We construct rotating boson stars and Myers-Perry black holes with scalar hair (MPBHsSH) as fully non-linear solutions of five dimensional Einstein gravity minimally coupled to a complex, massive scalar field. The MPBHsSH are, in general, regular on and outside the horizon, asymptotically flat, and possess angular momentum in a single rotation plane. They are supported by rotation and have no static limit. Such hairy BHs may be thought of as bound states of boson stars and singly spinning, vacuum MPBHs and inherit properties of both these building blocks. When the horizon area shrinks to zero, the solutions reduce to (in a single plane) rotating boson stars; but the extremal limit also yields a zero area horizon, as for singly spinning MPBHs. Similarly to the case of equal angular momenta, and in contrast to Kerr black holes with scalar hair, singly spinning MPBHsSH are disconnected from the vacuum black holes, due to a mass gap. We observe that for the general case, with two unequal angular momenta, the equilibrium condition for the existence of MPBHsSH is w = m1Ω1 + m2Ω2, where Ωi are the horizon angular velocities in the two independent rotation planes and w, mi, i = 1, 2, are the scalar field's frequency and azimuthal harmonic indices.

We present spinning Q-balls and boson stars in four dimensional anti-de Sitter spacetime. These are smooth, horizonless solutions for gravity coupled to a massive complex scalar field with a harmonic dependence on time and the azimuthal angle. Similar to the flat spacetime configurations, the angular momentum is quantized. We find that a class of solutions with a self-interaction potential has a limit corresponding to static solitons with axial symmetry only. An exact solution describing spherically symmetric Q-balls in a fixed AdS background is also discussed.Comment: 12 pages, 4 figure

We construct static, asymptotically flat black hole solutions with scalar hair. They evade the nohair theorems by having a scalar potential which is not strictly positive. By including an azimuthal winding number in the scalar field ansatz, we find hairy black hole solutions which are static but axially symmetric only. These solutions possess a globally regular limit, describing scalar solitons. A branch of axially symmetric black holes is found to possess a positive specific heat.Introduction.-The energy conditions are an important ingredient of various significant results in general relativity [1]. Essentially, they imply that some linear combinations of the energy-momentum tensor of the matter fields should be positive, or at least non-negative. However, over the last decades, it has become increasingly obvious that these conditions can be violated, even at the classical level. Remarkably enough, the violation may occur also for the simplest case of a scalar field (see e.g.[2] for a discussion of these aspects).Once we give up the energy conditions (and in particular the weak one), a number of results in the literature show that the asymptotically flat black holes may possess scalar hair 1 , which otherwise is forbidden by a number of well-known theorems [4]. Restricting to the simplest case of a minimally coupled scalar field with a scalar potential which is not strictly positive, this includes both analytical [5] Interestingly, in the limit of zero event horizon radius, some of these hairy black holes describe globally regular, particle-like objects, the so-called 'scalarons' [10]. At the same time, a complex scalar field is known for long time to possess non-topological solitonic solutions [12], even in the absence of gravity. These are the Q-balls introduced by Coleman in [13]. Such configuration owe their existence to a harmonic time dependence of the scalar field and possess a positive energy density.However, as argued below, the Q-balls can be reinterpreted as non-gravitating scalarons. The scalar field is static in this case and has a potential which takes negative values as well. As expected, the scalarons possess gravitating generalizations. However, different from the standard Q-ball case [14], their regular origin can be replaced with an event horizon. In this work we study such solutions for the simple case of a massive complex scalar field with a negative quartic self-interaction term in the potential. Apart from spherically symmetric configurations, we construct solitons and hairy black hole solutions which are static but axially symmetric only.The model.-Let us consider the action of a self-interacting complex scalar field Φ coupled to Einstein gravity in four spacetime dimensions,

Vortons can be viewed as (flat space-) field theory analogs of black rings in general relativity. They are made from loops of vortices, being sustained against collapse by the centrifugal force. In this work we discuss such configurations in the global version of Witten's U(1)xU(1) theory. We first consider solutions in a flat spacetime background and show their non-uniqueness. The inclusion of gravity leads to new features. In particular, an ergoregion can occur. Also, similar to boson stars, we show that the vortons exist only in a limited frequency range. The coupling to gravity gives rise to a spiral-like frequency dependence of the mass and charge. New solutions of the model describing 'semitopological vortons' and 'di-vortons' are also discussed.Comment: 34 pages, 19 figure

Recently, various families of black holes (BHs) with synchronised hair have been constructed. These are rotating BHs surrounded, as fully non-linear solutions of the appropriate Einstein-matter model, by a non-trivial bosonic field in synchronised rotation with the BH horizon. Some families bifurcate globally from a bald BH (e.g. the Kerr BH), whereas others bifurcate only locally from a bald BH (e.g. the D = 5 Myers-Perry BH). It would be desirable to understand how generically synchronisation allows hairy BHs to bifurcate from bald ones. However, the construction and scanning of the domain of existence of the former families of BHs can be a difficult and time consuming (numerical) task. Here, we first provide a simple perturbative argument to understand the generality of the synchronisation condition. Then, we observe that the study of Q-clouds is a generic tool to establish the existence of BHs with synchronised hair bifurcating (globally or locally) from a given bald BH without having to solve the fully non-linear coupled system of Einstein-matter equations. As examples, we apply this tool to establish the existence of synchronised hair around D = 6 Myers-Perry BHs, D = 5 black rings and D = 4 Kerr-AdS BHs, where D is the spacetime dimension. The black rings case provides an example of BHs with synchronised hair beyond spherical horizon topology, further establishing the generality of the mechanism.