Fuzzy and discrete black hole models

TL;DR

This paper explores quantum Riemannian geometry models of black holes with fuzzy and discrete spheres, revealing significant differences from classical solutions and connecting noncommutative geometries to higher-dimensional black hole behaviors.

Contribution

It introduces novel black hole models with fuzzy and discrete spheres, demonstrating how noncommutativity alters classical black hole properties and solutions.

Findings

Fuzzy sphere models lead to a dimension jump resembling 5D black holes.

Discrete circle models replicate 4D Schwarzschild black hole interiors.

Fuzzy sphere cosmology yields Friedmann equations similar to 4D closed universe.

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.