Noncommutative spherically symmetric spacetimes at semiclassical order

Working within the recent formalism of Poisson-Riemannian geometry, we completely solve the case of generic spherically symmetric metric and spherically symmetric Poisson-bracket to find a unique answer for the quantum differential calculus, quantum metric and quantum Levi-Civita connection at semiclassical order O(lambda). Here lambda is the deformation parameter, plausibly the Planck scale. We find that r, t, dr, dt are all forced to be central, i.e. undeformed at order lambda, while for each value of r, t we are forced to have a fuzzy sphere of radius r with a unique differential calculus which is necessarily nonassociative at order lambda(2). We give the spherically symmetric quantisation of the FLRW cosmology in detail and also recover a previous analysis for the Schwarzschild black hole, now showing that the quantum Ricci tensor for the latter vanishes at order lambda. The quantum Laplace-Beltrami operator for spherically symmetric models turns out to be undeformed at order lambda while more generally in Poisson-Riemannian geometry we show that it deforms to rectangle f + lambda/2 omega(alpha beta)(Ric(alpha)(gamma) - S-gamma;alpha)((del) over cap (beta)df)(gamma) + O(lambda(2)) in terms of the classical Levi-Civita connection (del) over cap, the contorsion tensor S, the Poisson-bivector omega and the Ricci curvature of the Poisson-connection that controls the quantum differential structure. The Majid-Ruegg spacetime [x, t] = lambda x with its standard calculus and unique quantum metric provides an example with nontrivial correction to the Laplacian at order lambda.