Scalar fields around compact objects are of interest for scalar-tensor theories of gravity and dark matter models consisting of a massive scalar, e.g. axions. We study the behaviour of a scalar field around a Kerr black hole with non trivial asymptotic boundary conditions - both non zero density and non zero angular momentum. Starting from an initial radially homogeneous configuration, a scalar cloud is accreted, which asymptotes to known stationary configurations over time. We study the cloud growth for different parameters including black hole spin, scalar field mass, and the scalar field density and angular momentum far from the black hole. We characterise the transient growth of the mass and angular momentum in the cloud, and the spatial profile of the scalar around the black hole, and relate the results of fully non-linear simulations to an analytic perturbative expansion. We also highlight the potential for these accreted clouds to create monochromatic gravitational wave signals - similar to the signals from superradiant clouds, although significantly weaker in amplitude.
We compute the external moduli and dilaton hair to $O(α')$ in the framework of the one-loop corrected superstring effective action for a rotating black hole background.
We construct electrically charged Kerr black holes (BHs) with scalar hair. Firstly, we take an uncharged scalar field, interacting with the electromagnetic field only indirectly, via the background metric. The corresponding family of solutions, dubbed Kerr-Newman BHs with ungauged scalar hair, reduces to (a sub-family of) Kerr-Newman BHs in the limit of vanishing scalar hair and to uncharged rotating boson stars in the limit of vanishing horizon. It adds one extra parameter to the uncharged solutions: the total electric charge. This leading electromagnetic multipole moment is unaffected by the…
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 study the horizon geometry of Kerr black holes (BHs) with scalar synchronised hair, a family of solutions of the Einstein-Klein-Gordon system that continuously connects to vacuum Kerr BHs. We identify the region in parameter space wherein a global isometric embedding in Euclidean 3-space, $\mathbb{E}^3$, is possible for the horizon geometry of the hairy BHs. For the Kerr case, such embedding is possible iff the horizon dimensionless spin $j_H$ (which equals the total dimensionless spin, $j$), the sphericity $\mathfrak{s}$ and the horizon linear velocity $v_H$ are smaller than critical value…
East and Pretorius (arXiv:1704.04791) have successfully evolved, using fully non-linear numerical simulations, the superradiant instability of the Kerr black hole (BH) triggered by a massive, complex vector field. Evolutions terminate in stationary states of a vector field condensate synchronised with a rotating BH horizon. We show these end points are fundamental states of Kerr BHs with synchronised Proca hair. Motivated by the "experimental data" from these simulations we suggest a universal (i.e. field-spin independent), analytic model for the subset of BHs with sychronised hair that posses…
For scalar perturbations of an extreme Reissner-Nordstr{\" o}m black hole we show numerically that the Ori pre-factor equals the Aretakis conserved charge. We demonstrate a linear relation of a generalized Ori pre-factor -- a certain expression obtained from the late-time expansion or the perturbation field at finite distances -- and the Aretakis conserved charge for a family of scalar or gravitational perturbations of an extreme Kerr black hole, whose members vary only in the radial location of the center of the initial packet. We infer that it can be established that there is an Aretakis con…
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…
For an observer, the Black Hole (BH) shadow is the BH's apparent image in the sky due to the gravitational lensing of nearby radiation, emitted by an external source. A recent class of solutions dubbed Kerr BHs with scalar hair possess smaller shadows than the corresponding Kerr BHs and, under some conditions, novel exotic shadow shapes can arise. Thus, these hairy BHs could potentially provide new shadow templates for future experiments such as the Event Horizon Telescope. In order to obtain the shadows, the backward ray-tracing algorithm is briefly introduced, followed by numerical examples …
Kerr black holes with synchronised hair [arXiv:1403.2757, arXiv:1603.02687] are a counter example to the no hair conjecture, in General Relativity minimally coupled to simple matter fields (with mass $μ$) obeying all energy conditions. Since these solutions have, like Kerr, an ergoregion it has been a lingering possibility that they are afflicted by the superradiant instability, the same process that leads to their dynamical formation from Kerr. A recent breakthrough [arXiv:1711.08464] confirmed this instability and computed the corresponding timescales for a sample of solutions. We discuss ho…
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.
The recently obtained hairy Kerr black holes, due to additional sources or surrounding fluid, like dark matter, with conserved energy-momentum tensor, have a deviation $α$ and primary hair $l_0$, apart from rotation parameter $a$ and mass $M$. In the wake of the \textit{Event Horizon Telescope} (\textit{EHT}) observations of the supermassive black hole M87*, a recent surge in interest in black hole shadows suggests comparing the black holes in general relativity (GR) and modified theories of gravity (MoG) to assess these models' differences. Motivated by this, we take on an extensive study of …
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…
Recent studies have made key progress on the black hole/solitonic solutions of the Einstein-Proca system. Firstly, fully non-linear dynamical evolutions of the Kerr black hole superradiant instability, triggered by a Proca field, have shown the formation of a new equilibrium state, a spinning black hole with synchronised Proca hair. Secondly, non-linear evolutions of spinning Proca stars have established that they are dynamically stable, unlike their scalar cousins. Thirdy, separability of the Proca equation on the Kerr background has been achieved. Motivated by these results, in this paper we…
We present a novel way to investigate scalar field profiles around black holes with an accretion disc for a range of models where the Compton wavelength of the scalar is large compared to other length scales. By analysing the problem in "Weyl" coordinates, we are able to calculate the scalar profiles for accretion discs in the static Schwarzschild, as well as rotating Kerr, black holes. We comment on observational effects.
We present a family of solutions of Einstein's gravity minimally coupled to a complex, massive scalar field, describing asymptotically flat, spinning black holes with scalar hair and a regular horizon. These hairy black holes (HBHs) are supported by rotation and have no static limit. Besides mass M and angular momentum J, they carry a conserved, continuous Noether charge Q measuring the scalar hair. HBHs branch off from the Kerr metric at the threshold of the superradiant instability and reduce to spinning boson stars in the limit of vanishing horizon area. They overlap with Kerr black holes f…
We study solutions of the Einstein-Skyrme model. Firstly we consider test field Skyrmions on the Kerr background. These configurations -- hereafter dubbed Skerrmions -- can be in equilibrium with a Kerr black hole (BH) by virtue of a synchronisation condition. We consider two sectors for Skerrmions. In the sector with non-zero baryon charge, Skerrmions are akin to the known Skyrme solutions on the Schwarzschild background. These `topological' configurations reduce to flat spacetime Skyrmions in a vanishing BH mass limit; moreoever, they never become "small" perturbations on the Kerr background…
The maximal ADM mass for (mini-)boson stars (BSs) -- gravitating solitons of Einstein's gravity minimally coupled to a free, complex, mass $μ$, Klein-Gordon field -- is $M_{\rm ADM}^{\rm max}\sim M_{Pl}^2/μ$. Adding quartic self-interactions to the scalar field theory, described by the Lagrangian $\mathcal{L}_I=λ|Ψ|^4$, the maximal ADM mass becomes $M_{\rm ADM}^{\rm max}\sim \sqrtλM_{Pl}^3/μ^2$. Thus, for mini-BSs, astrophysically interesting masses require ultra-light scalar fields, whereas self-interacting BSs can reach such values for bosonic particles with Standard Model range masses. We i…
In the context of complex scalar field coupled to Einstein gravity theory, we present a novel family of solutions of Kerr black holes with excited-state scalar hair inspired by the work of Herdeiro and Radu in [Phys.\ Rev.\ Lett.\ {\bf 112}, 221101 (2014)], which can be regarded as numerical solutions of rotating compact objects with excited scalar hair, including boson stars and black holes. In contrast to Kerr black holes with ground state scalar hair, we find that the first-excited Kerr black holes with scalar hair have two types of nodes, including radial $n_r=1$ and angular $n_θ=1$ nodes.…
Using backwards ray tracing, we study the shadows of Kerr black holes with scalar hair (KBHsSH). KBHsSH interpolate continuously between Kerr BHs and boson stars (BSs), so we start by investigating the lensing of light due to BSs. Moving from the weak to the strong gravity region, BSs - which by themselves have no shadows - are classified, according to the lensing produced, as: $(i)$ non-compact, which yield no multiple images; $(ii)$ compact, which produce an increasing number of Einstein rings and multiple images of the whole celestial sphere; $(iii)$ ultra-compact, which possess light rings…
We have recently reported (arXiv:1403.2757) the existence of Kerr black holes with scalar hair in General Relativity minimally coupled to a massive, complex scalar field. These solutions interpolate between boson stars and Kerr black holes. The latter have a well known topologically S^2 ergo-surface (ergo-sphere) whereas the former develop a S^1*S^1 ergo-surface (ergo-torus) in a region of parameter space. We show that hairy black holes always have an ergo-region, and that this region is delimited by either an ergo-sphere or an ergo-Saturn -- i.e. a S^2\oplus (S^1*S^1) ergo-surface. In the pha…
We continue our study on the capabilities of present and future X-ray missions to test the nature of astrophysical black hole candidates via X-ray reflection spectroscopy and distinguish Kerr black holes from other solutions of 4-dimensional Einstein's gravity in the presence of a matter field. Here we investigate the case of Kerr black holes with Proca hair [1]. The analysis of a sample of these configurations suggests that even extremely hairy black holes can mimic the iron line profile of the standard Kerr black holes, and, at least for the configurations of our study, we find that current …
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…
Distorting Kerr Images with Parity-Odd Scalar Hair
South China University of Technology（华南理工大学）；Shanghai Jiao-Tong University（上海交通大学）；University of Jinan（济南大学）；Beijing Normal University（北京师范大学）；School of Physical Science and Technology, Ningbo University（宁波大学物理科学与技术学院）；Peking University（北京大学）
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 clo…
Recent times witnessed a surge of interest in strong gravitational lensing by black holes due to the Event Horizon Telescope (EHT) results, which suggest comparing the black hole lensing in general relativity and modified gravity theories. This may help us to assess the phenomenological differences between these models. A Kerr black hole is also a solution to some alternative theories of gravity, while recently obtained modified Kerr black holes (hairy Kerr black holes), which evade the no-hair theorem, are due to additional sources from surrounding fluid, like dark matter, having conserved en…
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…
Research Centre for Theoretical Physics and Astrophysics, Institute of Physics, Silesian University in Opava（奥帕瓦西莱斯亚纳大学物理研究所理论与天体物理学研究中心）；Universidad de Tarapacá（塔拉帕卡大学）
Self-interacting boson stars have been shown to alleviate the astrophysically low maximal mass of their non-self-interacting counterparts. We report some physical features of spinning self-interacting boson stars, namely their compactness, the occurence of ergo-regions and the scalar field profiles, for a sample of values of the coupling parameter. The results agree with the general picture that these boson stars are comparatively less compact than the non-self-interacting ones. We also briefly discuss the effect of scalar self-interactions on the properties of Kerr black holes with scalar hai…
This work proposes a set of equations that can be used to numerically compute spacetimes containing a stationary black hole. The formalism is based on the 3+1 decomposition of General Relativity with maximal slicing and spatial harmonic gauge. The presence of the black hole is enforced using the notion of apparent horizon in equilibrium. This setting leads to the main result of this paper: a set of boundary conditions describing the horizon and that must be used when solving the 3+1 equations. Those conditions lead to a choice of coordinates that is regular even on the horizon itself. The whol…
Kerr black holes with scalar hair are solutions of the Einstein-Klein-Gordon field equations describing regular (on and outside an event horizon), asymptotically flat black holes with scalar hair (arXiv:1403.2757). These black holes 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-ti…
We present new equilibrium solutions of stationary models of magnetized thick disks (or tori) around Kerr black holes with synchronised scalar hair. The models reported here largely extend our previous results based on constant radial distributions of the specific angular momentum along the equatorial plane. We introduce a new way to prescribe the distribution of the disk's angular momentum based on a combination of two previous proposals and compute the angular momentum distribution outside the equatorial plane by resorting to the construction of von Zeipel cylinders. We find that the effect …
According to the no-hair theorem, an astrophysical black hole is uniquely described by only two quantities, the mass and the spin. In this series of papers, we investigate a framework for testing the no-hair theorem with observations of black holes in the electromagnetic spectrum. We formulate our approach in terms of a parametric spacetime which contains a quadrupole moment that is independent of both mass and spin. If the no-hair theorem is correct, then any deviation of the black-hole quadrupole moment from its Kerr value has to be zero. We analyze in detail the properties of this quasi-Ker…
Kerr black holes with synchronised scalar hair and azimuthal harmonic index $m>1$ are constructed and studied. The corresponding domain of existence has a broader frequency range than the fundamental $m=1$ family; moreover, larger ADM masses, $M$ and angular momenta $J$ are allowed. Amongst other salient features, non-uniqueness of solutions for fixed global quantities is observed: solutions with the same $M$ and $J$ co-exist, for consecutive values of $m$, and the ones with larger $m$ are always entropically favoured. Our analysis demonstrates, moreover, the qualitative universality of variou…
Thanks to the release of the extraordinary EHT image of shadow attributed to the M87* supermassive black hole (SMBH), we have a novel window to assess the validity of fundamental physics in the strong-field regime. Motivated by this, we consider Johannsen \& Psaltis metric parameterized by mass, spin, and an additional dimensionless hair parameter $ε$. This parametric framework in the high rotation regimes provides a well-behaved bed to the strong-gravity test of the no-hair theorem (NHT) using the EHT data. Incorporating the $ε$ into the standard Kerr spacetime enrich it in the sense that, de…
We investigate the gravitational lensing by spinning Proca stars and the shadows and lensing by Kerr black holes (BHs) with synchronised Proca hair, discussing both theoretical aspects and observational constraints from the Event Horizon Telescope (EHT) M87* and Sgr A* data. On the theoretical side, this family of BHs interpolates between Kerr-like solutions -- exhibiting a similar optical appearance to that of Kerr BHs -- to very non-Kerr like solutions, exhibiting exotic features such as cuspy shadows, egg-like shadows and ghost shadows. We interpret these features in terms of the structure …
Recently, a family of hairy black holes in 4-dimensional Einstein gravity minimally coupled to a complex, massive scalar field was discovered~\cite{hbh}. Besides the mass $M$ and spin angular momentum $J$, these objects are characterized by a Noether charge $Q$, measuring the amount of scalar hair, which is not associated to a Gauss law and cannot be measured at spatial infinity. Introducing a dimensionless scalar hair parameter $q$, ranging from 0 to 1, we recover (a subset of) Kerr black holes for $q=0$ and a family of rotating boson stars for $q=1$. In the present paper, we explore the poss…
In a recent letter, arXiv:1509.00021, it was shown that the lensing of light around rotating boson stars and Kerr black holes with scalar hair can exhibit chaotic patterns. Since no separation of variables is known (or expected) for geodesic motion on these backgrounds, we examine the 2D effective potentials for photon trajectories, to obtain a deeper understanding of this phenomenon. We find that the emergence of stable light rings on the background spacetimes, allows the formation of "pockets" in one of the effective potentials, for open sets of impact parameters, leading to an effective tra…
We perform an observational test of no-hair theorem using quasi-periodic oscillations within the relativistic precession model. Two well motivated metrics we apply are Kerr-Q and Hartle-Thorne metrics in which the quadrupole is the parameter that possibly encodes deviations from the Kerr black hole. The expressions for the quasi-periodic frequencies are derived before comparing the models with the observation. We encounter a degeneracy in constraining spin and quadrupole parameters that makes it difficult to measure their values. In particular, we here propose a novel test of no-hair theorem b…
In this work, taking the QED effect into account, we investigate the shadows of the Kerr black holes immersed in uniform magnetic fields through the numerical backward ray-tracing method. We introduce a dimensionless parameter $Λ$ to characterize the strength of magnetic fields and studied the influence of magnetic fields on the Kerr black hole shadows for various spins of the black holes and inclination angles of the observers. In particular, we find that the photon "hairs" appear near the left edge of the shadow in the presence of magnetic fields. The photon hairs may be served as a signatur…
A Kerr black hole sporting cosmic string hair is studied in the context of the abelian Higgs model vortex. It is shown that a such a system displays much richer phenomenology than its static Schwarzschild or Reissner--Nordstrom cousins, for example, the rotation generates a near horizon electric field. In the case of an extremal rotating black hole, two phases of the Higgs hair are possible: Large black holes exhibit standard hair, with the vortex piercing the event horizon. Small black holes on the other hand, exhibit a flux-expelled solution, with the gauge and scalar field remaining identic…
We show that the integral method used to prove the no-hair theorem for Black Holes (BH's) in spherically symmetric and static spacetimes within the framework of general relativity with matter composed by a complex-valued scalar-field does not lead to a straightforward conclusion about the absence of hair in the stationary and rotating (axisymmetric) scenario. We argue that such a failure can be used to justify in a simple and heuristic way the existence of non-trivial boson clouds or hair found numerically by Herdeiro and Radu [1,2] and analytically by Hod in the test field limit [3-5]. This i…
We have already derived the Criteria for thermal stability of charged rotating quantum black holes, for horizon areas that are large relative to the Planck area. The derivation is done by using results of loop quantum gravity and equilibrium statistical mechanics of the Grand Canonical ensemble. It is also shown that in four dimensional spacetime, quantum ADS Kerr-Newman Black hole is thermally stable within certain range of its' parameters. In this paper, the expectation values of fluctuations and correlations among horizon area, charge and angular momentum of stable quantum ADS black hole ar…
The "no-hair" theorem states that astrophysical black holes are fully characterised by just two numbers: their mass and spin. The gravitational-wave emission from a perturbed black-hole consists of a superposition of damped sinusoids, known as \textit{quasi-normal modes}. Quasi-normal modes are specified by three integers $(\ell,m,n)$: the $(\ell, m)$ integers describe the angular properties and $(n)$ specifies the (over)tone. If the no-hair theorem holds, the frequencies and damping times of quasi-normal modes are determined uniquely by the mass and spin of the black hole, while phases and am…
We extend existing treatments of black hole solutions in String Gravity to include moduli fields. We compute the external moduli and dilaton hair, as well as of their associated axions,to $O(α')$ in the framework of the loop corrected superstring effective action for a Kerr-Newman black hole background.
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. We find an exact 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 t…
Testing the true nature of black holes - the no-hair hypothesis - will become increasingly more precise in the next few years as new observational data is collected in both the gravitational wave channel and the electromagnetic channel. In this paper we consider numerically generated spacetimes of Kerr black holes with synchronised scalar hair and build stationary models of magnetized thick disks (or tori) around them. Our approach assumes that the disks are not self-gravitating, they obey a polytropic equation of state, the distribution of their specific angular momentum is constant, and they…
The superradiant instability of Kerr-AdS black holes is studied by numerically solving the full 3+1 dimensional Einstein equations. We find that the superradiant instability results in a two stage process with distinct initial and secondary instabilities. At the end of the secondary instability the geometry oscillates at several distinct fundamental frequencies -- a multi-oscillating black hole. The multi-oscillating black hole is remarkably close to a black resonator, albeit with a bit of gravitational hair. During the hairy black resonator epoch, the evolution of the horizon area is consiste…
We address the astrophysical imaging of a family of deformed Kerr black holes (BHs). These are stationary, asymptotically flat black hole (BH) spacetimes, that are solutions of General Relativity minimally coupled to a massive, complex scalar field: Kerr BHs with scalar hair (KBHsSH). Such BHs bifurcate from the vacuum Kerr solution and can be regarded as a horizon within a rotating boson star. In a recent letter, it was shown that KBHsSH can exhibit very distinct shadows from the ones of their vacuum counterparts. The setup therein, however, considered the light source to be a celestial spher…
In this work we analyze some judiciously chosen solutions of Kerr Black Holes with Scalar Hair (KBHsSH) of special interest for Gravitational Wave (GW) events originated from Extreme Mass Ratio Inspirals (EMRIs). Because of the off-center distribution of energy density, these spacetimes are warped in such a way that not all metric functions behave monotonically on the equatorial plane as in the exterior region of Kerr black holes (KBHs). This has great impact on the orbital parameters, which in turn affects the imprints on signals descendant from EMRIs in a adiabatic evolution. By investigatin…
