Some Results on Causal Modalities in General Spacetimes

TL;DR

This paper extends the analysis of causal modal logics from Minkowski spacetime to general smooth spacetimes, showing that the 'after formula' holds universally and exploring the logical complexity of different spacetime dimensions.

Contribution

It proves the 'after formula' applies to all smooth spacetimes and introduces a modal formula distinguishing two-dimensional from higher-dimensional spacetimes.

Findings

The 'after formula' holds in all smooth spacetimes.

Two-dimensional spacetimes have more expressive modal logic.

Logical properties relate to physical causal structures.

Abstract

Causality is one of the fundamental structures of spacetimes, determining the possible behaviour and propagation of physical information. Causal structure can be analysed through the various modal logics it induces. The modal logics for the chronological and causal relations of the archetypal Minkowski spacetime have been classified. However, only partial results have been achieved for the strict variant of the causal relation, known as the after relation. Towards classification, it was shown by Shapirovsky and Shehtman that the after modality in Minkowski space satisfies a formula we call the 'after formula'. The present work continues this analysis towards arbitrary spacetimes. In particular, we prove that the after modality in any smooth spacetime satisfies the after formula. We introduce a related modal formula that demonstrates that the logic of two-dimensional spacetimes are more expressive than higher-dimensional ones. Lastly, we study the interrelation between the logical properties and physical properties along the causal ladder.