Topological Change of Kerr Compact Object Shadows: Properties of Unstable Spherical Photon Orbits Around Kerr Naked Singularities*

The Event Horizon Telescope (EHT) recently resolved the supermassive black holes (SMBHs) at the center of M87 (M87*) and the Milky Way (Sgr A*) with the event horizon scale resolution. The geometric features of these observed photon rings have been used to test modified gravity theories and alternative models to SMBHs. Numerous studies have demonstrated that naked singularities, gravitational singularities without event horizons, can also project shadow-like regions. Thus, the EHT observations are still compatible with the possibility of Sgr A* being a naked singularity. This has raised theoretical interests, particularly the weak cosmic censorship hypothesis, which forbids the visibility of singularities for an observer at infinity. In this work, we analytically project the unstable spherical photon orbits around Kerr naked singularities (KNSs) to infinity, which we define to be the "shadow" of KNSs. We demonstrate that this shadow can be close, open, or vanishing, depending on the spins and observational inclination angles, and study the critical parameters where the shadow changes its topology. We then compare these analytical results to numerical ray-tracing simulations. This topological change in the shadow is a new phenomenon for KNS. By using EHT observations, it can be used to test KNSs as alternative models to SMBHs. Besides, we investigate other properties of null geodesics of KNSs that are distinctive from black holes, such as stable photon orbits and regions inside the KNS photon ring.