How short can stationary charged scalar hair be?

TL;DR

This paper proves that stationary charged scalar fields around Kerr-Newman black holes cannot be arbitrarily compact, establishing a universal lower bound on the size of these scalar 'clouds' independent of their physical properties.

Contribution

It introduces a universal lower bound on the effective length of stationary charged scalar clouds around Kerr-Newman black holes, independent of scalar field parameters.

Findings

Derived a lower bound (r_field - r_+) / (r_+ - r_-) > 1/s^2 for scalar clouds.

The bound is universal, independent of scalar field mass, charge, or angular momentum.

Applied to near-extremal Kerr-Newman black holes with linearized scalar fields.

Abstract

It is by now well established that charged rotating Kerr-Newman black holes can support bound-state charged matter configurations which are made of minimally coupled massive scalar fields. We here prove that the externally supported stationary charged scalar configurations {\it cannot} be arbitrarily compact. In particular, for linearized charged massive scalar fields supported by charged rotating near-extremal Kerr-Newman black holes, we derive the remarkably compact lower bound $(r_{\text{field}}-r_+)/(r_+-r_-)>1/s^2$ on the effective lengths of the external charged scalar `clouds' [here $r_{\text{field}}$ is the radial peak location of the stationary scalar configuration, and $\{s\equiv J/M^2, r_{\pm}\}$ are respectively the dimensionless angular momentum and the horizon radii of the central supporting Kerr-Newman black hole]. Remarkably, this lower bound is universal in the sense that it is independent of the physical parameters (proper mass, electric charge, and angular momentum) of the supported charged scalar fields.