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

We derive proper-time Lyapunov exponent $(λ_{p})$ and coordinate-time Lyapunov exponent $(λ_{c})$ for a regular Hayward class of black hole. The proper-time corresponds to $τ$ 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 $\frac{λ_{p}}{λ_{c}} = \frac{(r_σ^{3} + 2 l^{2} m )}{\sqrt{(r_σ^{2} + 2 l^{2} m )^{3}- 3 m r_σ^{5}}}$ for time-like geodesics. In the limit of $l=0$ that means for Schwarzschild black hole this ratio reduces to $\frac{λ_{p}}{λ_{c}} = \sqrt{\frac{r_σ}{(r_σ-3 m)}}$. 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.

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