Quasinormal modes and shadows of a new family of Ayón-Beato-García black holes

Abstract

We obtain a type of Ay\'on-Beato-Garc\'{\i}a (ABG) related black hole solutions with five parameters: the mass $m$, the charge $q$, and three dimensionless parameters $\ensuremath{\alpha}$, $\ensuremath{\beta}$ and $\ensuremath{\gamma}$ associated with nonlinear electrodynamics. We find that this type of black hole is regular under the conditions: $\ensuremath{\alpha}\ensuremath{\gamma}\ensuremath{\ge}6$, $\ensuremath{\beta}\ensuremath{\gamma}\ensuremath{\ge}8$, and $\ensuremath{\gamma}>0$. Here we focus on the saturated case: $\ensuremath{\alpha}=6/\ensuremath{\gamma}$ and $\ensuremath{\beta}=8/\ensuremath{\gamma}$, such that only three parameters $m$, $q$ and $\ensuremath{\gamma}$ remain, which leads to a new family of ABG black holes. For such a family of black holes, we investigate the influence of the charge $q$ and the parameter $\ensuremath{\gamma}$ on the horizon radius and the Hawking temperature. In addition, we calculate the quasinormal mode frequencies of massless scalar field perturbations by using the sixth-order WKB approximation method and the unstable circular null geodesic method in the eikonal limit. On the one hand, our results show that the increase of the charge $q$ makes the scalar waves decay faster at first and then slowly except for the case of $\ensuremath{\gamma}=2$. On the other hand, they show that the increase of the parameter $\ensuremath{\gamma}$ makes the scalar waves decay at first sharply and then slowly. In particular, $\ensuremath{\gamma}=1$ can be regarded as the critical value for the transition from an unstable configuration to a stable one. Finally, we compute the shadow radius for the new family of ABG black holes and use the shadow data of the $M8{7}^{*}$ black hole detected by the Event Horizon Telescope to provide an upper limit on the charge $q$ of the new black holes. We find that the increase of the charge $q$ makes the shadow radius decrease monotonically, while the increase of the parameter $\ensuremath{\gamma}$ makes the shadow radius increase at first rapidly and then almost remain unchanged, especially the parameter $\ensuremath{\gamma}$ has a significant impact on the shadow radius when it is less than one. Moreover, using the shadow data of the $M8{7}^{*}$ black hole, we find that the upper limit of the charge $q$ increases rapidly at first and then slowly but does not exceed the mass of the $M8{7}^{*}$ black hole at last when the parameter $\ensuremath{\gamma}$ is increasing and going to infinity, and that the data restrict the frequency range of the fundamental mode with $l=1$ to $1.4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}\text{ }\text{ }\mathrm{Hz}\ensuremath{\sim}1.9\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}\text{ }\text{ }\mathrm{Hz}$.

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