The ring-shaped shadow of a rotating naked singularity with a complete photon sphere*

Abstract

We investigate the shadows of Konoplya-Zhidenko naked singularity. In the spacetime of Konoplya-Zhidenko naked singularity, not only can unstable retrograde light ring (LR) exist, but also unstable prograde LR, leading to the formation of a complete photon sphere (PS). Due to the absence of an event horizon, a dark disc-shaped shadow does not appear; instead, a ring-shaped shadow is observed. The ring-shaped shadow appears as an infinite number of relativistic Einstein rings in the image of the naked singularity. For some parameter values, only the unstable retrograde LR exists, resulting in an incomplete unstable PS and consequently giving rise to the arc-shaped shadow for Konoplya-Zhidenko naked singularity. The shadow of Konoplya-Zhidenko naked singularity gradually shifts to the right as the rotation parameter $a$ increases, and gradually becomes smaller as the deformation parameter $|\eta|$ increases. Moreover, the stable LRs and stable photon spherical orbits can also exist in Konoplya-Zhidenko naked singularity spacetime, but they have no effect on the image of the naked singularity. This study demonstrates that rotating naked singularity can exhibit not only an arc-shaped shadow but also a ring-shaped shadow.