Some Results on Causal Modalities in General Spacetimes

TL;DR

The paper investigates how causal structure in general smooth spacetimes can be captured with modal logic, extending known Minkowski-space results to arbitrary spacetimes. It centers on the after relation $\alpha$ and proves that the after formula $a\alpha f$ is valid across spacetimes, while introducing a dimension-specific variant $a\alpha_2 f$ that is strictly more expressive in two dimensions. A new causal-ladder property, causally non-totally vicious (cNTV), is defined to separate reflexivity and irreflexivity in $\alpha$ and to assess its impact on modal logics. The work also connects logical properties to physical causal features, showing how modal theories change as one ascends or descends the causal ladder, and demonstrates a distinct separation of 2D spacetimes from higher dimensions via the after formulas and density notions. These results lay groundwork for more expressive logics and multi-time-dimensional analyses of spacetime causality with potential applications in general relativity and foundational physics.

Abstract

Causality is one of the fundamental structures of spacetimes, it determines the possible behaviour and propagation of physical information through different relations. Causal structure can be analysed through the various modal logics it induces. The modal logics for the standard 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, also known as the after relation. The present work continues this analysis towards arbitrary spacetimes. By utilizing the definition of the causal relations through causal paths, we can lift known results about the modal logics of Minkowski spacetime to general spacetimes. In particular, for the after relation, we show that a previously studied formula within the logics of Minkowski spacetime holds in arbitrary spacetimes. 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, a classification of causal structures according to a hierarchy of physically relevant properties.

Figures (12)

• Figure 1: Two-dimensional Minkowski spacetime.
• Figure 2: After relations in $\mathbb{R}^3$.
• Figure 3: The (Simple) Causal Ladder
• Figure 4: Spacetimes for Lemma \ref{['lem:cntvnotchron']}.
• Figure 5: New version of the NTV end of the causal ladder. Again, all implications are strict.

Theorems & Definitions (64)

• Example 2.1: Minkowski spacetime
• Lemma 2.2
• Definition 2.3
• Lemma 2.4: penrose1972TechniquesDifferentialTopologykronheimer1967StructureCausalSpaces
• Proposition 2.5: Push-up Rule penrose1972TechniquesDifferentialTopology
• Remark 2.6
• Definition 2.7
• Example 2.8
• Example 2.9
• Lemma 2.10