The Trinity of black hole correspondences: Shadows-quasinormal modes-graybody factors and cautionary remarks

Correspondences between apparently distant concepts are ubiquitous in theoretical physics. In the context of Black Holes (BHs), Quasi-Normal Modes (QNMs) were shown to be linked to both shadows, in the so-called eikonal limit, and Gray-Body Factors (GBFs), using the WKB approximation. We test the accuracy of the QNM-GBF correspondence in the context of the Hawking radiation for static and rotating black hole configurations, with particular attention to the superradiant regime. Our analysis reveals the correspondence failure to accurately reproduce the Hawking spectrum due to divergences. Furthermore, we bridge the gap between BH shadows and GBFs by drawing a correspondence between such quantities in the case of generic static and spherically symmetric spacetime configurations. The shadow-GBF correspondence is tested for some case studies, including regular BHs, and its limitations and applicability are discussed. This study opens new perspectives by introducing a new correspondence and remarking on the caution needed when considering these connections.