Transverse stable causality in Lorentzian foliations

Abstract

We introduce and investigate a notion analogous to stable causality for Lorentzian foliations, considerably extending the framework of transverse causality initiated in a recent work. It is well-known that in spacetime geometry stable causality is characterized by several equivalent conditions, such as the stability of non-existence of causal loops under metric perturbations, the existence of smooth temporal functions, and relation-theoretic formulations via the so-called Seifert and $K^+$ relations. We define natural transversal analogues of these distinct formulations and establish partial logical implications between them in the foliation setting. Whether and to what extent the full equivalences can be established remains an open question for general foliations, partly due to the eventual non-Hausdorff nature and other topological complexities of arbitrary leaf spaces. Furthermore, we demonstrate that for the important class of \textit{simple} foliations, which are defined by certain submersions, the equivalences of almost all the transverse analogues of stable causality are indeed recovered.