Internal structure of Hayward black holes

TL;DR

This paper studies the internal structure of Hayward black holes by analyzing scalar field collapse, revealing how inner horizons behave under different scalar perturbations and identifying a universal scaling law near critical collapse.

Contribution

It provides the first detailed analysis of scalar field collapse in Hayward spacetime, showing horizon stability, formation of singularities, and a universal scaling law for the inner horizon radius.

Findings

Inner horizon remains stable under weak scalar perturbations.

Strong scalar fields cause the inner horizon to shrink to zero, forming a spacelike singularity.

The inner horizon radius exhibits a universal scaling law near critical collapse, with an exponent approximately 0.5.

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: ${r_{-}}\propto{|p - {p_*}|^\gamma}$, with a critical exponent $\gamma\approx 0.5$.