Dynamical formation of a Reissner-Nordstr\"om black hole with scalar hair in a cavity

TL;DR

This paper uses numerical relativity to study how charged scalar fields with self-interactions evolve around Reissner-Nordström black holes in a cavity, leading to hairy black holes and revealing complex non-linear phenomena like suppressed superradiance and explosive mode decay.

Contribution

It introduces the effects of scalar field self-interactions on superradiant instabilities and the formation of hairy black holes in a cavity, extending previous linear analyses.

Findings

Self-interactions suppress exponential growth of superradiance.

Scalar energy exhibits non-monotonic behavior due to self-interactions.

Explosive decay modes correspond to modes leaving the superradiant regime.

Abstract

In a recent letter, we presented numerical relativity simulations, solving the full Einstein--Maxwell--Klein-Gordon equations, of superradiantly unstable Reissner-Nordstr\"om black holes (BHs), enclosed in a cavity. Low frequency, spherical perturbations of a charged scalar field, trigger this instability. The system's evolution was followed into the non-linear regime, until it relaxed into an equilibrium configuration, found to be a $\textit{hairy}$ BH: a charged horizon in equilibrium with a scalar field condensate, whose phase is oscillating at the (final) critical frequency. Here, we investigate the impact of adding self-interactions to the scalar field. In particular, we find sufficiently large self-interactions suppress the exponential growth phase, known from linear theory, and promote a non-monotonic behaviour of the scalar field energy. Furthermore, we discuss in detail the influence of the various parameters in this model: the initial BH charge, the initial scalar perturbation, the scalar field charge, mass, and the position of the cavity's boundary (mirror). We also investigate the "explosive" non-linear regime previously reported to be akin to a bosenova. A mode analysis shows that the "explosions" can be interpreted as the decay into the BH of modes that exit the superradiant regime.