Can Rotating Black Holes Have Short Hairs?

TL;DR

This paper investigates whether rotating black holes can have detectable 'hairs' extending beyond the event horizon, finding that the no-short hair theorem holds in some cases but may not be universally applicable due to rotation effects.

Contribution

It extends the no-short hair theorem to certain classes of rotating black holes and identifies conditions under which it may or may not hold.

Findings

The no-short hair property holds for specific rotating black hole solutions in non-vacuum GR.

Rotation-induced effects can invalidate the no-short hair theorem in a theory-agnostic context.

Additional criteria are necessary to generalize the theorem to other theories and rotating horizonless objects.

Abstract

Despite the no-hair theorem, several notable hairy black hole (BH) solutions exist in both General Relativity and modified gravity theories. For such hairs to be detectable, they must extend sufficiently beyond the event horizon. This idea has been rigorously formalized by the no-short hair theorem, which dictates that all existing hairs of a static spherically symmetric BH must extend at least to the innermost light ring (LR). However, the theorem's applicability to the astrophysically relevant rotating BHs remains elusive as yet. To address this gap, we examine its validity for rotating BHs in the Konoplya-Rezzolla-Zhidenko-Stuchl\'ik and Johannsen classes. Interestingly, for Klein-Gordon separable BHs in these classes that are solutions of non-vacuum GR, the no-short hair property continues to hold. However, unlike in static cases, this result may not apply in a theory-agnostic fashion due to the rotation-induced repulsive effects. Consequently, we identify a minimal set of additional criteria on the metric and matter content needed for such a generalization in other theories. Our study marks an important first step toward establishing general results on the extent of rotating BH hairs, reinforcing their observational detections. Further extension of this novel result for rotating horizonless objects is also discussed.