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 ${\mathbb{Z}}_n$, 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 ${\mathbb{Z}}_n\to S^1$. We also study the 3D FLRW model on $\mathbb{R}\times S^2$ with $S^2$ an expanding fuzzy sphere and find that the Friedmann equation for the expansion is the classical 4D one for a closed $\mathbb{R}\times S^3$ universe.

32 pages AMSlatex, 5 figures. This is a sequel to arXiv:2005.13999 (gr-qc) (which just appeared in Class. Quantum Gravity.)

References in corpus (3)

• Loop quantum cosmology of k=1 FRW models
• Gravitational Decoherence
• Quantum Gravity and Riemannian Geometry on the Fuzzy Sphere

Cited by in corpus (7)

• Quantum geodesic flows and curvature
• Quantum geodesics on $\lambda$-Minkowski spacetime
• Geometric Dirac operator on the fuzzy sphere
• Quantum Kaluza-Klein theory with $M_2(\mathbb{C})$
• Quantum Riemannian geometry of the discrete interval and q-deformation
• Yang-Mills field from fuzzy sphere quantum Kaluza-Klein model
• Fuzzy Schwarzschild (2+1)-spacetime