A noncommutative geometric approach to the quantum structure of spacetime

TL;DR

This paper reviews a noncommutative geometric framework applied to quantum spacetime, including models like quantum Schwarzschild and gravitational waves, aiming to capture Planck-scale spacetime features.

Contribution

It develops a noncommutative Riemannian geometry over Moyal algebras and applies it to construct quantum versions of classical spacetimes.

Findings

Constructed noncommutative models of classical spacetimes

Analyzed quantum Schwarzschild and gravitational wave spacetimes

Provided a framework for understanding quantum spacetime structure

Abstract

Together with collaborators, we introduced a noncommutative Riemannian geometry over Moyal algebras and systematically developed it for noncommutative spaces embedded in higher dimensions in the last few years. The theory was applied to construct a noncommutative version of general relativity, which is expected to capture some essential structural features of spacetime at the Planck scale. Examples of noncommutative spacetimes were investigated in detail. These include quantisations of plane-fronted gravitational waves, quantum Schwarzschild spacetime and Schwarzschild-de Sitter spacetime, and a quantun Tolman spacetime which is relevant to gravitational collapse. Here we briefly review the theory and its application in the study of quantum structure of spacetime.