Gravity/thermodynamics correspondence via black hole shadows

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.

Figures (10)

• Figure 1: Parameter regions corresponding to the number of horizons. In the regions I, II, III, and IV, the KZ black holes have one, two, three, and none horizon(s).
• Figure 2: The shadows of spinning KZ black holes with the inclination angle $\theta_0=\pi/2$. In certain cases, the cuspy shadows are exhibited. (a) $a/M=0.9$, $\eta/M^3=-0.1$. (b) $a/M=0.9$, $\eta/M^3=0.08$. (c) $a/M=0.9$, $\eta/M^3=0.2$. (d) $a/M=1.1$, $\eta/M^3=0.19$. (e) $a/M=1.1$, $\eta/M^3=0.22$. (f) $a/M=1.1$, $\eta/M^3=0.25$. (g) $a/M=1.2$, $\eta/M^3=0.3$. (h) $a/M=1.2$, $\eta/M^3=0.33$. (i) $a/M=1.2$, $\eta/M^3=0.4$.
• Figure 3: Shadow, reduced angular momentum and Carter constant for the spinning KZ black hole with $a/M=1.6$ and $\eta/M^3=0.45$. The solid and dashed lines correspond to the unstable and stable spherical orbits, respectively. (a) Shadow shape. (b) The reduced angular momentum $\xi$. (c) The reduced Carter constant $\sigma$. The regions in light blue color and red color are for the unstable and stable spherical orbits. The cusps C and E correspond to the extremal point of $\xi$ and $\sigma$. However, $\sigma$ has an extra extremal point located at point G. Points B and F denote the self-intersection point, where the cuspy shadow connects with the main shadow. Not that these points correspond to different spherical orbits. Points B, D, and F share the same value of $\alpha/M$.
• Figure 4: The derivatives of $V_{eff}$, $\xi$, and $\sigma$.
• Figure 5: The argument of the tangent vector $v$. (a) $a/M=0.9$. (b) $a/M=1.2$. A sudden change $\pi/2$ of the argument occurs for case (b) with $\eta/M^3$=0.30 and 0.33.