Stability of the rotating string loops in Kerr spacetime

TL;DR

The paper investigates the stability of circular, rotating relativistic elastic string loops in Kerr spacetime, analyzing equatorial and polar perturbations for arbitrary equations of state. By formulating the string dynamics on a Kerr background, deriving the equilibrium configurations, and solving the linearized perturbation equations, it obtains analytical expressions for the fundamental frequencies $\omega_r$, $\omega_\varphi$, and $\omega_\theta$ that describe loop oscillations. The results reveal that polar perturbations can be unstable for certain values of the Kerr parameter $a$ and transverse speed of sound $s^2$, though stability can be recovered for large radius $R$ and higher azimuthal modes; equatorial perturbations can also exhibit stable regions for the fundamental mode, indicating a nuanced stability landscape in Kerr compared to Schwarzschild. Overall, the work extends Schwarzschild analyses to Kerr, provides explicit frequency formulas, and delineates parameter regimes where rotating string loops remain linearly stable.

Abstract

We study stability of the circular string loops rotating in the equatorial plane of Kerr spacetime against equatorial and polar perturbations. We consider motion of such string loops for different modes in the case of arbitrary equation of state. We also obtain analytical expression for the fundamental frequencies of the string loop oscillations under equatorial and polar perturbations.

Figures (8)

• Figure 1: Equilibrium radius as a function of angular velocity for different values of the transverse speed of sound, $M=1$: left panel $a=0.2$, right panel $a=20$.
• Figure 2: Radial profiles of the fundamental frequency $\omega_r=\omega_\varphi$ for different values of the transverse speed of sound, $M=1$: left panel $a=0.2$, right panel $a=20$.
• Figure 3: Radial profiles of the fundamental frequency $\omega_\theta$ for different values of the transverse speed of sound, $M=1$: left panel $a=0.2$, right panel $a=20$.
• Figure 4: Contour plot of the global minimum of $\text{tanh}\left(\Delta_{\tilde{q}_j}\right)$ as a function of $s^2$ and $a$ for $R=4$, $R=7$ and $R=11$, $M=1$, $k=1$ (yellow - positive values of the global minimum of $\text{tanh}\left(\Delta_{\tilde{q}_j}\right)$, blue - negative ones).
• Figure 5: Contour plot of $\text{tanh}\left(\Delta_{\tilde{q}_1}\right)$ as a function of $s^2$ and $a$ for $R=4$, $R=7$ and $R=11$, $M=1$, $k=1$ (yellow - positive values of the global minimum of $\text{tanh}\left(\Delta_{\tilde{q}_j}\right)$, blue - negative ones).