Exploring the possibility of testing the no-hair theorem with Minkowski-deformed regular hairy black holes via photon rings

Abstract In this paper, we investigate the optical appearance of a regular static spherically symmetric hairy black hole within the context of Minkowski deformation governed by the parameter $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> . This black hole describes a hairy black hole with geometric deformations in the radial and temporal metric components, parameterized by $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> . The optical appearance of the black hole, illuminated by a static thin accretion disk in three toy emission function models, exhibits distinctive shadows and photon rings. Our findings reveal that for static spherically symmetric hairy Minkowski-deformed regular black holes, the event horizon radius $$r_h$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>r</mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:math> , photon sphere radius $$r_{ph}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>r</mml:mi> <mml:mrow> <mml:mi>ph</mml:mi> </mml:mrow> </mml:msub> </mml:math> , critical impact parameter $$b_{ph}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>b</mml:mi> <mml:mrow> <mml:mi>ph</mml:mi> </mml:mrow> </mml:msub> </mml:math> , and innermost stable circular orbit radius $$r_{isco}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>r</mml:mi> <mml:mrow> <mml:mi>isco</mml:mi> </mml:mrow> </mml:msub> </mml:math> all have a positive correlation with $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> . These parameters affect the null geodesic trajectories, shadows, and the optical appearance of the photon rings illuminated by a static thin accretion disk around the black hole. Utilizing this, we obtained the optical appearance of this black hole when observed in the forward direction of the optical and geometric thin accretion disk models. In all three models, the radius of the photon ring images also shows a positive correlation with the parameter $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> . Therefore, under the static spherically symmetric hairy Minkowski-deformed regular black hole, there is no degeneracy of photon rings dependent on the parameter $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> . Theoretically, this allows for the distinction of different spacetime metrics of hairy Minkowski-deformed regular black holes, providing a potential method for testing the no-hair theorem through future observations of black hole photon rings.