Internal structure of Hayward black holes

TL;DR

The paper investigates the internal structure of Hayward regular black holes under neutral scalar-field collapse, asking whether the inner (Cauchy) horizon remains stable or gives way to a spacelike singularity. It uses a spherically symmetric Einstein–nonlinear electrodynamics framework with a massless scalar field in Kruskal-like coordinates, initializing collapse with a Gaussian profile and evolving the system via a leapfrog scheme while enforcing time-symmetric data. The results reveal three regimes: weak perturbations preserve a finite inner horizon with divergent mass inflation near $r_-$, strong perturbations drive $r_- \to 0$ and produce a Schwarzschild-like singular interior, and near criticality the inner horizon obeys a universal scaling $r_{-} \propto |p - p_{*}|^{\gamma}$ with $\gamma \approx 0.5$. These findings highlight rich nonlinear and critical dynamics inside regular black holes and point to potential observational implications, while linking to known critical behavior in charged black hole interiors.

Abstract

Regular black holes, free of central singularities, provide an ideal laboratory for probing the geometric structure of spacetime. The global structure of some regular black holes, e.g. Hayward black hole, features an event horizon and a Cauchy horizon, raising fundamental questions about the latter's stability. In this work, we investigate collapse of a scalar field in Hayward spacetime. Under weak scalar perturbations, the inner horizon maintains a stable finite radius. In the circumstance of a strong scalar field, the inner horizon shrinks to zero volume, accompanied by the formation of a spacelike singularity. The Hayward geometry is effectively converted into a Schwarzschild-like geometry. Furthermore, the strength of the scalar field governs the contraction dynamics of the inner horizon. As the parameter $p$ of the initial profile for the scalar field approaches the critical threshold ${p_*}$, the radius of the inner horizon ${r_{-}}$ exhibits a universal scaling behavior: ${r_{-}}\propto{|p - {p_*}|^γ}$, with a critical exponent $γ\approx 0.5$.

Figures (7)

• Figure 1: The evolutions for $r$, $\sigma$ and $\varphi$ under a weak scalar field with $A=0.045$. (b) A detailed view of the evolution of $r$, illustrating the influence of the scalar field on its dynamics. Each line represents the variation of $r$, $\sigma$, and $\varphi$ with respect to $x$ at fixed time.
• Figure 2: Dynamical evolution of black hole characteristics along the $x=0.5$ slice. (a) Evolution of $r$ with $A = 0$, recovering the original Hayward black hole solution where ${r_ + }$ and ${r_ - }$ denote the initial outer and inner horizons respectively. (b) Evolution of $r$ with $A = 0.045$, showing the inner horizon’s contraction due to the scalar field. (c) Evolution of the Misner–Sharp mass $M$ along the slice $x=0.5$, exhibiting divergence over time for $A = 0.045$.
• Figure 3: The evolutions for $r$, $\sigma$ and $\varphi$ in a strong scalar field with $A=0.055$. (b) A detailed view of the evolution of $r$, illustrating the influence of the scalar field on its dynamics. Each line represents the variation of $r$, $\sigma$, and $\varphi$ with respect to $x$ at fixed time.
• Figure 4: Dynamical evolution of black hole characteristics along the $x=1.2$ slice. (a) Evolution of $r$ with $A = 0.055$, illustrating the disappearance of the inner horizon. Here, ${r_ + }$ and ${r_ - }$ represent the outer and inner horizons of the original Hayward black hole. (b) Evolution of the Misner–Sharp mass $M$, highlighting the mass inflation phenomenon. (c), (d): Evolution of $\sigma$ and $\varphi$, both of which diverge as $r$ approaches zero.
• Figure 5: Kretschmann scalar $K$ as $r$ approaches zero. (a) Initial state. (b) Final state.