We consider a gravitating system of vanishing cosmological constant consisting of an electromagnetic field and a scalar field coupled to the Einstein tensor. A Reissner-Nordström black hole undergoes a second-order phase transition to a hairy black hole of generally anisotropic hair at a certain critical temperature which we compute. The no-hair theorem is evaded due to the coupling between the scalar field and the Einstein tensor. Within a first order perturbative approach we calculate explicitly the properties of a hairy black hole configuration near the critical temperature and show that it is energetically favorable over the corresponding Reissner-Nordström black hole.
We study the Einstein-Born-Infeld (EBI) theory where the hairy black hole was found as the EBI black hole by using Hod's idea. The hair extends beyond the photon-sphere of the EBI black hole spacetimes. We show that the region beyond the photon-sphere involves more than half of the total hair's mass, which respects Hod's conjectured bound.
We investigate the existence of scalar hair in the background of a reflecting star. The kinetic part of the scalar field is nonminimally coupled to the Einstein tensor. For both the Dirichlet and the Neumann boundary conditions, we study the no-hair theorem. For the massless case, we show that there is no scalar hair, and in the massive case conditions leading to the no-hair theorem are specified.
We establish the existence of hairy black holes in su(N) Einstein-Yang-Mills theories, described by N-1 parameters, corresponding to the nodes of the gauge field functions.
We obtain a newly charged BBMB (Bocharova-Bronnikov-Melnikov-Bekenstein) black hole solution from the Einstein-Weyl-Maxwell-conformal (conformally coupled) scalar theory with a positive Weyl coupling parameter $m^2_2$ numerically. This solution has a primary scalar hair, compared to a secondary scalar hair in the charged BBMB black hole solution. The limiting case of $m_2^2\to\infty$ leads to the series form of the charged BBMB black hole solution. However, for $m^2_2<0$, we could not obtain any asymptotically flat black hole solution with scalar hair.
We study a new family of self-gravitating axially-symmetric, rotating boson stars in the two-component Friedberg-Lee-Sirlin model minimally coupled to Einstein gravity in four spacetime dimensions. In the flat space limit the boson stars tend to the corresponding Q-balls. Subject to the usual synchronization condition, the model admits spinning hairy black hole solutions with two different types of scalar hair. We here investigate parity-even and parity-odd boson stars and their associated hairy black holes. We explore the domain of existence of the solutions and address some of their physical…
Previous studies showed that, in the presence of a simple and well-motivated self-interaction scalar potential, asymptotically flat and spherical black holes can carry minimally coupled and charged scalar cloud/hair in Einstein-Maxwell gravity. We extend these studies to Einstein-Born-Infeld gravity to consider the effect of nonlinearity of the electromagnetic field. Series of spherical cloudy/hairy black hole solutions are constructed numerically. Results show that increasing the Born-Infeld coupling constant $b$ will make the domain of existence of the solution shrink or even disappear when …
An exact hairy asymptotically locally AdS black hole solution with a flat horizon in the Einstein-nonlinear sigma model system in (3+1) dimensions is constructed. The ansatz for the nonlinear $SU(2)$ field is regular everywhere and depends explicitly on Killing coordinates, but in such a way that its energy-momentum tensor is compatible with a metric with Killing fields. The solution is characterized by a discrete parameter which has neither topological nor Noether charge associated with it and therefore represents a hair. A $U(1)$ gauge field interacting with Einstein gravity can also be incl…
In this paper, we are interested in a model of exact asymptotically flat charged hairy black holes in the background of dilaton potential. We study the weak gravitational lensing in the spacetime of hairy black hole in Einstein-Maxwell theory with a non-minimally coupled dilaton and its non-trivial potential. In doing so, we use the optical geometry of the flat charged hairy black hole for some range of parameter $γ$. For this purpose, by using Gauss-Bonnet theorem, we obtain the deflection angle of photon in a spherically symmetric and asymptotically flat spacetime. Moreover, we also investig…
We study hair mass distributions in noncommutative Einstein-Born-Infeld hairy black holes with non-zero cosmological constants. We find that the larger noncommutative parameter makes the hair easier to condense in the near horizon area. We also show that Hod's lower bound can be evaded in the noncommutative gravity. However, for large black holes with a non-negative cosmological constant, Hod's lower hair mass bound almost holds in the sense that nearly half of the hair lays above the photonsphere.
We obtain scalar hairy black holes from Einstein-Maxwell-conformally coupled scalar (EMCS) theory with the scalar coupling parameter $α$ to the Maxwell term. In case of $α=0$, the $α=0$ EMCS theory provides constant (charged) scalar hairy black hole and charged BBMB (Bocharova-Bronnikov-Melnikov-Bekenstein) black hole where the former is stable against full perturbations, while the latter remains unstable because it belongs to an extremal black hole. It is noted that for $α\not=0$, the unstable Reissner-Nordström black holes without scalar hair imply infinite branches of $n=0(α\ge 8.019),1(α\g…
Recently, a no inner (Cauchy) horizon theorem for static black holes with non-trivial scalar hairs has been proved in Einstein-Maxwell-scalar theories. In this paper, we extend the theorem to the static black holes in Einstein-Maxwell-Horndeski theories. We study the black hole interior geometry for some exact solutions and find that the spacetime has a (space-like) curvature singularity where the black hole mass gets an extremum and the Hawking temperature vanishes. We discuss further extensions of the theorem, including general Horndeski theories from disformal transformations.
In Einstein-Maxwell theory black holes are uniquely determined by their mass, their charge and their angular momentum. This is no longer true in Einstein-Yang-Mills theory. We discuss sequences of neutral and charged SU(N) Einstein-Yang-Mills black holes, which are static spherically symmetric and asymptotically flat, and which carry Yang-Mills hair. Furthermore, in Einstein-Maxwell theory static black holes are spherically symmetric. We demonstrate that, in contrast, SU(2) Einstein-Yang-Mills theory possesses a sequence of black holes, which are static and only axially symmetric.
Inspired by possibility that the Schwarzschild black hole may not be the unique spherically symmetric vacuum solution to generalisations of general relativity, we consider black holes in pure fourth order higher derivative gravity treated as an effective theory. Such solutions may be of interest in addressing the issue of higher derivative hair or during the later stages of black hole evaporation. Non-Schwarzschild solutions have been studied but we have put earlier results on a firmer footing by finding a systematic asymptotic expansion for the black holes and matching them with known numeric…
In the context of the Einstein-scalar-Gauss-Bonnet theory, with a general coupling function between the scalar field and the quadratic Gauss-Bonnet term, we investigate the existence of regular black-hole solutions with scalar hair. Based on a previous theoretical analysis, that studied the evasion of the old and novel no-hair theorems, we consider a variety of forms for the coupling function (exponential, even and odd polynomial, inverse polynomial, and logarithmic) that, in conjunction with the profile of the scalar field, satisfy a basic constraint. Our numerical analysis then always leads …
The stability of a new class of hairy black hole solutions in the coupled system of Einstein-Yang-Mills-Higgs is examined, generalising a method suggested by Brodbeck and Straumann and collaborators, and Volkov and Gal'tsov. The method maps the algebraic system of linearised radial perturbations of the various field modes around the black hole solution into a coupled system of radial equations of Schrödinger type. No detailed knowledge of the black hole solution is required, except from the fact that the boundary conditions at the physical space-time boundaries (horizons) must be such so as to…
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 explore charged black hole solutions in Einstein-Gauss-Bonnet gravity in five dimensions, with a charged scalar hair. We interpret such hairy black holes as the final state of the superradiant instability previously reported for this system. We explore the relation of the hairy black hole solutions with the non-backreacting quasibound states and scalar clouds, as well as with the boson star solutions.
The spinning-hairy black holes that occur in Einstein gravity supplemented by a doublet of complex scalar fields are constructed within an extension of the model by a $U(1)$ gauge symmetry involving a massless vector potential. The hairy black holes then acquire an electric charge and a magnetic moment; their domain of existence is discussed in terms of the gauge coupling constant.
We study a hairy black hole solution in the dilatonic Einstein-Gauss-Bonnet theory of gravitation, in which the Gauss-Bonnet term is non-minimally coupled to the dilaton field. Hairy black holes with spherical symmetry seem to be easily constructed with a positive Gauss-Bonnet coefficient $α$ within the coupling function, $f(ϕ) = αe^{γϕ}$, in an asymptotically flat spacetime, i.e., no-hair theorem seems to be easily evaded in this theory. Therefore, it is natural to ask whether this construction can be expanded into the case with the negative coefficient $α$. In this paper, we present numerica…
The Einstein-Gauss-Bonnet gravity in five dimensions is extended by scalar fields and the corresponding equations are reduced to a system of non-linear differential equations. A large family of regular solutions of these equations is shown to exist. Generically, these solutions are spinning black holes with scalar hairs. They can be characterized (but not uniquely) by an horizon and an angular velocity on this horizon. Taking particular limits the black holes approach boson star or become extremal, in any case the limiting configurations remain hairy.
We study mass bounds of Maxwell fields in RN black holes and genuine hair in Einstein-Born-Infeld black holes with various cosmological constants. It shows that the Maxwell field serves as a good probe to disclose the hair distribution described with the event horizon and the photonsphere. And we find that the Hod's lower bound obtained in asymptotically flat space also holds in the asymptotically dS Einstein-Born-Infeld hairy black holes. In contrast, the Hod's lower bound can be invaded in the asymptotically AdS Einstein-Born-Infeld hairy black holes. It implies that the AdS boundary could m…
Recently, spontaneous scalarization of charged black holes has attracted a great deal of attention and motivated several studies of Einstein-Maxwell-scalar models. These studies have, however, only considered a massless and non-self-interacting scalar field. In this work a more realistic treatment of the problem is considered by studying the effects of scalar field mass and self-interacting terms on spontaneous scalarization in Einstein-Maxwell-scalar models, and on the string theory motivated dilatonic black hole. We assess the domains of existence of the black hole solutions, thermodynamic p…
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…
In regularized 4-dimensional Einstein-Gauss-Bonnet (EGB) gravity derived from a Kaluza-Klein reduction of higher-dimensional EGB theory, we study the existence and stability of black hole (BH) solutions on a static and spherically symmetric background. We show that asymptotically-flat hairy BH solutions realized for a spatially-flat maximally symmetric internal space are unstable against linear perturbations for any rescaled GB coupling constant. This instability is present for the angular propagation of even-parity perturbations both in the vicinity of an event horizon and at spatial infinity…
We studied, both analytically and numerically, an Abelian Higgs vortex in the spacetime of stationary axisymmetric black hole being the solution of the low-energy limit of the heterotic string theory, acted as hair on the black hole background. Taking into account the gravitational backreaction of the vortex we show that its influence is far more subtle than causing only a conical defect. As a consequence of its existence, we found that the ergosphere was shifted as well as event horizon angular velocity was affected by its presence. Numerical simulations reveal the strong dependence of the vo…
We investigate the stability analysis of Schwarzschild black hole in Einstein-scalar-Gauss-Bonnet (ESGB) theory because the instability of Schwarzschild black hole without scalar hair implies the Gauss-Bonnet black hole with scalar hair. The linearized scalar equation is compared to the Lichnerowicz-Ricci tensor equation in the Einstein-Weyl gravity. It turns out that the instability of Schwarzschild black hole in ESGB theory is interpreted as not the tachyonic instability, but the Gregory-Laflamme instability of black string. In the small mass regime of $1/λ<1.174/r_+$, the Schwarzschild solu…
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,…
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.
We obtain an exact static solution to Einstein-Power-Maxwell (EPM) theory in $(2+1)$ dimensional AdS spacetime, in which the scalar field couples to gravity in a non-minimal way. After considering the scalar potential, a stable system leads to a constraint on the power parameter $k$ of Maxwell field. The solution contains a curvature singularity at the origin and is non-conformally flat. The horizon structures are identified, which indicates the physically acceptable lower bound of mass in according to the existence of black hole solutions. Especially for the cases with $k>1$, the lower bound …
Apart from the Kerr-Newman black holes, the hairy black holes in general relativity have been widely investigated in the gravity and cosmology. In this paper, we extend the Sorce-Wald method to prove the weak cosmic censorship conjecture (WCCC) of the static and spherically symmetric hairy black holes in the Einstein gravity without using the explicit expressions of the metric and lagrangian for matter fields. We examine the WCCC in the collision process without requiring the spherically symmetry of the perturbation matter fields. After assuming the stability condition of spacetime and applyin…
General relativity admits a plethora of exact compact object solutions. The augmentation of Einstein's action with non-minimal coupling terms leads to modified theories with rich structure, which, in turn, provide non-trivial solutions with intriguing phenomenology. Thus, assessing their viability under generic fluctuations is of utmost importance for gravity theories. We consider static and spherically-symmetric solutions of a Horndeski subclass which includes a massless scalar field non-minimally coupled to the Einstein tensor. Such theory possesses second-order field equations and admits an…
We compute the quasinormal frequencies for scalar perturbations of hairy black holes in four-dimensional Einstein-Maxwell-dilaton theory assuming a non-trivial scalar potential for the dilaton field. We investigate the impact on the spectrum of the angular degree, the overtone number, the charges of the black hole as well as the magnitude of the scalar potential. All modes are found to be stable. Our numerical results are summarized in tables, and for better visualization, we show them graphically as well.
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…
We study separatively the quasinormal modes (QNM) of electromagnetic perturbations around three-dimensional anti-de Sitter(AdS) black holes in Jordan and Einstein frames, which are related by the conformal transformations and a redefinition of a scalar field. We find that, in the Jordan frame, the imaginary parts of QNM frequencies can reflect the thermodynamical stabilities of hairy black holes, including the possible phase transition between the hairy black hole and BTZ black hole, disclosed by examining the corresponding free energies. Similar results are also uncovered in the Einstein fram…
We discuss a phase transition of a Reissner-Nordström black hole to a hairy black hole in asymptotically flat spacetime. The hair is due to a massive charged scalar field. The no-hair theorem is evaded thanks to a derivative coupling of the scalar field to the Einstein tensor. The resulting hairy configuration is spherically symmetric. We solve the equations analytically near the transition temperature and show that the hair is concentrated near the horizon decaying exponentially away from it.
We show that, unlike vacuum General Relativity, Einstein-scalar theories allow balanced static, neutral, asymptotically flat, double-black hole solutions, for scalar field models minimally coupled to gravity, with appropriate self-interactions. These are scalar hairy versions of the double-Schwarzschild (or Bach-Weyl) solution, but regular on and outside the two (topologically spherical) horizons. The balancing repulsive force is provided by the scalar field. An explicit illustration is presented, using a Weyl-type construction adapted to numerical solutions, requiring no partial linearisation…
We investigate the interior of the Einstein-Gauss-Bonnet charged black-hole with scalar hair. We find a variety of dynamical epochs, with the particular important feature that the Cauchy horizon is not present. This makes the violation of the no-hair theorem a possible tool to understand how might the strong cosmic censorship conjecture work.
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…
We present spherically symmetric black hole solutions for Einstein gravity coupled to anisotropic matter. We show that these black holes have arbitrarily short hair, and argue for stability by showing that they can arise from dynamical collapse. We also show that a recent `no short hair' theorem does not apply to these solutions.
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 …
Starting from the static, spherically symmetric black hole solutions in massless Einstein-Maxwell-dilaton (EMD) theories, we build a "skeleton" action, that is, we phenomenologically replace black holes by an appropriate effective point particle action, which is well suited to the formal treatment of the many-body problem in EMD theories. We find that, depending crucially on the value of their scalar cosmological environment, black holes can undergo steep "scalarization" transitions, inducing large deviations to the general relativistic two-body dynamics, as shown, for example, when computing …
We discuss the uniqueness of the static black hole in the Einstein gravity with a conformally coupled scalar field. In particular, we prove the uniqueness of the region outside of the photon surface, not event horizon.
Einstein-Gauss-Bonnet-gravity (EGB) coupled minimally to a $U(1)$ gauged, massive scalar field possesses -- for appropriate choices of the $U(1)$ charge -- black hole solutions that carry charged scalar hair if the frequency of the harmonic time-dependence of the scalar field is equal to the upper bound on the superradiant frequency. The existence of these solutions has first been discussed in \cite{Grandi:2017zgz}. In this paper, we demonstrate that the critical value of the scalar charge results from the requirement of non-extremality of the charged black hole solutions and the fact that the…
The time-like and null-like geodesics around compact objects are some of the best tools to classify and understand the structure of a space-time. In this paper, we study the null geodesics around charged static dilaton black holes in Einstein-Maxwell-dilaton gravity. The physical parameters for non-radial geodesics including the effective potential, effective force, radius of the photon sphere and impact parameter are obtained and effects of the charge parameter and dilaton coupling constant on these quantities are studied. Possible photon motions for different values of the impact parameter a…
The specific nonlinear vector $σ$-model coupled to Einstein gravity is investigated. The model arises in the studies of the gravitating matter in non-commutative geometry. The static spherically symmetric spacetimes are identified by direct solving of the field equations. The asymptotically flat black hole with the ``non-commutative'' vector hair appears for the special choice of the integration constants, giving thus another counterexample to the famous ``no-hair'' theorem.
We analyze (3+1)-dimensional black-hole space-times in spontaneously broken Yang-Mills gauge theories that have been recently presented as candidates for an evasion of the scalar-no-hair theorem. Although we show that in principle the conditions for the no-hair theorem do not apply to this case, however we prove that the `spirit' of the theorem is not violated, in the sense that there exist instabilities, in both the sphaleron and gravitational sectors. The instability analysis of the sphaleron sector, which was expected to be unstable for topological reasons, is performed by means of a variat…
We construct static axially symmetric black holes in SU(2) Einstein-Yang-Mills-Higgs theory. Located in between a monopole-antimonopole pair, these black holes possess magnetic dipole hair. The difference of their mass and their horizon mass equals the mass of the regular monopole-antimonopole solution, as expected from the isolated horizon framework.
The strong gravitational lensing for charged black holes with scalar hair in Einstein-Maxwell-Dilaton theory are studied. We find, with the increase of scalar hair, that the radius of the photon sphere, minimum impact parameter, angular image position and relative magnitude increase, while the deflection angle and angular image separation decrease. Our results can be reduced to those of the Schwarzschild black hole in two cases, one of them is that the scalar hair disappears, the other is that the coupling constants take particular values with arbitrary scalar hair.
