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

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 η imprints in the wave dynamics of a scalar perturbation. In the weak coupling, we find that with the increase of the coupling constant η 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≠0 the instability occurs when η is larger than a certain threshold value ηc 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.