Echoes of axial gravitational perturbations in stars of uniform density

This study investigates the echoes in axial gravitational perturbations in compact objects. Accordingly, we propose an alternative scheme of the finite difference method implemented in two coordinate systems, where the initial conditions are placed on the axis of the tortoise coordinate with appropriate boundary conditions that fully respect the causality. The scheme is then employed to study the temporal profiles of the quasinormal oscillations in the Schwarzschild black hole and uniform-density stars. When presented as a two-dimensional evolution profile, the resulting ringdown waveforms in the black hole metric are split into reflected and transmitted waves as the initial perturbations evolve and collide with the peak of the effective potential. Meanwhile, for compact stars, quasinormal oscillations might be characterized by echoes. Consistent with the causality arguments, the phenomenon is produced by the gravitational waves bouncing between the divergent potential at the star's center and the peak of the effective potential. The implications of the present study are also discussed herein.