Quasinormal modes of the generalized Ayón-Beato–García black hole in scalar-tensor-vector gravity

We obtain the solution of a generalized Ay\'on-Beato--Garc\'{\i}a (ABG) black hole with the nonlinear tensor field ${B}_{\ensuremath{\mu}\ensuremath{\nu}}$ in scalar-tensor-vector gravity (STVG), called the generalized ABG STVG black hole. This black hole is endowed with four parameters, the black hole mass $M$, the parameter $\ensuremath{\alpha}$ associated with STVG theory, and the two parameters $\ensuremath{\lambda}$ and $\ensuremath{\beta}$ that are related to the dipole and quadrupole moments of the nonlinear tensor field, respectively. By analyzing the characteristics of the black hole, we find that the generalized ABG STVG black hole is regular when $\ensuremath{\lambda}\ensuremath{\ge}3$ and $\ensuremath{\beta}\ensuremath{\ge}4$, and we study the effects of the parameters $\ensuremath{\alpha}$, $\ensuremath{\lambda}$, and $\ensuremath{\beta}$ on the black hole horizon. We calculate the quasinormal mode frequencies of the odd parity gravitational perturbation for the generalized ABG STVG black hole by using the sixth order WKB approximation method and simultaneously using the null geodesic method at the eikonal limit. The results show that the increase of the parameters $\ensuremath{\alpha}$ and $\ensuremath{\lambda}$ makes the gravitational waves decay slowly, while the increase of the parameter $\ensuremath{\beta}$ makes the gravitational waves decay fast at first, then slowly. In addition, we verify that the improved correspondence between the real part of quasinormal frequencies at the eikonal limit and the shadow radius is valid for the generalized ABG STVG black hole.