Using backwards ray tracing, we study the shadows of Kerr black holes with scalar hair (KBHSH). KBHSH interpolate continuously between Kerr BHs and boson stars (BSs), so we start by investigating the lensing of light due to BSs. Moving from the weak to the strong gravity region, BSs-which by themselves have no shadows-are classified, according to the lensing produced, as (i) noncompact, which yield not multiple images, (ii) compact, which produce an increasing number of Einstein rings and multiple images of the whole celestial sphere, and (iii) ultracompact, which possess light rings, yielding an infinite number of images with (we conjecture) a self-similar structure. The shadows of KBHSH, for Kerr-like horizons and noncompact BS-like hair, are analogous to, but distinguishable from, those of comparable Kerr BHs. But for non-Kerr-like horizons and ultracompact BS-like hair, the shadows of KBHSH are drastically different: novel shapes arise, sizes are considerably smaller, and multiple shadows of a single BH become possible. Thus, KBHSH provide quantitatively and qualitatively new templates for ongoing (and future) very large baseline interferometry observations of BH shadows, such as those of the Event Horizon Telescope.

Bekenstein proved that in Einstein's gravity minimally coupled to one (or many) real, Abelian, Proca field, stationary black holes (BHs) cannot have Proca hair. Dropping Bekenstein's assumption that matter inherits spacetime symmetries, we show this model admits asymptotically flat, stationary, axisymmetric, regular on and outside an event horizon BHs with Proca hair, for an even number of real (or an arbitrary number of complex) Proca fields. To establish it, we start by showing that a test, complex Proca field can form bound states, with real frequency, around Kerr BHs: stationary Proca clouds. These states exist at the threshold of superradiance. It was conjectured in [1,2], that the existence of such clouds at the linear level implies the existence of a new family of BH solutions at the non-linear level. We confirm this expectation and explicitly construct examples of such Kerr black holes with Proca hair (KBHsPH). For a single complex Proca field, these BHs form a countable number of families with three continuous parameters (ADM mass, ADM angular momentum and Noether charge). They branch off from the Kerr solutions that can support stationary Proca clouds and reduce to Proca stars [3] when the horizon size vanishes. We present the domain of existence of one family of KBHsPH, as well as its phase space in terms of ADM quantities. Some physical properties of the solutions are discussed; in particular, and in contrast with Kerr BHs with scalar hair, some spacetime regions can be counter-rotating with respect to the horizon. We further establish a no-Proca-hair theorem for static, spherically symmetric BHs but allowing the complex Proca field to have a harmonic time dependence, which shows BHs with Proca hair in this model require rotation and have no static limit. KBHsPH are also disconnected from Kerr-Newman BHs with a real, massless vector field. * herdeiro@ua.pt

In a recent paper [P. V. P. Cunha, C. A. R. Herdeiro, E. Radu, and H. F. Runarsson, Phys. Rev. Lett. 115, 211102 (2015).], it was shown that the lensing of light around rotating boson stars and Kerr black holes with scalar hair can exhibit chaotic patterns. Since no separation of variables is known (or expected) for geodesic motion on these backgrounds, we examine the 2D effective potentials for photon trajectories, to obtain a deeper understanding of this phenomenon. We find that the emergence of stable light rings on the background spacetimes allows the formation of "pockets" in one of the effective potentials, for open sets of impact parameters, leading to an effective trapping of some trajectories, dubbed "quasibound orbits." We conclude that pocket formation induces chaotic scattering, although not all chaotic orbits are associated to pockets. These and other features are illustrated in a gallery of examples, obtained with a new ray-tracing code, PYHOLE, which includes tools for a simple, simultaneous visualization of the effective potential, together with the spacetime trajectory, for any given point in a lensing image. An analysis of photon orbits allows us to further establish a positive correlation between photon orbits in chaotic regions and those with more than one turning point in the radial direction; we recall that the latter is not possible around Kerr black holes. Moreover, we observe that the existence of several light rings around a horizon (several fundamental orbits, including a stable one), is a central ingredient for the existence of multiple shadows of a single hairy black hole. We also exhibit the lensing and shadows by Kerr black holes with scalar hair, observed away from the equatorial plane, obtained with PYHOLE.

The maximal Arnowitt-Deser-Misner (ADM) mass for (mini)boson stars (BSs)-gravitating solitons of Einstein's gravity minimally coupled to a free, complex, mass μ, Klein-Gordon field-is M max ADM ∼ M 2 Pl =μ. Adding quartic self-interactions to the scalar field theory, described by the Lagrangian L I ¼ λjΨj 4 , the maximal ADM mass becomesPl =μ 2 . Thus, for mini-BSs, astrophysically interesting masses require ultralight scalar fields, whereas self-interacting BSs can reach such values for bosonic particles with Standard Model range masses. We investigate how these same self-interactions affect Kerr black holes with scalar hair (KBHsSH) [C. A. R. Herdeiro and E. Radu, Kerr Black Holes with Scalar Hair, Phys. Rev. Lett. 112, 221101 (2014).], which can be regarded as (spinning) BSs in stationary equilibrium with a central horizon. Remarkably, whereas the ADM mass scales in the same way as for BSs, the horizon mass M H does not increases with the coupling λ, and, for fixed μ, it is maximized at the "Hod point," corresponding to the extremal Kerr black hole obtained in the vanishing hair limit. This mass is always M max H ∼ M 2 Pl =μ. Thus, introducing these self-interactions, the black hole spacetimes may become considerably "hairier" but the trapped regions cannot become "heavier." We present evidence that this observation also holds in a model with L I ¼ βjΨj 6 − λjΨj 4 ; if it extends to general scalar field models, KBHsSH with astrophysically interesting horizon masses require ultralight scalar fields. Their existence, therefore, would be a smoking gun for such (beyond the Standard Model) particles.

Confined scalar fields, either by a mass term or by a mirror-like boundary condition, have unstable modes in the background of a Kerr black hole. Assuming a time dependence as e −iωt , the growth time-scale of these unstable modes is set by the inverse of the (positive) imaginary part of the frequency, Im(ω), which reaches a maximum value of the order of Im(ω)M ∼ 10 −5 , attained for a mirror-like boundary condition, where M is the black hole mass. In this paper we study the minimally coupled Klein-Gordon equation for a charged scalar field in the background of a Reissner-Nordström black hole and show that the unstable modes, due to a mirror-like boundary condition, can grow several orders of magnitude faster than in the rotating case: we have obtained modes with up to Im(ω)M ∼ 0.07. We provide an understanding, based on an analytic approximation, to why the instability in the charged case has a smaller timescale than in the rotating case. This faster growth, together with the spherical symmetry, makes the charged case a promising model for studies of the fully non-linear development of superradiant instabilities.