Noncommutative spherically symmetric spacetimes at semiclassical order

TL;DR

This paper develops a semiclassical quantum geometry framework for spherically symmetric spacetimes, revealing constraints on the quantum differential calculus and providing explicit quantizations of cosmological and black hole models.

Contribution

It provides a complete solution for quantum differential calculus, metric, and Levi-Civita connection in spherically symmetric Poisson-Riemannian geometry at order , including new insights into quantum corrections.

Findings

r,t,dr,dt are undeformed at order 

Fuzzy sphere structure emerges at each radius r

Quantum Ricci tensor vanishes for Schwarzschild at order 

Abstract

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 $\lambda$.