Gravity/thermodynamics correspondence via black hole shadows

TL;DR

This paper explores the intricate features of black hole shadows, establishing a formal correspondence with thermodynamic phase transitions, and classifies shadow topologies with analytical and topological tools.

Contribution

It introduces a topological classification of cuspy black hole shadows and establishes a gravity/thermodynamics correspondence linking shadow features to thermodynamic behavior.

Findings

Classified black hole shadows into rectangular and 8-shape topologies.

Mapped shadow boundary features to thermodynamic swallowtail behavior.

Derived critical exponents consistent with mean-field universality class.

Abstract

The shadow of a black hole serves as a pristine window into the strong-gravity regime, with cuspy feature emerging as a smoking-gun signature of physics beyond the Kerr paradigm. In this paper, we extend the work of [arXiv:2601.15612 [gr-qc]] and study the detailed properties of the cuspy shadow by using the parametric expressions of the shadow boundary. From a topological perspective, we provide a rigorous topological classification of these shadows, categorizing them into distinct ``rectangular" and ``8-shape" topologies. Crucially, we establish a formal gravity/thermodynamics correspondence by mapping the cuspy shadow to the swallowtail behavior observed in thermodynamic free energy. We demonstrate that the self-intersection of the shadow boundary, marking a geometric phase transition, can be precisely determined through three independent but equivalently thermodynamic-like approaches. Furthermore, we analytically derive the critical exponents governing the emergence of these cusps, revealing that they are consistent with the mean-field universality class. Our results suggest that the observational features of black hole shadows are deeply rooted in the underlying gravitational thermodynamics, offering a novel framework to probe the fundamental nature of spacetime.