Extremal Kerr-Newman black holes with extremely short charged scalar hair

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 linearized stationary charged scalar configurations in their exterior regions.

. One may therefore regard the no-short-hair relation (1) as a more modest alternative (at least within the spherically-symmetric sector of hairy black-hole configurations) to the original [1,2] no hair conjecture.
It is important to stress the fact that the formal proof provided in [24] for the existence of the no-short-hair property (1) is restricted to the relatively simple case of spherically-symmetric static hairy black-hole spacetimes. The main goal of the present paper is to challenge the validity of this no-short-hair relation (1) beyond the regime of sphericallysymmetric static black holes. To that end, we shall study analytically the physical properties of non-spherically symmetric non-static Kerr-Newman black holes linearly coupled to stationary (rather than static) charged massive scalar fields.

II. COMPOSED BLACK-HOLE-SCALAR-FIELD CONFIGURATIONS.
It is important to emphasize that while existing no-hair theorems [7] rigorously rule out the existence of static regular hairy black-hole-scalar-field configurations, they do not rule out the existence of non-static scalar field configurations in the exterior regions of black-hole spacetimes.
In fact, recent analytical [27] and numerical [28] explorations of the Einstein-scalar and Einstein-Maxwell-scalar equations have revealed that stationary configurations of massive scalar fields (with or without electric charge) can be supported in the exterior spacetime regions of non-spherically symmetric rotating black holes.
These non-static spatially regular black-hole-scalar-field configurations [27,28] owe their existence to the well established phenomenon of superradiant scattering [29][30][31] of bosonic fields in black-hole spacetimes [32]. In particular, these exterior stationary field configurations have azimuthal frequencies ω field which coincide with the critical (threshold) frequency ω c for superradiant scattering in the charged rotating black-hole spacetime [33]: where [4- 6,34] Ω H = a r 2 + + a 2 and Φ H = are the angular velocity and the electric potential of the Kerr-Newman black hole, and {m, q} are respectively the azimuthal harmonic index and charge coupling constant of the scalar field [35].
The resonance condition (2) guarantees that the orbiting scalar field is not absorbed by the black hole [27,28,36]. In addition, for an asymptotically flat black hole to be able to support stationary (that is, non-decaying) field configurations in its exterior region, the bounded fields must be prevented from radiating their energy to infinity. For massive fields the required confinement mechanism is naturally provided by the mutual gravitational attraction between the central black hole and the orbiting massive bosonic configuration. In particular, bound-state (that is, asymptotically decaying) eigenfunctions of a scalar field of mass µ [37] are characterized by low frequency modes in the regime [see Eq. (17) below] ω 2 < µ 2 . (4) As discussed above, the no-short-hair property (1) was rigorously proved in [24] for the particular case of sphericallysymmetric static hairy black-hole configurations. The main goal of the present paper is to challenge the validity of this relation beyond the regime of spherically-symmetric static black holes. To that end, we shall here study analytically the physical properties of the (non-static non-spherically symmetric) composed Kerr-Newman-charged-massive-scalarfield configurations [27,28] in the large-mass regime [38] M µ ≫ 1 .
Before proceeding, it is worth emphasizing that the composed black-hole-scalar-field configurations that we shall analyze here are not genuine hairy black-hole configurations. In particular, the exterior charged massive scalar fields will be treated at the linear level. We shall therefore use the term 'clouds' to describe these linearized exterior scalar configurations (the term 'hair' usually describes non-linear exterior fields). As we shall show below, the main advantage of the present approach lies in the fact that the composed black-hole-linearized-scalar-field system is amenable to an analytical treatment [39].

III. DESCRIPTION OF THE SYSTEM.
We study a physical system which is composed of a charged massive scalar field Ψ linearly coupled to an extremal charged rotating Kerr-Newman black hole of mass M , electric charge Q, and angular-momentum per unit mass a [40]. Using the Boyer-Lindquist coordinate system, the spacetime metric is described by the line element [4-6] where ∆ ≡ r 2 − 2M r + a 2 + Q 2 and ρ ≡ r 2 + a 2 cos 2 θ. Extremal Kerr-Newman black holes are characterized by the relation where r H is the radius of the degenerate black-hole horizon. The dynamics of a scalar field Ψ of mass µ and charge coupling constant q [37] in the Kerr-Newman black-hole spacetime is governed by the Klein-Gordon wave equation where A ν is the electromagnetic potential of the charged black-hole spacetime. Substituting the field decomposition [41] into the Klein-Gordon wave equation (8) and using the Kerr-Newman metric components (6), one obtains two coupled ordinary differential equations [see Eqs. (10) and (15) below] for the radial and angular components, R lm (r) and S lm (θ), of the scalar eigenfunction Ψ. The angular equation for the familiar spheroidal harmonic eigenfunctions S lm (θ; m, sǫ) is given by [42][43][44][45][46][47] Here is the dimensionless spin (angular-momentum) of the Kerr-Newman black hole, and the dimensionless physical quantity measures the deviation of the scalar field mass µ from the resonant oscillation frequency ω c [see Eq.
The discrete set of angular eigenvalues {K lm (sǫ)} is determined from the regularity requirement of the corresponding angular eigenfunctions (spheroidal harmonics) S lm (θ; sǫ) at the two poles θ = 0 and θ = π. We shall henceforth consider equatorial scalar modes in the eikonal regime in which case the angular eigenvalues are given by [48][49][50] The radial eigenfunctions R lm are determined by the Teukolsky radial equation [42,43] where Note that the angular eigenvalues {K lm (sǫ)} [51] couple the radial equation (15) to the angular equation (10). The bound-state resonances of the composed black-hole-charged-massive-scalar-field system are characterized by exponentially decaying (bounded) radial eigenfunctions at asymptotic infinity [27,28,52]: (17) with ǫ 2 > 0 [53]. In addition, regular scalar configurations are characterized by finite radial eigenfunctions. In particular, The physically motivated boundary conditions (17) and (18), together with the resonance condition (2), single out a discrete set of eigen field-masses [54] {µ n (M, Q, a, l, m, q)} n=∞ n=0 (along with the associated radial eigenfunctions) which characterize the stationary bound-state resonances of the composed Kerr-Newman-charged-massive-scalar-field system.

IV. THE EFFECTIVE BINDING POTENTIAL OF THE BLACK-HOLE SPACETIME
Before solving the radial Teukolsky equation (15), it proves useful to analyze the spatial behavior of the effective radial potential which binds the charged massive scalar field to the charged rotating Kerr-Newman black hole. To that end, it is convenient to define the new radial function in terms of which the radial Teukolsky equation (15) can be written in the form of a Schrödinger-like wave equation where is a dimensionless radial coordinate. The effective radial potential in (20) is given by where and The boundary condition (18) together with the relation (19) dictate at the black-hole horizon. Thus, the effective radial potential (22) of the Schrödinger-like wave equation (20) must be infinitely repulsive at x = 0 [55]: In the case (27), and provided κ > 0, the effective radial potential (22) takes the form of a trapping potential well which can support the stationary bound-state resonances of the composed black-hole-charged-massive-scalar-field system. In particular, in this case the binding potential (22) has one minimum which is located at In the present section we shall derive a (remarkably simple) analytical formula for the discrete spectrum of field masses, {µ n (M, Q, a, l, m, q)} n=∞ n=0 , which characterize the bound-state resonances (the stationary charged scalar clouds) of the composed Kerr-Newman-charged-massive-scalar-field system The solution of the radial equation (15) can be expressed in the simple form [27,46]: where M (a, b, z) is the confluent hypergeometric function [46] and {C 1 , C 2 } are normalization constants. The notation (β → −β) in (29) means "replace β by −β in the preceding term." In order to obtain the resonance equation which characterizes the bound-state resonances of the composed Kerr-Newman-charged-massive-scalar-field system, we shall now examine the asymptotic spatial behaviors of the radial eigenfunction R(x) in the limits x → 0 and x → ∞: (1) The behavior of the radial eigenfunction (29) in the near-horizon x ≪ 1 region is given by [46] R The boundary condition (18), when applied to the near-horizon behavior (30) of the radial eigenfunction, implies that regular bound-state scalar configurations are characterized by the relations Note that (31) is consistent with our previous conclusion (27).
(2) The asymptotic x → ∞ behavior of the radial eigenfunction (29) at spatial infinity is given by [46] R The boundary condition (17), when applied to the asymptotic behavior (32) of the radial eigenfunction, implies that the coefficient of the exploding exponent e ǫx in (32) should vanish. This yields the characteristic resonance equation for the stationary bound-state resonances of the composed Kerr-Newman-charged-massive-scalar-field system. Taking cognizance of Eqs. (2), (12) and (23), one can write κ in the form where [58] α and [59] γ ≡ qQs m .
In the present section we shall challenge the validity of the no-short-hair bound (1) beyond the regime of sphericallysymmetric static black-hole spacetimes [64]. To this end, we shall now evaluate the effective lengths (radii) of the non-spherically symmetric non-static charged scalar clouds which characterize the extremal charged rotating Kerr-Newman black-hole spacetimes.
Taking cognizance of Eqs. (31) and (33), one can express the radial eigenfunctions (29) which characterize the stationary bound-state charged scalar clouds in the form where L (2β) n (x) are the generalized Laguerre polynomials [65] and A is a normalization constant. In particular, the ground-state (n = 0) radial eigenfunction is given by the compact expression [66] The radial eigenfunction (50), which characterizes the fundamental (ground-state) charged scalar cloud, has a global maximum at Note that the expression (51) for the location of the peak of the ground-state (n = 0) radial eigenfunction (50) agrees with the previously found expression (28) [67] for the location of the minimum of the effective binding potential (22). Using the relation (37), one can write (51) in the form Substituting (38) and (45) into (52), one finds This expression for the peak location of the radial eigenfunction (50) provides a quantitative measure for the characteristic size (length) of the fundamental bound-state charged scalar cloud. At this point, it is interesting to note that neutral scalar clouds, which are characterized by the simple relation [25,68] x peak (γ = 0; s) = 1 − 2s 2 respect the no-short-hair lower bound (1). To see this, we recall that the radii of the equatorial (l = m ≫ 1) null circular geodesics which characterize the charged rotating Kerr-Newman black-hole spacetimes are given by the relation [4] This equation can be solved analytically for near-extremal black holes. In particular, one finds which implies [see Eq. (21)] x null (s) → 1 − 2s for 0 ≤ s ≤ 1/2 ; 0 for 1/2 < s ≤ 1 in the extremal M 2 − a 2 − Q 2 → 0 limit.
Taking cognizance of Eqs. (54) and (57), one finds the characteristic inequality x peak (γ = 0; s) > x null (s) (58) for neutral scalar clouds in the entire range s ∈ (0, 1 √ 2 ) [68]. We therefore conclude that neutral scalar clouds conform to the no-short-hair lower bound (1). In particular, for the neutral scalar clouds one finds min s {x peak (γ = 0; s)/x null (s)} = 10 + 6 √ 3 ≃ 20.392 for s = ( √ 3 − 1)/2 ≃ 0.366. Let us now return to the more general case of charged scalar clouds. From the expression (53) for x peak (γ; s) one deduces that, for a given value of the black-hole angular momentum s, x peak (γ; s) is a decreasing function of the dimensionless physical parameter γ. In particular, the expression (53) for x peak (γ; s) reveals the remarkable fact that the exterior charged scalar clouds can be made arbitrary compact. In particular, one finds [69] It is worth emphasizing that the relation (59) is valid for extremal Kerr-Newman black holes in the entire range s ∈ (0, 1) of the black-hole rotation parameter. This fact implies that all [70] charged and rotating extremal Kerr-Newman black holes can support extremely compact charged scalar configurations in their exterior regions [71]. Taking cognizance of Eqs. (57) and (59), one finds the important inequality for charged scalar clouds in the entire range s ∈ (0, 1 2 ). We therefore conclude that charged scalar clouds may violate the no-short-hair lower bound (1) [64,72].

VII. SUMMARY.
A no-short-hair theorem for spherically-symmetric static black holes was proved in [24]. This theorem reveals that, if a spherically-symmetric static black hole has hair, then this hair (i.e. the external fields) must extend beyond the null circular geodesic (the"photonsphere") of the corresponding black-hole spacetime [see Eq. (1)].
The main goal of the present study was to challenge the validity of this no-short-hair property beyond the regime of static spherically-symmetric hairy black-hole configurations. To that end, we have studied analytically the physical properties of extremal charged rotating Kerr-Newman black holes linearly coupled to non-spherically symmetric stationary (rather than static) charged massive scalar fields.
In particular, for given parameters {M, Q, J} of the central Kerr-Newman black hole, we have identified the critical value of the field charge coupling constant, q = q * (s) [see Eqs. (36) and (59)], which minimizes the effective radial lengths of the exterior stationary charged scalar configurations. This allowed us to prove that the (non-static, nonspherically symmetric) composed Kerr-Newman-charged-massive-scalar-field configurations can violate the no-shorthair lower bound (1). In particular, it was shown that extremal Kerr-Newman black holes in the entire parameter space s ∈ (0, 1) can support extremely compact stationary charged scalar clouds (made of linearized charged massive scalar fields with the property r field → r H ) in their exterior regions. Specifically, these short-range Kerr-Newmancharged-massive-scalar-field configurations are characterized by the simple relation [see Eqs. (48) and (59)] It is interesting to note that, in order to support these extremely compact field configurations, the extremal black hole must have both angular-momentum and electric charge [73]. Finally, it is worth emphasizing again that we have treated here the exterior charged massive scalar fields at the linear level. As we have shown, the main advantage of this approach lies in the fact that the composed Kerr-Newmanlinearized-charged-scalar-field system is amenable to an analytical treatment. We believe that it would be highly important to use numerical techniques [28] in order to generalize our results to the regime of non-linear exterior fields.