Shadow Formation Conditions Beyond the Kerr Black Hole Paradigm

A compact object illuminated by background radiation produces a dark silhouette. The edge of the silhouette or shadow (alternatively, the apparent boundary or the critical curve) is commonly determined by the presence of the photon sphere (or photon shell in the case of rotating spacetime), corresponding to the maximum of the effective potential for null geodesics. While this statement stands true for Kerr black holes, here we remark that the apparent boundary (as defined by Bardeen) forms under a more general condition. We demonstrate that a shadow forms if the effective potential of null geodesics has a positive finite upper bound and includes a region where photons are trapped or scattered. Our framework extends beyond conventional solutions, including but not limited to naked singularities. Furthermore, we clarify the difference between the apparent boundary of a dark shadow and the bright ring on the screen of a distant observer. These results provide a unified theoretical basis for interpreting observations from the Event Horizon Telescope (EHT) and guiding future efforts towards extreme-resolution observations of compact objects.