Strong gravitational lensing effects of the rotating short-haired black hole and constraints from EHT observations

Abstract For the short hairs that have a significant impact only near the event horizon, studying their strong gravitational lensing effects is of great significance for revealing the properties of these hairs. In this study, we systematically investigated the strong gravitational lensing effects in the rotating short-haired black hole and constrained its hair parameter $$Q_m$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Q</mml:mi> <mml:mi>m</mml:mi> </mml:msub> </mml:math> . Specifically, $$Q_m$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Q</mml:mi> <mml:mi>m</mml:mi> </mml:msub> </mml:math> causes the event horizon radius, photon – orbit radius, and impact parameter to be lower than those of the Kerr black hole. Regarding 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 the spin parameter $$a$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>a</mml:mi> </mml:math> increases, $$\bar{a}$$ <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> shows an increasing trend, while $$\bar{b}$$ <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> shows a decreasing trend. In the observational simulations of M87* and Sgr A*, the angular position and angular separation of the relativistic image increase with the increase of $$a$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>a</mml:mi> </mml:math> , while the magnification of the image shows an opposite trend. The existence of $$Q_m$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Q</mml:mi> <mml:mi>m</mml:mi> </mml:msub> </mml:math> only intensifies these trends, while parameter k suppresses such tendencies. More importantly, the rotating short-haired black hole exhibits a significant difference in time delay compared to other black hole models. Especially in the simulation of M87*, the time delay deviation between the rotating short-haired black hole and the Kerr black hole, as well as the Kerr–Newman black hole, can reach dozens of hours. Through a comparative analysis with the observational data from the EHT, we effectively constrain the parameter space of the rotating short-haired black hole. The results indicate that this model has potential application prospects in explaining cosmic black hole phenomena and provides a possible theoretical basis for differentiating between different black hole models.