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arXiv 1205.5656gr-qcastro-ph.CO
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基于李雅普诺夫指数的Reissner Nordstrøm时空的稳定性分析和拟正规模

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

Parthapratim Pradhan

备注：
Accepted in Pramana, 07/09/2015
发表信息：
Pramana, June 2016, 87:5
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摘要：

本文给出了Reissner Nordstrøm（RN）黑洞（BH）的真时$(τ)$主李雅普诺夫指数（$λ_{p}$）和坐标时（$t$）主李雅普诺夫指数（$λ_{c}$）.我们还计算了它们的比值。对于RN时空，它表明，该比率是$\frac{λ_{p}}{λ_{c}}=\frac{r_{0}}{\sqrt{r_{0}^2-3Mr_{0}+2Q^2}}$类时圆测地线和Schwarzschild BH它是$\frac{λ_{p}}{λ_{c}}=\frac{\sqrt{r_{0}}}{\sqrt{r_{0}-3M}}$。我们进一步表明，它们的比例$\frac{λ_{p}}{λ_{c}}$可能会有所不同，从轨道到轨道。例如，Schwarzschild BH在最内侧稳定圆轨道（ISCO）上的比值为$\frac{λ_{p}}{λ_{c}}\mid_{r_{ISCO}=6M}=\sqrt{2}$，在边缘束缚圆轨道（MBCO）上的比值计算为$\frac{λ_{p}}{λ_{c}}\mid_{r_{mb}=4M}=2$。类似地，对于极值RN BH，ISCO处的比率为$\frac{λ_{p}}{λ_{c}}\mid_{r_{ISCO}=4M}=\frac{2\sqrt{2}}{\sqrt{3}}$。我们还进一步分析了测地稳定性通过这个指数。通过计算李雅普诺夫指数，证明了在程函极限下，RN BH的拟简正模的实部和虚部由不稳定零圆测地线的频率和不稳定时标给出.

英文摘要：

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 $\frac{λ_{p}}{λ_{c}}=\frac{r_{0}}{\sqrt{r_{0}^2-3Mr_{0}+2Q^2}}$ for time-like circular geodesics and for Schwarzschild BH it is $\frac{λ_{p}}{λ_{c}}=\frac{\sqrt{r_{0}}}{\sqrt{r_{0}-3M}}$. We further show that their ratio $\frac{λ_{p}}{λ_{c}}$ may vary from orbit to orbit. For instance, Schwarzschild BH at innermost stable circular orbit(ISCO), the ratio is $\frac{λ_{p}}{λ_{c}}\mid_{r_{ISCO}=6M}=\sqrt{2}$ and at marginally bound circular orbit (MBCO) the ratio is calculated to be $\frac{λ_{p}}{λ_{c}}\mid_{r_{mb}=4M}=2$. Similarly, for extremal RN BH the ratio at ISCO is $\frac{λ_{p}}{λ_{c}}\mid_{r_{ISCO}=4M}=\frac{2\sqrt{2}}{\sqrt{3}}$. 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.

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