In a distinguishing spacetime the horismos relation generates the causal relation

TL;DR

In a distinguishing spacetime, the paper proves that the horismos relation alone can generate the entire causal relation, highlighting a fundamental link between local and global causal structures.

Contribution

It demonstrates that in distinguishing spacetimes, the horismos relation generates the causal relation, a result that cannot be extended to weaker distinctions.

Findings

The horismos relation generates the causal relation in distinguishing spacetimes.

Such spacetimes are necessarily non-total imprisoning.

The result is sharp and cannot be extended to weaker distinctions.

Abstract

It is proved that in a distinguishing spacetime the horismos relation E^+=J^+\I^+ generates the causal relation J^+. In other words two causally related events are joined by a chain of horismotically related events, or again, the causal relation is the smallest transitive relation containing the horismos relation. The result is sharp in the sense that distinction can not be weakened to future or past distinction. Finally, it is proved that a spacetime in which the horismos relation generates the causal relation is necessarily non-total imprisoning.