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

Understand

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.