Inside black holes with synchronized hair

TL;DR

This paper investigates the internal structure of rotating black holes with synchronized hair, showing that such solutions lack a smooth inner horizon and exhibit diverging curvature inside, challenging traditional black hole models.

Contribution

It provides the first non-linear analysis of the interior geometry of hairy black holes, demonstrating the absence of a regular inner horizon and the development of divergent curvature.

Findings

No regular inner horizon exists for these solutions.

The spacetime curvature diverges before reaching an inner horizon.

Interior geometry remains smooth near the event horizon but becomes singular inward.

Abstract

Recently, various examples of asymptotically flat, rotating black holes (BHs) with synchronized hair have been explicitly constructed, including Kerr BHs with scalar or Proca hair, and Myers-Perry BHs with scalar hair and a mass gap, showing there is a general mechanism at work. All these solutions have been found numerically, integrating the fully non-linear field equations of motion from the event horizon outwards. Here, we address the spacetime geometry of these solutions inside the event horizon. Firstly, we provide arguments, within linear theory, that there is no regular inner horizon for these solutions. Then, we address this question fully non-linearly, using as a tractable model five dimensional, equal spinning, Myers-Perry hairy BHs. We find that, for non-extremal solutions: $(1)$ the inside spacetime geometry in the vicinity of the event horizon is smooth and the equations of motion can be integrated inwards; $(2)$ before an inner horizon is reached, the spacetime curvature grows (apparently) without bound. In all cases, our results suggest the absence of a smooth Cauchy horizon, beyond which the metric can be extended, for hairy BHs with synchronized hair.