Fuzzy and discrete black hole models

Abstract

Using quantum Riemannian geometry, we solve for a Ricci=0 static spherically-symmetric solution in 4D, with the $S^2$ at each $t,r$ a noncommutative fuzzy sphere, finding a dimension jump with solutions having the time and radial form of a classical 5D Tangherlini black hole. Thus, even a small amount of angular noncommutativity leads to radically different radial behaviour, modifying the Laplacian and the weak gravity limit. We likewise provide a version of a 3D black hole with the $S^1$ at each $t,r$ now a discrete circle $\Bbb Z_n$, with the time and radial form of the inside of a classical 4D Schwarzschild black hole far from the horizon. We study the Laplacian and the classical limit $\Bbb Z_n\to S^1$. We also study the 3D FLRW model on $\Bbb R\times S^2$ with $S^2$ an expanding fuzzy sphere and find that the Friedmann equation for the expansion is the classical 4D one for a closed $\Bbb R\times S^3$ universe.