Recent no-hair theorems have revealed the intriguing fact that horizonless stars with compact reflecting surfaces {\it cannot} support non-linear matter configurations made of scalar, vector, and tensor fields. In the present paper we extend the regime of validity of these no-hair theorems by explicitly proving that spherically symmetric compact reflecting stars cannot support static configurations made of {\it massive} scalar fields with {\it non}-minimal coupling to gravity. Interestingly, our no-hair theorem is valid for {\it generic} values of the dimensionless field-curvature coupling parameter $ξ$.
We provide a remarkably compact proof that spherically symmetric neutral black holes cannot support static nonminimally coupled massless scalar fields. The theorem is based on causality restrictions imposed on the energy-momentum tensor of the fields near the regular black-hole horizon.
It has recently been proved that horizonless compact stars with reflecting boundary conditions {\it cannot} support spatially regular matter configurations made of minimally coupled scalar fields, vector fields, and tensor fields. In the present paper we extend this intriguing no-hair property to the physically interesting regime of scalar fields with {\it nonminimal} coupling to gravity. In particular, we prove that static spherically symmetric configurations made of nonminimally coupled massless scalar fields cannot be supported by compact reflecting stars.
It is proved that spherically symmetric asymptotically flat neutral black holes cannot support spatially regular static configurations made of massive scalar fields with non-minimal coupling to gravity. Interestingly, our compact no-hair theorem is valid for generic values of the dimensionless physical parameter $ξ$ which quantifies the strength of coupling between the scalar field and the spacetime curvature.
In a recent paper, Hod proved that spherically symmetric Dirichlet reflecting compact stars cannot support static nonminimally coupled scalar fields. In the present paper, we study the validity of no hair theorems for compact stars with Neumann surface boundary conditions. We find that Neumann compact stars cannot support static massive scalar field hairs with a generic dimensionless nonminimal coupling parameter.
Self-gravitating scalar fields with nonminimal coupling to gravity and having a quartic self-interaction are considered in the domain of outer communications of a static black hole. It is shown that there is no value of the nonminimal coupling parameter $ζ$ for which nontrivial static black hole solutions exist. This result establishes the correctness of Bekenstein ``no-scalar-hair'' conjecture for quartic self-interactions.
We prove three theorems in general relativity which rule out classical scalar hair of static, spherically symmetric, possibly electrically charged black holes. We first generalize Bekenstein's no--hair theorem for a multiplet of minimally coupled real scalar fields with not necessarily quadratic action to the case of a charged black hole. We then use a conformal map of the geometry to convert the problem of a charged (or neutral) black hole with hair in the form of a neutral self--interacting scalar field nonminimally coupled to gravity to the preceding problem, thus establishing a no--hair th…
It is proved that spherically symmetric compact reflecting objects cannot support static bound-state configurations made of scalar fields whose self-interaction potential $V(ψ^2)$ is a monotonically increasing function of its argument. Our theorem rules out, in particular, the existence of massive scalar hair outside the surface of a spherically symmetric compact reflecting star.
Static, spherically symmetric, traversable wormholes, induced by massless, nonminimally coupled scalar fields in general relativity, are shown to be unstable under spherically symmetric perturbations. The instability is related to blowing-up of the effective gravitational constant on a certain sphere.
Introducing a new form of scalar potential $V(ϕ)$, we derive a proper form of the rotating black hole solution in three-dimensional Einstein gravity with nonminimally coupled scalar field and find that the first law of thermodynamics of this new rotating hairy black hole can be protected, where the scalar field parameter $B$ is constrained to relate to the black hole size. We also disclose the Hawking-Page phase transition between this rotating hairy black holes and the pure thermal radiation. Moreover, we study phase transitions between this rotating hairy black hole and rotating BTZ black ho…
We present a compact theorem which reveals the fact that static spatially regular massive scalar fields with nonminimal coupling to gravity cannot form spherically symmetric asymptotically flat horizonless matter configurations. In particular, the no-go theorem rules out the existence of boson stars made of static scalar fields with generic values of the physical parameter $ξ$ which quantifies the coupling between the spacetime curvature and the massive bosonic fields.
We apply a recently developed 2+1+1 decomposition of spacetime, based on a nonorthogonal double foliation for the study of spherically symmetric, static black hole solutions of Horndeski scalar-tensor theory. Our discussion proceeds in an effective field theory (EFT) of modified gravity approach, with the action depending on metric and embedding scalars adapted to the nonorthogonal 2+1+1 decomposition. We prove that the most generic class of Horndeski Lagrangians compatible with observations can be expressed in this EFT form. By studying the first order perturbation of the EFT action we derive…
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 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.
We investigate the existence of the skyrmion field in the background of an asymptotically flat stationary reflecting star. For this purpose, we consider the Einstein Skyrme system for which there is a skyrmion hair in the black hole case. We discuss spherically symmetric skyrmions and employ the hedgehog ansatz for the skyrmion field. We show that, in contrast to the black hole there is no skyrmion hair for a reflecting star.
It is shown that the conclusion regarding the existence of a scalar hair for a black hole in a nonminimally coupled self interacting scalar tensor theory can be drawn from the that for a scalar field minimally coupled to gravity by means of a conformal transformation.
The recently proved `no short hair' theorem asserts that, if a spherically-symmetric static black hole has hair, then this hair (the external fields) must extend beyond the null circular geodesic (the "photonsphere") of the corresponding black-hole spacetime: $r_{\text{field}}>r_{\text{null}}$. In this paper we provide compelling evidence that the bound can be {\it violated} by {\it non}-spherically symmetric hairy black-hole configurations. To that end, we analytically explore the physical properties of cloudy Kerr-Newman black-hole spacetimes -- charged rotating black holes which support lin…
We discuss with a rather critical eye the current situation of black hole (BH) solutions in $f(R)$ gravity and shed light about its geometrical and physical significance. We also argue about the meaning, existence or lack thereof of a Birkhoff's theorem in this kind of modified gravity. We focus then on the analysis and quest of $non-trivial$ (i.e. hairy) $asymptotically\,\,flat$ (AF) BH solutions in static and spherically symmetric (SSS) spacetimes in vacuum having the property that the Ricci scalar does $not$ vanish identically in the domain of outer communication. To do so, we provide and e…
The no-hair theorem by Mayo and Bekenstein states that there exists no non-extremal static and spherical charged black hole endowed with hair in the form of a charged scalar field with a self-interaction potential. In our recent work, we showed that the effect of a scalar mass term is important at an asymptotic infinity, which was omitted to prove the no-hair theorem. In this paper, we demonstrate that there actually exists static and spherical charged scalar hair, dubbed as Q-hair, around charged black holes, by taking into account the backreaction to the metric and gauge field. We also discu…
We study scalar condensation in the background of asymptotically flat spherically symmetric regular Dirichlet stars. We assume that the scalar field decreases as the star surface is approached. Under these circumstances, we prove a no hair theorem for neutral regular compact stars. We also extend the discussion to charged regular compact stars and find an upper bound for the charged star radius. Above the upper bound, the scalar hair cannot exist. Below the upper bound, we numerically obtain solutions of scalar hairy charged stars.
When the Higgs field is nonminimally coupled to gravity, there exists a family of spherically symmetric particlelike solutions to the field equations. These monopoles are the only globally regular and asymptotically flat distributions with finite energy of the Higgs field around compact objects. Moreover, spontaneous scalarization is strongly amplified for specific values of their mass and compactness.
Originally regarded as forbidden, black hole ``hair'' are fields associated with a stationary black hole apart from the gravitational and electromagnetic ones. Several stable stationary black hole solutions with gauge or Skyrme field hair are known today within general relativity. We formulate here a ``no scalar--hair'' conjecture, and adduce some new theorems that almost establish it by ruling out - for all but a small parameter range - scalar field hair of spherical black holes in general relativity, whether the field be self--interacting, coupled to an Abelian gauge field, or nonminimally c…
We prove under certain assumptions no-hair theorems for non-canonical self-gravitating static multiple scalar fields in spherically symmetric spacetimes. It is shown that the only static, spherically symmetric and asymptotically flat black hole solutions consist of the Schwarzschild metric and a constant multi-scalar map. We also prove that there are no static, horizonless, asymptotically flat, spherically symmetric solutions with static scalar fields and a regular center. The last theorem shows that the static, asymptotically flat, spherically symmetric reflecting compact objects with Neumann…
This paper consolidates noscalar hair theorem for a charged spherically symmetric black hole in four dimension in general relativity as well as in all scalar tensor theories, both minimally and nonminimally coupled, when the effective Newtonian constant of gravity is positive. However, there is an exception when the matter field itself is coupled to the scalar field, such as in dilaton gravity.
We consider a generic scalar-tensor theory involving a shift-symmetric scalar field and minimally coupled matter fields. We prove that the Noether current associated with shift-symmetry vanishes in regular, spherically symmetric and static spacetimes. We use this fact to prove the absence of scalar hair for spherically symmetric and static stars in Horndeski and beyond theories. We carefully detail the validity of this no-hair theorem.
We investigate the static, spherically symmetric black hole solutions in the quasi-dilaton model and its generalizations, which are scalar extended dRGT massive gravity with a shift symmetry. We show that, unlike generic scalar extended massive gravity models, these theories do not admit static, spherically symmetric black hole solutions until the theory parameters in the dRGT potential is fine-tuned. When fine-tuned, the geometry of the static, spherically symmetric black hole is necessarily that of general relativity and the quasi-dilaton field is constant across the spacetime. The fine-tuni…
In a recent paper Cañate (CQG, {\bf 35}, 025018 (2018)) proved a no hair theorem to static and spherically symmetric or stationary axisymmetric black holes in general $f(R)$ gravity. The theorem applies for isolated asymptotically flat or asymptotically de Sitter black holes and also in the case when vacuum is replaced by a minimally coupled source having a traceless energy momentum tensor. This theorem excludes the case of pure quadratic gravity, $f(R) = R^2$. In this paper we use the scalar tensor representation of general $f(R)$ theory to show that there are no hairy black hole in pure $R^2…
In this work we construct a static and spherically symmetric black hole geometry supported by a family of generic mono-parametric sources thorough the Gravitational Decoupling. The parameter characterizing the matter sector can be interpreted as hair which cannot be associated to any global charge. Although the solution is constructed by demanding the weak energy condition, we find that the resulting matter sector satisfies all the energy conditions at and outside the horizon. We study the effect of the hair on the periastron advance and the gravitational lensing of the black hole. We estimate…
The elegant `no short hair' theorem states that, if a spherically-symmetric static black hole has hair, then this hair must extend beyond 3/2 the horizon radius. In the present paper we provide evidence for the failure of this theorem beyond the regime of spherically-symmetric static black holes. In particular, we show that rotating black holes can support extremely short-range stationary scalar configurations (linearized scalar `clouds') in their exterior regions. To that end, we solve analytically the Klein-Gordon-Kerr-Newman wave equation for a linearized massive scalar field in the regime …
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…
A static spherically symmetric black hole usually turns out to be either a Schwarzschild black hole or a Reissner-Nordström black hole. This result was summarised by Ruffini and Wheeler as the so-called no hair conjecture which states that for a spherically symmetric black hole only the information about mass ($M$) and electric charge ($e$) of the black hole is available for an external observer. In this work, we calculate the emission rate of charged particles from an asymptotically flat charged spherically symmetric black hole endowed with a scalar hair using a semi-classical tunneling forma…
If a black hole has hair, how short can this hair be? A partial answer to this intriguing question was recently provided by the 'no-short hair' theorem which asserts that the external fields of a spherically-symmetric electrically neutral hairy black-hole configuration must extend beyond the null circular geodesic which characterizes the corresponding black-hole spacetime. One naturally wonders whether the no-short hair inequality $r_{\text{hair}}>r_{\text{null}}$ is a generic property of all electrically neutral hairy black-hole spacetimes? In this paper we provide evidence that the answer to…
In a recent paper, Hod started a study on no scalar hair theorem for asymptotically flat spherically symmetric neutral horizonless reflecting compact stars. In fact, Hod's approach only rules out massive scalar fields. In the present paper, for massless scalar fields outside neutral horizonless reflecting compact stars, we provide a rigorous mathematical proof on no hair theorem. We show that asymptotically flat spherically symmetric neutral horizonless reflecting compact stars cannot support exterior massless scalar field hairs.
Boson Stars in Teleparallel Gravity with a Nonminimally Coupled Field: The Violation of Energy Conditions and Gravitational Waveforms from EMRIs
Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics of Gansu Province, School of Physical Science and Technology, Lanzhou University（兰州大学物理科学与技术学院）；School of Physical Science and Technology, Southwest University（西南大学物理科学与技术学院）
The `no-hair' conjecture claims that for a spherically symmetric black hole, only the information regarding the mass and charge of the black hole is available to an external observer. However, there are numerous counterexamples to the `no-hair' conjecture. In this work, we consider a particular counter-example to the `no-hair' conjecture in (3+1) dimensions, namely, a static spherically symmetric charged black hole with a scalar hair. We provide semi-analytic bounds on the greybody factors and study the sparsity of Hawking radiation of mass-less uncharged scalar fields. Our results show that t…
We study the nonminimally coupled complex scalar field within the framework of teleparallel gravity. Coupling of the field nonminimally to the torsion scalar destroys the Lorentz invariance of the theory in the sense that the resulting equations of motion depend on the choice of a tetrad. For the assumed static spherically symmetric spacetime, we find a tetrad which leads to a self-consistent set of equations, and we construct the self-gravitating configurations of the scalar field---boson stars. The resulting configurations develop anisotropic principal pressures and satisfy the dominant ener…
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 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 discus all possible spherically symmetric black hole type solutions to an N=2 supergravity model with SO(3) gauging. The solutions consist of a one parameter family illustrating that the no-hair theorem does not hold and an isolated solution that a supersymmetric analogue of a coloured black hole.
A static, spherically symmetric, asymptotically flat spacetime may allow for circular, closed null-geodesics which are said to belong to a photon sphere. In the context of gravitational lensing in the strong deflection regime, the presence of a photon sphere leads to an unbounded angle of deflection of light (multiple turns) and formation of relativistic images. In this paper, we show that photon spheres may form in some configurations of boson stars constructed with a free massive complex scalar field nonminimally coupled to gravity. Assuming that the boson star is transparent to light, photo…
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 study the Vainshtein screening mechanism in Horndeski theories in the presence of a scalar field $ϕ$ nonminimally and kinetically coupled to ordinary matter field. A general interacting Lagrangian describing this coupling is characterized by energy transfer $f_1$ and momentum exchange $f_2$. For a spherically symmetric configurations on top of the cosmological background, we investigate the static perturbations in linear and nonlinear regimes with respect to the scalar field perturbation. In the former regime, the parametrized post-Newtonian parameter generally deviates from unity as long a…
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.
A class of exact rotating black hole solutions of gravity nonminimally coupled to a self-interacting scalar field in arbitrary dimensions is presented. These spacetimes are asymptotically locally anti-de Sitter manifolds and have a Ricci-flat event horizon hiding a curvature singularity at the origin. The scalar field is real and regular everywhere and its effective mass, coming from the nonminimal coupling with the scalar curvature, saturates the Breitenlohner-Freedman bound for the corresponding spacetime dimension. The rotating black hole is obtained by applying an improper coordinate trans…
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.
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.
Devoured by a Hairy Gargantua: Probing Massive Scalar Charges with Non-minimal Curvature Coupling with Extreme-Mass-Ratio Inspirals
Beijing Institute of Mathematical Sciences and Applications（北京国际数学研究中心）；DAMTP, Centre for Mathematical Sciences, University of Cambridge（剑桥大学数学科学中心 DAMTP）
该研究探讨与曲率非最小耦合的超轻标量场对黑洞及极端质量比旋近的影响，通过建模旋近过渡发现引力波相位偏移，还提出可利用 LISA 测量曲率耦合与标量质量的高精度。
We study static spherically symmetric black hole solutions with a linearly time-dependent scalar field and discuss their linear stability in the shift- and reflection-symmetric subclass of quadratic degenerate higher-order scalar-tensor (DHOST) theories. We present the explicit forms of the reduced system of background field equations for a generic theory within this subclass. Using the reduced equations of motion, we show that in several cases the solution is forced to be of the Schwarzschild or Schwarzschild-(anti-)de Sitter form. We consider odd-parity perturbations around general static sp…
The intermediate and late-time behaviour of massive Dirac hair in the static spherically symmetric black hole spacetime was studied. It was revealed that the intermediate asymptotic pattern of decay of massive Dirac spinor hair is dependent on the mass of the field under consideration as well as the multiple number of the wave mode. The long-lived oscillatory tail observed at timelike infinity in the considered background decays slowly as t^{-5/6}.
We consider a galileon field coupled to gravity. The standard no-hair theorems do not apply, because of the galileon's peculiar derivative interactions. We prove that, nonetheless, static spherically symmetric black holes cannot sustain non-trivial galileon profiles. Our theorem holds regardless of whether there are non-minimal couplings between the galileon and gravity of the covariant galileon type.
We analyze in detail the geometry and dynamics of the cosmological region arising in spherically symmetric black hole solutions of the Einstein-Maxwell-scalar field system with a positive cosmological constant. More precisely, we solve, for such a system, a characteristic initial value problem with data emulating a dynamic cosmological horizon. Our assumptions are fairly weak, in that we only assume that the data approaches that of a subextremal Reissner-Nordström-de Sitter black hole, without imposing any rate of decay. We then show that the radius (of symmetry) blows up along any null ray pa…
The trace anomaly for nonminimally coupled scalars in spherically reduced gravity obtained by Bousso and Hawking (hep-th/9705236) is incorrect. We explain the reasons for the deviations from our correct (published) result which is supported by several other recent papers.
