Relativistic simultaneity and causality

TL;DR

This paper explores two types of relativistic simultaneity—spacelike and lightlike—analyzing their geometric properties and conditions for their existence as foliations in spacetime.

Contribution

It provides a geometric analysis of Landau and horismos submanifolds, establishing conditions for their spacelike or lightlike nature and for forming foliations in spacetime.

Findings

Landau submanifolds can be spacelike under certain conditions

Horismos submanifolds are always lightlike

Conditions for foliations by simultaneity submanifolds are established

Abstract

We analyze two types of relativistic simultaneity associated to an observer: the spacelike simultaneity, given by Landau submanifolds, and the lightlike simultaneity (also known as observed simultaneity), given by past-pointing horismos submanifolds. We study some geometrical conditions to ensure that Landau submanifolds are spacelike and we prove that horismos submanifolds are always lightlike. Finally, we establish some conditions to guarantee the existence of foliations in the space-time whose leaves are these submanifolds of simultaneity generated by an observer.