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.
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 introduce a systematic and direct procedure to generate hairy rotating black holes by deforming a spherically symmetric seed solution. We develop our analysis in the context of the gravitational decoupling approach, without resorting to the Newman-Janis algorithm. As examples of possible applications, we investigate how the Kerr black hole solution is modified by a surrounding fluid with conserved energy-momentum tensor. We find non-trivial extensions of the Kerr and Kerr-Newman black holes with primary hair. We prove that a rotating and charged black hole can have the same horizon as Kerr'…
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 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…
We discuss the formation of scalar hair on Gauss-Bonnet solitons and black holes in 5-dimensional Anti-de Sitter (AdS) space-time. We present new results on the static case and point out further details. We find that the presence of the Gauss-Bonnet term has an influence on the pattern of soliton solutions for small enough values of the electric charge. We also discuss rotating Gauss-Bonnet black holes with and without scalar hair.
Regular hairy black holes through gravitational decoupling method
College of Physics Science and Technology, Hebei University（河北大学物理科学与技术学院）；Hebei Key Lab of Optic-Electronic Information and Materials, Hebei University（河北大学河北省光电信息材料重点实验室）；National-Local Joint Engineering Laboratory of New Energy Photoelectric Devices, Hebei University（河北大学新能源光电器件国家地方联合工程实验室）；Key Laboratory of High-pricision Computation and Application of Quantum Field Theory of Hebei Province, Hebei University（河北大学量子场论高精度计算与应用重点实验室）
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 …
We study Kerr black holes with synchronised non-trivial parity-odd massive scalar hair in four dimensional asymptotically flat space-time. These axially symmetric stationary spinning solutions of the minimally coupled Einstein-Klein-Gordon theory provide yet another example of bound states in synchronous rotation with the event horizon. We discuss the properties of these parity-odd hairy black holes and boson stars and exhibit their domain of existence. Considering the ergo-regions of these hairy black holes, we show that apart from the previously discussed ergo-sphere and ergo-Saturn, they su…
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…
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 construct static axially symmetric black holes in multi-Skyrmion configurations coupled to Einstein gravity in four dimensional asymptotically flat space-time. In a simplest case the event horizon is located in-between a Skyrmion-anti-Skyrmion pair, other solutions represent black holes with gravitationally bounded chains of Skyrmions and anti-Skyrmions placed along the axis of symmetry in alternating order. We discuss the properties of these hairy black holes and exhibit their domain of existence.
We study spherically symmetric soliton solutions in a model with a conformally coupled scalar field as well as in full conformal gravity. We observe that a new type of limiting behaviour appears for particular choices of the self-coupling of the scalar field, i.e. the solitons interpolate smoothly between the Anti-de Sitter vacuum and an uncharged configuration. Furthermore, within conformal gravity the qualitative approach of a limiting solution does not change when varying the charge of the scalar field - contrary to the Einstein-Hilbert case. However, it changes with the scalar self-couplin…
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 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…
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 …
The properties of light rays around compact objects surrounded by a plasma are affected by both strong gravitational fields described by a general-relativistic spacetime and by a dispersive and refractive medium, characterized by the density distribution of the plasma. We study these effects employing the relativistic Hamiltonian formalism under the assumption of stationarity and axisymmetry. The necessary and sufficient conditions on the metric and on the plasma frequency are formulated, such that the rays can be analytically determined from a fully separated Hamilton-Jacobi equation. We demo…
The recently introduced Isolated Horizons (IH) formalism has become a powerful tool for realistic black hole physics. In particular, it generalizes the zeroth and first laws of black hole mechanics in terms of quasi-local quantities and serves as a starting point for quantum entropy calculations. In this note we consider theories which admit hair, and analyze some new results that the IH provides, when considering solitons and stationary solutions. Furthermore, the IH formalism allows to state uniqueness conjectures (i.e. horizon `no-hair conjectures') for the existence of solutions.
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,…
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 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 a very interesting paper, Hod has proven that the equatorial null circular geodesic provides the extreme orbital period to circle a kerr black hole, which is closely related to the Fermat's principle. In the present paper, we extend the discussion to kerr black holes with scalar field hair. We show that the circle with the extreme orbital period is still identical to the null circular geodesic. Our analysis also implies that the Hod's theorem may be a general property in any axially symmetric spacetime with reflection symmetry on the equatorial plane.
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 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…
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 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…
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…
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 axially symmetric solutions of the Einstein-Maxwell-Klein-Gordon equations describing spinning gauged boson stars in a 3+1 dimensional asymptotically AdS spacetime. These smooth horizonless solutions possess an electric charge and a magnetic dipole moment, their angular momentum being proportional to the electric charge. A special class of solutions with a self-interacting scalar field, corresponding to static axially symmetric solitons with a nonzero magnetic dipole moment, is also investigated.
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 construct and analyse Kerr black holes (BHs) with synchronised axionic hair. These are the BH generalisations of the recently constructed rotating axion boson stars arXiv:2005.05982. Such BHs are stationary, axially symmetric, asymptotically flat solutions of the complex Einstein-Klein-Gordon theory with a QCD axion-like potential. They are regular everywhere on and outside the event horizon. The potential is characterised by two parameters: the mass of the axion-like particle, $m_a$ and the decay constant $f_a$. The limit $f_a \rightarrow \infty$ recovers the original example of Kerr BHs w…
This preprint mainly reflects the new chapter we are prepearing for the second edition of our book "Gravitational Solitons" (V. Belinski and E. Verdaguer). However, here it is written in the form of more or less self-consistent paper dedicated to the mathematical theory of black holes immersed in the external electromagnetic field. The purpose of this development is to describe the procedure by which one can construct a number of axially symmetric black holes or naked singularities arranged along the symmetry axis and immersed in an (also axially symmetric) external electromagnetic field.
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 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…
Matter fields don't necessarily have to share the symmetries with the spacetime they live in. When this happens, we speak of the symmetry inheritance of fields. In this paper we classify the obstructions of symmetry inheritance by the scalar fields, both real and complex, and look more closely at the special cases of stationary and axially symmetric spacetimes. Since the symmetry noninheritance is present in the scalar fields of boson stars and may enable the existence of the black hole scalar hair, our results narrow the possible classes of such solutions. Finally, we define and analyse the s…
We discuss charged and static solutions in a shift-symmetric scalar-tensor gravity model including a negative cosmological constant. The solutions are only approximately Anti-de Sitter (AdS) asymptotically. While spherically symmetric black holes with scalar-tensor hair do exist in our model, the uncharged spherically symmetric scalar-tensor solitons constructed recently cannot be generalised to include charge. We point out that this is due to the divergence of the electric monopole at the origin of the coordinate system, while higher order multipoles are well-behaved. We also demonstrate that…
It is known that there is no three-dimensional analog of de Sitter black holes. I show that the analog does exist when non-Gaussian (i.e., ring-type) smearings of point matter hairs are considered. This provides a new way of constructing black hole solutions from hairs. I find that the obtained black hole solutions are quite different from the usual large black holes in that there are i) large to small black hole transitions which may be considered as inverse Hawking-Page transitions and ii) soliton-like (i.e., non-perturbative) behaviors. For Gaussian smearing, there is no black hole but a gr…
