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

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 $$\eta $$ η of the gravitational field, the mass M, the electric charge Q and the coupling constant $$\nu $$ ν between the U(1) field and the dilaton field, we find that the shadow radii, the Lyapunov exponent $$\lambda $$ λ and the coordinate angular velocity $$\Omega _{c}$$ Ω c only significantly affected by $$\nu $$ ν if the Q is close to the extremal value. Especially, near the $$\nu =1$$ ν = 1 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 $$\nu $$ ν is large enough. In the eikonal limit, the real and imaginary parts are proved to be approximated by $$\Omega _{c}$$ Ω c and $$\lambda $$ λ , respectively.