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 $\hat{I}˜$ satisfies $\hat{I}{˜}^{\hat{I}¼\hat{I}½}{\nabla }_{\hat{I}}½\hat{I}{˜}^{\ddot{I}\ddot{I}ƒ}=0$, 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.