A Covariant Information-Density Cutoff in Curved Space-Time

TL;DR

This paper extends sampling theory to curved space-time, offering a mathematical framework that links continuous and discrete descriptions at the Planck scale, with implications for quantum gravity and string theory.

Contribution

It introduces a covariant information-density cutoff in curved space-time, bridging continuous and lattice theories in a covariant manner.

Findings

Provides a mathematical tool for space-time analysis at the Planck scale

Establishes equivalence between differentiable and lattice theories in curved space-time

Connects sampling theory with generalized uncertainty relations in quantum gravity

Abstract

In information theory, the link between continuous information and discrete information is established through well-known sampling theorems. Sampling theory explains, for example, how frequency-filtered music signals are reconstructible perfectly from discrete samples. In this Letter, sampling theory is generalized to pseudo-Riemannian manifolds. This provides a new set of mathematical tools for the study of space-time at the Planck scale: theories formulated on a differentiable space-time manifold can be completely equivalent to lattice theories. There is a close connection to generalized uncertainty relations which have appeared in string theory and other studies of quantum gravity.