The lensing effect of quantum-corrected black hole and parameter constraints from EHT observations

Abstract The quantum-corrected black hole model demonstrates significant potential in the study of gravitational lensing effects. By incorporating quantum effects, this model addresses the singularity problem in classical black holes. In this paper, we investigate the impact of the quantum correction parameter on the lensing effect based on the quantum-corrected black hole model. Using the black holes $$M87^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>M</mml:mi><mml:msup><mml:mn>87</mml:mn><mml:mo>∗</mml:mo></mml:msup></mml:mrow></mml:math> and $$Sgr A^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>S</mml:mi><mml:mi>g</mml:mi><mml:mi>r</mml:mi><mml:msup><mml:mi>A</mml:mi><mml:mo>∗</mml:mo></mml:msup></mml:mrow></mml:math> as our subjects, we explore the influence of the quantum correction parameter on angular position, Einstein ring, and time delay. Additionally, we use data from the Event Horizon Telescope observations of black hole shadows to constrain the quantum correction parameter. Our results indicate that the quantum correction parameter significantly affects the lensing coefficients $$\bar{a}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mover><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mo>¯</mml:mo></mml:mrow></mml:mover></mml:math> and $$\bar{b}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mover><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mo>¯</mml:mo></mml:mrow></mml:mover></mml:math> , as well as the Einstein ring. The 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> and brightness ratio S of the relativistic image exhibit significant changes,with deviations on the order of magnitude of $$\sim 1\,\upmu \! \textrm{as}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>∼</mml:mo><mml:mn>1</mml:mn><mml:mspace/><mml:mi>μ</mml:mi><mml:mspace/><mml:mtext>as</mml:mtext></mml:mrow></mml:math> and $$\sim 0.01\,\upmu \! \textrm{as}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>∼</mml:mo><mml:mn>0.01</mml:mn><mml:mspace/><mml:mi>μ</mml:mi><mml:mspace/><mml:mtext>as</mml:mtext></mml:mrow></mml:math> , respectively. The impact of the quantum correction parameter on the time delay $$\Delta T_{21}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>Δ</mml:mi><mml:msub><mml:mi>T</mml:mi><mml:mn>21</mml:mn></mml:msub></mml:mrow></mml:math> is particularly significant in the $$M87^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>M</mml:mi><mml:msup><mml:mn>87</mml:mn><mml:mo>∗</mml:mo></mml:msup></mml:mrow></mml:math> black hole, with deviations reaching up to several tens of hours. Using observational data from the Event Horizon Telescope(EHT) of black hole shadows to constrain the quantum correction parameter, the constraint range under the $$M87^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>M</mml:mi><mml:msup><mml:mn>87</mml:mn><mml:mo>∗</mml:mo></mml:msup></mml:mrow></mml:math> black hole is $$0\le \frac{\alpha }{M^2}\le 1.4087$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>0</mml:mn><mml:mo>≤</mml:mo><mml:mfrac><mml:mi>α</mml:mi><mml:msup><mml:mi>M</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mfrac><mml:mo>≤</mml:mo><mml:mn>1.4087</mml:mn></mml:mrow></mml:math> and the constraint range under the $$Sgr A^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>S</mml:mi><mml:mi>g</mml:mi><mml:mi>r</mml:mi><mml:msup><mml:mi>A</mml:mi><mml:mo>∗</mml:mo></mml:msup></mml:mrow></mml:math> black hole is $$0.9713\le \frac{\alpha }{M^2}\le 1.6715$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>0.9713</mml:mn><mml:mo>≤</mml:mo><mml:mfrac><mml:mi>α</mml:mi><mml:msup><mml:mi>M</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mfrac><mml:mo>≤</mml:mo><mml:mn>1.6715</mml:mn></mml:mrow></mml:math> . Although the current resolution of the EHT limits the observation of subtle differences, future high-resolution telescopes are expected to further distinguish between the quantum-corrected black hole and the Schwarzschild black hole, providing new avenues for exploring quantum gravitational effects.