Probing Loop Quantum Gravity black holes through gravitational lensing

TL;DR

This paper investigates strong and weak gravitational lensing by charged loop quantum gravity black holes (LQGBHs) within the Borges et al. polymerisation framework, where a symmetric transition surface replaces the RN singularity and a Planck-scale remnant emerges. By deriving photon geodesics and employing Bozza–Tsukamoto formalism, it shows that LQG corrections enhance the weak-field deflection and typically enlarge the strong-field image separation relative to RN, while the photon-sphere radius and critical impact parameter remain RN-like. The authors compute lensing observables for Sgr A* and M87*, finding that angular separations and magnifications predicted by LQG lie within the sensitivity of next-generation VLBI facilities, enabling potential constraints on the polymerisation parameter $\delta_b$ and Barbero-Immirzi parameter $\gamma$. The work highlights a concrete observational pathway to test quantum gravity in the strong-field regime and suggests extensions to rotating LQGBHs and multi-messenger connections with gravitational-wave signatures. These results strengthen the case for using horizon-scale lensing as a probe of quantum gravity.

Abstract

We investigate strong gravitational lensing by a charged loop quantum gravity (LQG) black hole obtained through the polymerisation scheme of Borges \textit{et al.} \cite{Borges:2023fog}. These effective geometries replace the Reissner--Nordström singularity with a symmetric transition surface and admit an extremal, cold remnant determined by the minimal area gap in LQG. In turn, we derive the null geodesic equations, investigate the photon effective potential, and obtain expressions for the photon-sphere radius and critical impact parameter. We compute the weak-field deflection angle and Einstein ring size, highlighting the deviations induced by the polymerisation parameter and the Barbero--Immirzi parameter. In the strong-field regime, we compute the strong deflection coefficients $(\bar{a},\bar{b})$ and evaluate the lensing observables $θ_\infty$, $s$, and $r_{\rm mag}$. Unlike the Reissner--Nordström case, the LQG corrections enhance the deflection angle and increase the angular separation of relativistic images, with deviations growing as the geometry approaches the LQG remnant limit. We further compute the corresponding observables for Sgr~A* and M87*, finding that the quantum-gravity modifications lie within the potential sensitivity of next-generation VLBI facilities. For M87*, the angular separation $s\in(0.05712,0.19123)\,μ\text{as}$, while it is $s\in(0.07595,0.25426)\,μ\text{as}$ for Sgr A*. The relative flux ratio is found to lie in the range, $r_{\rm mag}\in(4.49272,5.96397)$. Our analysis demonstrates that LQG-induced corrections leave characteristic strong and weak-lensing imprints, offering a promising observational pathway to probe quantum gravity using near-future high-resolution observations.

Figures (6)

• Figure 1: The deviation in the weak--field deflection angle, $\delta\alpha_{D}(u) = \alpha_{D}(u) - \alpha_{D}(u)|_{\rm RN}$, for LQGBHs relative to the RN case. The Barbero--Immirzi parameter is fixed to $\gamma = \sqrt{3}/6$. The positive deviation increases with the charge $Q$, reflecting the enhancement of the deflection due to LQG polymerisation effects.
• Figure 2: The angular radius of the Einstein ring $\theta_{E}$ plotted as a function of the charge $Q$ for RN and charged LQGBHs. The red dotted line marks the observed value $\theta^{\rm obs}_{E} = 2.85$ arcsec and the range of y-axis denoting the $1\sigma$ uncertainty. LQG corrections lead to an increase in $\theta_{E}$ with $Q$, in contrast to the decreasing trend in the RN case.
• Figure 3: The strong--field deflection angle $\alpha_{D}(u)$ as a function of the impact parameter $u$ for various values of the electric charge $Q$. Both RN (dashed curves) and charged LQGBHs (solid curves) show divergence at $u = u_{\rm ps}$, but LQGBHs exhibit consistently larger deflection for $u > u_{\rm ps}$. We fix the Barbero--Immirzi parameter to $\gamma = \sqrt{3}/6$.
• Figure 4: The strong lensing observable $s = \theta_{1} - \theta_{\infty}$ is displayed as a function of the charge $Q$ for Sgr A$^{*}$ (left) and M87$^{*}$ (right). Results are shown for RN and charged LQGBHs, with $\gamma = 0.2375$ and $\gamma = \sqrt{3}/6$. LQG corrections enlarge the angular separation $s$, making relativistic images more widely separated than in the RN case.
• Figure 5: The strong--lensing observable $r_{\rm mag} = \mu_{1} / \sum_{n=2}^{\infty} \mu_{n}$ is plotted as a function of the charge $Q$ for RN and charged LQGBHs with $\gamma = 0.2375$ and $\gamma = \sqrt{3}/6$. LQG corrections reduce $r_{\rm mag}$, indicating that the first relativistic image becomes less dominant in brightness compared to higher-order images.