Stability Analysis of Geodesics and Quasinormal Modes of a Dual Stringy Black Hole Via Lyapunov Exponents

We investigate the stability of both timelike as well as null circular geodesics in the vicinity of a dual (3+1) dimensional stringy black hole (BH) spacetime by using an excellent tool so-called Lyapunov exponent. The proper time ($\tau$) Lyapunov exponent ($\lambda_{p}$) and coordinate time ($t$) Lyapunov exponent~($\lambda_{c}$) are explicitly derived to analyze the stability of equatorial circular geodesics for the stringy BH spacetime with \emph{electric charge} parameter ($\alpha$) and \emph{magnetic charge} parameter~($Q$). By computing these exponents for both the cases of BH spacetime, it is observed that the coordinate time Lyapunov exponent of magnetically charged stringy BH for both timelike and null geodesics are independent of magnetic charge parameter $(Q)$. The variation of the ratio of Lyapunov exponents with radius of timelike circular orbits ($r_{0}/M$) for both the cases of stringy BH are presented. The behavior of instability exponent for null circular geodesics with respect to charge parameters ($\alpha$ and $Q$) are also observed for both the cases of BH. Further, by establishing a relation between quasinormal modes (QNMs) and parameters related to null circular geodesics (like angular frequency and Lyapunov exponent), we deduced the QNMs (or QNM frequencies) for a massless scalar field perturbation around \emph{both} the cases of stringy BH spacetime in the eikonal limit. The variation of scalar field potential with charge parameters and angular momentum of perturbation ($l$) are visually presented and discussed accordingly.