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

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 xₚₛ x ps , the critical impact parameter uₚₛ u ps , the deflection angle α D(θ ) α D ( θ ) and angular position θ ∞ θ ∞ also increases. Meanwhile, the angular separation s decreases, and relative magnitude rmag r 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 θ ∞ θ ∞ ∈ ∈ (16.4, 39.8) μ as, while for M87* ∈ ∈ (12.33, 29.9) μ as. The angular separation s , for SMBHs Sgr A* and M87*, differs significantly, with values ranging ∈ ∈ (0.008–0.376) μ as for Sgr A* and ∈ ∈ (0.006–0.282) μ as for M87*. The deviations of the lensing observables Δ θ ∞ Δ θ ∞ and Δ s Δ s for LMRBH ( $$a=0.80,l=2.0$$ a = 0.80 , l = 2.0 ) from Kerr black holes can reach up to 10.22\, 10.22 μ as and 0.241~ 0.241 μ as for Sgr A*, and 7.683~ 7.683 μ as and 0.181~ 0.181 μ as for M87*. The relative magnitude rmag r mag ∈ ∈ (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 M8