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

TL;DR

This paper analyzes axial gravitational perturbations of a uniform-density star in Anti-de Sitter spacetime. By reformulating the perturbation in tortoise coordinates, the radial domain becomes finite, causing the perturbations to behave as standing waves in a one-dimensional equivalent potential well. The authors use shooting methods and finite-difference time-domain simulations to obtain real normal-mode frequencies for various angular quantum numbers and densities, demonstrating standing-wave patterns and surface echo phenomena for Gaussian initial data. They also compare with a Pöschl-Teller approximation, noting its limited accuracy due to potential discontinuities, and discuss the implications for AdS/CFT-related physics and numerical efficiency in a finite domain setting.

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.

Figures (11)

• Figure 1: the relation between $P_i(r)$ and $\Lambda$
• Figure 2: The metric of uniform density stars in Anti-de Sitter spacetime, where $P(r)$ and $\rho(r)$ are the pressure and the density of the star respectively, and the vertical dashed line indicates the position of the star's surface.
• Figure 3: Axial gravitational potentials and tortoise coordinates of uniform density star in Anti-de Sitter spacetime, where the vertical dashed line indicates the position of the star's surface.
• Figure 4: Axial gravitational normal modes with their respective eigenfunctions and eigenvalues, where the vertical dashed line indicates the position of the star's surface.
• Figure 5: The relationship between the axial gravitational normal mode frequencies $\omega$, the cosmological constant $\Lambda$, and the Schwarzschild radius $r_\text{Sch}$, where $\beta=-\frac{r_\text{Sch}}{r_s}\Lambda r_\text{Sch}$, and Schwarzschild radius $r_\text{Sch}=2G_nM$ satisfies $f_o(r_\text{Sch})=0$, and the vertical dashed line indicates the position of the star's surface.