Multioscillating black holes
PPrepared for submission to JHEP
KUNS-2856
Multioscillating black holes
Takaaki Ishii, Keiju Murata, Jorge E. Santos, Benson Way Department of Physics, Kyoto University, Kitashirakawa Oiwake-cho, Kyoto 606-8502,Japan Department of Physics, College of Humanities and Sciences, Nihon University, Sakura-josui, Tokyo 156-8550, Japan DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road,Cambridge CB3 0WA, UK Departament de F´ısica Qu`antica i Astrof´ısica, Institut de Ci`encies del CosmosUniversitat de Barcelona, Mart´ı i Franqu`es, 1, E-08028 Barcelona, Spain
E-mail: [email protected] , [email protected] , [email protected] , [email protected] Abstract:
We study rotating global AdS solutions in five-dimensional Einsteingravity coupled to a multiplet complex scalar within a cohomogeneity-1 ansatz. Theonset of the gravitational and scalar field superradiant instabilities of the Myers-Perry-AdS black hole mark bifurcation points to black resonators and hairy Myers-Perry-AdS black holes, respectively. These solutions are subject to the other (gravi-tational or scalar) instability, and result in hairy black resonators which contain bothgravitational and scalar hair. The hairy black resonators have smooth zero-horizonlimits that we call graviboson stars. In the hairy black resonator and graviboson so-lutions, multiple scalar components with different frequencies are excited, and hencethese are multioscillating solutions. The phase structure of the solutions are exam-ined in the microcanonical ensemble, i.e. at fixed energy and angular momenta. It isfound that the entropy of the hairy black resonator is never the largest among them.We also find that hairy black holes with higher scalar wavenumbers are entropicallydominant and occupy more of phase space than those of lower wavenumbers. a r X i v : . [ h e p - t h ] J a n ontents j A.1 Equations of motion 23A.2 Boundary conditions at infinity 24A.3 Boundary conditions at the origin for horizonless solutions 25A.4 Boundary conditions at the horizon 26A.5 Physical quantities 28A.6 Interpolation of hairy black resonator data 29
B Instability of hairy MPAdS 30C Perturbative solution of boson star 31
C.1 Higher order perturbation 31C.2 Large- j expansion of the perturbative solution 33– 1 – Introduction
Through superradiant scattering, energy can be extracted from rapidly rotating blackholes (see [1] for a review). In global anti-de Sitter space (AdS), the reflectingboundary causes these black holes to be unstable to the superradiant instability [2–11], whose ultimate endpoint remains an open problem [12–15]. Because small blackholes typically have high angular frequency, the ultimate configuration of low-energystates in AdS likewise remains unknown.Early work on this problem involved studying perturbations of the Kerr-AdSand Myers-Perry-AdS black holes (the higher-dimensional analog of Kerr-AdS [16–19], see [20] for a review), where quasi-normal spectra were obtained in [10, 11]. Forspecific modes, the onset of the superradiant instability occurs at a frequency where (cid:61) ( ω ) = 0, but (cid:60) ( ω ) (cid:54) = 0. This suggests the existence of a time-periodic black holewith a single helical Killing vector that branches from these onsets. Such black holes,called black resonators , which can be viewed as black holes with a gravitational hair,were later constructed in [13], where it was found that they have higher entropy(horizon area) than the corresponding Kerr-AdS black hole with the same mass andangular momentum.It is therefore entropically permissible for Kerr-AdS black holes to evolve towardsblack resonators. However, though black resonators are stable to the mode thatgenerated them, they are still rapidly rotating and hence remain unstable to other,typically higher, superradiant modes [12–15, 21]. It appears, therefore, that theinstability leads to a cascade with higher and higher modes growing in time. If thereis indeed an unceasing energy cascade towards higher modes, there will eventuallybe a significant amount of energy placed in sub-Planckian length scales, which canbe viewed as a violation of the weak cosmic censorship conjecture [22].To date, there is only a single study of time-evolution involving the rotationalsuperradiance of AdS [15]. In Kerr-AdS, there is typically one unstable mode thatdominates the dynamics at early times. Evolution then proceeds towards a blackresonator until the instabilities of the black resonator itself begin to take over anddrive the continuing evolution. The evolution in [15] was not continued further dueto numerical limitations.Because of the lack of symmetries and the long time-scales involved, the studyof the superradiant instability is a significant numerical challenge. It is thereforefortunate that a simplification in a more limited setting has been found. By movingto five dimensions and allowing both angular momenta to be equal, black resonatorswith a cohomogeneity-1 ansatz (i.e. the metric functions depend only on a singlevariable, and the solution can be obtained by solving ODEs) were constructed in[23]. The scalar, electromagnetic, and gravitational quasinormal modes of these blackresonators were studied shortly thereafter in [24]. As anticipated, black resonators For the superradiant instability of an electromagnetic perturbation isolated from these modes, – 2 –re unstable to higher modes.These higher-mode instabilities of black resonators also have their individualonsets, from which new black resonators having multiple frequencies could be gen-erated. We will refer such solutions as multi black resonators. Here, we set outto construct such multi black resonator solutions and to study their relationshipwith black resonators and Myers-Perry-AdS black holes. Ideally, we would studymulti black resonators that are generated by gravitational perturbations of blackresonators. However, these perturbations break too many symmetries of the orig-inal black resonator solution. We therefore focus on multi black resonators thatare generated by scalar fields. Ordinarily, a scalar field would also break most ofthese symmetries, but we will rely on a multiplet scalar constructed using WignerD-matrices, from which the cohomogeneity-1 structure can be preserved. A scalardoublet version of such a system was previously studied in [9, 26] . The higherscalar multiplets we introduce can coexist with oscillations of the metric, but do notcontribute additional extra oscillating frequencies to the metric.Even in this limited setting, the full space of solutions is intricate. There areMyers-Perry-AdS black holes, black holes with scalar hair, black resonators, and nowhairy black resonators, which are a kind of multi black resonator. In addition, thereare boson stars, geons, and now graviboson stars, which are all horizonless solutionsthat serve as the zero-size limit of hairy Myers-Perry-AdS, black resonators, andhairy black resonators, respectively. All of these solutions compete thermodynami-cally when they share the same energy and angular momenta. We will compute thefull phase diagram of this system. Perhaps surprisingly, we find that hairy blackresonators are never dominant in such a phase diagram.Another advantage of the multiplet scalar model is that different nonlinear so-lutions generated by different mode-instabilities can be consistently compared withone another, while maintaining a cohomogeneity-1 ansatz. This will allow us to showthat black holes generated from higher scalar mode instabilities have higher entropyand occupy a larger region of phase space than those from lower modes. Similar con-clusions reached by previous work were only argued by extrapolating perturbativecalculations, and here we are able to perform full nonlinear calculations and computethe actual phase boundaries.This paper is structured as follows. In the next section, we review some basicproperties of isometries of S and Wigner D-matrices that we use in the constructionof our ansatz. Then in section 3, we describe details of our ansatz. Sections 4 and5 review the Myers-Perry-AdS solution, geons, and black resonators in this ansatz,which were studied in [23, 24]. Section 6 discusses hairy black holes and boson starswhich are higher multiplet versions of those in [9]. Then, in section 7, we present the a cohomogeneity-1 photonic black resonator was also constructed in [25]. See also [27, 28] for a study covering different number of dimensions. – 3 –ntirely new hairy black resonator and graviboson star solutions. The entire phasediagram of all solutions then pieced together in section 8, and then we compare theresults to a higher wavenumber in section 9. We then finish with some concludingremarks in section 10. The appendix contains technical details.
Our metric ansatz will contain deformations of an S whose perturbations can natu-rally be written in terms of Wigner D-matrices D jmk ( θ, φ, χ ). Let us therefore beginwith a review of the Wigner D-matrices, which we will later use for designing acohomogeneity-1 scalar field ansatz. (See Refs. [7, 24, 29–33] for an introduction tothe Wigner D-matrix and its applications in gravitational perturbation theory.)We focus on the SO (4) (cid:39) SU (2) L × SU (2) R isometry of S , whose metric canbe written as dΩ = 14 ( σ + σ + σ ) , (2.1)where σ i ( i = 1 , ,
3) are 1-forms defined by σ = − sin χ d θ + cos χ sin θ d φ ,σ = cos χ d θ + sin χ sin θ d φ ,σ = d χ + cos θ d φ . (2.2)These satisfy the SU (2) Maurer-Cartan equation d σ i = (1 / (cid:15) ijk σ j ∧ σ k . The coor-dinate ranges are 0 ≤ θ < π , 0 ≤ φ < π , and 0 ≤ χ < π , and have a twistedperiodicity ( θ, φ, χ ) (cid:39) ( θ, φ + 2 π, χ + 2 π ) (cid:39) ( θ, φ, χ + 4 π ). The Killing vectors gen-erating SU (2) L and SU (2) R , denoted by ξ i and ¯ ξ i , respectively, are given by ξ = cos φ∂ θ + sin φ sin θ ∂ χ − cot θ sin φ∂ φ ,ξ = − sin φ∂ θ + cos φ sin θ ∂ χ − cot θ cos φ∂ φ ,ξ = ∂ φ , (2.3)and ¯ ξ = − sin χ∂ θ + cos χ sin θ ∂ φ − cot θ cos χ∂ χ , ¯ ξ = cos χ∂ θ + sin χ sin θ ∂ φ − cot θ sin χ∂ χ , ¯ ξ = ∂ χ . (2.4)Note that ¯ ξ i are the dual vectors of σ i : ( σ i ) α ( ¯ ξ j ) α = δ ij ( α = θ, φ, χ ).Using language from quantum mechanics, we can define the “angular momen-tum” operators L i = iξ i , R i = i ¯ ξ i , (2.5)– 4 –hich satisfy the commutation relations [ L i , L j ] = i(cid:15) ijk L k and [ R i , R j ] = − i(cid:15) ijk R k .These operators are Hermitian under the inner product on the S ,( f, g ) ≡ (cid:90) π d θ (cid:90) π d χ (cid:90) π d χ sin θf ∗ ( θ, φ, χ ) g ( θ, φ, χ ) . (2.6)Under the SU (2) L and SU (2) R , the 1-forms introduced in Eq. (2.2) transformas L i σ j = 0 , R i σ j = − i(cid:15) ijk σ k , (2.7)where the operations of L i and R i are defined by Lie derivatives. From the firstequation of (2.7), one can see that σ i are invariant under SU (2) L . For this reason,they are called SU (2)-invariant 1-forms. The second equation means that R i generatethe three-dimensional rotation of the “vector” ( σ , σ , σ ). In particular, R generates U (1) R ⊂ SU (2) R , which corresponds to rotation in the ( σ , σ )-plane. The invarianceof dΩ under SU (2) L × SU (2) R can be easily checked by using Eq. (2.7).The generators L i and R i share the same Casimir operator: L ≡ L + L + L = R + R + R , and the set of commutative operators is given by ( L , L z , R z ). TheWigner D-matrix D jmk ( θ, φ, χ ) is defined to be the eigenfunction of these operators: L D jmk = j ( j + 1) D jmk , L z D jmk = mD jmk , R z D jmk = kD jmk , (2.8)where the ranges of the quantum numbers ( j, m, k ) are j = 0 , / , , / , . . . ,m = − j, − j + 1 , . . . , j ,k = − j, − j + 1 , . . . , j . (2.9)The Wigner D-matrices are orthogonal under the inner product (2.6),( D j (cid:48) m (cid:48) k (cid:48) , D jmk ) = 2 π j + 1 δ mm (cid:48) δ kk (cid:48) δ jj (cid:48) . (2.10)We can also define the ladder operators L ± = L x ± iL y and R ± = R y ± iR x ,which shift the “orbital angular momenta” of D jmk as L + D jmk = ε m +1 D j ( m +1) k , L − D jmk = ε m D j ( m − k ,R + D jmk = (cid:15) k +1 D jm ( k +1) , R − D jmk = (cid:15) k D jm ( k − , (2.11)where ε m = (cid:112) ( j + m )( j − m + 1) and (cid:15) k = (cid:112) ( j + k )( j − k + 1).The Wigner D-matrices satisfy a convenient formula for summation, j (cid:88) m = − j ( D jmk (cid:48) ) ∗ D jmk = δ k (cid:48) k . (2.12)– 5 –his can be proved easily using the ladder operators. Using Eqs. (2.8) and (2.11), wefind L i ( (cid:80) jm = − j ( D jmk (cid:48) ) ∗ D jmk ) = 0. Hence, the left hand side of (2.12) is a constant.Integrating this equation over the S as Eq. (2.6) and using Eq. (2.10), we find thatthe constant is δ kk (cid:48) .For later convenience, we introduce a (2 j + 1)-component vector (cid:126)D k by (cid:126)D k = D jm = j,k D jm = j − ,k ... D jm = − j,k . (2.13)Although (cid:126)D k also depends on the index j , we suppress it for notational simplicitybecause we will generally keep j fixed once the content of the scalar field is specified.In this notation, Eq. (2.12) is simply written as (cid:126)D ∗ k (cid:48) · (cid:126)D k = δ k (cid:48) k . (2.14) We now describe the ansatz for resonating cohomogeneity-1 spacetimes. We will showthat the energy-momentum tensor of the matter field we introduce is consistent withthe symmetries of the metric and therefore that the equations of motion reduce to aconsistent set of ordinary differential equations.We consider the following five-dimensional Einstein-scalar system with a negativecosmological constant: S = 116 πG (cid:90) d x √− g (cid:18) R + 12 L − ∂ µ (cid:126) Π ∗ · ∂ µ (cid:126) Π (cid:19) , (3.1)where (cid:126) Π denotes a (2 j + 1)-component complex scalar multiplet, G is the five-dimensional Newton constant, and L is the AdS radius. Hereafter, we set L = 1.For the metric, we take the cohomogeneity-1 ansatz [23]d s = − (1 + r ) f ( r )d τ + d r (1 + r ) g ( r )+ r { α ( r ) σ + 1 α ( r ) σ + β ( r )( σ + 2 h ( r )d τ ) } , (3.2)where σ i were defined in (2.2). For the scalar multiplet, we take (cid:126) Π( τ, r, θ, φ, χ ) = (cid:88) k ∈ K Φ k ( r ) (cid:126)D k ( θ, φ, χ ) , (3.3)– 6 –here Φ k ( r ) are real scalar fields, and K is defined by K = { j, j − , j − , · · · , − j } ( j : integer) ,K = { j, j − , j − , · · · , − j + 1 } ( j : half integer) . (3.4)We first comment on the metric ansatz (3.2) before later addressing the scalar.This metric ansatz preserves SU (2) L but breaks SU (2) R . If α ( r ) = 1, then a U (1) R ⊂ SU (2) R symmetry generated by R is restored. To see this, we can use the fact that σ + σ = d θ + sin θ d φ = dΩ , (3.5)which is independent of χ .In the metric (3.2), we also assume invariance under two discrete transformations P and P defined by P ( τ, χ, φ ) = ( − τ, − χ, − φ ) , P ( τ, χ, φ ) = ( τ, χ + π, φ ) . (3.6)The 1-forms (d τ, σ , σ , σ ) are transformed by P and P as P (d τ, σ , σ , σ ) = ( − d τ, − σ , σ , − σ ) ,P (d τ, σ , σ , σ ) = (d τ, − σ , − σ , σ ) . (3.7)Because of the invariance under P and P , cross terms such as σ σ do not appearin Eq.(3.2).By examining boundary conditions, it turns out that the metric ansatz (3.2) istaken to be in a frame where asymptotic infinity is rotating. We will search for blackholes with a Killing horizon generated by ∂ τ . This condition in turn enforces that f ( r h ) = g ( r h ) = 0. For black hole solutions with α ( r ) (cid:54) = 1 we must also satisfy h ( r h ) = 0 (see appendix A.4). Meanwhile, h ( r ) approaches a constant value Ω atinfinity r → ∞ , and the asymptotic form of the metric becomes ds (cid:39) − r d τ + d r r + r { σ + σ + ( σ + 2Ωd τ ) } , (3.8)from which we see that the boundary metric is R ( τ ) × S , but with rotation in the σ directions. This is the rotating frame at infinity.The meaning of Ω becomes clear by moving to the non-rotating frame at infinity,which is the natural frame for interpreting conserved charges and other quantities ofthe black hole. We can switch to the non-rotating frame by applying the followingcoordinate transformation: d t = d τ , d ψ = d χ + 2Ωd τ . (3.9) If α ( r ) (cid:54) = 1, the U (1) R isometry χ → χ + const . is broken, and therefore there is no continuousshift of h ( r ) that does not change the metric. – 7 –n the new frame, the horizon generator is written as ∂∂τ = ∂∂t + Ω ∂∂ ( ψ/ . (3.10)Therefore, Ω corresponds to the angular velocity of the horizon. When Ω >
1, thenorm of ∂ τ , g ττ , becomes positive at infinity. This implies that there is no globaltime-like Killing vector in the domain of outer communications, and therefore thespacetime is non-stationary for Ω >
1. If α (cid:54) = 1, the components of the new metrictransformed from Eq. (3.2) become explicitly time dependent. Thus our metriccan be said to describe time periodic solutions, even though the metric ansatz iscohomogeneity-1. In our metric ansatz, we can therefore distinguish between time-periodic solutions (with α (cid:54) = 1) and solutions that are stationary (with α = 1).Now we comment on the scalar field (3.3). This ansatz is precisely the “doublestepping” ansatz ( k decreases by 2 in the sum) introduced in the perturbative analysisof black resonators and geons [24]. Indeed, the modes in K decouple from those ofits complement K c = { j, j − , · · · , − j } \ K . This fact ultimately stems from thediscrete isometry P of the metric (3.2). Under P , the Klein-Gordon equation for (cid:126) Πcan be decomposed into even and odd parts. More specifically, because (cid:126)D k ∝ e − ikχ ,the Wigner D-matrices with k ∈ K and k ∈ K c acquire different phase factors of ±
1. The Klein-Gordon equations for k ∈ K and k ∈ K c are therefore decoupled. Inthis paper, we will consider only k ∈ K .Note also that we have defined our ansatz with fixed j and 2 j + 1 multiplet.However, a solution with a particular j is also a solution to the theory with largermultiplets than 2 j + 1, simply by setting the extra components of the multiplet tozero. Solutions with different j can therefore be consistently compared with oneanother.Finally, we show that the scalar field ansatz (3.3) is consistent with the metric(3.2), i.e. the Einstein and Klein-Gordon equations reduce to a consistent set ofODEs. The Einstein equations from Eq. (3.1) are given by G µν − g µν = T µν , wherethe energy-momentum tensor is T µν = T ( µν ) − g µν T , T µν = ∂ µ (cid:126) Π ∗ · ∂ ν (cid:126) Π , T = g µν T µν . (3.11)To derive the explicit expression of the energy-momentum tensor for Eq. (3.3),it is convenient to introduce 1-forms σ ± defined by σ ± = 12 ( σ ∓ iσ ) = 12 e ∓ iχ ( ∓ i d θ + sin θ d φ ) (3.12) We use a canonically normalized angular coordinate ψ/ ∈ [0 , π ). See also Ref. [34] for a five-dimensional cohomogeneity-1 geometry with periodic time depen-dence in asymptotically Poincar´e AdS space with a S direction. – 8 –nd use the basis e a = { d τ, d r, σ + , σ − , σ } ( a = τ, r, + , − , e a = { ∂ τ , ∂ r , e + , e − , ∂ χ } where e ± = ¯ ξ ± i ¯ ξ = ± R ∓ = e ± iχ ( ± i∂ θ + 1sin θ ∂ φ − cot θ∂ χ ) . (3.13)One can check that e aµ e µb = δ ab .In this basis, the derivatives of the scalar field can be evaluated by using Eqs. (2.8)and (2.11) as ∂ τ (cid:126) Π = 0 , ∂ r (cid:126) Π = (cid:88) k ∈ K Φ (cid:48) k (cid:126)D k , ∂ + (cid:126) Π = (cid:88) k ∈ K (cid:15) k Φ k (cid:126)D k − ,∂ − (cid:126) Π = − (cid:88) k ∈ K (cid:15) k +1 Φ k (cid:126)D k +1 , ∂ (cid:126) Π = − i (cid:88) k ∈ K k Φ k (cid:126)D k , (3.14)where ∂ a ≡ e µa ∂ µ .Some components of T ab vanish because of the double stepping coupling (3.4).For example, we find T r + = (cid:88) k,k (cid:48) ∈ K (cid:15) k Φ (cid:48) k (cid:48) Φ k (cid:126)D ∗ k (cid:48) · (cid:126)D k − = (cid:88) k,k (cid:48) ∈ K (cid:15) k Φ (cid:48) k (cid:48) Φ k δ k (cid:48) ,k − = 0 , (3.15)where in the second equality we used Eq. (2.14), and the last one follows from thefact that k (cid:48) and k − k (3.4).To evaluate the non-vanishing components of T ab , we also use the orthogonalityof the Wigner D-matrices (2.14). The upshot is that the energy momentum tensoris given by T ( ab ) e a e b = (cid:88) k ∈ K (cid:20) Φ (cid:48) k d r − (cid:15) k − (cid:15) k Φ k − Φ k ( σ + σ − )+ ( (cid:15) k + (cid:15) k +1 )Φ k σ + σ − + k Φ k σ (cid:21) . (3.16)This result, invariant under SU (2) L , is consistent with the spacetime (3.2), and theEinstein and Klein-Gordon equations reduce to a consistent set of coupled ODEs.The explicit form of the equations of motion is summarized in appendix A, wheretechnical details in solving the equations are also explained.In our ansatz of the scalar field (3.3), the conserved current of the complex scalarfield J µ is given by J µ d x µ ≡ Im[ (cid:126) Π · ∂ µ (cid:126) Π ∗ ]d x µ = (cid:88) k ∈ K k Φ k σ . (3.17)We have J ψ (cid:54) = 0 if and only if Φ k (cid:54) = 0 for k (cid:54) = 0. This indicates that there is arotating flow of the scalar field and it carries angular momentum when the scalarfield has non-trivial ψ -dependence. – 9 – Figure 1 . Energy of geons E geon as a function of the angular momentum J . If α ( r ) = 1 in Eq. (3.2), there is an exact solution describing a rotating black holewith both angular momenta set equal. It is part of the Myers-Perry-AdS family ofsolutions, which we will abbreviate MPAdS. In our ansatz, the metric functions are g ( r ) = 1 − µ (1 − a ) r (1 + r ) + 2 a µr (1 + r ) , β ( r ) = 1 + 2 a µr ,h ( r ) = Ω − µar + 2 a µ , f ( r ) = g ( r ) β ( r ) , α ( r ) = 1 . (4.1)The event horizon r = r h is located at the largest root of g ( r h ) = 0, and the isometrygroup of this solution is R ( τ ) × SU (2) L × U (1) R .The solution is parametrised by µ and a , with µ = 0 corresponding to pure AdS.As written, Ω is merely a gauge parameter that allows us to move between rotatingand non-rotating frames at infinity. For consistency of notation and convenience, weset Ω = 2 µar h + 2 a µ (4.2)so that Ω is the angular velocity of the horizon. For this choice, we have h ( r h ) = 0and, thus, ∂ τ becomes the horizon generator. MPAdS solutions are bounded byextremality, which occurs at Ω extr = (cid:112) r h √ r h . (4.3)Before discussing perturbations of MPAdS, let us briefly describe a family ofsolutions called geons. Geons are horizonless, nonlinear extensions of gravitationalnormal modes of pure AdS. Within our ansatz, these normal modes are given by a– 10 – x t r e m e M P A dS Gravitational perturbationScalar fi eld perturbations Extreme MPAdSGravitational perturbationScalar field perturbation
Figure 2 . (Left) Onset of the superradiant instability of MPAdS against gravitationalperturbation (red) and scalar field perturbations with j = k = 9 / , , r h )-plane. Above the red and blue curves, MPAdS is unstable to gravitational and scalarperturbations, respectively. The extreme limit of MPAdS is shown by the black curve.(Right) The same data as the left panel are shown in the ( E − E geon , J )-plane, though forthe scalar field perturbations only the onset of the mode with j = k = 9 / perturbation of the form α = 1+ δα ( r ) about AdS, with Ω appearing as an eigenvalue.Geons therefore carry angular momentum, with Ω as an angular frequency. Fig. 1 isthe energy of geons E geon as a function of the angular momentum J . Later in thispaper, we will primarily use the difference of the energy E from E geon in figures forbetter visibility. Near vacuum AdS, the geons have frequency parameter Ω (cid:39) / E geon (cid:39) (3 / J .Now let us return to perturbations of MPAdS black holes. When the MPAdSblack hole has sufficiently high angular frequency, it is unstable to superradianceagainst both gravitational and scalar field perturbations. Among the unstable su-perradiant modes is one that breaks only the U (1) R isometry of the metric. Its onsetmode of the gravitational perturbation can be found in our ansatz by perturbing themetric function as α ( r ) = 1 + δα ( r ) about MPAdS and linearising the equation ofmotion (A.5) in δα ( r ). The onset mode of the scalar field perturbation is just givenby the probe scalar field satisfying the Klein-Gordon equation (A.5). In the MPAdSbackground, modes with different k decouple in the Klein-Gordon equation. We willfocus on modes with k = j , as these are the most dominant.In Fig. 2, we show results for gravitational and scalar field onsets with j = k =9 / j = k = 5. In the left figure, we show the onset of the instabilities inthe (Ω , r h )- and ( E − E geon , J )-planes. The scalar and gravitational onset curvesintersect at green dots. For reference, we also show where MPAdS is extremal, andwhere Ω = 1. Recall that MPAdS is unstable for Ω >
1, but not necessarily to the– 11 – xtreme MPAdS
Figure 3 . Entropy of black resonators as a function of ( E − E geon , J ). The upper solidcurve indicates the onset of the gravitational superradiant instability of the MPAdS, wherethe black resonators branch off. The extreme MPAdS is shown by a black curve. In theupper-right region below the onset curve, we simply do not have numerical data. j = k = 9 / j = k = 5 modes.In the right figure, we show the same results in the ( E − E geon , J )-plane, butonly show the result for the scalar for j = k = 9 / E − . J as the vertical axis for visibility. As mentioned in the previous section, MPAdS is unstable to superradiance againstgravitational perturbations. A new family of cohomogeneity-1 solutions with α ( r ) (cid:54) =1 branches off from the onset of the instability [23]. These onsets were given by thered curves labelled “gravitational perturbations” in Fig. 2. Because α (cid:54) = 1 for thisnew family, these black holes are time-periodic as seen in the non-rotating frame atinfinity and are known as black resonators. These black resonators have R × SU (2) L isometries. Like MPAdS black holes, these black resonators are a two-parameterfamily. The horizonless limit of black resonators are geons [12, 35–37], which wehave already described in the previous section and in Fig. 1. Like black resonators,geons also have R × SU (2) L isometries and are time-periodic in the non-rotatingframe at infinity. – 12 –n Fig. 3, the entropy of black resonators S is shown by the colour map as afunction of ( E − E geon , J ). The extreme MPAdS is shown by a black curve. OnlyMPAdS exists in the upper side of the black curve. The black resonators branch offfrom the onset of the superradiant instability shown by the solid curve on the upperedge of the plotted region. In the upper-right white region below the onset curve,we do not have numerical data for black resonators. For a large J , the MPAdS atthe onset is very close to the extremality, and numerical construction of the blackresonator becomes difficult. In the limit of geons, E − E geon →
0, the entropy alsoapproaches zero. In Ref. [23], it has been found that the angular velocity of the blackresonator and geon always satisfied Ω >
1. Therefore, the black resonator and geonare superradiant and non-stationary.
Thus far, we have discussed MPAdS, black resonators, and geons, which are allsolutions that satisfy (cid:126)
Π = 0. Now we turn to solutions with (cid:126) Π (cid:54) = 0, beginning withthose with α = 1.A special case of our ansatz (3.3) is given by the scalar field with “single- k ”: (cid:126) Π( τ, r, θ, φ, χ ) = Φ k ( r ) (cid:126)D k ( θ, φ, χ ) (no summation) . (6.1)In this case, the matter stress tensor Eq. (3.16) reduces to T ( ab ) e a e b = Φ (cid:48) k d r + 14 ( (cid:15) k + (cid:15) k +1 )Φ k ( σ + σ ) + k Φ k σ , (6.2)where we returned to σ , from σ ± . In this expression, the coefficients of σ and σ coincide, and therefore it has the invariance under U (1) R ∈ SU (2) R generated by theangular momentum operator R . The metric (3.2) therefore also has this isometry,which implies that α ( r ) = 1 in the single- k case.Superradiant instabilities can be induced by scalar fields as well as gravitationalfields. These onset curves were shown earlier for the scalar field with j = k = 9 / j = k = 5 by the blue curves in Fig. 2. For the scalar field, we refer to thesolutions branching off from the onset of the superradiant instability as the hairyMPAdS black holes , as they contain scalar hair. Because black resonators can beinterpreted as black holes with gravitational hair, hairy black holes are the scalarfield counterparts to black resonators.A hairy MPAdS black hole becomes a boson star in the horizonless limit. Bosonstars can also be described as nonlinear scalar normal modes of pure AdS. Bosonstars are therefore the scalar field counterparts to geons. We obtain the perturbativesolution of the boson star near pure AdS in appendix C, where we also discuss thelarge- j limit for the perturbative solution. In the following, we focus on the scalarfield with k = j , which is the most relevant mode for the superradiant instability for– 13 – -0.100.10.20.30.40.50.6 0 2 4 6 8 10 12 (a) Energy of boson star Turning pointsof hairy MPAdSOnset of scalar fi eldinstability of MPAdS B o s o n s t a r (b) Entropy of hairy MPAdS M u l t i - v a l u e d Hairy MPAdS E x t r e m e M P A dS Turning points of hairy MPAdS
Onset of scalar field insability of MPAdS
Boson star (c) Domain of hairy MPAdS
Figure 4 . (a) Energy of boson stars as a function of the angular momentum J . Theexistence of multiple turning points are shown in the insets for j = 1 /
2. (b) Entropy ofthe hairy MPAdS for k = j = 9 /
2, parametrized by J and E − E geon . The solid curve onthe left edge indicates the onset of the scalar field superradiant instability of the MPAdS,from which the hairy MPAdS branches off. (c) Domain of existence of the hairy MPAdS.In orange and red regions, the hairy MPAdS exists. In the red region, the entropy andother physical quantities become multi-valued because of the turning points. a given j . We note that the hairy MPAdS black hole with j = 1 / Our treatment (6.1) gives a generalization to j ≥ J . The curves correspond to j = 1 / , , / , · · · , E and J agree with perturbation theory about pureAdS. More specifically, the scalar field (6.1) with k = j has the lowest normal mode The ansatz (6.1) for j = 1 / – 14 –t Ω = 1 + 2 /j , from which the boson star branches off. For small J , the energyof the boson star is given by E (cid:39) (1 + 2 /j ) J . Comparing this with that of thegravitational geon E geon (cid:39) (3 / J , one finds that E − E geon in small J is O ( J ) for j = 4 and negative for j ≥ / J is increased, there are turning points in the energy of boson stars, indicatingthe change of stability of the boson stars. It is common for solutions past the turningpoints to be unstable [39–42]. In the insets of the figure, we zoom in on the curvefor j = 1 / E − . J is used in the vertical axis for visibility). The curve isfolded multiple times.The entropy of the hairy MPAdS for j = 9 / j = 5,see section 9. The hairy MPAdS branches off from the onset of the scalar fieldsuperradiant instability denoted by the solid curve on the left edge of the plot region.Turning points of the energy also exist for hairy MPAdS as shown by the green curve,and as a result the entropy and other physical quantities of the hairy MPAdS becomemulti-valued. We see Ω >
1, and this indicates that the hairy MPAdS and bosonstar are superradiant and non-stationary.Fig. 4(c) shows the domain of existence of the hairy MPAdS in the ( E − E geon , J )-plane. In orange and red regions, the hairy MPAdS exists. In particular, in the redregion, there are multiple solutions for a fixed ( E, J ), and physical quantities aremulti-valued. Our numerical calculation indicates that the curve of turning pointsterminates at the black dot: the intersecting point between the extreme MPAdS andthe onset of scalar field superradiant instability of MPAdS. A part of the turningpoints is shown by the dashed blue curve. We drew this part by interpolation of ournumerical data.
Finally, we consider the most general case in our ansatz (3.2) and (3.3) that has both α ( r ) (cid:54) = 1 and (cid:126) Π( r ) (cid:54) = 0.Ref. [24] has located the onset of the superradiant instability of black resonatorsfor scalar fields. At the onset of the instability, there is a τ -independent perturbationof the scalar field in the form of Eq. (3.3). This is expected to lead to a new familyof black resonator solutions with a nontrivial scalar hair.Similarly, the hairy MPAdS black hole is expected to be unstable against grav-itational perturbations with α ( r ) (cid:54) = 1 (see appendix B), and a new family of blackresonator solutions is expected to branch from the onset of this instability. The normal mode frequency for the scalar field with the quantum numbers ( j, k ) is given byΩ = (2 + j + n ) /k where n is the radial overtone number. The lowest mode has n = 0. One canalso see that k = j gives the lowest | Ω | . – 15 – Boson star
Hairy black resonator
Onset of instabilityof black resonator O n s e t o f i n s t a b ili t y o f h a i r y M P A d S Onset of gravitational instability of MPAdS G r a v i b o s o n s t a r (a) Domain of hairy black resonator Black resonatorHairy MPAdSHairy black resonator (b) Entropy
Black resonatorHairy black resonatorHairy MPAdS (c) Cross section at J = 0 . Figure 5 . (a) Domain of existence of the hairy black resonator. It exists inside the “tri-angle” surrounded by purple curves. The equal-entropy-curve between the hairy MPAdSand black resonator is plotted by the orange curve. The angular velocity is shown by thecolor map. (b) Entropies of the black resonator (blue), hairy MPAdS (orange), and hairyblack resonator (green) for j = 9 /
2. The entropy of the hairy black resonator is never thelargest. (c) Cross section of (a) at J = 0 . We can therefore have scalar hair on black resonators, and gravitational “resonator”-type excitations on hairy MPAdS black holes. It turns out both of these excitationsare part of the same family of solutions, which we call hairy black resonators . Hairyblack resonators have their own family of horizonless solutions, which we refer to as graviboson stars , as they resemble a combination of a geon and a boson star.Fig. 5(a) shows the domain of existence of the hairy black resonator. The hairyblack resonator exists in the coloured “triangular” domain surrounded by the purplecurves. The colour map corresponds to the angular velocity of the horizon. The topand bottom-left edges of the triangle correspond to the onset of instability of thehairy MPAdS and black resonator, respectively. The hairy MPAdS is unstable in– 16 –he upper side of the top purple curve. The black resonator is unstable in the lowerside of the bottom-left purple curve. The bottom-right edge is the horizonless limitof the hairy black resonator: graviboson star. We also show the equal-entropy-curvebetween the hairy MPAdS and black resonator by the orange curve. The angularvelocity of the hairy black resonator always satisfies Ω >
1. Therefore, the hairyblack resonator is also superradiant and non-stationary.We can now compare the entropy of the hairy black resonators to that of blackresonators and hairy black holes. Fig. 5(b) gives a summary of the entropies of theblack resonator (blue), hairy MPAdS (orange), and hairy black resonator (green)for j = 9 /
2, which is the smallest j for which black resonators can be unstable.Fig. 5(c) corresponds to its slice at J = 0 .
8. We find that the entropy of the hairyblack resonator is never the largest among the available solutions. Instead, the mostentropic solution is either a hairy MPAdS black hole or a black resonator (or MPAdS,but only in regions where none of the other solutions exist). The entropy changescontinuously, but the field configurations are discontinuous across this transition.The hairy black resonator and graviboson star can be interpreted as a simpleexample of a multi-oscillating solution. Recall that the Wigner D-matrix depends on χ as D k ( θ, φ, χ ) ∝ e − ikχ . Then, in the the non-rotating frame at infinity (3.9), thescalar field of the hairy black resonator can be written as (cid:126) Π( t, r, θ, φ, ψ ) = (cid:88) k ∈ K e ik Ω t Φ k ( r ) (cid:126)D k ( θ, φ, ψ ) . (7.1)This has the eigenfrequencies ω = 2 k Ω ( k ∈ K ). Since the solution has periodictime dependence on several frequencies, it is multi-oscillating. In Ref. [43], multi-oscillating boson stars with non-commensurate frequencies have been constructed bysolving partial differential equations. In this paper, resonating solutions are obtainedby solving ordinary differential equations although the frequencies are commensurate.One important difference in our solutions from those of [43] is the presence ofa horizon. In order for solutions to remain steady-state (i.e. independent of τ ),fields cannot pass through the horizon. This restricts the frequency of the fields tobe multiple of the angular frequency of the horizon, and hence any multi-oscillatingsolutions must have commensurate frequencies. We expect non-commensurate multi-oscillating geons and boson stars to exist within this theory (3.1), but they wouldneither fall within our ansatz nor be the horizonless limit to a black hole. Finally, we put all the solutions together in a phase diagram in Fig. 6(a). We take j = 9 / E, J ). We use E − E geon as the vertical axis forvisibility. – 17 – T u r n i ng p o i n t s o f h a i r y M P A dS GeonBoson star E x t r e m e M P A dS Onset of instability ofhairy MPAdS andblack resonatorOnset of gravitational and scalar field instability of MPAdS G r a v i b o s o n s t a r (a) Phase diagram MPAdSBlack resonatorHairyMPAdS (b) Black hole having maximal entropy
Figure 6 . (a) Phase diagram of asymptotically AdS solutions in Einstein-multipletcomplex scalar fields system for j = 9 /
2. (b) Black hole having maximal entropy amongthe MPAdS, hairy MPAdS, black resonator, and hairy black resonator for j = 9 / • The extreme MPAdS is located on the black curve. Regular MPAdS black holesexist above this curve, while the MPAdS develops a naked singularity below it. • The red curve and line are for the gravitational black resonators and geons.The curve on the top corresponds to the onset of the gravitational superradiantinstability of the MPAdS. The black resonators branch off from this curve tothe bottom. The horizontal red line in the bottom ( E = E geon ) expresses thefamily of the gravitational geons. The black resonators lie between the redcurve and line. • The blue curves are associated with the hairy MPAdSs and boson stars. Theupper-left part of the blue curve, from J = 0 to the black dot at J = 2 . k = j = 9 /
2. The onset coincides with the extreme MPAdS at the black dot,where the onset terminates. The family of hairy MPAdS black holes branchesoff from this curve.The other blue curve corresponds to the family of boson stars. There is amaximum in (
E, J ) at the top right, which is a turning point for the curve forthe boson star. A collection of turning points for hairy MPAdSs, denoted by ablue dashed curve, extends from the top-right tip to the left as r h is increased.It appears to extend toward the black dot. The hairy MPAdSs exist in theregion that is apparently enclosed by the blue curves for the onset and theboson star before the turning point, and the blue dashed curve. In the upper-right region enclosed by the upper boson star onset curve and the dashed curve,physical quantities of hairy MPAdSs become multi-valued.– 18 – The purple curves denote the boundary of the existing region for the hairyblack resonators and graviboson stars. The bottom-right side of the distortedpurple triangle curve is the locations of the family of graviboson stars. Thebottom-left side is the onset of the scalar field superradiant instability of theblack resonator [24]. The top side is the onset of the gravitational instabilityof hairy MPAdS. The hairy black resonators exist in the region enclosed by thepurple curves. • The orange curve gives the location where the entropies of the black resonatorsand hairy MPAdSs become equal. Across the transition, the entropy is con-tinuous, but the field configurations are discontinuous. The hairy MPAdS hasthe higher entropy to the right of this curve, while the other side is dominatedby the gravitational black resonators.Fig. 6(b) is the phase diagram of the black hole solutions with the maximum entropyin the (
E, J )-plane. The hairy black resonators never have the largest entropy andhence do not appear in this figure. j We can also obtain solutions for the multiplet complex scalar with j > /
2. Here,we consider j = 5. Then, we need to consider 11-component complex scalar fieldsat least. Note that by setting one of the scalar multiplet components to zero, the j = 9 / j = 5. So though we havedefined a scalar field ansatz for particular j ’s, the solutions with different j can beconsistently compared to one another, so long as we choose the scalar field to havea sufficiently large multiplet.In Ref. [24], it was shown that, for an integer j , the scalar field is decomposed intoeven and odd parity modes under the parity transformation P defined in Eq.(3.6).The even and odd parity modes satisfy Φ − k = Φ k and Φ − k = − Φ k ( k ∈ K ), respec-tively. In this section, we only consider the even parity mode.In Fig. 7(a), we compare the entropies of the hairy MPAdS for j = 9 / j = 5. We find that the solution for j = 5 has higher entropy than that for j = 9 / j (cid:48) + 1)-component complex scalarfield, a hairy MPAdS with j < j (cid:48) evolves into that with j = j (cid:48) by the superradiantinstability in the region of a small angular momentum if we assume SU (2) L spacetimesymmetry. (Note that the hairy MPAdS with j = j (cid:48) should be further unstable to SU (2) L -breaking perturbations [21].) In the case of α ( r ) = 1, the equations of motion are identical for Φ j ( r ) = cos λ φ ( r ), Φ − j ( r ) =sin λ φ ( r ) and Φ k = 0 ( | k | (cid:54) = j ) for any value of λ . The even and odd parity modes correspond to λ = ± π/
2, and both modes give the same equations of motion. – 19 – (a) Entropy for j = 9 / j = 5 -0.100.10.20.30.40.5 0 2 4 6 8 10 T u r n i ng p o i n t s o f h a i r y M P A dS GeonBoson star
Onset of instability ofhairy MPAdS andblack resonatorOnset of gravitational and scalar fi eld instability of MPAdS G r a v i b o s o n s t a r Extreme MPAdS (b) Phase diagram for j = 5 MPAdSBlack resonatorHairyMPAdS -0.100.10.20.30.40.5 0 1 2 3 4 5 6 7 8 (c) Black hole having maximal entropy
Figure 7 . (a) Entropy of the hairy MPAdS for j = 9 / j = 5 (green).(b) Phase diagram of asymptotically AdS solutions in Einstein-multiple complex scalarfields system for j = 5. (c) Black hole having maximal entropy among the MPAdS, hairyMPAdS, black resonator, and hairy black resonator for j = 5. Fig. 7(b) is the phase diagram of solutions with j = 5. For the explanation ofeach curve, see section 8. The diagram is qualitatively similar to that for j = 9 / j = 5. The region in which the hairy MPAdS with j = 5 entropicallydominates is bigger compared to the case of j = 9 /
2. This indicates that, the largerthe quantum number j is, the wider the region covered by the hairy MPAdS will bein Fig. 6(b). If we extrapolate to arbitrarily large j , the black resonator would neverdominate the phase diagram in a theory with an infinite number of complex scalarfields. – 20 – To summarise our results, we have studied asymptotically global AdS solutions ofEinstein gravity coupled to a (2 j +1) complex scalar multiplet within a cohomogeneity-1 ansatz. The following solutions are available within our ansatz: Myers-Perry-AdSblack holes, black resonators (black holes with gravitational hair), black holes withscalar hair, and hairy black resonators (black holes with both gravitational and scalarhair). The latter three of these branch from various superradiant instabilities andhave zero horizon limits that are geons, boson stars, and graviboson stars, respec-tively. The phase diagram of all of these solutions was shown in Figs. 6(a) and 7(b)for j = 9 / j = 5, respectively.The entropy of the hairy black resonator is never the largest among the threeavailable solutions as shown in Fig. 6(b) for j = 9 /
2. This seems natural in theview of the perturbative stability. Inside the triangular region enclosed by purplecurves in Fig. 6(a), both of the black resonator and hairy MPAdS is stable againstcorresponding perturbations. Thus, the black resonator and hairy MPAdS would notevolve into the hairy black resonator. This is consistent with the fact that the hairyblack resonator is entropically subdominant.Finally, we were able to compare both j = 9 / j = 5 solutions, and we findthat the j = 5 hairy MPAdS solutions are dominant and cover a larger portion ofphase space than those of j = 9 /
2. It is natural to expect that the trend continuesto higher j . We can make this claim stronger by the following argument. Thephase boundary between hairy MPAdS and black resonators always lies between twopoints: (1) The intersection between gravitational and scalar onsets of MPAdS, and(2) where the boson stars intersect geons. Results in Ref. [24] suggest that the point(1) is located at a higher angular momentum for a higher j . Also, by explicitlyconstructing boson stars for j ≤ /
2, we found that the same applies to the point(2) at least for j = 9 / , , / SU (2) L -symmetry in the spacetime, a time-dependent ansatz for this systemwould give a 1 + 1 dimensional evolution system. Curiously, though black resonatorsare unstable to superradiant scalar perturbations and hairy MPAdS are unstableto superradiant gravitational perturbations, the solutions that branch from theseinstabilities (namely, the hairy black resonators) are never entropically dominant.Therefore, these unstable black resonators or hairy MPAdS cannot evolve to hairyblack resonators. Instead, unstable black resonators will likely evolve towards hairyMPAdS, removing its gravitational hair. Similarly, unstable hairy MPAdS will evolvetowards black resonators, shedding its scalar hair. Hairy black resonators themselvescan (entropically) evolve to either black resonators or hairy MPAdS, most likely towhichever is dominant. In all of these case, scalars with k < j will be suppressed,and either the gravitational or k = j scalar field instability will survive.– 21 –t would be especially interesting to study the time evolution of a system withlarge j . As we have mentioned, a large j multiplet contains smaller j solutionswithin it, so the full system contains a tower of lower wavenumbers, many of whichare unstable modes in black resonators or MPAdS. These modes have different growthrates, with the largest wavenumber typically being the slowest. However, the hairyblack hole with the largest wavenumber is likely the most dominant entropically. Atime evolution would therefore tell us how these competing instabilities interact witheach other, and how a cascade to higher wavenumbers proceeds.Though the growth rates of high modes are extremely small and present a signif-icant numerical challenge, such a calculation seems more feasible in this 1+1 settingthan the full 3+1 setting of Kerr-AdS. Furthermore, the high-wavenumbers are asso-ciated with angular directions rather than the radial direction. As the 1+1 equationsdo not directly see the angular gradients, the numerical resolution can be kept rela-tively low, allowing faster time evolution due to the larger Courant number.A time-dependent ansatz for this system would give a 1+1 dimensional evolutionsystem. In the discrete isometries (3.7), we can only assume P -invariance for thetime-dependent spacetime, and the equations of motion require a non-trivial crossterm like γ ( t, r ) σ σ in the metric for consistency.Despite the fact that we can construct solutions with several values of j , thiscomes at a cost. In particular, we would like to remind the reader that for each valueof j we need 2 j +1 complex scalar fields to make our co-homogeneity one ansatz work.Furthermore, the phases of each of these scalars must be fine tuned so that the overalldependence in the angles do cancel. Perhaps more importantly, all these scalars areminimally coupled to gravity. One might ask whether such scalars are easy tocome by in consistent reductions from some higher dimensional supergravity theorysuch as type IIB and the answer appears to be no. To our knowledge the largestknown conjectured truncation of type IIB supergravity arises when consideringcompactifications of the form AdS × S , with the lower dimensional theory beingfive-dimensional N = 8 gauged supergravity comprising a total 42 scalars, 15 gaugefields and 12 form fields. However, these scalars appear to be non-minimally coupledto gravity, and to have very complicated potentials (see for instance [44]), thus givingvery little hope that our model will find a precise holographic realisation. Acknowledgments
We would like to thank Oscar Dias for useful conversations. The work of T. I. wassupported in part by JSPS KAKENHI Grant Number JP18H01214 and JP19K03871. One could potentially add a mass term, and much of our discussion would still go through. This has actually never been shown in full generality, partially because of the self dual conditionimposed on the Ramond-Ramond F5 form flux, even though interesting progress has been recentlymade in [44]. – 22 –he work of K. M. was supported in part by JSPS KAKENHI Grant NumberJP18H01214 and JP20K03976. BW acknowledges support from ERC AdvancedGrant GravBHs-692951 and MEC grant FPA2016-76005-C2-2-P. J. E. S. is sup-ported in part by STFC grants PHY-1504541 and ST/P000681/1. J. E. S. alsoacknowledges partial support from a J. Robert Oppenheimer Visiting Professorship.
A Technical details
In this appendix, we collect technical details for solving the Einstein-complex scalarmultiplet system.
A.1 Equations of motion
With our ansatz, the equations of motion are given by coupled ODEs. From Eqs. (3.2)and (3.16), the trace T defined in Eq. (3.11) can be computed as T = (cid:88) k ∈ K (cid:20) (1 + r ) g Φ (cid:48) k + 2 r (cid:18) α − α (cid:19) (cid:15) k − (cid:15) k Φ k − Φ k + 1 r (cid:18) α + 1 α (cid:19) ( (cid:15) k + (cid:15) k +1 )Φ k + 4 (cid:18) r β − h (1 + r ) f (cid:19) k Φ k (cid:21) . (A.1)For the metric ansatz (3.2), the Einstein equations G µν − g µν = T µν become f (cid:48) = 1 r (1 + r ) gα ( rβ (cid:48) + 6 β ) [4 r h ( α − β + r ( r + 1) g { r (1 + r ) f α (cid:48) β − r h (cid:48) α β − r ) f α β (cid:48) }− r ) f { r α β ( g −
1) + 3 gα β + ( α − αβ + 1) − α } ]+ 4 rf βrβ (cid:48) + 6 β T rr , (A.2) g (cid:48) = 16 r (1 + r ) f α β [ − r h ( α − β + r (1 + r ) g {− r (1 + r ) f α (cid:48) β + r h (cid:48) α β − ( − r (1 + r ) f (cid:48) + 2 f ) α β (cid:48) } + 4(1 + r ) f {− r α β ( g − − gα β + α + 4 α β − α β − α + 4 αβ + 1 } ] − r (1 + r ) f β [ r βT tt − r hβT t + 4( r h β − (1 + r ) f ) T ]+ 13 r (1 + r ) (cid:20)(cid:18) α − α (cid:19) ( T ++ + T −− ) − (cid:18) α + 1 α (cid:19) T + − (cid:21) , (A.3) h (cid:48)(cid:48) = 12 r (1 + r ) α βf g [8 f h ( α − − r (1 + r ) h (cid:48) α { r ( f g (cid:48) β − f (cid:48) gβ + 3 f gβ (cid:48) ) + 10 f gβ } ] + 4(2 hT − T t ) r (1 + r ) gβ , (A.4)– 23 – (cid:48)(cid:48) = 12 r (1 + r ) f αgβ [2 r ( r + 1) f gα (cid:48) β − r ( r + 1) αα (cid:48) { r (1 + r )( f gβ ) (cid:48) + 2(3 + 5 r ) f gβ }− α − { r h β ( α + 1) − (1 + r ) f α ( α − β ) − (1 + r ) f } ] − αr (1 + r ) g (cid:20)(cid:18) α + 1 α (cid:19) ( T ++ + T −− ) − (cid:18) α − α (cid:19) T + − (cid:21) , (A.5) β (cid:48)(cid:48) = 1(2 r (1 + r )) f gα β [ − r gh (cid:48) α β − rα β (cid:48) { r (1 + r )( f (cid:48) gβ + f g (cid:48) β − f gβ (cid:48) ) + 2(3 + 5 r ) f gβ }− f β ( α + α β − α β − α + αβ + 1)] − r (1 + r ) T − βr (1 + r ) g (cid:20)(cid:18) α − α (cid:19) ( T ++ + T −− ) − (cid:18) α + 1 α (cid:19) T + − (cid:21) . (A.6)For our scalar field ansatz (3.3), the Klein-Gordon equation (cid:3) (cid:126) Π = 0 gives | K | = (cid:98) j + 1 (cid:99) equations of the form L k Φ k + c k − Φ k − + c k +1 Φ k +2 = 0 , (A.7)where L k = (1 + r ) g d d r + (cid:20) r (cid:18) f (cid:48) f + g (cid:48) g + β (cid:48) β (cid:19) + 3 + 5 r r (cid:21) g dd r − (cid:15) k + (cid:15) k +1 r (cid:18) α + 1 α (cid:19) − k r β + 4 k h (1 + r ) f , (A.8)and c k = − (cid:15) k (cid:15) k +1 r (cid:18) α − α (cid:19) . (A.9)In (A.7), the mode coupling is “double-stepping” — the mode with k is coupled tothose with k ± r = r h (or the center r = 0 ifthe geometry is horizonless) to infinity r → ∞ by using the 4th order Runge-Kuttamethod. The boundary conditions for solving the equations are given below. A.2 Boundary conditions at infinity
We require the spacetime to be asymptotically AdS at infinity. In the rotating frameat infinity, the condition for the metric components is f, α, β → , h → Ω ( r → ∞ ) , (A.10)and then g → h ( r ) actually corresponds to the angular velocity of the horizon Ω as– 24 –xplained in section 3. We also require that the massless scalar field falls off atinfinity, Φ k → r → ∞ ) . (A.11)In the interpretation of the gauge/gravity duality, this means that there is no externalsource for the dual scalar operators in the boundary field theory. Thus, nontrivialscalar fields are spontaneously induced by the instability in the geometry.However, imposing f → r → ∞ is actually redundant because there isrescaling symmetry in Eq. (3.2). By a coordinate transformation τ → c τ for aconstant c , the line element is invariant if the metric components are rescaled as f ( r ) → f ( r ) c , h ( r ) → h ( r ) c . (A.12)Hence, if we obtain a solution with f ∞ ≡ f ( r = ∞ ) (cid:54) = 1, we can rescale it so thatthe new f satisfies f → c = √ f ∞ in the above scalingequation. Therefore, when we solve the equations of motion, we only need to impose α, β → r → ∞ ) , (A.13)and then we can apply the transformation (A.12) to obtain a rescaled solution sat-isfying f → A.3 Boundary conditions at the origin for horizonless solutions
For horizonless solutions, we require regularity at the origin of AdS r = 0. To avoida conical singularity at r = 0, we impose g, α, β → r → . (A.14)Then, from Eq. (A.7), the regular solution of the scalar field near r = 0 has thebehavior Φ k ∼ r j . (A.15)To handle this behavior at arbitrary j , we find it convenient to redefine the scalarfield as Φ k ( r ) = (cid:18) r r (cid:19) j Ψ k ( r ) (A.16)and solve the equations of motion for new variables X ≡ ( f, g, h, α, β, Ψ k ). They canbe expanded near r = 0 as X ( r ) = ∞ (cid:88) m =0 X m r m , (A.17)where g = α = β = 1 as Eq. (A.14). Substituting this expansion into Eqs. (A.2-A.7) and specifying f , h , α , β , Ψ k, as the input parameters, we can determine the– 25 –igher order coefficients X m order by order. (In practice, we evaluated X m for m ≤ f = 1 without loss of generality.To construct gravitational geons, we set the scalar field zero, Ψ k ( r ) = 0 for all k .Then, we have three free parameters to be specified at the origin, h , α , and β , whilethere are two boundary conditions (A.13) at infinity. Thus, the geons are obtainedas a one-parameter family. We set α (cid:54) = 0 as the input parameter and determine h and β by the shooting method so that Eq. (A.13) is satisfied. To obtain a family ofgeons, we start from a normal mode of pure AdS where h = 3 / β = 0 andturn on a tiny α . Once a solution is successfully obtained, we slightly vary the valueof α as well as using the previous result as the initial guess for the next solution.We repeat this process and construct the geon solutions shown in Fig. 1.For boson stars, we set α ( r ) = 1 and Ψ k ( r ) = 0 for k < j , which means that α = Ψ k 1. Similar to the case of geons,we set Ψ j, (cid:54) = 0 as the input and start the shooting method from the initial guessgiven by the scalar field normal mode in pure AdS with h = 1 + 2 /j and β = 0.For the construction of graviboson stars, we have | K | + 3 free parameters at theorigin h , α , β , Ψ k, , and | K | + 2 conditions (A.11) and (A.13) at infinity. We setΨ j, (cid:54) = 0 as the input and determine the other | K | + 2 parameters by the shooting.The normal mode frequencies ω for scalar fields in gravitational geon backgroundshave been computed in Ref. [24]. The onset for graviboson stars corresponds to ω = 0. We start from that point and turn on a tiny value of Ψ j, . A.4 Boundary conditions at the horizon For black hole solutions, boundary conditions are imposed at the horizon r = r h .The field variables Y ≡ ( f, g, h, α, β, Φ k ) can be expanded near the horizon as Y ( r ) = ∞ (cid:88) m =0 Y m ( r − r h ) m , (A.18)where f = g = 0. Substituting Eq. (A.18) into the equation of motion for h ( r )(A.4) and looking at the leading order, we obtain (cid:40) ( α − + 2 α (cid:88) k ∈ K k Φ k, (cid:41) h = 0 . (A.19)There are two possibilities for the solutions to this equation: (i) α = 1 and Φ k, = 0,and (ii) h = 0. The case (i) is nothing but the MPAdS solution. Indeed, the horizonvalue h can be arbitrary because of the recovered U (1) R isometry. For hairy andresonating solutions, we consider (ii). Let us redefine the field variables as f ( r ) = F ( r ) ˜ f ( r ) , g ( r ) = F ( r )˜ g ( r ) , h ( r ) = F ( r )˜ h ( r ) , (A.20)– 26 –here F ( r ) ≡ − r h /r . With the new variables, the near horizon expansion can begiven by Z ( r ) = ∞ (cid:88) m =0 Z m ( r − r h ) m , (A.21)where Z = ( ˜ f , ˜ g, ˜ h, α, β, Φ k ). This F ( r ) is chosen by hand so as to accommodatethe near horizon behavior and employ ˜ f , ˜ g , ˜ h (cid:54) = 0. Substituting Eq. (A.21)into Eqs. (A.2-A.7) and taking ˜ f , ˜ h , α , β , Φ k, , r h as the input parameters, wecan determine the higher order coefficients Z m . (Practically in our calculations, wetruncate the series to m ≤ f = 1 without loss ofgenerality because of the rescaling symmetry (A.12).For black resonators, the scalar fields are trivial, Φ k ( r ) = 0. There are four freeparameters ˜ h , α , β , r h at the horizon and two conditions (A.13) at infinity. Thuswe need to specify two input parameters at the horizon, for which we choose α and r h , and then ˜ h and β are determined by the shooting method. We start from theonset of the superradiant instability of the MPAdS evaluated in [7, 24], where α = 1,and turn on a small deformation as α − (cid:54) = 0. Once the shooting method convergesand a black resonator solution is obtained, we slightly increase α − α ( r ) = 1 and Φ k ( r ) = 0 for k < j . Therefore, wehave α = 1 and Φ k 1. We could also obtainthe solutions branching off from the scalar field superradiant instability of black res-onators evaluated in [24]. However, it turned out that the shooting method did notconverge nicely around the onset of the instability, and therefore we resort to theother option. This problem is discussed in appendix A.6. The choice of F ( r ) can be arbitrary, but we assume F ( r h ) = 0 and F (cid:48) ( r h ) (cid:54) = 0. – 27 – .5 Physical quantities Near the asymptotic infinity r → ∞ , the asymptotic solutions of the metric compo-nents and scalar fields are f ( r ) = 1 + c f r + · · · , g ( r ) = 1 + c f + c β r + · · · , h ( r ) = Ω + c h r + · · · ,α ( r ) = 1 + c α r + · · · , β ( r ) = 1 + c β r + · · · , Φ k ( r ) = c k r + · · · , (A.22)where c f , c h , c α , c β , and c k are the constants that are determined by matching theseries with the bulk, and the source of Φ k has already been set zero. As discussed insection 3, the asymptotic value of h ( r ) corresponds to the angular velocity Ω. Theconstant c k corresponds to the expectation value of the operator (cid:126) O dual to (cid:126) Π in theboundary theory as (cid:104) (cid:126) O(cid:105) = (cid:88) k ∈ K c k (cid:126)D k ( θ, φ, χ ) = (cid:88) k ∈ K c k e − k Ω t (cid:126)D k ( θ, φ, ψ ) . (A.23)This depends on both time and spatial coordinates on the boundary.Because the boundary source for the scalar field is absent, the boundary energy-momentum tensor T ij can be given by [45–48].8 πG T ij = − r C iρjσ n ρ n σ (cid:12)(cid:12)(cid:12)(cid:12) r = ∞ , (A.24)where i and j run over the coordinates on the AdS boundary, n µ is the unit normalto a bulk r -constant surface, and C µνρσ is the bulk Weyl tensor. Using Eq. (A.22),we obtain8 πG T ij d x i d x j = 12 ( c β − c f )d τ + 2 c h d τ ( σ + 2Ωd τ ) − c f + c β σ + σ )+ c α σ − σ ) + 18 ( − c f + 3 c β )( σ + 2Ωd τ ) . (A.25)This is written in the rotating frame at infinity ( τ, χ ). In the non-rotating frame( t, ψ ), the boundary stress tensor is rewritten as8 πG T ij d x i d x j = 12 ( c β − c f )d t + 2 c h d t ¯ σ − c f + c β σ + ¯ σ )+ c α ( e i Ω t ¯ σ + e − i Ω t ¯ σ − ) + 18 ( − c f + 3 c β )¯ σ , (A.26)where ¯ σ i ( i = 1 , , 3) are the invariant one-forms defined in the non-rotating frame: χ in (2.2) is replaced with ψ for ¯ σ i . The energy and angular momentum are given by E = (cid:90) dΩ T tt = π ( c β − c f )8 G , J = − (cid:90) dΩ T t ( ψ/ = − πc h G , (A.27)– 28 – .1250.130.1350.140.145 0.5 1 1.5 2 2.5 Interpolation by 2nd order polynomialHairy black resonatorOnset of instability ofblack resonator Figure 8 . Energy of the hairy black resonator as a function of the angular momentum J for a fixed horizon radius r h = 0 . where we define the angular momentum with respect to ψ/ ∈ [0 , π ).For black hole solutions, the entropy S and temperature T are given by S = π r h (cid:112) β ( r h )2 G , T = (1 + r h ) (cid:112) f (cid:48) ( r h ) g (cid:48) ( r h )4 π . (A.28)For simplicity, we set G = 1 in this paper. We can easily recover the dependenceon G by E → G E , J → G J , and S → G S . A.6 Interpolation of hairy black resonator data When we construct the hairy black resonator, we take the route to extend the solutionfrom the onset of instability of the hairy MPAdS. Near the onset, the shooting methodconverges well, and we were able to obtain hairy black resonators. As the solutionapproaches the onset of the instability of the black resonator, however, we find thatthe shooting method fails to converge. Here we argue that the void, however, canbe filled by interpolation. In Fig. 8, we show E of the hairy black resonator as afunction of J for a fixed horizon radius r h = 0 . 3. For visibility, we use E − . J asthe vertical axis. The purple points denote the numerical data, and the green point isthe onset of the scalar field instability of the black resonator. Between these points,we were not able to obtain numerical solutions. There might be other solutions withsimilar parameters at the horizon. One of the candidates is the black resonator withthe scalar hair with nontrivial radial overtones. Nevertheless, our data is fitted wellby a second order polynomial as shown in the black curve in the figure. We used theinterpolated data in the corresponding region in Fig. 5(b).– 29 – Instability of hairy MPAdS In this appendix, we consider the linear perturbation of the hairy MPAdS. To findthe onset of the instability, we only need to consider τ -independent perturbationsand find their normal modes. For the hairy MPAdS solutions, we have α ( r ) = 1 andΦ k ( r ) = 0 ( k < j ), while the other functions have nontrivial r -dependence. Aroundthis background, we perturb the variables as ( f, g, h, α, β, Φ k ) → ( f + δf, g + δg, h + δh, α + δα, β + δβ, Φ k + δ Φ k ) in Eqs. (A.2-A.7) and keep the linear order in theperturbations. It turns out that the perturbation variables δα and δ Φ j − decouplefrom the others. Their perturbation equations are given by δα (cid:48)(cid:48) = − (cid:26) ( f gβ ) (cid:48) f gβ + 2(3 + 5 r ) r (1 + r ) (cid:27) δα (cid:48) − r ) g (cid:26) β − r β + 2 h (1 + r ) f − T + − r (cid:27) δα − δT ++ + δT −− r (1 + r ) g , (B.1) L j − δ Φ j − = 4 (cid:112) j (2 j − r Φ j δα , (B.2)where T + − = − r (cid:20) (1 + r ) g Φ (cid:48) j + 4 j (cid:18) r β − h (1 + r ) f (cid:19) Φ j (cid:21) , (B.3) δT ±± = − (cid:112) j (2 j − j δ Φ j − − r (cid:20) (1 + r ) g Φ (cid:48) j + 4 jr Φ j + 4 j (cid:18) r β − h (1 + r ) f (cid:19) Φ j (cid:21) δα . (B.4)We wish to find normal mode solutions to these equations with the sourcelessboundary condition at infinity: δα → δ Φ j − → 0. On the black hole horizon,we require regularity. Substituting δα = a + a ( r − r h ) + · · · and δ Φ j − = φ + φ ( r − r h ) + · · · into Eqs. (B.1-B.2), we can obtain the regular series solution where a i and φ i for i ≥ a and φ . We set a = 1 to fix the scaleof the linear perturbation. Then, φ is left as the free parameter on the horizon.This parameter is tuned so that the boundary condition at infinity, δ Φ j − → 0, issatisfied. For a general hairy MPAdS background specified by the horizon valueof the background scalar field Φ j | r = r h , however, δα → δα is satisfied only at special values of Φ j | r = r h . Wemonitor the value of δα | r = ∞ by increasing Φ j | r = r h and search the value of Φ j | r = r h at which δα | r = ∞ = 0 is satisfied. We repeat this process for each horizon radius r h .For j = 9 / 2, the result of the onset is the upper purple curve in Fig. 5(a). FromFig. 2(a), we know that MPAdS at the branching points of hairy MPAdS becomesunstable against gravitational perturbation for r h (cid:38) . 37. This indicate that, in theupper side of the onset curve in Fig. 5(a), hairy MPAdS is unstable.– 30 – Perturbative solution of boson star In this appendix, we perturbatively construct the boson stars for small deformationsfrom pure AdS. This has been considered in similar contexts [12, 35, 36, 49–51]. Inparticular, we consider the large- j limit. C.1 Higher order perturbation We consider the higher order perturbation around global AdS asΦ j ( r ) = Φ ( r ) (cid:15) + Φ ( r ) (cid:15) + · · · (C.1)and f ( r ) g ( r ) h ( r ) β ( r ) = + f ( r ) g ( r ) h ( r ) β ( r ) (cid:15) + · · · . (C.2)We set α ( r ) = 1 for the boson star. The fundamental normal mode frequency of thetest scalar field in pure AdS is Ω = 1 + 2 /j .The equation in the first order in (cid:15) is given byΦ (cid:48)(cid:48) + 3 + 5 r r (1 + r ) Φ (cid:48) − { j ( j + 1) − (4 + 3 j ) r } r (1 + r ) Φ = 0 . (C.3)This can be solved by Φ ( r ) = φ ( r ) ≡ r ) (cid:18) r r (cid:19) j , (C.4)which is regular at the origin ( r = 0) and infinity ( r = ∞ ): φ ( r ) ∼ r j ( r → , φ ( r ) ∼ r ( r → ∞ ) . (C.5)The other solution to Eq. (C.3) is˜ φ ( r ) = φ ( r ) (cid:90) rr (1 + r ) φ . (C.6)However, this is singular at the origin and infinity as˜ φ ( r ) ∼ r − j − ( r → , ˜ φ ( r ) ∼ r → ∞ ) . (C.7)Therefore, we adopt φ (C.4) as the first order solution. The solution satisfies φ (cid:48) = 2( j − r ) r (1 + r ) φ . (C.8)– 31 –e will use this equation to eliminate φ (cid:48) in the following calculations.The equations in the second order in (cid:15) are h (cid:48)(cid:48) + 5 r h (cid:48) = 8 j ( j + 2) r (1 + r ) φ ,β (cid:48)(cid:48) + 3 + 5 r r (1 + r ) β (cid:48) − r (1 + r ) β = − j (2 j − r (1 + r ) φ ,g (cid:48)(cid:48) + 2(1 + 2 r ) r (1 + r ) g + 13(1 + r ) β (cid:48) + 103 r (1 + r ) β = − j + 2 r )3 r (1 + r ) φ ,f (cid:48) + 2(1 + 2 r ) r (1 + r ) g + 2 + 3 r r ) β (cid:48) + 23 r (1 + r ) β = − j − r )3 r (1 + r ) φ , (C.9)where, in the right hand side, we have eliminated φ (cid:48) by using Eq. (C.8). The solutionsto them can be decomposed into particular and homogeneous solutions as f g h β = f p2 g p2 h p2 β p2 + f hom2 g hom2 h hom2 β hom2 , (C.10)where h p2 = 2 j ( j + 1) (cid:20)(cid:90) r φ r (1 + r ) d r − r (cid:90) r r φ r d r (cid:21) ,β p2 = j (2 j − (cid:20) r (cid:90) r rF ( r ) φ d r − F ( r ) (cid:90) r φ r d r (cid:21) ,g p2 = − r ) (cid:20) β p2 + 8 r (cid:90) r { ( j + 2 r ) rφ + rβ p2 } d r (cid:21) ,f p2 = − r r ) β p2 − (cid:90) r r (1 + r ) (cid:26) (1 + 2 r ) g p2 + 23 ( j − r ) φ + 13(1 + r ) β p2 (cid:27) d r , (C.11)and h hom2 = C h , β hom2 = C β A ( r ) ,g hom2 = C β (4 r + 3) A ( r ) − r r ) , f hom2 = C f + C β r A ( r )3(1 + r ) . (C.12)Here, we defined A ( r ) ≡ − r + 2 ln(1 + r ) r . (C.13)Since we set the lower bound of the integral at r = 0, the regularity at the originis guaranteed in these expressions. Integration constants C β and C f will be chosen– 32 –o that β → f → r → ∞ ) are satisfied. C h will be determined from thethird order equation.The third order equation isΦ (cid:48)(cid:48) + 3 + 5 r r (1 + r ) Φ (cid:48) − { j ( j + 1) − (4 + 3 j ) r } r (1 + r ) Φ = S , (C.14)where S = 4 φ (1 + r ) (cid:20) ( j + 2) f − r − r − jr + j r g − j ( j + 2) h − j (1 + r ) + 2 r − jr β + j ( j − r ) r φ (cid:21) . (C.15)We have eliminated f (cid:48) , g (cid:48) and φ (cid:48) in the source term by using Eqs. (C.8) and (C.9).The solution in this order is written asΦ = C φ φ ( r ) − φ ( r ) (cid:90) r r (1 + r ) S ˜ φ ( r )d r + ˜ φ ( r ) (cid:90) r r (1 + r ) S φ ( r )d r . (C.16)The first and second terms are regular at the origin and infinity. The last term isalso regular at the origin but it behaves as ∼ (cid:90) ∞ r (1 + r ) S φ ( r )d r = 0 . (C.17)This equation determines C h as announced. C.2 Large- j expansion of the perturbative solution The integral expressions of the perturbative results obtained above can be evaluatedin the large- j limit in powers of 1 /j .For the large- j expansion, it is convenient to introduce a new coordinate x as r r = exp( − x /j ) . (C.18)In the new coordinate, r = 0 and ∞ correspond to x = ∞ and 0, respectively. Interms of x , the first order solution (C.4) is rewritten as φ = (1 − e − x /j ) e − x . (C.19)Because of the exponential factor e − x , this function is highly suppressed in x (cid:29) r ∼ 0. Assuming x (cid:46) 1, we can expand the prefactor (1 − e − x /j ) by 1 /j as φ = e − x (cid:18) x j − x j + 7 x j − x j + · · · (cid:19) . (C.20)– 33 –et us consider the large- j approximation of the second order solution. Forexample, the first term in h p2 (C.11) is written as (cid:90) r φ r (1 + r ) d r = 1 j (cid:90) ∞ x x (1 − e − x /j ) e − x d x (C.21)The integrand contains an exponential factor e − x . Hence, we assume x (cid:46) /j as (cid:90) ∞ x e − x (cid:18) x j − x j + 13 x j − x j + · · · (cid:19) d x = e − x (cid:20) j (2 x + 4 x + 6 x + 6 x + 3) − j (4 x + 10 x + 20 x + 30 x + 30 x + 15)+ 1396 j (4 x + 12 x + 30 x + 60 x + 90 x + 90 x + 45) − j (8 x + 28 x + 84 x + 210 x + 420 x + 630 x + 630 x + 315)+ · · · (cid:21) . (C.22)We can do the same procedure for the other integrals in Eq. (C.11). Since theintegrals always produce the exponential factor e − x , the functions of r outside ofthe integrals can be expanded by 1 /j after changing coordinates from r to x . As aresult, we have e x h p2 = 14 j ( x + 1)(2 x + 3 x + 3) − j (6 x + 16 x + 31 x + 36 x + 18)+ 148 j (30 x + 89 x + 216 x + 393 x + 450 x + 225) − j (36 x + 120 x + 343 x + 804 x + 1425 x + 1620 x + 810)+ 12880 j (516 x + 1930 x + 6336 x + 17805 x + 41040 x + 71775 x + 81270 x + 40635) + · · · , (C.23)– 34 – x β p2 = − j (cid:0) x + 1 (cid:1) (cid:0) x + 3 x + 3 (cid:1) + 18 j (6 x + 24 x + 47 x + 48 x + 18) − j (30 x + 161 x + 456 x + 3(259 + 24 y ) x + 666 x + 225)+ 196 j (36 x + 240 x + 887 x + 12(181 − y ) x + 3(1151 + 144 y ) x + 2520 x + 810) − j (516 x + 4090 x + 18576 x + 105(565 + 24 y ) x + 540(245 − y ) x + 675(299 + 40 y ) x + 129870 x + 40635) + · · · , (C.24) e x g p2 = − j x ( x − x + 5)+ 18 j x (8 x + 29 x + 5 x − (37 + 16 y ) x − − j x (52 x + 243 x + 169 x − (241 − y ) x − y ) x − j x (80 x + 454 x + 531 x − − y ) x − (1853 − y ) x − y ) x − · · · , (C.25) e x f p2 = 14 j ( x + 1)(2 x + 3 x + 3) − j (10 x + 18 x + 27 x + 30 x + 18)+ 148 j (78 x + 95 x + 150 x + 3(81 − y ) x + 342 x + 225) − j (140 x + 102 x + 105 x + 186 x + 3(191 − y ) x + 1170 x + 810)+ 12880 j (2916 x + 910 x − x − y ) x − x + 75(221 − y ) x + 56970 x + 40635) + · · · , (C.26)where we defined y = e x Ei (2 x ) , Ei ( z ) = (cid:90) ∞ e − tz t d t . (C.27)– 35 –t the origin and infinity, this behaves as y (cid:39) − γ − ln 2 − x ( x → , y (cid:39) x ( x → ∞ ) , (C.28)where γ is Euler’s constant. Near the infinity ( x = 0), the second order solutionsapproach h p2 → j − j + 7516 j − j + 90364 j + · · · , (C.29) β p2 → − j + 94 j − j + 13516 j − j + · · · , (C.30) g p2 → , (C.31) f p2 → j − j + 7516 j − j + 90364 j + · · · . (C.32)Therefore, from the boundary condition for the second order perturbation, β → f → r → ∞ , we can determine the integration constants C β and C f as C β = 34 j − j + 7516 j − j + 90364 j + · · · ,C f = − (cid:18) j − j + 7516 j − j + 90364 j + · · · (cid:19) . (C.33)To determine C h , we substitute the particular solutions (C.23-C.26) and homoge-neous solutions (C.12) into Eq. (C.17). Then, expanding the integrand by 1 /j aftermoving to the x -coordinate, we can carry out the integration. As the result, weobtain C h as C h = − j + 94 j − j (cid:18) e γ j ) (cid:19) + 214 j (cid:18) − e γ j ) (cid:19) − j (cid:18) − e γ j ) (cid:19) + · · · . (C.34)Near the infinity, the second order solutions are expanded as f = c f r + · · · , g = c g r + · · · , h = Ω + c h r + · · · , β = c β r + · · · , (C.35)– 36 –here c f = − j − j + 12 j (cid:18) 98 + ln(2 e γ j ) (cid:19) − j (cid:18) − e γ j ) (cid:19) + 258 j (cid:18) − e γ j ) (cid:19) + · · · ,c g = − j + 2 j (1 + ln(2 e γ j )) − j (cid:18) − 12 + ln(2 e γ j ) (cid:19) + 252 j (cid:18) − e γ j ) (cid:19) + · · · ,c h = − j + 58 j − j + 6532 j − j + · · · ,c β = 14 j − j + 32 j (cid:18) e γ j ) (cid:19) − j (cid:18) − e γ j ) (cid:19) + 758 j (cid:18) − e γ j ) (cid:19) + · · · . (C.36)The second order contribution of the normal mode frequency isΩ = − j (cid:18) − e γ j ) (cid:19) + 214 j (cid:18) − e γ j ) (cid:19) − j (cid:18) − e γ j ) (cid:19) + · · · . 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