Stability between foliations in general relativity

TL;DR

This paper introduces a new concept of stability between foliations in general relativity, characterizes regular stability via the Riemann curvature tensor, and explores their existence in various space-times.

Contribution

It defines a novel stability concept for foliations, provides a curvature-based characterization, and investigates their presence in specific space-times like Schwarzschild, Robertson-Walker, and pp-waves.

Findings

No regularly self-stable foliations of dimension >1 in Schwarzschild space-time.

No such foliations in Robertson-Walker space-times.

Existence of regularly self-stable foliations in pp-wave space-times.

Abstract

The aim of this paper is to study foliations that remain invariant by parallel transports along the integral curves of vector fields of another foliations. According to this idea, we define a new concept of stability between foliations. A particular case of stability (called regular stability) is studied, giving a useful characterization in terms of the Riemann curvature tensor. This characterization allows us to prove that there are no regularly self-stable foliations of dimension greater than 1 in Schwarzschild and Robertson-Walker space-times. Finally, we study the existence of regularly self-stable foliations in other space-times, like $pp$-wave space-times.