Investigation of Circular Photon Orbits in Naked Singularity Spacetimes from a Geometric Method

Abstract

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 on 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 crucial roles. We analyze the existence of circular photon orbits through the intrinsic geodesic curvature, and we then investigate the number 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 arbitrary spherically symmetric naked singularity spacetimes: the total number of circular photon orbits is an even integer ($N = 2k$), consisting of $k$ stable orbits and $k$ 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 count 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 geodesics is primarily governed by the presence of event horizons rather than spacetime singularities. Keywords: Circular Photon Orbit, Naked Singularity Spacetime, Optical Geometry, Geometric Curvatures