We explicitly derive the proper time (τ) principal Lyapunov exponent
(λp) and coordinate time (t) principal Lyapunov exponent
(λc) 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
λcλp=r02−3Mr0+2Q2r0 for
time-like circular geodesics and for Schwarzschild BH it is
λcλp=r0−3Mr0. We
further show that their ratio λcλp may vary from
orbit to orbit. For instance, Schwarzschild BH at innermost stable circular
orbit(ISCO), the ratio is
λcλp∣rISCO=6M=2 and at marginally
bound circular orbit (MBCO) the ratio is calculated to be
λcλp∣rmb=4M=2. Similarly, for extremal RN
BH the ratio at ISCO is
λcλp∣rISCO=4M=322.
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.Comment: Accepted in Pramana, 07/09/201