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

We study the dynamical evolution of a scalar field coupling to Einstein's tensor in the background of a Reissner-Nordstrom black hole. Our results show that the coupling constant eta imprints in the wave dynamics of a scalar perturbation. In the weak coupling, we find that with the increase of the coupling constant 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 not equal 0 the instability occurs when eta is larger than a certain threshold value 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.