Gravitational equal-area law and critical phenomena of cuspy black hole shadow

Abstract

The formation of a cusp on a black hole shadow is a striking signature of physics beyond the Kerr paradigm. We demonstrate that this morphological change fundamentally alters the shadow's topology with the topological charge flipping from 1 to -1. To analyze this topological transition, we introduce a gravitational equal-area law, analogous to Maxwell's construction in thermodynamics, and identify a critical point for cusp formation. Near this point, we uncover universal behavior characterized by a critical exponent 1/2, which places this gravitational lensing system within the mean-field universality class. These results establish a new framework for testing fundamental physics of black hole shadows, reframing the search for deviations from general relativity as a targeted hunt for a distinct topological and critical phenomenon.

Figures (5)

• Figure 1: Shadows cast by the KZ black hole with $a/M=2$. The deformation parameter takes values of $\eta/M^3=$1.5, 1.052, 0.6, and 0.5. Black dots denote the critical points where cusps are about to form. The bottom panels clearly show the resulting cuspy shadow structures.
• Figure 2: Argument of the normal vector $\hat{n}$ along the shadow boundary parameterized by $r/M$ for $a/M=2$. The deformation parameter $\eta/M^3=$1.5 and 0.5 are described by the top blue and bottom red curves, respectively. Note that the curve is segmented for $\eta/M^3=$0.5, and both the differences of the sudden change in the argument are $\Delta\theta=\pi$. As $\eta/M^3$ decreases, the length of the lower segment of the red curve increases.
• Figure 3: Gravitational equal-area law for the cuspy black hole shadow with $a/M=2$. (a) $\eta/M^3$=0.5. The two equal areas are each 3.316, with the slopes $\mathcal{F}_1$=-0.0169 and $\mathcal{F}_2$=0.8397. (b) $\eta/M^3$=0.6. The two equal areas are 2.755. (c) $\eta/M^3$=0.7. The two equal areas are 0.134. (d) $\eta/M^3$=0.8. The two equal areas are 0.064. The horizontal dashed lines are for $\alpha_*/M$=3.872, 3.469, 3.109, and 2.785 for (a)-(d), respectively.
• Figure 4: Numerical results of $\mathcal{F}_1$ (bottom curves) and $\mathcal{F}_2$ (top curves) with $a/M=2$ near the critical point. The critical value of $\mathcal{F}$ is 0.747. (a) $\mathcal{F}$ vs. $\alpha/M$. The critical point is at $\alpha_c/M$=2.088. (b) $\mathcal{F}$ vs. $\eta/M^3$. The critical point is at $\eta_c/M^3$=1.052.
• Figure 5: Critical behavior of $\Delta\mathcal{F}=\mathcal{F}_2-\mathcal{F}_1$ with $a/M=2$. Dots represent numerical results, and blue lines correspond to fitting curves. (a) $\Delta\mathcal{F}$ vs. $\ln(\frac{\alpha-\alpha_c}{M})$. The slope of the blue line is 0.495. (b) $\Delta\mathcal{F}$ vs. $\ln(\frac{\eta_c-\eta}{M^3})$. The slope of the blue line is 0.496.