Late-Time Evolution of the Phantom Scalar Perturbation in the Background of a Spherically Symmetric Static Black Hole

The late-time evolution of the phantom scalar perturbation is investigated in the spacetime of a four-dimensional spherically symmetric static black hole. It is revealed that the asymptotic tail of the phantom scalar field is dominated by the growth behavior t^-(l+3/2)e^μt, which depends on the multipole moment l and the field massμ but is independent of the mass M and charge Q of the black hole. This growth behavior is in strong contrast to the decaying tail of the usual massive scalar perturbation and shows that the external phantom scalar perturbation is unstable in the spherically symmetric static black hole spacetime.