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A background independent notion of causality

Antonio Capolupo, Aniello Quaranta

Abstract

We develop a notion of causal order on a generic manifold as independent of the underlying differential and topological structure. We show that sufficiently regular causal orders can be recovered from a distinguished algebra of sets, which plays a role analogous to that of topologies and $σ$ algebras. We then discuss how a natural notion of measure can be associated to the algebra of causal sets.

A background independent notion of causality

Abstract

We develop a notion of causal order on a generic manifold as independent of the underlying differential and topological structure. We show that sufficiently regular causal orders can be recovered from a distinguished algebra of sets, which plays a role analogous to that of topologies and algebras. We then discuss how a natural notion of measure can be associated to the algebra of causal sets.
Paper Structure (10 sections, 14 theorems, 62 equations)

This paper contains 10 sections, 14 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Causal structure on Manifolds
  3. A side on notation and terminology
  4. Reverse causality
  5. Abstracting the cone structure: $\nabla$ and $\Delta$ sets
  6. The abstract algebra of causal sets
  7. Causal order from the algebra of sets
  8. Causal measure
  9. Horizon Entropy
  10. Conclusions

Key Result

Proposition 2.6

On a manifold satisfying the crossing property, the intersection of two $\Delta$ sets is $\Delta$, the intersection of two $\nabla$ sets is $\nabla$.

Theorems & Definitions (47)

  • Definition 1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • proof
  • Proposition 2.7
  • proof
  • ...and 37 more