Eur. Phys. J. C (2023) 83: 1014
https://doi.org/10.1140/epjc/s10052-023-12205-3

Regular Article - Theoretical Physics

Strong gravitational lensing by loop quantum gravity motivated rotating black holes and EHT observations

¹ Centre for Theoretical Physics, Jamia Millia Islamia, 110025, New Delhi, India
² Astrophysics and Cosmology Research Unit, School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag 54001, 4000, Durban, South Africa

^b Shafphy@gmail.com

Received: 8 August 2023
Accepted: 28 October 2023
Published online: 8 November 2023

Abstract

We investigate gravitational lensing in the strong deflection regime by loop quantum gravity (LQG)-motivated rotating black hole (LMRBH) metrics with an additional parameter l besides mass M and rotation a. The LMRBH spacetimes are regular everywhere, asymptotically encompassing the Kerr black hole as a particular case and, depending on the parameters, describe black holes with one horizon only (BH-I), black holes with an event horizon and a Cauchy horizon (BH-II), black holes with three horizons (BH-III), or black holes with no horizons (NH) spacetime. It turns out that as the LQG parameter l increases, the unstable photon orbit radius $x_{\mathit{ps}}$, the critical impact parameter $u_{\mathit{ps}}$, the deflection angle ${\alpha }_D(\theta )$ and angular position ${\theta }_{\infty }$ also increases. Meanwhile, the angular separation s decreases, and relative magnitude $r_{\mathit{mag}}$ increases with increasing l for prograde motion but they show opposite behaviour for the retrograde motion. Using supermassive black holes (SMBH) Sgr A* and M87* as lenses, we compare the observable signatures of LMRBH with those of Kerr black holes. For Sgr A*, the angular position ${\theta }_{\infty }$$\in$ (16.4, 39.8) $\mathrm{\mu }$as, while for M87* $\in$ (12.33, 29.9) $\mathrm{\mu }$as. The angular separation s, for SMBHs Sgr A* and M87*, differs significantly, with values ranging $\in$ (0.008–0.376) $\mathrm{\mu }$as for Sgr A* and $\in$ (0.006–0.282) $\mathrm{\mu }$as for M87*. The deviations of the lensing observables $\mathrm{\Delta }{\theta }_{\infty }$ and $\mathrm{\Delta }s$ for LMRBH ($a=0.80,l=2.0$) from Kerr black holes can reach up to $10.22\mathrm{\mu }$as and $0.241\mathrm{\mu }$as for Sgr A*, and $7.683\mathrm{\mu }$as and $0.181\mathrm{\mu }$as for M87*. The relative magnitude $r_{\mathit{mag}}$$\in$ (0.047, 1.54). We estimate the time delay between the first and second relativistic images using twenty supermassive galactic centre black holes as lenses to find, for example, the time delay for Sgr A* and M87* can reach approximately 23.26 min and 33,261.8 min, respectively. Our analysis concludes that, within the $1\sigma$ region, a significant portion of the BH-I and BH-II parameter space agrees with the EHT results of M87* and Sgr A*. The possibility of LMRBH being a BH-III with three horizons has been almost ruled out, except for a small portion of parameter space, by ${\theta }_{\mathit{sh}}$ bounds of Sgr A* and M87* measured by EHT. In contrast, NH without a horizon is completely ruled out. We discover that the EHT results of Sgr A* place more stringent limits on the parameter space of LMRBH black holes than those established by the EHT results of M87*.