Stability analysis and quasinormal modes of Reissner Nordstr{\o}m
  Space-time via Lyapunov exponent
  We explicitly derive the proper time $(\tau)$ principal Lyapunov exponent
($\lambda_{p}$) and coordinate time ($t$) principal Lyapunov exponent
($\lambda_{c}$) for Reissner Nordstr{\o}m (RN) black hole (BH) . We also
compute their ratio. For RN space-time, it is shown that the ratio is
$\frac{\lambda_{p}}{\lambda_{c}}=\frac{r_{0}}{\sqrt{r_{0}^2-3Mr_{0}+2Q^2}}$ for
time-like circular geodesics and for Schwarzschild BH it is
$\frac{\lambda_{p}}{\lambda_{c}}=\frac{\sqrt{r_{0}}}{\sqrt{r_{0}-3M}}$. We
further show that their ratio $\frac{\lambda_{p}}{\lambda_{c}}$ may vary from
orbit to orbit. For instance, Schwarzschild BH at innermost stable circular
orbit(ISCO), the ratio is
$\frac{\lambda_{p}}{\lambda_{c}}\mid_{r_{ISCO}=6M}=\sqrt{2}$ and at marginally
bound circular orbit (MBCO) the ratio is calculated to be
$\frac{\lambda_{p}}{\lambda_{c}}\mid_{r_{mb}=4M}=2$. Similarly, for extremal RN
BH the ratio at ISCO is
$\frac{\lambda_{p}}{\lambda_{c}}\mid_{r_{ISCO}=4M}=\frac{2\sqrt{2}}{\sqrt{3}}$.
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.
