Almost commutative Riemannian geometry: wave operators

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.