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 $\eta$ of the gravitational field, the mass $M$, the electric charge $Q$ and the coupling constant $\nu$ between the $U(1)$ field and the dilaton field, we find that the shadow radii, the Lyapunov exponent $\lambda$ and the coordinate angular velocity $\Omega_{c}$ only significantly affected by $\nu$ if the $Q$ is close to the extremal value, especially when $\nu$ approaches to one. Furthermore, the QNMs are numerically computed by using the Hatsuda method and verify with the higher-order WKB approximations with the Pad\'e summation. We find that the QNMs are close to that of the low energy limit of the string theory when $\nu$ is large enough. In the eikonal limit, the real and imaginary parts are proved to be given by $\Omega_{c}$ and $\lambda$, respectively.