Geodesic stability and Quasi normal modes via Lyapunov exponent for Hayward Black Hole

TL;DR

This paper derives Lyapunov exponents for Hayward black holes to analyze geodesic stability, relates them to quasi-normal modes, and discusses photon spheres and shadows, extending stability analysis beyond Schwarzschild black holes.

Contribution

It introduces proper-time and coordinate-time Lyapunov exponents for Hayward black holes and connects these to quasi-normal modes and shadow properties, providing new insights into black hole stability.

Findings

Derived Lyapunov exponents for Hayward black holes.

Linked Lyapunov exponents to quasi-normal modes.

Analyzed photon sphere and shadow radius.

Abstract

We derive proper-time Lyapunov exponent $(\lambda_{p})$ and coordinate-time Lyapunov exponent $(\lambda_{c})$ for a regular Hayward class of black hole. The proper-time corresponds to $\tau$ and the coordinate time corresponds to $t$. Where $t$ is measured by the asymptotic observers both for for Hayward black hole and for special case of Schwarzschild black hole. We compute their ratio as $\frac{\lambda_{p}}{\lambda_{c}} = \frac{(r_{\sigma}^{3} + 2 l^{2} m )}{\sqrt{(r_{\sigma}^{2} + 2 l^{2} m )^{3}- 3 m r_{\sigma}^{5}}}$ for time-like geodesics. In the limit of $l=0$ that means for Schwarzschild black hole this ratio reduces to $\frac{\lambda_{p}}{\lambda_{c}} = \sqrt{\frac{r_{\sigma}}{(r_{\sigma}-3 m)}}$. Using Lyponuov exponent, we investigate the stability and instability of equatorial circular geodesics. By evaluating the Lyapunov exponent, which is the inverse of the instability time-scale, we show that, in the eikonal limit, the real and imaginary parts of quasi-normal modes~(QNMs) is specified by the frequency and instability time scale of the null circular geodesics. Furthermore, we discuss the unstable photon sphere and radius of shadow for this class of black hole.