黑洞光子球周围的发散反射
Divergent Reflections around the Photon Sphere of a Black Hole
摘要：
从黑洞事件视界外的任何位置，光到达观察者的轨迹都是无限多的。这些路径中的每一个都在围绕黑洞旋转的轨道数量和它们与最后一个光子轨道的接近程度上有所不同。简单的数值和扰动的零测地线方程的史瓦西黑洞的解析解，我们将重申如何每个额外的轨道是一个因素$e^{2 π}$接近黑洞的光学边缘。因此，黑洞的表面和任何背景光都将在最后一个光子轨道周围以指数级变薄的切片中无限镜像。此外，所介绍的形式主义证明了如何在强场极限的光的整个轨迹规定的发散和收敛指数。最后，将指数族的存在性推广到Kerr黑洞的赤道平面，并导出了指数族对自旋的依赖关系。由此，证明了后续图像之间的距离分别针对逆行图像和逆行图像增大和减小。在极端旋转的Kerr黑洞的极限下，不存在对数发散。
英文摘要：
From any location outside the event horizon of a black hole there are an infinite number of trajectories for light to an observer. Each of these paths differ in the number of orbits revolved around the black hole and in their proximity to the last photon orbit. With simple numerical and a perturbed analytical solution to the null-geodesic equation of the Schwarzschild black hole we will reaffirm how each additional orbit is a factor $e^{2 π}$ closer to the black hole's optical edge. Consequently, the surface of the black hole and any background light will be mirrored infinitely in exponentially thinner slices around the last photon orbit. Furthermore, the introduced formalism proves how the entire trajectories of light in the strong field limit is prescribed by a diverging and a converging exponential. Lastly, the existence of the exponential family is generalized to the equatorial plane of the Kerr black hole with the exponentials dependence on spin derived. Thereby, proving that the distance between subsequent images increases and decreases for respectively retrograde and prograde images. In the limit of an extremely rotating Kerr black hole no logarithmic divergence exists for prograde trajectories.
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