Almost commutative Riemannian geometry: wave operators

TL;DR

This paper develops a noncommutative geometric framework for (pseudo)-Riemannian manifolds, constructing wave operators and exploring their properties, with applications to noncommutative black hole models and spacetime structures.

Contribution

It introduces a new noncommutative differential structure on manifolds, linking curvature to algebraic properties and extending classical wave operators to noncommutative settings.

Findings

The braiding obeys braid relations only when the manifold is flat.

The braiding squared equals identity only for Einstein manifolds.

Constructed wave operator on noncommutative Schwarzschild black hole shows finite redshift at the horizon.

Abstract

Associated to any (pseudo)-Riemannian manifold $M$ of dimension $n$ is an $n+1$-dimensional noncommutative differential structure $(\Omega^1,\extd)$ on the manifold, with the extra dimension encoding the classical Laplacian as a noncommutative `vector field'. We use the classical connection, Ricci tensor and Hodge Laplacian to construct $(\Omega^2,\extd)$ and a natural noncommutative torsion free connection $(\nabla,\sigma)$ on $\Omega^1$. We show that its generalised braiding $\sigma:\Omega^1\tens\Omega^1\to \Omega^1\tens\Omega^1$ obeys the quantum Yang-Baxter or braid relations only when the original $M$ is flat, i.e their failure is governed by the Riemann curvature, and that $\sigma^2=\id$ only when $M$ is Einstein. We show that if $M$ has a conformal Killing vector field $\tau$ then the cross product algebra $C(M)\rtimes_\tau\R$ viewed as a noncommutative analogue of $M\times\R$ has a natural $n+2$-dimensional calculus extending $\Omega^1$ and a natural spacetime Laplacian now directly defined by the extra dimension. The case $M=\R^3$ recovers the Majid-Ruegg bicrossproduct flat spacetime model and the wave-operator used in its variable speed of light preduction, but now as an example of a general construction. As an application we construct the wave operator on a noncommutative Schwarzschild black hole and take a first look at its features. It appears that the infinite classical redshift/time dilation factor at the event horizon is made finite.