No-short scalar hair theorem for spinning acoustic black holes in a photon-fluid model

TL;DR

This paper proves a no-short hair theorem for stationary scalar clouds around spinning acoustic black holes in a photon-fluid model, establishing a lower bound on the effective length of these clouds.

Contribution

It provides the first analytical proof of a no-short hair theorem for acoustic black hole scalar clouds, linking their size to the black hole's horizon and null geodesic.

Findings

Effective lengths of scalar clouds are bounded from below by the golden ratio times the horizon radius.

The scalar clouds cannot be arbitrarily short, respecting a specific inequality.

The theorem applies to co-rotating stationary density fluctuations in photon-fluid black hole models.

Abstract

It has recently been revealed that spinning black holes of the photon-fluid model can support acoustic `clouds', stationary density fluctuations whose spatially regular radial eigenfunctions are determined by the $(2+1)$-dimensional Klein-Gordon equation of an effective massive scalar field. Motivated by this intriguing observation, we use {\it analytical} techniques in order to prove a no-short hair theorem for the composed acoustic-black-hole-scalar-clouds configurations. In particular, it is proved that the effective lengths of the stationary bound-state co-rotating acoustic scalar clouds are bounded from below by the series of inequalities $r_{\text{hair}}>{{1+\sqrt{5}}\over{2}}\cdot r_{\text{H}}>r_{\text{null}}$, where $r_{\text{H}}$ and $r_{\text{null}}$ are respectively the horizon radius of the supporting black hole and the radius of the co-rotating null circular geodesic that characterizes the acoustic spinning black-hole spacetime.