Recent numerical relativity simulations within the Einstein--Maxwell--(charged-)Klein-Gordon (EMcKG) system have shown that the non-linear evolution of a superradiantly unstable Reissner-Nordström black hole (BH) enclosed in a cavity, leads to the formation of a BH with scalar hair. Perturbative evidence for the stability of such hairy BHs has been independently established, confirming they are the true endpoints of the superradiant instability. The same EMcKG system admits also charged scalar soliton-type solutions, which can be either stable or unstable. Using numerical relativity techniques, we provide evidence that the time evolution of some of these $\textit{unstable}$ solitons leads, again, to the formation of a hairy BH. In some other cases, unstable solitons evolve into a (bald) Reissner-Nordström BH. These results establish that the system admits two distinct channels to form hairy BHs at the threshold of superradiance: growing hair from an unstable (bald) BH, or growing a horizon from an unstable (horizonless) soliton. Some parallelism with the case of asymptotically flat boson stars and Kerr BHs with scalar hair is drawn.
Using a recently developed perturbation formalism based on curvature quantities, we complete our investigation of the linear stability of black holes and solitons with Yang-Mills hair and a negative cosmological constant. We show that those solutions which have no linear instabilities under odd- and even-parity spherically symmetric perturbations remain stable under even-parity, linear, non-spherically symmetric perturbations. Together with the result from a previous work, we have therefore established the existence of stable hairy black holes and solitons with anti-de Sitter asymptotic.
We present new soliton and hairy black hole solutions of Einstein-non-Abelian-Proca theory in asymptotically anti-de Sitter space-time with gauge group ${\mathfrak {su}}(2)$. For static, spherically symmetric configurations, we show that the gauge field must be purely magnetic, and solve the resulting field equations numerically. The equilibrium gauge field is described by a single function $ω(r)$, which must have at least one zero. The solitons and hairy black holes share many properties with the corresponding solutions in asymptotically flat space-time. In particular, all the solutions we st…
Using a recently developed perturbation formalism based on curvature quantities, we investigate the linear stability of black holes and solitons with Yang-Mills hair and a negative cosmological constant. We show that those solutions which have no linear instabilities under odd- and even- parity spherically symmetric perturbations remain stable under odd-parity, linear, non-spherically symmetric perturbations.
We construct exact hairy AdS soliton solutions in Einstein-dilaton gravity theory. We examine their thermodynamic properties and discuss the role of these solutions for the existence of first order phase transitions for hairy black holes. The negative energy density associated to hairy AdS solitons can be interpreted as the Casimir energy that is generated in the dual filed theory when the fermions are antiperiodic on the compact coordinate.
A brief review of recent research on soliton and black hole solutions of Einstein's equations with nonlinear field sources is presented and some open questions are pointed out.
Properties of the horizon mass of hairy black holes are discussed with emphasis on certain subtle and initially unexpected features. A key property suggests that hairy black holes may be regarded as `bound states' of ordinary black holes without hair and colored solitons. This model is then used to predict the qualitative behavior of the horizon properties of hairy black holes, to provide a physical `explanation' of their instability and to put qualitative constraints on the end point configurations that result from this instability. The available numerical calculations support these predictio…
We study the stability of (4+1)-dimensional charged Gauss-Bonnet black holes and solitons. We observe an instability related to the condensation of a scalar field and construct explicit "hairy" black hole and soliton solutions of the full system of coupled field equations. We investigate the cases of a massless scalar field as well as that of a tachyonic scalar field. The solitons with scalar hair exist for a particular range of the charge and the gauge coupling. This range is such that for intermediate values of the gauge coupling a "forbidden band" of charges for the hairy solitons exists. W…
Scalar fields minimally coupled to General Relativity in three dimensions are considered. For certain families of self-interaction potentials, new exact solutions describing solitons and hairy black holes are found. It is shown that they fit within a relaxed set of asymptotically AdS boundary conditions, whose asymptotic symmetry group coincides with the one for pure gravity and its canonical realization possesses the standard central extension. Solitons are devoid of integration constants and their (negative) mass, fixed and determined by nontrivial functions of the self-interaction couplings…
Hairy black holes and gravitational solitons in three dimensions for the new massive gravity theory proposed by Bergshoeff, Hohm and Townsend (BHT) are considered at the special case when there is a unique maximally symmetric solution. Following the Brown-York approach with suitable counterterms, it is shown that the soliton possesses a fixed negative mass which coincides with the one of AdS spacetime regardless the value of the integration constant that describes it. Hence, the soliton can be naturally regarded as a degenerate ground state labeled by a single modulus parameter. The Euclidean …
General Relativity coupled to a self-interacting scalar field in three dimensions is shown to admit exact analytic soliton solutions, such that the metric and the scalar field are regular everywhere. Since the scalar field acquires slow fall-off at infinity, the soliton describes an asymptotically AdS spacetime in a relaxed sense as compared with the one of Brown and Henneaux. Nevertheless, the asymptotic symmetry group remains to be the conformal group, and the algebra of the canonical generators possesses the standard central extension. For this class of asymptotic behavior, the theory also …
We study the existence of soliton and black hole solutions of four-dimensional su(N) Einstein-Yang-Mills theory with a negative cosmological constant. We prove the existence of non-trivial solutions for any integer N, with N-1 gauge field degrees of freedom. In particular, we prove the existence of solutions in which all the gauge field functions have no zeros. For fixed values of the parameters (at the origin or event horizon, as applicable) defining the soliton or black hole solutions, if the magnitude of the cosmological constant is sufficiently large, then the gauge field functions all hav…
We study black hole solutions of Einstein gravity coupled to a specific global symmetry breaking Goldstone model described by an O(3) isovector scalar field in four spacetime dimensions. Our configurations are static and spherically symmetric, approaching at infinity a Minkowski spacetime background. A set of globally regular, particle-like solutions are found in the limit of vanishing event horizon radius. These configurations can be viewed as 'regularised' global monopoles, since their mass is finite and the spacetime geometry has no deficit angle. As an unusual feature, we notice the existe…
A numerical analysis shows that a class of scalar-tensor theories of gravity with a scalar field minimally and nonminimally coupled to the curvature allows static and spherically symmetric black hole solutions with scalar-field hair in asymptotically flat spacetimes. In the limit when the horizon radius of the black hole tends to zero, regular scalar solitons are found. The asymptotically flat solutions are obtained provided that the scalar potential $V(ϕ)$ of the theory is not positive semidefinite and such that its local minimum is also a zero of the potential, the scalar field settling asym…
We investigate the stability of spherically symmetric, purely magnetic, soliton and black hole solutions of four-dimensional ${\mathfrak {su}}(N)$ Einstein-Yang-Mills theory with a negative cosmological constant $Λ$. These solutions are described by $N-1$ magnetic gauge field functions $ω_{j}$. We consider linear, spherically symmetric, perturbations of these solutions. The perturbations decouple into two sectors, known as the sphaleronic and gravitational sectors. For any $N$, there are no instabilities in the sphaleronic sector if all the magnetic gauge field functions $ω_{j}$ have no zeros,…
Can a black hole that suffers a superradiant instability evolve towards a 'hairy' configuration which is stable? We address this question in the context of Einstein-charged scalar field theory. First, we describe a family of static black hole solutions which possess charged scalar-field hair confined within a mirror-like boundary. Next, we derive a set of equations which govern the linear, spherically symmetric perturbations of these hairy solutions. We present numerical evidence which suggests that, unlike the vacuum solutions, the (single-node) hairy solutions are stable under linear perturb…
We find hairy black holes of Einstein-Maxwell theory with a complex scalar field that is confined inside a box in a Minkowski background. These regular hairy black holes are asymptotically flat and thus the presence of the box or mirror allows to evade well-known no-hair theorems. We also find the Israel surface stress tensor that the confining box must have to obey the energy conditions. In the zero horizon radius limit, these hairy black holes reduce to a regular asymptotically flat hairy soliton. We find our solutions using perturbation theory. At leading order, a hairy black hole can be se…
We investigate the existence of black hole and soliton solutions to four dimensional, anti-de Sitter (adS), Einstein-Yang-Mills theories with general semisimple connected and simply connected gauge groups, concentrating on the so-called "regular" case. We here generalise results for the asymptotically flat case, and compare our system with similar results from the well-researched adS $\mathfrak{su}(N)$ system. We find the analysis differs from the asymptotically flat case in some important ways: the biggest difference is that for $Λ<0$, solutions are much less constrained as $r\rightarrow\inft…
The Isolated Horizons (IH) formalism, together with a simple phenomenological model for colored black holes has been used to predict non-trivial formulae that relate the ADM mass of the solitons and hairy Black Holes of Gravity-Matter system on the one hand, and several horizon properties of the black holes in the other. In this article, the IH formalism is tested numerically for spherically symmetric solutions to an Einstein-Higgs system where hairy black holes were recently found to exist. It is shown that the mass formulae still hold and that, by appropriately extending the current model, o…
Einstein gravity minimally coupled to a scalar field with a two-parameter Higgs-like self-interaction in three spacetime dimensions is recast in terms of a Chern-Simons form for the algebra $g^{+}\oplus g^{-}$ where, depending on the sign of the self-interaction couplings, $g^{\pm}$ can be $so(2,2)$, $so(3,1)$ or $iso(2,1)$. The field equations can then be expressed through the field strength of non-flat composite gauge fields, and conserved charges are readily obtained from boundary terms in the action that agree with those of standard Chern-Simons theory for pure gravity, but with non-flat c…
We present a spinning hairy black hole in gravity minimally coupled to a self-interacting real scalar field in three spacetime dimensions. The black hole is characterized by having a single horizon which encloses a curvature singularity and the scalar field is regular everywhere. The mass and angular momentum are shown to be finite. The presence of a scalar field with a slower fall-off at infinity leads an anti-de Sitter asymptotic behavior which differs from the one found by Brown and Henneaux, but has the same symmetry group as in pure gravity. A scalar soliton, which is a finite mass regula…
Echoes of a hairy black hole from gravitational decoupling
Guizhou University of Finance and Economics（贵州财经大学）；Eastern Mediterranean University（东地中海大学）；Universit degli Studi di Salerno（萨莱诺大学）；Istituto Nazionale di Fisica Nucleare - Gruppo Collegato di Salerno - Sezione di Napoli（意大利国家核物理研究所 - 萨莱诺协作组 - 那不勒斯分部）；Guizhou Minzu University（贵州民族大学）；Guizhou University（贵州大学）
The Einstein-Klein-Gordon Lagrangian is supplemented by a non-minimal coupling of the scalar field to specific geometric invariants : the Gauss-Bonnet term and the Chern-Simons term. The non-minimal coupling is chosen as a general quadratic polynomial in the scalar field and allows - depending on the parameters - for large families of hairy black holes to exist. These solutions are characterized, namely, by the number of nodes of the scalar function. The fundamental family encompasses black holes whose scalar hairs appear spontaneously and solutions presenting shift-symmetric hairs. When suppl…
We study the thermodynamic behavior of static and spherically symmetric hairy black holes in massive gravity. In this case, the black hole is surrounded in a spherical cavity with a fixed temperature on the surface. It is observed that these black holes have a phase transition similar to the liquid-gas phase transition of a Van der Waals fluid. Also, by treating the cosmological constant $Λ$ as a thermodynamic pressure $P$, we study the thermodynamic behavior of charged anti-de Sitter black holes in an ensemble with a pressure of $P$ and an electric potential $Φ$ as the natural variables. A se…
We construct static, asymptotically flat black hole solutions with scalar hair. They evade the no-hair 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.
We investigate static, planar, solutions of Einstein-scalar gravity admitting an anti-de Sitter (AdS) vacuum. When the squared mass of the scalar field is positive and the scalar potential can be derived from a superpotential, minimum energy theorems indicate the existence of a scalar soliton. On the other hand, for these models, no-hair theorems forbid the existence of hairy black brane solutions with AdS asymptotics. By considering a specific example (an exact integrable model which has the form of a Toda molecule) and by deriving explicit exact solution, we show that these models allow for …
We study higher-dimensional soliton and hairy black hole solutions of the Einstein equations non-minimally coupled to a scalar field. The scalar field has no self-interaction potential but a cosmological constant is included. Non-trivial solutions exist only when the cosmological constant is negative and the constant governing the coupling of the scalar field to the Ricci scalar curvature is positive. At least some of these solutions are stable when this coupling constant is not too large.
We present different Taub-NUT/Bolt-anti de Sitter (AdS) solutions in a shift-symmetric sector of Horndeski theory of gravity possessing nonminimal kinetic coupling of scalar fields to the Einstein tensor. In four dimensions, we find locally and asymptotically locally AdS solutions possessing nontrivial scalar field. In higher dimensions, analytical Taub-NUT/Bolt-AdS p-branes and solitons are obtained, supported by the existence of p Horndeski scalar fields with axionic profile. The thermodynamical properties are studied through Euclidean methods and it is found that the first law of thermodyna…
We study the existence of hairy black holes in the generalized Einstein-Skyrme model. It is proven that in the BPS model limit there are no hairy black hole solutions, although the model admits gravitating (and flat space) solitons. Furthermore, we find strong evidence that a necessary condition for the existence of black holes with Skyrmionic hair is the inclusion of the Skyrme term $\mathcal{L}_4$. As an example, we show that there are no hairy black holes in the $\mathcal{L}_2+\mathcal{L}_6+\mathcal{L}_0$ model and present a new kind of black hole solutions with compact Skyrmion hair in the…
We investigate dyonic black hole and dyon solutions of four-dimensional $SU(N)$ Einstein-Yang-Mills theory with a negative cosmological constant. We derive a set of field equations in this case, and prove the existence of non-trivial solutions to these equations for any integer $N$, with $2N-2$ gauge degrees of freedom. We do this by showing that solutions exist locally at infinity, and at the event horizon for black holes and the origin for solitons. We then prove that we can patch these solutions together regularly into global solutions that can be integrated arbitrarily far into the asympto…
We construct hairy black hole solutions that merge with the anti-de Sitter (AdS$_4$) Reissner-Nordström black hole at the onset of superradiance. These hairy black holes have, for a given mass and charge, higher entropy than the corresponding AdS$_4$-Reissner-Nordström black hole. Therefore, they are natural candidates for the endpoint of the charged superradiant instability. On the other hand, hairy black holes never dominate the canonical and grand-canonical ensembles. The zero-horizon radius of the hairy black holes is a soliton (i.e. a boson star under a gauge transformation). We construct…
We study small charged black holes in global AdS spacetime in the presence of a charged massless minimally coupled scalar field. In a certain parameter range these black holes suffer from well known superradiant instabilities. We demonstrate that the end point of the resultant tachyon condensation process is a hairy black hole which we construct analytically in a perturbative expansion in the black hole radius. At leading order our solution is a small undeformed RNAdS black hole immersed into a charged scalar condensate that fills the AdS `box'. These hairy black hole solutions appear in a two…
We show that, independently of the scalar field potential and of specific asymptotic properties of the spacetime (asymptotically flat, de Sitter or anti-de Sitter), any static, spherically symmetric or planar, black hole or soliton solution of the Einstein theory minimally coupled to a real scalar field with a general potential is mode stable under linear odd-parity perturbations. To this end, we generalize the Regge-Wheeler equation for a generic self-interacting scalar field, and show that the potential of the relevant Schrödinger operator can be mapped, by the so-called S-deformation, to a …
We consider the $\mathrm{O}(3)$ non-linear sigma-model, composed of three real scalar fields with a standard kinetic term and with a symmetry breaking potential in four spacetime dimensions. We show that this simple, geometrically motivated model, admits both self-gravitating, asymptotically flat, non-topological solitons and hairy black holes, when minimally coupled to Einstein's gravity, $without$ the need to introduce higher order kinetic terms in the scalar fields action. Both spherically symmetric and spinning, axially symmetric solutions are studied. The solutions are obtained under a an…
We investigate stationary rotationally symmetric solutions of a particular truncation of Horndeski theory in three dimensions, including a non-minimal scalar kinetic coupling to the curvature. After discussing the special case of a vanishing scalar charge, which includes most of the previously known solutions, we reduce the general case to an effective mechanical model in a three-dimensional target space. We analyze the possible near-horizon behaviors, and conclude that black hole solutions with degenerate horizons and constant curvature asymptotics may exist if the minimal and non-minimal sca…
We study soliton and black hole solutions of Einstein charged scalar field theory in cavity. We examine the effect of introducing a scalar field mass on static, spherically symmetric solutions of the field equations. We focus particularly on the spaces of soliton and black hole solutions, as well as studying their stability under linear, spherically symmetric perturbations of the metric, electromagnetic field, and scalar field.
We find the phase diagram of solutions of the charged black hole bomb system. In particular, we find the static hairy black holes of Einstein-Maxwell-Scalar theory confined in a Minkowski box. We impose boundary conditions such that the scalar field vanishes at and outside a cavity of constant radius. These hairy black holes are asymptotically flat with a scalar condensate floating above the horizon. We identify four critical scalar charges which mark significant changes in the qualitative features of the phase diagram. When they coexist, hairy black holes always have higher entropy than the R…
We consider the static charged black hole bomb system, originally designed for a (uncharged) rotating superradiant system by Press and Teukolsky. A charged scalar field confined in a Minkowski cavity with a Maxwell gauge field has a quantized spectrum of normal modes that can fit inside the box. Back-reacting non-linearly these normal modes, we find the hairy solitons, a.k.a boson stars (depending on the chosen $U(1)$ gauge), of the theory. The scalar condensate is totally confined inside the box and, outside it, we have the Reissner-Nordström solution. The Israel junction conditions at the bo…
We show that scalar hair can be added to rotating, vacuum black holes of general relativity. These hairy black holes (HBHs) clarify a lingering question concerning gravitational solitons: if a black hole can be added at the centre of a boson star, as it typically can for other solitons. We argue that it can, but only if it is spinning. The existence of such HBHs is related to the Kerr superradiant instability triggered by a massive scalar field. This connection leads to the following conjecture: a (hairless) black hole which is afflicted by the superradiant instability of a given field must al…
We present new regular solutions of Einstein-charged scalar field theory in a cavity. The system is enclosed inside a reflecting mirror-like boundary, on which the scalar field vanishes. The mirror is placed at the zero of the scalar field closest to the origin, and inside this boundary our solutions are regular. We study the stability of these solitons under linear, spherically symmetric perturbations of the metric, scalar and electromagnetic fields. If the radius of the mirror is sufficiently large, we present numerical evidence for the stability of the solitons. For small mirror radius, som…
We present new soliton and hairy black hole solutions of su(N) Einstein-Yang-Mills theory in asymptotically anti-de Sitter space. These solutions are described by N+1 independent parameters, and have N-1 gauge field degrees of freedom. We examine the space of solutions in detail for su(3) and su(4) solitons and black holes. If the magnitude of the cosmological constant is sufficiently large, we find solutions where all the gauge field functions have no zeros. These solutions are of particular interest because we anticipate that at least some of them will be linearly stable.
The stability of black holes and solitons in d-dimensional Anti-de Sitter space-time against scalar field condensation is discussed. The resulting solutions are "hairy" black holes and solitons, respectively. In particular, we will discuss static black hole solutions with hyperbolic, flat and spherical horizon topology and emphasize that two different type of instabilities exist depending on whether the scalar field is charged or uncharged, respectively. We will also discuss the influence of Gauss-Bonnet curvature terms. The results have applications within the AdS/CFT correspondence and descr…
The thermodynamic properties of a static and spherically symmetric hairy black hole solution arising in massive gravity with spontaneous Lorentz breaking are investigated. The analysis is carried out by enclosing the black hole in a spherical cavity whose surface is maintained at a fixed temperature $T$. It turns out that the ensemble is well-defined only if the "hair" parameter $Q$ characterizing the solution is conserved. Under this condition we compute some relevant thermodynamic quantities, such as the thermal energy and entropy, and we study the stability and phase structure of the ensemb…
We study the stability of charged solitons in 5-dimensional Anti-de Sitter (AdS) space-time. We show that for appropriate choices of the parameters of the model these solutions become unstable to form scalar hair. We find that the existence of charged solitons with scalar hair depends crucially on the charge and the mass of the scalar field. We investigate the dependence of the spectrum of solutions on the mass of the scalar field in detail. For positive mass of the scalar field the hairy solitons can be interpreted as charged boson stars. We find that for sufficiently small value of the charg…
In this work, taking advantage of the Generalized Hedgehog Ansatz, we construct new self-gravitating solitons in a cavity with mirror-like boundary conditions for the SU(2) Non-linear Sigma Model and Skyrme model. For spherically symmetric spacetimes, we are able to reduce the system to three independent equations that are numerically integrated. There are two branches of well-behaved solutions. The first branch is defined for arbitrary values of the Skyrme coupling and therefore also leads to a gravitating soliton in the Non-linear Sigma Model, while the second branch exists only for non-vani…
Press and Teukolsky famously introduced the concept of a black hole bomb system: a scalar field scattering a Kerr black hole confined inside a mirror undergoes superradiant amplification that keeps repeating due to the reflecting boundary conditions at the mirror. A similar charged black hole bomb system exists if we have a charged scalar field propagating in a Reissner-Nordstrom black hole confined inside a box. We point out that scalar fields propagating in such a background are unstable not only to superradiance but also to a mechanism known as the near-horizon scalar condensation instabili…
We study the stability of static as well as of rotating and charged black holes in (4+1)-dimensional Anti-de Sitter space-time which possess spherical horizon topology. We observe a non-linear instability related to the condensation of a charged, tachyonic scalar field and construct "hairy" black hole solutions of the full system of coupled Einstein, Maxwell and scalar field equations. We observe that the limiting solution for small horizon radius is either a hairy soliton solution or a singular solution that is not a regular extremal solution. Within the context of the gauge/gravity duality t…
