Dynamical evolution of a scalar field coupling to Einstein’s tensor in the Reissner-Nordström black hole spacetime

Abstract

We study the dynamical evolution of a scalar field coupling to Einstein's tensor in the background of a Reissner-Nordstr\"om black hole. Our results show that the coupling constant $\ensuremath{\eta}$ imprints in the wave dynamics of a scalar perturbation. In the weak coupling, we find that with the increase of the coupling constant $\ensuremath{\eta}$ the real parts of the fundamental quasinormal frequencies decrease and the absolute values of imaginary parts increase for fixed charge $q$ and multipole number $l$. In the strong coupling, we find that for $l\ensuremath{\ne}0$ the instability occurs when $\ensuremath{\eta}$ is larger than a certain threshold value ${\ensuremath{\eta}}_{c}$ which deceases with the multipole number $l$ and charge $q$. However, for the lowest $l=0$, we find that there does not exist such a threshold value and the scalar field always decays for arbitrary coupling constant.