We present a family of solutions of Einstein's gravity minimally coupled to a complex, massive scalar field, describing asymptotically flat, spinning black holes with scalar hair and a regular horizon. These hairy black holes (HBHs) are supported by rotation and have no static limit. Besides mass $M$ and angular momentum $J$, they carry a conserved, continuous Noether charge $Q$ measuring the scalar hair. HBHs branch off from the Kerr metric at the threshold of the superradiant instability and reduce to spinning boson stars in the limit of vanishing horizon area. They overlap with Kerr black holes for a set of ($M$, $J$) values. A single Killing vector field preserves the solutions, tangent to the null geodesic generators of the event horizon. HBHs can exhibit sharp physical differences when compared to the Kerr solution, such as $J/{M}^{2}>1$, a quadrupole moment larger than ${J}^{2}/M$, and a larger orbital angular velocity at the innermost stable circular orbit. Families of HBHs connected to the Kerr geometry should exist in scalar (and other) models with more general self-interactions.

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.

Extended scalar-tensor Gauss-Bonnet (ESTGB) gravity has been recently argued to exhibit spontaneous scalarization of vacuum black holes (BHs). A similar phenomenon can be expected in a larger class of models, which includes, e.g., Einstein-Maxwell scalar (EMS) models, where spontaneous scalarization of electrovacuum BHs should occur. EMS models have no higher curvature corrections, a technical simplification over ESTGB models that allows us to investigate, fully nonlinearly, BH scalarization in two novel directions. First, numerical simulations in spherical symmetry show, dynamically, that Reissner-Nordström (RN) BHs evolve into a perturbatively stable scalarized BH. Second, we compute the nonspherical sector of static scalarized BH solutions bifurcating from the RN BH trunk. Scalarized BHs form an infinite (countable) number of branches and possess a large freedom in their multipole structure. Unlike the case of electrovacuum, the EMS model admits static, asymptotically flat, regular on and outside the horizon BHs without spherical symmetry and even without any spatial isometries, which are thermodynamically preferred over the electrovacuum state. We speculate on a possible dynamical role of these nonspherical scalarized BHs.

We establish that massive complex Abelian vector fields (mass μ) can form gravitating solitons, when minimally coupled to Einstein's gravity. Such Proca stars (PSs) have a stationary, everywhere regular and asymptotically flat geometry. The Proca field, however, possesses a harmonic time dependence (frequency w), realizing Wheeler's concept of geons for an Abelian spin 1 field. We obtain PSs with both a spherically symmetric (static) and an axially symmetric (stationary) line element. The latter form a countable number of families labelled by an integer m∈Z+. PSs, like (scalar) boson stars, carry a conserved Noether charge, and are akin to the latter in many ways. In particular, both types of stars exist for a limited range of frequencies and there is a maximal ADM mass, Mmax, attained for an intermediate frequency. For spherically symmetric PSs (rotating PSs with m=1,2,3), Mmax≃1.058MPl2/μ (Mmax≃1.568,2.337,3.247MPl2/μ), slightly larger values than those for (mini-)boson stars. We establish perturbative stability for a subset of solutions in the spherical case and anticipate a similar conclusion for fundamental modes in the rotating case. The discovery of PSs opens many avenues of research, reconsidering five decades of work on (scalar) boson stars, in particular as possible dark matter candidates