Shadows and photon rings of a quantum black hole

TL;DR

This paper analyzes the shadows and photon rings of a quantum black hole model in loop quantum gravity, deriving analytical formulas for the light ring and shadow radius, and comparing shadow images with classical Schwarzschild black holes.

Contribution

It provides the first analytical approximations for light and shadow radii of a quantum black hole in loop quantum gravity and compares shadow images across different illumination models.

Findings

Quantum corrections slightly alter the shadow and photon ring radii.

Shadow images of the quantum black hole can be distinguished from Schwarzschild black holes.

Analytical formulas for deflection angles near the photon ring are derived.

Abstract

Recently, a black hole model in loop quantum gravity has been proposed by Lewandowski, Ma, Yang and Zhang (Phys. Rev. Lett. \textbf{130}, 101501 (2023)). The metric tensor of the quantum black hole (QBH) is a suitably modified Schwarzschild one. In this paper, we calculate the radius of light ring and obtain the linear approximation of it with respect to the quantum correction parameter $\alpha$: $r_{l} \simeq 3 M - \frac{\alpha}{9 M}$. We then assume the QBH is backlit by a large, distant plane of uniform, isotropic emission and calculate the radius of the black hole shadow and its linear approximation: $r_{s} = 3 \sqrt{3} M - \frac{\alpha}{6 \left(\sqrt{3} M\right)}$. We also consider the photon ring structures in the shadow when the impact parameter $b$ of the photon approaches to a critical impact parameter $b_{\textrm{c}}$, and obtain a formula for estimating the deflection angle, which is $\varphi_{\textrm{def}} = - \frac{\sqrt{2}}{\omega r_{l}^2}\log{\left(b - b_c\right) + \widetilde{C}(b)}$. We also numerically plot the images of shadows and photon rings of the QBH in three different illumination models and compare them with that of a Schwarzschild in each model. It is found that we could distinguish the quantum black hole with a Schwarzschild black hole by the shadow images in certain specific illumination model.