Focus issue: Hairy black holes

A linear coupling between a scalar field and the Gauss–Bonnet invariant is the only known interaction term between a scalar and the metric that: respects shift symmetry; does not lead to higher order equations; inevitably introduces black hole hair in asymptotically flat, 4-dimensional spacetimes. Here we focus on the simplest theory that includes such a term and we explore the dynamical formation of scalar hair. In particular, we work in the decoupling limit that neglects the backreaction of the scalar onto the metric and evolve the scalar configuration numerically in the background of a Schwarzschild black hole and a collapsing dust star described by the Oppenheimer–Snyder solution. For all types of initial data that we consider, the scalar relaxes at late times to the known, static, analytic configuration that is associated with a hairy, spherically symmetric black hole. This suggests that the corresponding black hole solutions are indeed endpoints of collapse.

We here review asymptotically flat rotating black holes in the presence of non-Abelian gauge fields. Like their static counterparts these black holes are no longer uniquely determined by their global charges. In the case of pure SU(2) Yang–Mills fields, the rotation generically induces an electric charge, while the black holes do not carry a magnetic charge. When a Higgs field is coupled, rotating black holes with monopole hair arise in the case of a Higgs triplet, while in the presence of a complex Higgs doublet the black holes carry sphaleron hair. The inclusion of a dilaton allows for Smarr type mass formulae.

Black holes (BHs) in general relativity (GR) are very simple objects. This property, that goes under the name of 'no-hair', has been refined in the last few decades and admits several versions. The simplicity of BHs makes them ideal testbeds of fundamental physics and of GR itself. Here we discuss the no-hair property of BHs, how it can be measured in the electromagnetic or gravitational window, and what it can possibly tell us about our Universe.

We review black hole and star solutions for Horndeski theory. For non-shift symmetric theories, black holes involve a Kaluza–Klein reduction of higher dimensional Lovelock solutions. On the other hand, for shift symmetric theories of Horndeski and beyond Horndeski, black holes involve two classes of solutions: those that include, at the level of the action, a linear coupling to the Gauss–Bonnet term and those that involve time dependence in the galileon field. We analyze the latter class in detail for a specific subclass of Horndeski theory, discussing the general solution of a static and spherically symmetric spacetime. We then discuss stability issues, slowly rotating solutions as well as black holes coupled to matter. The latter case involves a conformally coupled scalar field as well as an electromagnetic field and the (primary) hair black holes thus obtained. We review and discuss the recent results on neutron stars in Horndeski theories.

Bekenstein proved that in Einstein's gravity minimally coupled to one (or many) real, Abelian, Proca field, stationary black holes (BHs) cannot have Proca hair. Dropping Bekenstein's assumption that matter inherits spacetime symmetries, we show this model admits asymptotically flat, stationary, axi-symmetric, regular on and outside an event horizon BHs with Proca hair, for an even number of real (or an arbitrary number of complex) Proca fields. To establish it, we start by showing that a test, complex Proca field can form bound states, with real frequency, around Kerr BHs: stationary Proca clouds. These states exist at the threshold of superradiance. It was conjectured in [1, 2], that the existence of such clouds at the linear level implies the existence of a new family of BH solutions at the nonlinear level. We confirm this expectation and explicitly construct examples of such Kerr BHs with Proca hair (KBHsPH). For a single complex Proca field, these BHs form a countable number of families with three continuous parameters (ADM mass, ADM angular momentum and Noether charge). They branch off from the Kerr solutions that can support stationary Proca clouds and reduce to Proca stars [3] when the horizon size vanishes. We present the domain of existence of one family of KBHsPH, as well as its phase space in terms of ADM quantities. Some physical properties of the solutions are discussed; in particular, and in contrast with Kerr BHs with scalar hair, some spacetime regions can be counter-rotating with respect to the horizon. We further establish a no-Proca-hair theorem for static, spherically symmetric BHs but allowing the complex Proca field to have a harmonic time dependence, which shows BHs with Proca hair in this model require rotation and have no static limit. KBHsPH are also disconnected from Kerr–Newman BHs with a real, massless vector field.

According to the general-relativistic no-hair theorem, astrophysical black holes depend only on their masses and spins and are uniquely described by the Kerr metric. Mass and spin are the first two multipole moments of the Kerr spacetime and completely determine all other moments. The no-hair theorem can be tested by measuring potential deviations from the Kerr metric which alter such higher-order moments. In this review, I discuss tests of the no-hair theorem with current and future observations of such black holes across the electromagnetic spectrum, focusing on near-infrared observations of the supermassive black hole at the Galactic center, pulsar-timing and very-long baseline interferometric observations, as well as x-ray observations of fluorescent iron lines, thermal continuum spectra, variability, and polarization.

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 ${r}_{\mathrm{hair}}\gt {r}_{\mathrm{null}}$ is a generic property of all electrically neutral hairy black-hole spacetimes. In this paper we provide evidence that the answer to this interesting question may be positive. In particular, we prove that the recently discovered cloudy Kerr black-hole spacetimes—non-spherically symmetric non-static black holes which support linearized massive scalar fields in their exterior regions—also respect this no-short hair lower bound. Specifically, we analytically derive the lower bound ${r}_{\mathrm{field}}/{r}_{+}\gt {r}_{+}/{r}_{-}$ on the effective lengths of the external bound-state massive scalar clouds (here ${r}_{\mathrm{field}}$ is the peak location of the stationary bound-state scalar fields and r_± are the horizon radii of the black hole). Remarkably, this lower bound is universal in the sense that it is independent of the physical parameters (proper mass and angular harmonic indices) of the exterior scalar fields. Our results suggest that the lower bound ${r}_{\mathrm{hair}}\gt {r}_{\mathrm{null}}$ may be a general property of asymptotically flat electrically neutral hairy black-hole configurations.

The continuum-fitting and the iron line methods are leading techniques capable of probing the spacetime geometry around astrophysical black hole candidates and testing the no-hair theorem. In the present paper, we review the two approaches, from the astrophysical models and their assumptions, to the constraining power with present and future facilities.

We review the black hole (BH) solutions of the ghost-free massive gravity theory and its bimetric extension, and outline the main results on the stability of these solutions against small perturbations. Massive (bi)-gravity accommodates exact BH solutions, analogous to those of general relativity (GR). In addition to these solutions, hairy BHs—solutions with no correspondent in GR—have been found numerically, whose existence is a natural consequence of the absence of Birkhoff's theorem in these theories. The existence of extra propagating degrees of freedom, makes the stability properties of these BHs richer and more complex than those of GR. In particular, the bi-Schwarzschild BH exhibits an unstable spherically symmetric mode, while the bi-Kerr geometry is also generically unstable, both against the spherical mode and against superradiant instabilities. If astrophysical BHs are described by these solutions, the superradiant instability of the Kerr solution imposes stringent bounds on the graviton mass.

Matter fields do not 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 symmetry noninheritance contributions to the Komar mass and angular momentum of the black hole scalar hair.

Kerr black holes (BHs) with scalar hair are solutions of the Einstein–Klein–Gordon field equations describing regular (on and outside an event horizon), asymptotically flat BHs with scalar hair (Herdeiro and Radu 2014 Phys. Rev. Lett. 112 221101). These BHs interpolate continuously between the Kerr solution and rotating boson stars in D = 4 spacetime dimensions. Here we provide details on their construction, discussing properties of the ansatz, the field equations, the boundary conditions and the numerical strategy. Then, we present an overview of the parameter space of the solutions, and describe in detail the space–time structure of the BH's exterior geometry and of the scalar field for a sample of reference solutions. Phenomenological properties of potential astrophysical interest are also discussed, and the stability properties and possible generalizations are commented on. As supplementary material to this paper we make available numerical data files for the sample of reference solutions discussed, for public use (see stacks.iop.org/cqg/32/144001/mmedia).

The interplay between black holes and fundamental fields has attracted much attention over the years from both physicists and mathematicians. In this paper we study analytically a physical system that is composed of massive scalar fields linearly coupled to a rapidly rotating Kerr black hole. Using simple arguments, we first show that the coupled black hole-scalar field system may possess stationary bound-state resonances (stationary scalar 'clouds') in the bounded regime $1\lt \mu /m{\Omega }_{\rm{H}}\lt \sqrt{2}$, where μ and m are respectively the mass and azimuthal harmonic index of the field, and ${\Omega }_{\rm{H}}$ is the angular velocity of the black-hole horizon. We then show explicitly that these two bounds on the dimensionless ratio $\mu /m{\Omega }_{\rm{H}}$ can be saturated in the asymptotic $m\to \infty$ limit. In particular, we derive a remarkably simple analytical formula for the resonance mass spectrum of the stationary bound-state scalar clouds in the regime $M\mu \gg 1$ of large field masses: ${\mu }_{\;n}=\sqrt{2}m{\Omega }_{\rm{H}}\left[1-\frac{\pi (\mathcal{R}+n)}{m| \mathrm{ln}\tau | }\right]$, where τ is the dimensionless temperature of the rapidly rotating (near extremal) black hole, $\mathcal{R}\lt 1$ is a constant, and $n=0,1,2,\ldots$ is the resonance parameter. In addition, it is shown that, contrary to the flat-space intuition, the effective lengths of the scalar field configurations in the curved black-hole spacetime approach a finite asymptotic value in the large mass $M\mu \gg 1$ limit. In particular, we prove that in the large mass limit, the characteristic length scale of the scalar clouds scales linearly with the black-hole temperature.

Superradiant instabilities of spinning black holes (BHs) can be used to impose strong constraints on ultralight bosons, thus turning BHs into effective particle detectors. However, very little is known about the development of the instability and whether its nonlinear time evolution accords to the linear intuition. For the first time, we attack this problem by studying the impact of gravitational-wave (GW) emission and gas accretion on the evolution of the instability. Our quasi-adiabatic, fully-relativistic analysis shows that: (i) GW emission does not have a significant effect on the evolution of the BH, (ii) accretion plays an important role, and (iii) although the mass of the scalar cloud developed through superradiance can be a sizeable fraction of the BH mass, its energy-density is very low and backreaction is negligible. Thus, massive BHs are well described by the Kerr geometry even if they develop bosonic clouds through superradiance. Using Monte Carlo methods and very conservative assumptions, we provide strong support to the validity of the linearized analysis and to the bounds of previous studies.

We show how to correctly account for scalar accretion onto black holes in scalar field models of dark energy by a consistent expansion in terms of a slow roll parameter. At leading order, we find an analytic solution for the scalar field within our Hubble volume, which is regular on both black hole and cosmological event horizons, and compute the back reaction of the scalar on the black hole, calculating the resulting expansion of the black hole. Our results are independent of the relative size of black hole and cosmological event horizons. We comment on the implications for more general black hole accretion, and the no hair theorems.

We present a survey of the known cosmological and black hole solutions in ghost-free bigravity and massive gravity theories. These can be divided into three classes. First, there are solutions with proportional metrics, which are the same as in General Relativity with a cosmological term, which can be positive, negative or zero. Secondly, for spherically symmetric systems, there are solutions with non-bidiagonal metrics. The g-metric fulfils Einstein equations with a positive cosmological term and a matter source, while the f-metric is anti-de Sitter. The third class contains solutions with bidiagonal metrics, and these can be quite complex. The time-dependent solutions describe homogeneous (isotropic or anisotropic) cosmologies which show a late-time self-acceleration or other types of behavior. The static solutions describe black holes with a massive graviton hair, and also globally regular lumps of energy. None of these are asymptotically flat. Including a matter source gives rise to asymptotically flat solutions which exhibit the Vainshtein mechanism of recovery of General Relativity in a finite region.

Physic in curved spacetime describes a multitude of phenomena, ranging from astrophysics to high-energy physics (HEP). The last few years have witnessed further progress on several fronts, including the accurate numerical evolution of the gravitational field equations, which now allows highly nonlinear phenomena to be tamed. Numerical relativity simulations, originally developed to understand strong-field astrophysical processes, could prove extremely useful to understand HEP processes such as trans-Planckian scattering and gauge–gravity dualities. We present a concise and comprehensive overview of the state-of-the-art and important open problems in the field(s), along with a roadmap for the next years.

We review the black hole (BH) solutions of the ghost-free massive gravity theory and its bimetric extension, and outline the main results on the stability of these solutions against small perturbations. Massive (bi)-gravity accommodates exact BH solutions, analogous to those of general relativity (GR). In addition to these solutions, hairy BHs—solutions with no correspondent in GR—have been found numerically, whose existence is a natural consequence of the absence of Birkhoff's theorem in these theories. The existence of extra propagating degrees of freedom, makes the stability properties of these BHs richer and more complex than those of GR. In particular, the bi-Schwarzschild BH exhibits an unstable spherically symmetric mode, while the bi-Kerr geometry is also generically unstable, both against the spherical mode and against superradiant instabilities. If astrophysical BHs are described by these solutions, the superradiant instability of the Kerr solution imposes stringent bounds on the graviton mass.

Matter fields do not 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 symmetry noninheritance contributions to the Komar mass and angular momentum of the black hole scalar hair.

Kerr black holes (BHs) with scalar hair are solutions of the Einstein–Klein–Gordon field equations describing regular (on and outside an event horizon), asymptotically flat BHs with scalar hair (Herdeiro and Radu 2014 Phys. Rev. Lett. 112 221101). These BHs interpolate continuously between the Kerr solution and rotating boson stars in D = 4 spacetime dimensions. Here we provide details on their construction, discussing properties of the ansatz, the field equations, the boundary conditions and the numerical strategy. Then, we present an overview of the parameter space of the solutions, and describe in detail the space–time structure of the BH's exterior geometry and of the scalar field for a sample of reference solutions. Phenomenological properties of potential astrophysical interest are also discussed, and the stability properties and possible generalizations are commented on. As supplementary material to this paper we make available numerical data files for the sample of reference solutions discussed, for public use (see stacks.iop.org/cqg/32/144001/mmedia).

The interplay between black holes and fundamental fields has attracted much attention over the years from both physicists and mathematicians. In this paper we study analytically a physical system that is composed of massive scalar fields linearly coupled to a rapidly rotating Kerr black hole. Using simple arguments, we first show that the coupled black hole-scalar field system may possess stationary bound-state resonances (stationary scalar 'clouds') in the bounded regime $1\lt \mu /m{\Omega }_{\rm{H}}\lt \sqrt{2}$, where μ and m are respectively the mass and azimuthal harmonic index of the field, and ${\Omega }_{\rm{H}}$ is the angular velocity of the black-hole horizon. We then show explicitly that these two bounds on the dimensionless ratio $\mu /m{\Omega }_{\rm{H}}$ can be saturated in the asymptotic $m\to \infty$ limit. In particular, we derive a remarkably simple analytical formula for the resonance mass spectrum of the stationary bound-state scalar clouds in the regime $M\mu \gg 1$ of large field masses: ${\mu }_{\;n}=\sqrt{2}m{\Omega }_{\rm{H}}\left[1-\frac{\pi (\mathcal{R}+n)}{m| \mathrm{ln}\tau | }\right]$, where τ is the dimensionless temperature of the rapidly rotating (near extremal) black hole, $\mathcal{R}\lt 1$ is a constant, and $n=0,1,2,\ldots$ is the resonance parameter. In addition, it is shown that, contrary to the flat-space intuition, the effective lengths of the scalar field configurations in the curved black-hole spacetime approach a finite asymptotic value in the large mass $M\mu \gg 1$ limit. In particular, we prove that in the large mass limit, the characteristic length scale of the scalar clouds scales linearly with the black-hole temperature.

Superradiant instabilities of spinning black holes (BHs) can be used to impose strong constraints on ultralight bosons, thus turning BHs into effective particle detectors. However, very little is known about the development of the instability and whether its nonlinear time evolution accords to the linear intuition. For the first time, we attack this problem by studying the impact of gravitational-wave (GW) emission and gas accretion on the evolution of the instability. Our quasi-adiabatic, fully-relativistic analysis shows that: (i) GW emission does not have a significant effect on the evolution of the BH, (ii) accretion plays an important role, and (iii) although the mass of the scalar cloud developed through superradiance can be a sizeable fraction of the BH mass, its energy-density is very low and backreaction is negligible. Thus, massive BHs are well described by the Kerr geometry even if they develop bosonic clouds through superradiance. Using Monte Carlo methods and very conservative assumptions, we provide strong support to the validity of the linearized analysis and to the bounds of previous studies.

We show how to correctly account for scalar accretion onto black holes in scalar field models of dark energy by a consistent expansion in terms of a slow roll parameter. At leading order, we find an analytic solution for the scalar field within our Hubble volume, which is regular on both black hole and cosmological event horizons, and compute the back reaction of the scalar on the black hole, calculating the resulting expansion of the black hole. Our results are independent of the relative size of black hole and cosmological event horizons. We comment on the implications for more general black hole accretion, and the no hair theorems.

We present a survey of the known cosmological and black hole solutions in ghost-free bigravity and massive gravity theories. These can be divided into three classes. First, there are solutions with proportional metrics, which are the same as in General Relativity with a cosmological term, which can be positive, negative or zero. Secondly, for spherically symmetric systems, there are solutions with non-bidiagonal metrics. The g-metric fulfils Einstein equations with a positive cosmological term and a matter source, while the f-metric is anti-de Sitter. The third class contains solutions with bidiagonal metrics, and these can be quite complex. The time-dependent solutions describe homogeneous (isotropic or anisotropic) cosmologies which show a late-time self-acceleration or other types of behavior. The static solutions describe black holes with a massive graviton hair, and also globally regular lumps of energy. None of these are asymptotically flat. Including a matter source gives rise to asymptotically flat solutions which exhibit the Vainshtein mechanism of recovery of General Relativity in a finite region.

Physic in curved spacetime describes a multitude of phenomena, ranging from astrophysics to high-energy physics (HEP). The last few years have witnessed further progress on several fronts, including the accurate numerical evolution of the gravitational field equations, which now allows highly nonlinear phenomena to be tamed. Numerical relativity simulations, originally developed to understand strong-field astrophysical processes, could prove extremely useful to understand HEP processes such as trans-Planckian scattering and gauge–gravity dualities. We present a concise and comprehensive overview of the state-of-the-art and important open problems in the field(s), along with a roadmap for the next years.