Spacetime could be simultaneously continuous and discrete in the same way that information can

TL;DR

This paper explores the idea that spacetime might be both continuous and discrete simultaneously, using mathematical analogies from information theory and spectral geometry to support the concept.

Contribution

It extends previous work on sampling theorems for curved spaces to include Lorentzian manifetime, proposing a unified view of spacetime's structure.

Findings

Spacetime can be sampled and reconstructed up to a cutoff scale.

Methods from spectral geometry apply to Lorentzian manifolds.

Supports the idea of a dual continuous-discrete nature of spacetime.

Abstract

There are competing schools of thought about the question of whether spacetime is fundamentally either continuous or discrete. Here, we consider the possibility that spacetime could be simultaneously continuous and discrete, in the same mathematical way that information can be simultaneously continuous and discrete. The equivalence of continuous and discrete information, which is of key importance in information theory, is established by Shannon sampling theory: of any bandlimited signal it suffices to record discrete samples to be able to perfectly reconstruct it everywhere, if the samples are taken at a rate of at least twice the bandlimit. It is known that physical fields on generic curved spaces obey a sampling theorem if they possess an ultraviolet cutoff. Most recently, methods of spectral geometry have been employed to show that also the very shape of a curved space (i.e., of a Riemannian manifold) can be discretely sampled and then reconstructed up to the cutoff scale. Here, we develop these results further, and we here also consider the generalization to curved spacetimes, i.e., to Lorentzian manifolds.