Surface geometry and shadow of a Schwarzschild black hole with a halo

We studied the surface geometry and shadows of a Schwarzschild black hole with a halo containing quadrupolar and octopolar terms.It was found that the quadrupole term causes a Schwarzschild black hole to be prolate at the quadrupole strength Q<0 and oblate at Q>0,and the octopole term causes to the shadow stretch upward at the octopolar strength ngth O<0 and downward at O>0.The shadow of a Schwarzschild black hole with a halo stretches and squeezes along the horizontal direction at Q<0 and Q>0,respectively.Meanwhile,the black hole shadow shifts upward at O<0 and downward at O>0.We present the light rays that form the shadow boundary to explain the emergence of the extraordinary patterns of the black hole shadow with quadrupole and octopole terms.From the observable width W,height H,oblateness K,and distortion parameterδcof the black hole shadow,we can determine the quadrupole and octopolar strengths of a Schwarzschild black hole with a halo.Our results show that the quadrupolar and octopolar terms yield a series of interesting patterns for the shadow of a Schwarzschild black hole with a halo.

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

We investigate the shadows of the Konoplya-Zhidenko naked singularity.In the spacetime of the Konoplya-Zhidenko naked singularity,not only an unstable retrograde light ring(LR)but also an unstable prograde LR exists,leading to the formation of a complete photon sphere(PS).Due to the absence of an event horizon,a dark discshaped 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 thus giving rise to an arc-shaped shadow for the Konoplya-Zhidenko naked singularity.The shadow of the Konoplya-Zhidenko naked singularity gradually shifts to the right as the rotation parameter a increases and gradually becomes smaller as the deformation parameter |η| increases.Moreover,stable LRs and stable photon spherical orbits can exist in the Konoplya-Zhidenko naked singularity spacetime,but they have no effect on the image of the naked singularity.This study demonstrates that a rotating naked singularity can exhibit not only an arc-shaped shadow but also a ring-shaped shadow.