Quantum Spacetimes from General Relativity?

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 $\mathrm{\Theta }$ satisfies ${\mathrm{\Theta }}^{\mu \nu }{\mathrm{\nabla }}_{\nu }{\mathrm{\Theta }}^{\rho \sigma }=0$, 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.