Strong gravitational lensing by static black holes in effective quantum gravity

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 $$\zeta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ζ</mml:mi> </mml:math> . For $$\zeta = 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ζ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , 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 $$|\delta \theta _{\infty }|$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>δ</mml:mi> </mml:mrow> <mml:msub> <mml:mi>θ</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> <mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> </mml:mrow> </mml:math> of black holes in EQG from Schwarzschild black hole, for SMBHs Sgr A* and M87*, can reach as much as $$1.75~\upmu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1.75</mml:mn> <mml:mspace/> <mml:mi>μ</mml:mi> </mml:mrow> </mml:math> as and $$1.32~\upmu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1.32</mml:mn> <mml:mspace/> <mml:mi>μ</mml:mi> </mml:mrow> </mml:math> as, while $$|\delta s|$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>δ</mml:mi> <mml:mi>s</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> </mml:math> 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 $$|\delta r_{mag}|$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>δ</mml:mi> </mml:mrow> <mml:msub> <mml:mi>r</mml:mi> <mml:mrow> <mml:mi>mag</mml:mi> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> </mml:mrow> </mml:math> reaching up to 2.02. The time delays between the second and first images, denoted $$|\delta T_{2,1}|$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>δ</mml:mi> </mml:mrow> <mml:msub> <mml:mi>T</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , 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 $$\theta _{sh}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>θ</mml:mi> <mml:mrow> <mml:mi>sh</mml:mi> </mml:mrow> </mml:msub> </mml:math> of Sgr A* and M87*, within the 1- $$\sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>σ</mml:mi> </mml:math> region, limits the parameter $$\zeta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ζ</mml:mi> </mml:math> . Our analysis concludes that EQG black holes are consistent with the EHT observations within a finite parameter space.