The shadow and quasinormal modes of the asymptotically flat hairy black holes with a dilaton potential

Abstract In this article, the shadow and the quasinormal modes (QNMs) of an exact asymptotically flat hairy electrically charged black hole solution with a dilaton potential are investigated. Using the constraint equation among the integration constant $$\eta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>η</mml:mi> </mml:math> of the gravitational field, the mass M , the electric charge Q and the coupling constant $$\nu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ν</mml:mi> </mml:math> between the U (1) field and the dilaton field, we find that the shadow radii, the Lyapunov exponent $$\lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> </mml:math> and the coordinate angular velocity $$\Omega _{c}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Ω</mml:mi> <mml:mi>c</mml:mi> </mml:msub> </mml:math> only significantly affected by $$\nu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ν</mml:mi> </mml:math> if the Q is close to the extremal value. Especially, near the $$\nu =1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ν</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> critical point the phenomenon exhibits singular enhancement. Furthermore, the QNMs are numerically computed by using the Hatsuda method and verified with the higher-order WKB approximations with the Padé summation. We find that the QNMs are close to that of the low energy limit of the string theory when $$\nu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ν</mml:mi> </mml:math> is large enough. In the eikonal limit, the real and imaginary parts are proved to be approximated by $$\Omega _{c}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Ω</mml:mi> <mml:mi>c</mml:mi> </mml:msub> </mml:math> and $$\lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> </mml:math> , respectively.