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

Abstract

We explicitly derive the proper time $(\tau )$ principal Lyapunov exponent (${\lambda }_p$) and coordinate time ($t$) principal Lyapunov exponent (${\lambda }_c$) for Reissner Nordstrøm (RN) black hole (BH) . We also compute their ratio. For RN space-time, it is shown that the ratio is $\frac{{\lambda }_p}{{\lambda }_c}=\frac{r_0}{\sqrt{r_0^2-3M{r}_0+2Q^2}}$ 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.

Accepted in Pramana, 07/09/2015

References in corpus (9)

• Inspiral, merger and ringdown of unequal mass black hole binaries: a multipolar analysis
• Circular motion of neutral test particles in Reissner-Nordström spacetime
• Black Hole Mergers and Unstable Circular Orbits
• Lyapunov indices with two nearby trajectories in a curved spacetime
• Relativistic chaos is coordinate invariant
• Free motion around black holes with discs or rings: between integrability and chaos - III
• Lyapunov Exponent and Charged Myers Perry Spacetimes
• Adjusting chaotic indicators to curved spacetimes
• Extremal Limits and Kerr Spacetime

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