1205.5656 (Parthapratim Pradhan)

Lyapunov Exponent and Reissner Nordstrøm Black Hole    [PDF]

Parthapratim Pradhan

We explicitly derive the principal Lyapunov Exponent \emph{in terms of the radial equation of ISCO}(Innermost Stable Circular Orbit) for Spherically symmetry (Schwarzschild, Reissner Nordstr{\o}m) black-hole space-times. Using it, we show that the ISCO occurs at $r_{ISCO}=4M$ for extremal Reissner Nordstr{\o}m black-hole and $r_{ISCO}=6M$ for Schwarzschild black-hole. We elucidate the connection between Lyapunov Exponent and \emph{Geodesic Deviation Equation}. We also compute the \emph{Kolmogorov-Sinai(KS)} entropy which measures the rate of exponential divergence between two trajectories(geodesics)via Lyapunov Exponent. We further prove that ISCO is characterized by the \emph{greatest} possible orbital period i.e. $T_{ISCO}>T_{photon}$ among all types of circular geodesics(both time-like and null, geodesic and non-geodesic) as measured by the asymptotic observers. Therefore, ISCO provide the \emph{slowest way} to circle the black hole.

View original: http://arxiv.org/abs/1205.5656