slicing (3+1) and threading (1+3) decompositions of spacetime in terms of foliations. It is well-known how to decompose the metric and connection in the slicing picture; this is at the heart of any initial-value problem in general relativity. We describe here the analogous problem in the threading picture, recovering the recent results of Perjes on parametric manifolds.