Binary black hole shadows, chaotic scattering and the Cantor set
Binary black hole shadows, chaotic scattering and the Cantor set
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DOI：
10.1088/0264-9381/33/17/175001
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发表时间：
2016-03
影响因子：
3.5
通讯作者：
Jake O. Shipley;S. Dolan
中科院分区：
文献类型：
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作者：
Jake O. Shipley;S. Dolan
我们使用静态平衡状态下的两个极带电黑洞模型（Majumdar-Papapetrou 解）研究了二元黑洞阴影的定性特征。我们的观点是，二元时空是混沌散射的自然范例，因为它们允许多个基本零轨道，因此存在一组不可数无限的永久零轨道，这些轨道在初始数据中产生散射奇点。受三盘模型的启发，我们开发了适当的符号动力学来描述双黑洞时空上的平面零测地线。我们证明一维（1D）黑洞阴影可以通过类似于构造康托集的迭代过程来构造；因此一维阴影是自相似的。接下来，我们研究非平面射线，以了解角动量如何影响基本零轨道的存在和属性。通过 2D 阴影进行切片，我们观察到三种类型的 1D 阴影：规则的、类康托尔的和高度混乱的。从类康托尔轨道到常规轨道的转变发生在角动量禁止外部基本轨道的情况下。高度混沌的部分与一个意想不到的特征相关：稳定且有界的零轨道，存在于两个质量相等的黑洞周围，它们之间的距离为 a 1 < a < 2 a 1 ，其中 a 1 = 4 M / 27 。为了说明这种可能性是如何出现的，我们定义了某个势函数并对其驻点进行了分类。我们推测二维阴影的高度混沌部分具有和田性质。最后，我们考虑通过事件视界遵循零测地线的可能性，以及最大扩展时空中的混沌。
We investigate the qualitative features of binary black hole shadows using the model of two extremally charged black holes in static equilibrium (a Majumdar–Papapetrou solution). Our perspective is that binary spacetimes are natural exemplars of chaotic scattering, because they admit more than one fundamental null orbit, and thus an uncountably infinite set of perpetual null orbits which generate scattering singularities in initial data. Inspired by the three-disc model, we develop an appropriate symbolic dynamics to describe planar null geodesics on the double black hole spacetime. We show that a one-dimensional (1D) black hole shadow may be constructed through an iterative procedure akin to the construction of the Cantor set; thus the 1D shadow is self-similar. Next, we study non-planar rays, to understand how angular momentum affects the existence and properties of the fundamental null orbits. Taking slices through 2D shadows, we observe three types of 1D shadow: regular, Cantor-like, and highly chaotic. The switch from Cantor-like to regular occurs where outer fundamental orbits are forbidden by angular momentum. The highly chaotic part is associated with an unexpected feature: stable and bounded null orbits, which exist around two black holes of equal mass M separated by a 1 < a < 2 a 1 , where a 1 = 4 M / 27 . To show how this possibility arises, we define a certain potential function and classify its stationary points. We conjecture that the highly chaotic parts of the 2D shadow possess the Wada property. Finally, we consider the possibility of following null geodesics through event horizons, and chaos in the maximally extended spacetime.