Stationary Clouds Research Articles

Stationary scalar clouds around Kerr-Newman black holes

Abstract This study investigates scalar clouds around Kerr-Newman black holes within the Einstein-Maxwell-scalar model. Tachyonic instabilities are identified as the driving mechanism for scalar cloud formation. Employing the spectral method, we numerically compute wave functions and parameter space existence domains for both fundamental and excited scalar cloud modes. Our analysis demonstrates that black hole spin imposes an upper limit on the existence of scalar clouds, with excited modes requiring stronger tachyonic instabilities for their formation. These findings lay the groundwork for exploring the nonlinear dynamics and astrophysical implications of scalar clouds.

Spinning Proca-Higgs balls, stars and hairy black holes

Recently, spherical and static flat space solitons (balls) and self-gravitating, everywhere regular, asymptotically flat solitons (stars) were constructed in an Einstein-Proca-Higgs model [1], where a complex vector field gains mass by coupling to a real scalar field with a Higgs-type potential. The Proca-Higgs model serves as a UV completion of a complex Proca model with self-interactions. Here, we construct and examine the mathematical and physical properties of rotating configurations. In particular, rotation allows horizon-bearing solutions, including stationary clouds surrounding Kerr black holes and their non-linear continuation into black holes with Proca-Higgs hair.

Axisymmetric, stationary collisionless gas clouds trapped in a Newtonian potential

The properties of an axisymmetric, stationary gas cloud surrounding a massive central object are discussed. It is assumed that the gravitational field is dominated by the central object which is modeled by a nonrelativistic rotationally-symmetric potential. Further, we assume that the gas consists of collisionless, identical massive particles that follow bound orbits in this potential. Several models for the one-particle distribution function are considered and the essential formulae that describe the relevant macroscopical observables, such as the particle and energy densities, pressure tensor, and the kinetic temperature are derived. The asymptotic decay of the solutions at infinity is discussed and we specify configurations with finite total mass, energy and (zero or non-zero) angular momentum. Finally, our configurations are compared to their hydrodynamic analogs. In an accompanying paper, the equivalent general relativistic problem is discussed, where the central object consists of a Schwarzschild black hole.

Quasinormal modes, quasibound states, scalar clouds, and superradiant instabilities of a Kerr-like black hole

We use the continued fraction method to determine the eigenfrequencies associated with a scalar field around a Kerr-like black hole. The Kerr-like metric considered in this article is a subclass of the general parametrization of axisymmetric black holes proposed by Konoplya, Rezzolla and Zhidenko. In addition to its mass $M$ and specific angular momentum $a$, the black hole depends on a third parameter $\eta$, called the deformation parameter. We investigate how the deformation parameter affects the quasinormal modes and the quasibound states of a massive scalar field around the black hole. In particular, we compute the time scales associated with the superradiant instabilities of the scalar field in such a spacetime. The properties of stationary scalar clouds that could be formed by these instabilities are also discussed.

Dirac Clouds around Dilatonic Black Holes.

Dirac cloud is in absence in general relativity since the superradiance mechanism fails to work for Dirac fields. For the first time, we find a novel mechanism to support Dirac clouds, which is independent on superradiance mechanism. We study quasibound states of Dirac particles around a charged spherical black hole in dilatonic gravity. We find that the quasibound states become real bound states when the central black hole becomes extremal. We make an intensive study of the energy spectrum of the stationary clouds for different fine structure constant μM and reveal the existence condition of these clouds. Our result strongly implies that extreme dilatonic black holes behave as elementary particles.

No-short scalar hair theorem for spinning acoustic black holes in a photon-fluid model

It has recently been revealed that spinning black holes of the photon-fluid model can support acoustic `clouds', stationary density fluctuations whose spatially regular radial eigenfunctions are determined by the $(2+1)$-dimensional Klein-Gordon equation of an effective massive scalar field. Motivated by this intriguing observation, we use {\it analytical} techniques in order to prove a no-short hair theorem for the composed acoustic-black-hole-scalar-clouds configurations. In particular, it is proved that the effective lengths of the stationary bound-state co-rotating acoustic scalar clouds are bounded from below by the series of inequalities $r_{\text{hair}}>{{1+\sqrt{5}}\over{2}}\cdot r_{\text{H}}>r_{\text{null}}$, where $r_{\text{H}}$ and $r_{\text{null}}$ are respectively the horizon radius of the supporting black hole and the radius of the co-rotating null circular geodesic that characterizes the acoustic spinning black-hole spacetime.

Stationary scalar clouds supported by rapidly-rotating acoustic black holes in a photon-fluid model

It has recently been proved that, in the presence of vortex flows, the fluctuation dynamics of a rotating photon-fluid model is governed by the Klein-Gordon equation of an effective massive scalar field in a $(2+1)$-dimensional acoustic black-hole spacetime. Interestingly, it has been demonstrated numerically that the rotating acoustic black hole, like the familiar Kerr black-hole spacetime, may support spatially regular stationary density fluctuations (linearized acoustic scalar `clouds') in its exterior regions. In particular, it has been shown that the composed rotating-acoustic-black-hole-stationary-scalar-field configurations of the photon-fluid model exist in the narrow dimensionless regime $\alpha\equiv\Omega_0/m\Omega_{\text{H}}\in(1,\alpha_{\text{max}})$ with $\alpha_{\text{max}}\simeq 1.08$ [here $\Omega_{\text{H}}$ is the angular velocity of the black-hole horizon and $\{\Omega_0,m\}$ are respectively the effective proper mass and the azimuthal harmonic index of the acoustic scalar field]. In the present paper we use analytical techniques in order to explore the physical and mathematical properties of the acoustic scalar clouds of the photon-fluid model in the regime $\Omega_{\text{H}}r_{\text{H}}\gg1$ of rapidly-spinning central supporting acoustic black holes. In particular, we derive a remarkably compact analytical formula for the discrete resonance spectrum $\{\Omega_0(\Omega_{\text{H}},m;n)\}$ which characterizes the stationary bound-state acoustic scalar clouds of the photon-fluid model. Interestingly, it is proved that the critical (maximal) mass parameter $\alpha_{\text{max}}$, which determines the regime of existence of the composed acoustic-black-hole-stationary-bound-state-massive-scalar-field configurations, is given by the exact dimensionless relation $\alpha_{\text{max}}=\sqrt{{{32}\over{27}}}$.

Black holes, stationary clouds and magnetic fields

As the electron in the hydrogen atom, a bosonic field can bind itself to a black hole occupying a discrete infinite set of states. When (i) the spacetime is prone to superradiance and (ii) a confinement mechanism is present, some of such states are infinitely long–lived. These equilibrium configurations, known as stationary clouds, are states “synchronized” with a rotating black hole's event horizon. For most, if not all, stationary clouds studied in the literature so far, the requirements (i)–(ii) are independent of each other. However, this is not always the case. This paper shows that massless neutral scalar fields can form stationary clouds around a Reissner–Nordström black hole when both are subject to a uniform magnetic field. The latter simultaneously enacts both requirements by creating an ergoregion (thereby opening up the possibility of superradiance) and trapping the scalar field in the black hole's vicinity. This leads to some novel features, in particular, that only black holes with a subset of the possible charge to mass ratios can support stationary clouds.

Acoustic black-hole bombs and scalar clouds in a photon-fluid model

Massive bosonic fields in the background of a Kerr black hole can either trigger superradiant instabilities (black-hole bombs) or form equilibrium configurations corresponding to pure bound states, known as stationary scalar clouds. Here, similar phenomena are shown to emerge in the fluctuation dynamics of a rotating photon-fluid model. In the presence of suitable vortex flows, the density fluctuations are governed by the massive Klein-Gordon equation on a ($2+1$) curved spacetime, possessing an ergoregion and an event horizon. We report on superradiant instabilities originating from quasibound phonon states trapped by the vortex background and, remarkably, on the existence of stationary modes in synchronous rotation with the horizon. These represent the acoustic counterpart of astrophysical scalar clouds. Our system offers a promising platform for analogue gravity experiments on superradiant instabilities of massive bosons and black-hole-field equilibrium configurations.

Stationary scalar and vector clouds around Kerr–Newman black holes

Massive bosons in the vicinity of Kerr–Newman black holes can form pure bound states when their phase angular velocity fulfills the synchronization condition, i.e. at the threshold of superradiance. The presence of these stationary clouds at the linear level is intimately linked to the existence of Kerr black holes with synchronized hair at the nonlinear level. These configurations are very similar to the atomic orbitals of the electron in a hydrogen atom. They can be labeled by four quantum numbers: [Formula: see text], the number of nodes in the radial direction; [Formula: see text], the orbital angular momentum; [Formula: see text], the total angular momentum; and [Formula: see text], the azimuthal total angular momentum. These synchronized configurations are solely allowed for particular values of the black holes mass, angular momentum and electric charge. Such quantization results in an existence surface in the three-dimensional parameter space of Kerr–Newman black holes. The phenomenology of stationary scalar clouds has been widely addressed over the last years. However, there is a gap in the literature concerning their vector cousins. Following the separability of the Proca equation in Kerr(–Newman) spacetime, this paper explores and compares scalar and vector stationary clouds around Kerr and Kerr–Newman black holes, extending previous research.

Black holes with synchronised Proca hair: linear clouds and fundamental non-linear solutions

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. Thirdly, separability of the Proca equation on the Kerr background has been achieved. Motivated by these results, in this paper we reconsider Kerr black holes with synchronised Proca hair. The separability of the Proca equation on the Kerr background allows us to examine the stationary Proca clouds in greater detail, in particular their dependence on the different quantum numbers. These stationary clouds occur at a set of existence lines in the Kerr parameter space, from which the black holes with synchronised Proca hair bifurcate. We construct the domain of existence of these black holes, comparing the fundamental states missed in the original study with the first excited states and with the cousin scalar model, giving illustrative examples of Kerr-like and non- Kerr-like BHs. In the vanishing event horizon limit, these hairy black holes connect to the fundamental states of spinning Proca stars, which include the dynamically stable solutions.

Stationary scalar clouds outside charged reflecting compact stars

Recently, Hod showed that neutral stationary scalar field clouds can exist outside neutral reflecting compact stars. In the present paper, we extend the discussion by considering charged stationary scalar fields in the background of charged reflecting compact stars. If stationary scalar clouds exist, we analytically show that the field frequency belongs to an interval. For certain discrete field frequency values in the interval, we obtain numerical solutions of scalar field clouds. We also study effects of model parameters on discrete frequency.

Kerr-MOG black holes with stationary scalar clouds

We establish the existence of stationary clouds of massive test scalar fields around Kerr-MOG black holes. By solving the Klein–Gordon equation numerically, we present the existence lines of the clouds in the parameter space of the Kerr-MOG black holes, and investigate the effect of the MOG parameter on the rich structure of scalar clouds. We observe that the MOG parameter leads to the split of the existence lines for the scalar clouds, and the larger MOG parameter makes it possible for the clouds to exist in the case of the lower background angular velocity. Numerical results are compared with the analytical formula obtained by an asymptotic matching method, and we find that both results are consistent with each other. In particular, it is shown that the larger MOG parameter, the better agreement between analytical and numerical results. This implies that the matching method is a powerful analytical tool to investigate the scalar clouds existing in the Kerr-MOG black holes. Moreover, we obtain the location of the existence lines and show that the clouds are concentrated at the larger radial position for the Kerr-MOG black holes when compared to the Kerr black holes.

Spherical electro-vacuum black holes with resonant, scalar Q-hair

The asymptotically flat, spherical, electro-vacuum black holes (BHs) are shown to support static, spherical configurations of a gauged, self-interacting, scalar field, minimally coupled to the geometry. Considering a Q-ball type potential for the scalar field, we dub these configurations Q-clouds, in the test field approximation. The clouds exist under a resonance condition, at the threshold of (charged) superradiance. This is similar to the stationary clouds supported by Kerr BHs, which exist for a synchronisation condition, at the threshold of (rotational) superradiance. In contrast with the rotating case, however, Q-clouds require the scalar field to be massive and self-interacting; no similar clouds exist for massive but free scalar fields. First, considering a decoupling limit, we construct Q-clouds around Schwarzschild and Reissner–Nordström BHs, showing there is always a mass gap. Then, we make the Q-clouds backreact, and construct fully non-linear solutions of the Einstein–Maxwell-gauged scalar system describing spherical, charged BHs with resonant, scalar Q-hair. Amongst other properties, we observe there is non-uniqueness of charged BHs in this model and the Q-hairy BHs can be entropically preferred over Reissner–Nordström, for the same charge to mass ratio; some Q-hairy BH solutions can be overcharged. We also discuss how some well known no-hair theorems in the literature, applying to electro-vacuum plus minimally coupled scalar fields, are circumvented by this new type of BHs.

Computational analysis of shock-induced flow through stationary particle clouds

We investigate the shock-induced flow through random particle arrays using particle-resolved Large Eddy Simulations for different incident shock wave Mach numbers, particle volume fractions and particle sizes. We analyze trends in mean flow quantities and the unresolved terms in the volume averaged momentum equation, as we vary the three parameters. We find that the shock wave attenuation and certain mean flow trends can be predicted by the opacity of the particle cloud, which is a function of particle size and particle volume fraction. We show that the Reynolds stress field plays an important role in the momentum balance at the particle cloud edges, and therefore strongly affects the reflected shock wave strength. The Reynolds stress was found to be insensitive to particle size, but strongly dependent on particle volume fraction. It is in better agreement with results from simulations of flow through particle clouds at fixed mean slip Reynolds numbers in the incompressible regime, than with results from other shock wave particle cloud studies, which have utilized either inviscid or two-dimensional approaches. We propose an algebraic model for the streamwise Reynolds stress based on the observation that the separated flow regions are the primary contributions to the Reynolds stress.

Synchronized stationary clouds in a static fluid

The existence of stationary bound states for the hydrodynamic velocity field between two concentric cylinders is established. We argue that rotational motion, together with a trapping mechanism for the associated field, is sufficient to mitigate energy dissipation between the cylinders, thus allowing the existence of infinitely long lived modes, which we dub stationary clouds. We demonstrate the existence of such stationary clouds for sound and surface waves when the fluid is static and the internal cylinder rotates with constant angular velocity Ω. These setups provide a unique opportunity for the first experimental observation of synchronized stationary clouds. As in the case of bosonic fields around rotating black holes and black hole analogues, the existence of these clouds relies on a synchronization condition between Ω and the angular phase velocity of the cloud.

Skyrmions around Kerr black holes and spinning BHs with Skyrme hair

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 non-linearities of the Skyrme model are crucial for all such Skerrmions. In the non-topological sector, on the other hand, Skerrmions have no analogue on the Schwarzschild background. Non-topological Skerrmions carry not baryon charge and bifurcate from a subset of Kerr solutions defining an existence line. Therein the appropriate truncation of the Skyrme model yield a linear scalar field theory containing a complex plus a real field, both massive and decoupled, and the Skerrmions reduce to the known stationary scalar clouds around Kerr BHs. Moreover, non-topological Skerrmions trivialise in the vanishing BH mass limit. We then discuss the backreaction of these Skerrmions, that yield rotating BHs with synchronised Skyrme hair, which continously connect to the Kerr solution (self-gravitating Skyrmions) in the non-topological (topological) sector. In particular, the non-topological hairy BHs provide a non-linear realisation, within the Skyrme model, of the synchronous stationary scalar clouds around Kerr.

Stationary scalar clouds around a BTZ black hole

We establish the existence of stationary clouds of massive test scalar fields around BTZ black holes. These clouds are zero-modes of the superradiant instability and are possible when Robin boundary conditions (RBCs) are considered at the AdS boundary. These boundary conditions are the most general ones that ensure the AdS space is an isolated system, and include, as a particular case, the commonly considered Dirichlet or Neumann-type boundary conditions (DBCs or NBCs). We obtain an explicit, closed form, resonance condition, relating the RBCs that allow the existence of normalizable (and regular on and outside the horizon) clouds to the system's parameters. Such RBCs never include pure DBCs or NBCs. We illustrate the spatial distribution of these clouds, their energy and angular momentum density for some cases. Our results show that BTZ black holes with scalar hair can be constructed, as the non-linear realization of these clouds.

Reissner–Nordström black holes with non-Abelian hair

We consider d⩾4 Einstein–(extended-)Yang–Mills theory, where the gauge sector is augmented by higher order terms. Linearising the (extended) Yang–Mills equations on the background of the electric Reissner–Nordström (RN) black hole, we show the existence of normalisable zero modes, dubbed non-Abelian magnetic stationary clouds. The non-linear realisation of these clouds bifurcates the RN family into a branch of static, spherically symmetric, electrically charged and asymptotically flat black holes with non-Abelian hair. Generically, the hairy black holes are thermodynamically preferred over the RN solution, which, in this model, becomes unstable against the formation of non-Abelian hair, for sufficiently large values of the electric charge.

Stationary scalar clouds around maximally rotating linear dilaton black holes

We investigate the wave dynamics of a charged massive scalar field propagating in a maximally rotating (extremal) linear dilaton black hole geometry. We prove the existence of a discrete and infinite family of resonances describing non-decaying stationary scalar configurations (clouds) enclosing these rapidly rotating black holes. The results obtained signal the potential stationary scalar field distributions (dark matter) around the extremal linear dilaton black holes. In particular, we analytically compute the effective heights of those clouds above the center of the black hole.