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 (Ω¹, d) 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 (Ω², d) and a natural noncommutative torsion free connection ${(\nabla,\sigma)}$ on Ω¹. We show that its generalised braiding ${\sigma:\Omega^1\otimes\Omega^1\to \Omega^1\otimes\Omega^1}$ 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 σ ² = id only when M is Einstein. We show that if M has a conformal Killing vector field τ then the cross product algebra ${C(M)\rtimes_\tau\mathbb{R}}$ viewed as a noncommutative analogue of ${M\times\mathbb{R}}$ has a natural n + 2-dimensional calculus extending Ω¹ and a natural spacetime Laplacian now directly defined by the extra dimension. The case ${M=\mathbb{R}^3}$ recovers the Majid-Ruegg bicrossproduct flat spacetime model and the wave-operator used in its variable speed of light prediction, 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.