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arXiv 2008.11022gr-qc
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基于李雅普诺夫指数的Hayward黑洞的测地稳定性和准正规模

Geodesic stability and Quasi normal modes via Lyapunov exponent for Hayward Black Hole

Monimala Mondal, Parthapratim Pradhan, Farook Rahaman, Indrani Karar

备注：
To appear in Mod.Phys.Lett.A, 19 pages and 6 figures
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摘要：

我们导出了正则Hayward类黑洞的本征时间李雅普诺夫指数$(λ_{p})$和坐标时间李雅普诺夫指数$(λ_{c})$.固有时间对应于$τ$，坐标时间对应于$t$。其中$t$是由渐近观测者对Hayward黑洞和特殊情形Schwarzschild黑洞测量的。对于类时测地线，我们将它们的比值计算为$\frac{λ_{p}}{λ_{c}} = \frac{(r_σ^{3} + 2 l^{2} m )}{\sqrt{(r_σ^{2} + 2 l^{2} m )^{3}- 3 m r_σ^{5}}}$。在$l=0$的极限下，这意味着对于史瓦西黑洞，这个比率减少到$\frac{λ_{p}}{λ_{c}} = \sqrt{\frac{r_σ}{(r_σ-3 m)}}$。利用Lyponuov指数，研究了赤道圆测地线的稳定性和不稳定性。通过计算不稳定时间尺度的倒数李雅普诺夫指数，我们证明了在程函极限下，准简正模的实部和虚部由零圆测地线的频率和不稳定时间尺度决定。进一步讨论了这类黑洞的不稳定光子球和阴影半径。

英文摘要：

We derive proper-time Lyapunov exponent $(λ_{p})$ and coordinate-time Lyapunov exponent $(λ_{c})$ for a regular Hayward class of black hole. The proper-time corresponds to $τ$ and the coordinate time corresponds to $t$. Where $t$ is measured by the asymptotic observers both for for Hayward black hole and for special case of Schwarzschild black hole. We compute their ratio as $\frac{λ_{p}}{λ_{c}} = \frac{(r_σ^{3} + 2 l^{2} m )}{\sqrt{(r_σ^{2} + 2 l^{2} m )^{3}- 3 m r_σ^{5}}}$ for time-like geodesics. In the limit of $l=0$ that means for Schwarzschild black hole this ratio reduces to $\frac{λ_{p}}{λ_{c}} = \sqrt{\frac{r_σ}{(r_σ-3 m)}}$. Using Lyponuov exponent, we investigate the stability and instability of equatorial circular geodesics. By evaluating the Lyapunov exponent, which is the inverse of the instability time-scale, we show that, in the eikonal limit, the real and imaginary parts of quasi-normal modes~(QNMs) is specified by the frequency and instability time scale of the null circular geodesics. Furthermore, we discuss the unstable photon sphere and radius of shadow for this class of black hole.

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