Dynamical evolution of a vector field perturbation coupling to Einstein tensor

We have investigated the wave dynamics of a vector field perturbation coupling to Einstein tensor in the four-dimensional Reissner-Nordstr\"{o}m black hole spacetime. Our results show that besides the dependence on the coupling between the vector field and Einstein tensor, the wave dynamic equation of the vector field perturbation strongly depends on the parity of the perturbation itself, which is quite different from that of the usual vector field perturbation without the coupling in the four-dimensional spacetime. Moreover, we also find that the vector field perturbation with odd parity grows with exponential rate if the coupling strength is stronger than certain a critical value. However, the vector field perturbation with even parity always decays in the Reissner-Nordstr\"{o}m black hole spacetime.