Families of spectral triples and foliations of space(time)

TL;DR

This paper extends the concept of spectral triples to model foliations of spacetime in both Riemannian and Lorentzian signatures, introducing new structures called product and Lorentzian spectral triples.

Contribution

It generalizes the classical Dirac operator reconstruction to noncommutative geometry, defining new spectral triples for Lorentzian spacetimes within the Krein space framework.

Findings

Reconstruction of Dirac operators from hypersurface families in classical case

Introduction of product spectral triples for Riemannian signature

Development of Lorentzian spectral triples as reverse Wick rotations

Abstract

We study a noncommutative analogue of a spacetime foliated by spacelike hypersurfaces, in both Riemannian and Lorentzian signatures. First, in the classical commutative case, we show that the canonical Dirac operator on the total spacetime can be reconstructed from the family of Dirac operators on the hypersurfaces. Second, in the noncommutative case, the same construction continues to make sense for an abstract family of spectral triples. In the case of Riemannian signature, we prove that the construction yields in fact a spectral triple, which we call a product spectral triple. In the case of Lorentzian signature, we correspondingly obtain a 'Lorentzian spectral triple', which can also be viewed as the 'reverse Wick rotation' of a product spectral triple. This construction of 'Lorentzian spectral triples' fits well into the Krein space approach to noncommutative Lorentzian geometry.