Black holes with synchronised Proca hair: linear clouds and fundamental non-linear solutions

Abstract

Recent studies have made key progress on the black hole/solitonic solutions of the Einstein-Proca system. Firstly, fully non-linear dynamical evolutions of the Kerr black hole superradiant instability, triggered by a Proca field, have shown the formation of a new equilibrium state, a spinning black hole with synchronised Proca hair. Secondly, non-linear evolutions of spinning Proca stars have established that they are dynamically stable, unlike their scalar cousins. Thirdly, separability of the Proca equation on the Kerr background has been achieved. Motivated by these results, in this paper we reconsider Kerr black holes with synchronised Proca hair. The separability of the Proca equation on the Kerr background allows us to examine the stationary Proca clouds in greater detail, in particular their dependence on the different quantum numbers. These stationary clouds occur at a set of existence lines in the Kerr parameter space, from which the black holes with synchronised Proca hair bifurcate. We construct the domain of existence of these black holes, comparing the fundamental states missed in the original study with the first excited states and with the cousin scalar model, giving illustrative examples of Kerr-like and non- Kerr-like BHs. In the vanishing event horizon limit, these hairy black holes connect to the fundamental states of spinning Proca stars, which include the dynamically stable solutions.

Magnetized accretion disks around Kerr black holes with scalar hair: Constant angular momentum disks

“…1 The solutions studied here are the fundamental states of the minimal Einstein-Klein-Gordon model without self-interactions. Different generalizations can be obtained, including charged [29] and excited states in the same model [30,31], as well as cousin solutions in different scalar [32][33][34][35][36][37][38] or Proca models [39,40].…”

Equatorial timelike circular orbits around generic ultracompact objects

“…The spinning solutions therein, however, were excited states. The fundamental spinning solutions were discussed in [31,37].…”

A bound on energy extraction (and hairiness) from superradiance

“…Figure 1 shows the domain of existence of Kerr BHs with synchronised scalar (left panel) and vector (right panel) hair with (n, m) = (0, 1). A detailed comparison between the two families can be found in [37]. The light gray shaded region represents the domain of existence of Kerr BHs in Einstein's gravity, which satisfy the Kerr bound, i.e.…”

“…That is, the region of dynamically viable solutions, corresponding to those that can form by some mechanism and be sufficiently long-lived, can go beyond the region of interest discussed here. Secondly, BHsSH in the region of interest can be arbitrarily close to Kerr BHs and therefore are quite Kerr-like: they are more accurately described as event horizons surrounded by a bosonic cloud rather than bosonic stars with an event horizon at its center [37]. For instance, the areal radius of their shadow is at most 11% larger than that of comparable Kerr BHs [40].…”