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.
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Communicated by A. Connes
The author was supported by a Leverhulme Senior Research Fellowship.
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Majid, S. Almost Commutative Riemannian Geometry: Wave Operators. Commun. Math. Phys. 310, 569–609 (2012). https://doi.org/10.1007/s00220-012-1416-0
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DOI: https://doi.org/10.1007/s00220-012-1416-0