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

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 $\frac{{\lambda }_p}{{\lambda }_c}=\frac{(r_{\sigma }^3+2l^{2}m)}{\sqrt{(r_{\sigma }^2+2l^{2}m{)}^3-3m{r}_{\sigma }^5}}$ for time-like geodesics. In the limit of $l=0$ that means for Schwarzschild black hole this ratio reduces to $\frac{{\lambda }_p}{{\lambda }_c}=\sqrt{\frac{r_{\sigma }}{(r_{\sigma }-3m)}}$. 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.

arXiv commentTo appear in Mod.Phys.Lett.A, 19 pages and 6 figures

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