Quantum Spacetimes from General Relativity?

Abstract

We introduce a non-commutative product for curved spacetimes, that can be regarded as a generalization of the Rieffel (or Moyal-Weyl) product. This product employs the exponential map and a Poisson tensor, and the deformed product maintains associativity under the condition that the Poisson tensor $\Theta$ satisfies ${\Theta }^{\mu \nu }{\nabla }_{\nu }{\Theta }^{\rho \sigma }=0$, in relation to a Levi-Cevita connection. We proceed to solve the associativity condition for various physical spacetimes, uncovering non-commutative structures with compelling properties.

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