Marginal scalar and Proca clouds around Reissner-Nordstr\"om black holes

TL;DR

This paper investigates the existence and properties of marginal scalar and Proca clouds around Reissner-Nordström black holes, revealing that such long-lived bound states exist for scalar fields but not for Proca fields, with potential implications for black hole hair.

Contribution

It demonstrates the existence of marginal scalar and Proca clouds around Reissner-Nordström black holes and shows that Proca clouds do not form in the superradiant regime, extending understanding of black hole field configurations.

Findings

Scalar clouds can exist as long-lived bound states.

Proca clouds do not form in the superradiant regime.

Long-lived quasi-bound states have arbitrarily small decay rates.

Abstract

Massive scalar test fields around Kerr black holes can form quasi-bound states with complex frequencies. Some of these states decay in time, but some others -- in the superradiant regime -- grow, causing a superradiant instability. Precisely at the threshold between decaying and growing modes, there exist bound states with real frequency, known as scalar clouds. Fully non-linear counterparts of these clouds have been shown to yield Kerr black holes with scalar hair. Here, we consider massive, (electrically) charged, test scalar and Proca fields on the background of Reissner-Nordstr\"om black holes. By computing the quasi-bound state frequencies, we establish that no such states exist in the superradiant regime for the Proca field -- a similar behavior to that known for scalar fields. But decaying quasi-bound states with an arbitrarily small imaginary part of the frequency exist and thus which are arbitrarily long lived. In the limit of vanishing imaginary part of the frequency, the field does not trivialize and we dub the corresponding configurations as marginal (charged) scalar or Proca clouds, since they are only marginally bound. We comment on the possible existence of non-linear counterparts to these marginal clouds.