Axially symmetric static scalar solitons and black holes with scalar hair

Abstract

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,

Non-linear Q -clouds around Kerr black holes

“…static solitons with w = 0), in a model with a shifted scalar field mass, for a new potential U = U (Q -ball) − w 2 |Φ| 2 [31]. Then the redefined potential U is necessarily negative for some range of |Φ| which is realized by the solutions.…”

Static black holes with axial symmetry in asymptoticallyAdS4spacetime

“…The shape of the scalar field and the energy density in the bulk are rather similar to those shown in Appendix A 1 for the probe limit, and we shall not display them here. 9 As seen in Figs. 3 and 4, the deviation from sphericity of the bulk reflects itself in the holographic stress tensor hτ j i i.…”

“…The recent work [9] has shown the existence of both BH and soliton solutions of this model for the Λ ¼ 0 limiting case, for the same scalar potential (3.5).…”

“…for configurations in a Minkowski spacetime background), the existence of BHs with a single Killing vector field was studied some time ago for matter fields of YangMills type [8], whereas more recently models with scalar fields only attracted attention, as discussed e.g. in [9]. In particular, there are also spinning BHs, which possess a single Killing vector, like the spinning BHs with Yang-Mills hair [10].…”

Dynamics of axial symmetric system in self-interacting Brans–Dicke gravity

“…(2) indicates that both terms are sources of geometry (gravitation). There also exists a static field in the axial symmetric SBD model which has φ = φ(t 0 ) = constant with respect to cosmic time t 0 and generalizes the Einstein equations with an effective cosmological constant V (φ 0 ) [41][42][43][44]. These static field configurations lead to the dynamics of the non-static axial system.…”