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

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 $$x_{ps}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>x</mml:mi> <mml:mrow> <mml:mi>ps</mml:mi> </mml:mrow> </mml:msub> </mml:math> , the critical impact parameter $$u_{ps}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>u</mml:mi> <mml:mrow> <mml:mi>ps</mml:mi> </mml:mrow> </mml:msub> </mml:math> , the deflection angle $$\alpha _D(\theta )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>α</mml:mi> <mml:mi>D</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>θ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> and angular position $$\theta _{\infty }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>θ</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> </mml:math> also increases. Meanwhile, the angular separation s decreases, and relative magnitude $$r_{mag}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>r</mml:mi> <mml:mrow> <mml:mi>mag</mml:mi> </mml:mrow> </mml:msub> </mml:math> 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 $$\theta _{\infty }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>θ</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> </mml:math> $$\in $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>∈</mml:mo> </mml:math> (16.4, 39.8) $$\upmu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>μ</mml:mi> </mml:math> as, while for M87* $$\in $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>∈</mml:mo> </mml:math> (12.33, 29.9) $$\upmu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>μ</mml:mi> </mml:math> as. The angular separation s , for SMBHs Sgr A* and M87*, differs significantly, with values ranging $$\in $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>∈</mml:mo> </mml:math> (0.008–0.376) $$\upmu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>μ</mml:mi> </mml:math> as for Sgr A* and $$\in $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>∈</mml:mo> </mml:math> (0.006–0.282) $$\upmu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>μ</mml:mi> </mml:math> as for M87*. The deviations of the lensing observables $$\Delta \theta _{\infty }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:msub> <mml:mi>θ</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> </mml:mrow> </mml:math> and $$\Delta s$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:mi>s</mml:mi> </mml:mrow> </mml:math> for LMRBH ( $$a=0.80,l=2.0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0.80</mml:mn> <mml:mo>,</mml:mo> <mml:mi>l</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2.0</mml:mn> </mml:mrow> </mml:math> ) from Kerr black holes can reach up to $$10.22\,\upmu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>10.22</mml:mn> <mml:mspace/> <mml:mi>μ</mml:mi> </mml:mrow> </mml:math> as and $$0.241~\upmu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0.241</mml:mn> <mml:mspace/> <mml:mi>μ</mml:mi> </mml:mrow> </mml:math> as for Sgr A*, and $$7.683~\upmu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>7.683</mml:mn> <mml:mspace/> <mml:mi>μ</mml:mi> </mml:mrow> </mml:math> as and $$0.181~\upmu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0.181</mml:mn> <mml:mspace/> <mml:mi>μ</mml:mi> </mml:mrow> </mml:math> as for M87*. The relative magnitude $$r_{mag}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>r</mml:mi> <mml:mrow> <mml:mi>mag</mml:mi> </mml:mrow> </mml:msub> </mml:math> $$\in $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>∈</mml:mo> </mml:math> (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

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