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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 Reissner-Nordström black hole. Our results show that the 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 $l\neq0$ the instability occurs when $η$ is larger than a certain threshold value $η_c$ 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.