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 $(Ω^1,\extd)$ 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 $(Ω^2,\extd)$ and a natural noncommutative torsion free connection $(\nabla ,\sigma )$ on ${\Omega }^1$. We show that its generalised braiding $σ:Ω^1\tensΩ^1\to Ω^1\tensΩ^1$ 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 $\tau$ then the cross product algebra $C(M){⋊}_{\tau }\mathbb{R}$ viewed as a noncommutative analogue of $M\times \mathbb{R}$ has a natural $n+2$-dimensional calculus extending ${\Omega }^1$ and a natural spacetime Laplacian now directly defined by the extra dimension. The case $M={\mathbb{R}}^3$ 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.

39 pages, 4 pdf images. Removed previous Sections 5.1-5.2 to a separate paper (now ArXived) to meet referee length requirements. Corresponding slight restructure but no change to remaining content

References in corpus (2)

• Noncommutative Harmonic Analysis, Sampling Theory and the Duflo Map in 2+1 Quantum Gravity
• Nonassociative Riemannian Geometry by Twisting

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