Strong gravitational lensing by static black holes in effective quantum gravity

Yiyang Wang; Amnish Vachher; Qiang Wu; Tao Zhu; Sushant G. Ghosh

doi:10.1140/epjc/s10052-025-13970-z

2023 Impact factor 4.2

Particles and Fields

Open Access

Eur. Phys. J. C (2025) 85: 302
https://doi.org/10.1140/epjc/s10052-025-13970-z

Regular Article - Theoretical Physics

Strong gravitational lensing by static black holes in effective quantum gravity

Yiyang Wang¹^,2, Amnish Vachher³, Qiang Wu¹^,2^a, Tao Zhu¹^,2 and Sushant G. Ghosh³^,4

¹ Institute for Theoretical Physics and Cosmology, Zhejiang University of Technology, 310023, Hangzhou, China
² United Center for Gravitational Wave Physics (UCGWP), Zhejiang University of Technology, 310032, Hangzhou, China
³ Centre for Theoretical Physics, Jamia Millia Islamia, 110025, New Delhi, India
⁴ Astrophysics and Cosmology Research Unit, School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag 54001, 4000, Durban, South Africa

^a wuq@zjut.edu.cn

Received: 3 October 2024
Accepted: 20 February 2025
Published online: 15 March 2025

Abstract

We investigate strong gravitational lensing by two static black hole models (Model-1 and Model-2) within the Effective Quantum Gravity (EQG) framework, characterized by mass M and parameter $ζ$ $\zeta$ . For $ζ = 0$ $\zeta = 0$ , they reduce to the Schwarzschild solution, and depending on the parameters, they describe black holes with an event and Cauchy horizon (Model-1), a single horizon (Model-2), or no horizons. Using supermassive black holes (SMBHs), Sgr A* and M87*, as lenses and integrating theoretical predictions with recent Event Horizon Telescope (EHT) data, we identify significant differences in lensing signatures due to quantum corrections. For Model-1, the deviations of the lensing observables $| δ θ_{\infty} |$ $|\delta \theta _{\infty }|$ of black holes in EQG from Schwarzschild black hole, for SMBHs Sgr A* and M87*, can reach as much as $1.75 μ$ $1.75~\upmu$ as and $1.32 μ$ $1.32~\upmu$ as, while $| δ s |$ $|\delta s|$ is about 30.12 nas for Sgr A* and 22.63 nas for M87*. The flux ratio of the first image to all subsequent packed images indicates that EQG black hole images are brighter than their Schwarzschild counterparts, with a deviation in the brightness ratio $| δ r_{mag} |$ $|\delta r_{mag}|$ reaching up to 2.02. The time delays between the second and first images, denoted $| δ T_{2, 1} |$ $|\delta T_{2,1}|$ , exhibit substantial deviations from the GR counterpart, reaching up to 1.53 min for Sgr A* and 1159.9 min for M87*. The Event Horizon Telescope (EHT) constraint on $θ_{sh}$ $\theta _{sh}$ of Sgr A* and M87*, within the 1- $σ$ $\sigma$ region, limits the parameter $ζ$ $\zeta$ . Our analysis concludes that EQG black holes are consistent with the EHT observations within a finite parameter space.

© The Author(s) 2025

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