Einstein-Maxwell-scalar black holes with massive and self-interacting scalar hair
Pedro G. S. Fernandes
备注：
15 pages, 10 figures; V2: references and extra discussion added; V3: extra discussion added, including thorough radial stability analysis. Accepted for publication in Physics of the Dark Universe. arXiv admin note: text overlap with arXiv:1908.00037
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 preference and radial profiles. Then we discuss on the stability of the scalarized solutions, focusing on spherical perturbations, finding all studied solutions to be stable.
Treating Maxwell soft hair as a transition function that relates U(1) gauge fields living in the asymptotic region and near-horizon region, the U(1) gauge parameter $ϕ$ naturally becomes a good label of those Maxwell soft hair degrees of freedom. This interpretation also builds the connection between Maxwell soft hair and U(1) edge modes living in the intermediate region, which admits a well-defined effective action description. We study the statistical properties by Euclidean path integral, which concludes that the soft hair density of state increases with black hole temperature. Hawking radi…
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 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…
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…
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.
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 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 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…
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 …
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 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 …
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.
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…
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 …
Is it possible the no-hair paradigm is violated when a black hole undergoes an instability? Employing massless charged scalar perturbations, we address this question within the context of conformally invariant Einstein-Maxwell theory for some allowed $d$-dimensional ($d=4n+4$ with conformal parameter $n=0,1,2,...$ ) topological small AdS black-holes. We provide numerical analyses that show the non trivial scalar hairy black hole solutions to include planar and spherical horizon topologies in higher dimensions with an even conformal parameter $n$. It is also shown that the solutions presented h…
We analyze condensation behaviors of neutral scalar fields outside horizonless reflecting stars in the Einstein-Maxwell-scalar gravity. It was known that minimally coupled neutral scalar fields cannot exist outside horizonless reflecting stars. In this work, we consider non-minimal couplings between scalar fields and Maxwell fields, which is included to aim to trigger formations of scalar hairs. We analytically demonstrate that there is no hair theorem for small coupling parameters below a bound. For large coupling parameters above the bound, we numerically obtain regular scalar hairy configur…
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.
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…
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.
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…
We study the process of fully nonlinear dynamical scalarization starting from a charged black hole or a naked singularity in asymptotically flat spacetime in the Einstein-Maxwell-dilaton theory. Initially the dilaton field is negligible compared to the gravitational and the Maxwell field. Then the dilaton field experiences an immediate growth, later it oscillates with damping amplitude and finally settles down to a finite value. For a hairy black hole develops from an original Reissner-Nordström black hole, since the dilaton oscillation and decay are almost independent of the coupling paramete…
The cosmic no hair conjecture is tested in the spherically symmetric Einstein-Maxwell-dilaton~(EMD) system with a positive cosmological constant $Λ$. Firstly, we analytically show that once gravitational collapse occurs in the massless dilaton case, the system of field equations breaks down inevitably in outer communicating regions or at the boundary provided that a future null infinity ${\cal I}^+$ exists. Next we find numerically the static black hole solutions in the massive dilaton case and investigate their properties for comparison with the massless case. It is shown that their Abbott-De…
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 study standard Einstein-Maxwell theory minimally coupled to a complex valued and self-interacting scalar field. We demonstrate that new, previously unnoticed spherically symmetric, charged black hole solutions with scalar hair exist in this model for sufficiently large gravitational coupling and sufficiently small electromagnetic coupling. The novel scalar hair has the form of a spatially oscillating ''wave packet'' and back-reacts on the space-time such that both the Ricci and the Kretschmann scalar, respectively, possess qualitatively similar oscillations.
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.
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.
In this paper, we study thermodynamics and phase structure of asymptotically AdS hairy and Reissner-Nordström-AdS (RNAdS) black holes in the extended phase space, where the cosmological constant is interpreted as a thermal pressure. The RNAdS and hairy black holes are black hole solutions of an Einstein-Maxwell-scalar (EMS) model with a non-minimal coupling between the scalar and electromagnetic fields. The Smarr relation, the first law of thermodynamics and the free energy are derived for black hole solutions in the EMS model. Moreover, the phase structure of the RNAdS and hairy black holes i…
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…
We study the extended phase space thermodynamics for hairy AdS black hole solutions to Einstein-Maxwell-$Λ$ theory conformally coupled to a scalar field in five dimensions. We find these solutions to exhibit van der Waals behaviour in both the charged/uncharged cases, and reentrant phase transitions in the charged case. This is the first example of reentrant phase transitions in a five dimensional gravitational system which does not include higher curvature corrections.
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 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…
The scalarization of Reissner-Nordström black holes was recently proposed in the Einstein-Maxwell-scalar theory. Here, we show that the appearance of the scalarized Reissner-Nordström black hole is closely related to the Gregory-Laflamme instability of the Reissner-Nordström black hole without scalar hair.
It has been previously shown that a Gauss-Bonnet term non-minimally coupled to a scalar field produces a scalarised black hole solution, which can be considered as having secondary scalar hair, parametrised in terms of the black hole's mass and charge. In this paper we extend a previously investigated linear coupling of the form $f(ϕ)=ϕ$ to a non-minimally coupled Maxwell term, with the form $\frac{1}{8}F_{μν}F^{μν}+βϕF_{μν}F^{μν}$. By using numerical methods the solutions to the full differential equations are found, as well as a perturbative expansion in the $r\rightarrow\infty$ limit and a …
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…
In the present paper we investigate the general problem of uniqueness of the stationary black solutions in 5D Einstein-Maxwell-dilaton gravity with arbitrary dilaton coupling parameter containing the Einstein-Maxwell gravity as a particular case. We formulate and prove uniqueness theorems classifying the stationary black hole solutions in terms of their interval structure, electric and magnetic charges and the magnetic fluxes. The proofs are based on the nonpositivity of the Riemann curvature operator on the space of the potentials which imposes restrictions on the dilaton coupling parameter.
Searches for {\it dark matter} sector field imprints on the astrophysical phenomena are one of the most active branches of the current researches. Using numerical methods we elaborate the influence of {\it dark matter} on the emergence of black hole hair and formation of boson stars. We explore thermodynamics of different states of the system in Einstein-Maxwell-scalar {\it dark matter} theory with box boundary conditions. Finally we find that the presence of dark sector within the system diminishes a chance of formation of scalar hair around a black hole.
We perform the stability analysis on scalarized charged black holes in the Einstein-Maxwell-Scalar (EMS) theory by computing quasinormal mode spectrum. It is noted that the appearance of these black holes with scalar hair is closely related to the instability of Reissner-Nordström black holes without scalar hair in the EMS theory. The scalarized black hole solutions are classified by the node number of $n=0,1,2,\cdots$, where $n=0$ is called the fundamental branch and $n=1,2,\cdots$ denote the $n$ excited branches. Here, we show that the $n=1,2$ excited black holes are unstable against against…
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,…
