Causality in noncommutative two-sheeted space-times

TL;DR

This paper explores the causal relationships in noncommutative two-sheeted space-times using Lorentzian spectral triples, revealing inter-sheet causality and the effects of Dirac operator fluctuations.

Contribution

It introduces a method to analyze causality in noncommutative geometries of two-sheeted space-times, including the impact of Dirac operator fluctuations.

Findings

Causal relations can exist between the two sheets.

The framework applies to 2- and 4-dimensional globally hyperbolic spin manifolds.

Fluctuations of the Dirac operator induce a point-dependent distance between sheets.

Abstract

We investigate the causal structure of two-sheeted space-times using the tools of Lorentzian spectral triples. We show that the noncommutative geometry of these spaces allows for causal relations between the two sheets. The computation is given in details when the sheet is a 2- or 4-dimensional globally hyperbolic spin manifold. The conclusions are then generalised to a point-dependent distance between the two sheets resulting from the fluctuations of the Dirac operator.