AI 中文总结
本研究采用几何方法，结合光学几何框架探究裸奇点时空中的圆光子轨道，证明其总数为偶数且含等量稳定与不稳定轨道，发现轨道数由事件视界而非奇点决定。
AI 中文摘要
圆光子轨道在引力理论和天文观测中都发挥着关键作用。然而，裸奇点时空中圆光子轨道的性质仍未得到充分探索，值得深入研究。本工作致力于全面研究裸奇点时空中圆光子轨道的特征。值得注意的是，本研究采用几何方法来探究这些轨道，其中光学几何（Optical Geometry）框架及其内禀曲率起着关键作用。我们通过内禀测地线曲率分析圆光子轨道的存在性，随后研究稳定和不稳定圆轨道的数量，以及与圆光子轨道相关的拓扑不变量。通过考察各类裸奇点时空，我们得出适用于任意球对称裸奇点时空的关于圆光子轨道的一般性结论：圆光子轨道的总数为偶数（N = 2k），由k个稳定轨道和k个不稳定轨道以相等比例组成。本工作还提供了该结论的数学证明。此外，与其他类别的时空对比显示，裸奇点时空中圆光子轨道数量的结论与无裸奇点的致密天体时空中的结论一致，表明圆光子测地线的数量主要由事件视界的存在而非时空奇点决定。
英文摘要
Circular photon orbits play a pivotal role in both gravitational theories and astronomical observations. However, the properties of circular photon orbits in naked singularity spacetimes remain insufficiently explored and deserve in-depth investigation. The present work is dedicated to a comprehensive study of the features of circular photon orbits in naked singularity spacetimes. Notably, a geometric approach is employed to investigate these orbits, in which the framework of optical geometry together with its intrinsic curvatures plays a crucial role. We analyze the existence of circular photon orbits through the intrinsic geodesic curvature, and then examine the numbers of stable and unstable circular orbits, as well as the topological invariant associated with circular photon orbits. By examining various classes of naked singularity spacetimes, we obtain a general conclusion regarding circular photon orbits that holds for the most of spherically symmetric naked singularity spacetimes: the total number of circular photon orbits is an even integer ($\textit{N} = \text{2}\textit{k}$), consisting of $\textit{k}$ stable orbits and $\textit{k}$ unstable orbits in equal proportion. A mathematical proof of this conclusion is also provided in the present work. Furthermore, a comparison with other categories of spacetimes reveals that the conclusions regarding the number of circular photon orbits in naked singularity spacetimes agree with those obtained for compact object spacetimes without naked singularities, indicating that the number of circular photon orbits is primarily governed by the presence of event horizons rather than spacetime singularities. Keywords: Circular Photon Orbit, Naked Singularity Spacetime, Optical Geometry, Geometric Curvatures
Comments23 pages, 3 figures, 1 table, 3 appendices. V2: some gaps and ambiguous points have been clarified
