Fuzzy and discrete black hole models* *The first author was partially supported by CONACyT (México).

J N Argota-Quiroz,S Majid

doi:10.1088/1361-6382/abfea6

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 , 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 . We also study the 3D FLRW model on with S 2 an expanding fuzzy sphere and find that the Friedmann equation for the expansion is the classical 4D one for a closed Universe.

Highlights

The idea that quantum phase spaces but spacetime coordinates themselves could be noncommutative or ‘quantum’ due to quantum gravity effects has been around since the first days of quantum theory
Several flat quantum spacetimes were studied in the 1990s[20, 38, 28], but only recently has there emerged a constructive formalism of quantum Riemannian geometry[12] to more develop curved models[11, 36, 37, 39, 3, 29]
We have solved for the quantum Levi-Civita connection and found the quantum geometry for quantum metrics with each sphere at radius r, t replaced by a fuzzy sphere Cλ[S2]

Summary

Introduction

The idea that quantum phase spaces but spacetime coordinates themselves could be noncommutative or ‘quantum’ due to quantum gravity effects has been around since the first days of quantum theory. The novel feature will be to replace e−ımt by an exact reference solution of the Klein Gordon equation, and we explain first how this looks for a classical Schwarzschild black hole This appears to be rather different from well-known methods of quantum field theory on a curved background [14, 43, 42] but fits with the general idea of [10, 13] that a quantum geodesic flow is a Schroedinger-like evolution.

Recap of the fuzzy sphere and its classical limit

Expanding fuzzy sphere FLRW model

Black hole with the fuzzy sphere

Black hole with the discrete circle

Concluding remarks