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 $$\square f+{\lambda\over 2}\omega^{\alpha\beta}({\rm Ric}^\gamma{}_\alpha-S^\gamma{}_{;\alpha})(\widehat\nabla_\beta d f)_\gamma + O(\lambda^2)$$ in terms of the classical Levi-Civita connection $\widehat\nabla$, 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 $\lambda$.