Paper detail

Stability analysis and quasinormal modes of Reissner Nordstrøm Space-time via Lyapunov exponent

We explicitly derive the proper time $(τ)$ principal Lyapunov exponent ($λ_{p}$) and coordinate time ($t$) principal Lyapunov exponent ($λ_{c}$) for Reissner Nordstrøm (RN) black hole (BH) . We also compute their ratio. For RN space-time, it is shown that the ratio is $\frac{λ_{p}}{λ_{c}}=\frac{r_{0}}{\sqrt{r_{0}^2-3Mr_{0}+2Q^2}}$ for time-like circular geodesics and for Schwarzschild BH it is $\frac{λ_{p}}{λ_{c}}=\frac{\sqrt{r_{0}}}{\sqrt{r_{0}-3M}}$. We further show that their ratio $\frac{λ_{p}}{λ_{c}}$ may vary from orbit to orbit. For instance, Schwarzschild BH at innermost stable circular orbit(ISCO), the ratio is $\frac{λ_{p}}{λ_{c}}\mid_{r_{ISCO}=6M}=\sqrt{2}$ and at marginally bound circular orbit (MBCO) the ratio is calculated to be $\frac{λ_{p}}{λ_{c}}\mid_{r_{mb}=4M}=2$. Similarly, for extremal RN BH the ratio at ISCO is $\frac{λ_{p}}{λ_{c}}\mid_{r_{ISCO}=4M}=\frac{2\sqrt{2}}{\sqrt{3}}$. We also further analyse the geodesic stability via this exponent. By evaluating the Lyapunov exponent, it is shown that in the eikonal limit , the real and imaginary parts of the quasi-normal modes of RN BH is given by the frequency and instability time scale of the unstable null circular geodesics.