Dynamical evolution of phantom scalar perturbation in the background of Schwarzschild black String spacetime

Using Leaver's continue fraction and time domain method, we study the wave dynamics of phantom scalar perturbation in a Schwarzschild black string spacetime. We find that the quasinormal modes contain the imprint from the wavenumber $k$ of the fifth dimension. The late-time behaviors are dominated by the difference between the wavenumber $k$ and the mass $\mu$ of the phantom scalar perturbation. For $k<\mu$, the phantom scalar perturbation in the late-time evolution grows with an exponential rate as in the four-dimensional Schwarzschild black hole spacetime. While, for $k=\mu$, the late-time behavior has the same form as that of the massless scalar field perturbation in the background of a black hole. Furthermore, for $k>\mu$, the late-time evolution of phantom scalar perturbation is dominated by a decaying tail with an oscillation which is consistent with that of the usual massive scalar field. Thus, the Schwarzschild black string is unstable only against the phantom scalar perturbations which satisfy the wavelength $\lambda>2\pi/\mu$. These information can help us know more about the wave dynamics of phantom scalar perturbation and the properties of black string.