Strong lensing by GUP-improved black holes

TL;DR

The paper investigates quantum-gravity corrections to black holes by constructing a regular, asymptotically flat spacetime metric from an improved GUP framework and then applying Bozza's strong-field lensing method to compute light deflection near the photon sphere. The authors derive the full spacetime metric using a momentum-dependent deformation (bar-$\beta$) scheme, identify a horizon shift $R_H=R_s\sqrt{1-| ilde{Q}_b|/R_s^2}$, and show the solution is non-singular at $r\to0$. By computing the strong-lensing coefficients $c_1$ and $c_2$ and the observables $\theta_\infty$ and $s$ for M87$^*$ and Sgr A$^*$, they obtain the first observational bound on $| ilde{Q}_b|$, namely $0\leq | ilde{Q}_b|\leq 0.3$, while $| ilde{Q}_c|$ remains unconstrained. The results demonstrate that strong lensing in the near-photon-sphere regime is sensitive to GUP-induced corrections and can constrain quantum-gravity motivated parameters, linking horizon structure and shadow properties to observable lensing signatures. This provides a bridge between quantum gravity phenomenology and high-resolution black-hole imaging, with implications for future tests using rotating black holes and improved observational data.

Abstract

We employ Bozza's method to calculate the deflection angle of light in the presence of the strong gravitational field generated by an improved Schwarzschild-like black hole whose metric, regular throughout the entire spacetime, was derived using the improved Generalized Uncertainty Principle (GUP). This framework incorporates effective quantum gravity corrections that resolve the physical singularity inside the black hole, quantified by a model parameter $\vert \tilde{Q}_{c}\vert$. In addition, the event horizon, the photon sphere, and the shadow radius receive modifications characterized by a second model parameter $\tilde{Q}_{b}$. Using observational properties of the supermassive black holes Messier~87$\ast$ and Sagittarius~A$\ast$ reported by the Event Horizon Telescope, we derive constraints on the parameter $\vert \tilde{Q}_{b}\vert$, namely $0 \leq \vert \tilde{Q}_{b}\vert \leq 0.3$. To the best of our knowledge, these are the first constraints reported in the literature for this improved GUP parameter. Since $\vert \tilde{Q}_{c}\vert$ does not play a significant role in the correction of the shadow radius, it was not possible to impose restrictions on its allowed values. However, it is important to consider a non-zero $\vert \tilde{Q}_{c}\vert$ in order to avoid a black hole singularity.

Figures (5)

• Figure 1: Plots of Ricci and Kretschmann scalars corresponding to the improved metric, whose components are given by Eqs. \ref{['eq:Improv_goo_compont']}--\ref{['eq:Improv_g22_g33_compont']}, as well as the Kretschmann scalar corresponding to the usual Schwarzschild metric, denoted by $K_{class}.$ We note that both scalars corresponding to the improved metric reach a finite value at $r=0$, while the classical Kretschmann scalar is divergent at $r=0$; the classical Ricci scalar is zero. The vertical dashed line indicates the position of the modified horizon which is determined by $R_{H}=\sqrt{R_{s}^2-|\tilde{Q}_b|}$. It is worth mentioning that the Ricci scalar has units of $[\text{Length}]^{-2}$, while the Kretschmann scalar has units of $[\text{Length}]^{-4}$. For all curves we used $|\tilde{Q}_b|=0.3$, $|\tilde{Q}_c|=3$ and $R_{s}=1$.
• Figure 2: Schematic diagram for the deflection of light by a black hole. Photons approaching with the critical impact parameter $b_c$ enter a circular unstable orbit with radius $x_m$, the radius of the photon sphere.
• Figure 3: Parameters $c_1$ and $c_2$ that determine the strong deflection limit in Bozza's approximation, for $0\leq|\tilde{Q}_b|\leq0.5$, and $|\tilde{Q}_c|=10^{-1}$.
• Figure 4: Critical impact parameter, $\tilde{b}_c = b_c/R_s$, divided by the horizon of each solution, for $0\leq|\tilde{Q}_b|\leq0.5$.
• Figure 5: Deflection angle as a function of $x_{0} = r_0/{2M}$. The dashed line represents the effective black hole, while the solid purple line corresponds to the exact Schwarzschild solution. As $|\tilde{Q}_{b}|$ increases, the deflection angle $\alpha$ deviates from its Schwarzschild value, gradually moves further to the left.