Axial Gravitational Normal Modes of Uniform Density Star in Anti-de Sitter Spacetime

Abstract

The article studies the dynamical behavior of axial gravitational perturbations of homogeneous stars in Anti-de Sitter spacetime. Because the radial coordinate $r$ transforms into the tortoise coordinate $y = y(r)$, obeying $y(r=0)=0$ and $y(r\rightarrow\infty)=y_\text{max}<\infty$, the tortoise coordinates domain is finite, and the gravitational waves fail to reach out of the domain. Thus, from the perspective of tortoise coordinates, the perturbation equation of uniform density stars in Anti-de Sitter spacetime is analogous to the infinite deep potential well in quantum mechanics. Therefore, the perturbative behavior in this spacetime represents a standing wave vibration. Here we use shooting method and finite difference method to show the imagery of standing waves in this spacetime for $n=0,1,2$ as examples. On the other hand, by finite difference method, a Gaussian wave packet bounces back and forth within this potential well. Furthermore, due to the shape of the gravitational potential function $V$ within the range $0\le y\le y_\text{max}$, an echo phenomenon occurs at the surface of the star. As the energy of the wave packet remains within the potential well, each time it traverses the surface of the star, a reflective echo is generated. Consequently, after multiple reflections, the Gaussian wave packet cannot maintain its original shape and ultimately disperses.