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

Abstract

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 $({\lambda _{p}}/{\lambda _{c}})={r_{0}}/{\sqrt {{r_{0}^{2}}-3Mr_{0}+2Q^{2}}}$ for time-like circular geodesics and for Schwarzschild BH, it is $({\lambda _{p}}/{\lambda _{c}})={\sqrt {r_{0}}}/{\sqrt {r_{0}-3M}}$ . We further show that their ratio λ p /λ c may vary from orbit to orbit. For instance, for Schwarzschild BH at the innermost stable circular orbit (ISCO), the ratio is $({\lambda _{p}}/{\lambda _{c}})|_{r_{\text {ISCO}}=6M}=\sqrt {2}$ and at marginally bound circular orbit (MBCO) the ratio is calculated to be $({\lambda _{p}}/{\lambda _{c}})|_{r_{\mathrm {m}\mathrm {b}}=4M}=2$ . Similarly, for extremal RN BH, the ratio at ISCO is $({\lambda _{p}}/{\lambda _{c}})|_{r_{\text {ISCO}}=4M}={2\sqrt {2}}/{\sqrt {3}}$ . 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 quasinormal modes of RN BH is given by the frequency and instability time-scale of the unstable null circular geodesics.