On the Universal Cuspy Behavior in Black Hole Shadows

Abstract

This work investigates the universality of cusp formation in the shadows of compact objects. The emergence of cusps is accompanied by three interrelated phenomena: a topological charge transition, an equal-area law governing the self-intersecting structure, and universal critical scaling behavior. We demonstrate that, because these phenomena originate from the global morphology of the shadow, they are fundamentally independent of specific spacetime metric details and apply across diverse models. These features are systematically analyzed for the Kerr black hole endowed with a running Newton coupling. By extending our framework to rotating traversable wormholes, we confirm that the same universal behavior persists in more general compact objects. Our study uncovers the universality underlying cusp formation, offering a model-independent framework for identifying non-Kerr signatures in future black hole observations.

Figures (8)

• Figure 1: Black hole shadows for the KZ black hole with $a_* = 2$. For deformation parameter $\varepsilon>\varepsilon_c$, the shadow is smooth, while the shadow develops a cuspy structure for $\varepsilon<\varepsilon_c$. The critical case is shown in (b).
• Figure 2: Shadow cast by running-$G$ Kerr black hole. For $a_*<a_c\approx 2.343$, the shadow is the standard quasi-circular contour, as shown in (a). The critical behavior with $a_*=a_c$ is shown in (b), and for $a_*>a_c$, there are cuspy behaviors for the shadow as shown in (c) and (d).
• Figure 3: An illustration of the caluation of topological charge. The smooth part contains three segments: $A-B$, $B-C$ and $C-D$. Moreover, there are two exterior angles at points $B$ and $C$, each equals $\Delta \theta=-\pi$.
• Figure 4: The equal-area law for the cuspy shadow of the running-$G$ Kerr black hole. Above criticality $a>a_c$, there is an equal-area law in the $\alpha-\mathcal{F}$ diagram. At the critical point, the interval $[\mathcal{F}_1,\mathcal{F}_2]$ collapses to zero, and the discontinuous jump reduces to an inflection point.
• Figure 5: Critical behavior of the running-$G$ Kerr black hole. The critical exponents are $\zeta_1=\zeta_2=1/2$. The numerical results near the criticality (represented by red dots) exhibit a scaling behavior consistent with the critical exponents, as validated by their close proximity to the reference blue lines of slope $1/2$.