Hairy black resonators and the AdS4 superradiant instability

Paul M. Chesler(Dated: September 28, 2021)

The superradiant instability of Kerr-AdS black holes is studied by numerically solving the full3+1 dimensional Einstein equations. We find that the superradiant instability results in a two stageprocess with distinct initial and secondary instabilities. At the end of the secondary instability thegeometry oscillates at several distinct fundamental frequencies — a multi-oscillating black hole. Themulti-oscillating black hole is remarkably close to a black resonator, albeit with a bit of gravitationalhair. During the hairy black resonator epoch, the evolution of the horizon area is consistent withthe exponential approach to a constant. By employing different seed perturbations in the initialKerr-AdS geometry, we also demonstrate that the black resonator’s hair is not unique. In the dualquantum field theory description, rotation invariance is spontaneously broken and the energy densityis negative in some regions, signaling an exotic state of matter which does not relax to a stationaryconfiguration.

I. INTRODUCTION

Waves with suitably tuned frequency ω and azimuthalquantum number m scattering off rotating objects canbe amplified via superradiance [1, 2]. Of particular in-terest are rotating black holes, which contain ergoregionsfrom which it is possible to extract energy and angularmomentum [3, 4]. Half a century ago it was pointed outthat if waves can be confined with something akin toa mirror, so outgoing waves are reflected back inwards,then repeated interactions with a black hole can lead toexponential growth of the wave amplitude, potentiallyconverting a significant fraction of the black hole’s massinto radiation [5]. The dynamics of superradiant instabil-ities — also known as black hole bombs — and their finalstate are of considerable of interest to a variety of fieldsincluding early universe cosmology [6], particle physicsand gravitational wave astrophysics [7–10], astrophysi-cal jets [11], and phase transitions in holographic duality[12, 13]. For a detailed review of superradiance and blackhole bombs see Ref. [11].

Perhaps the purest manifestation of a black hole bombis that of the AdS4 superradiant instability. The dynam-ics of the system are governed by the 3+1 dimensionalvacuum Einstein’s equations with a negative cosmolog-ical constant and asymptotically global AdS boundaryconditions. The geometry contains a time-like bound-ary (conformally equivalent to sphere), which serves asa mirror from which gravitational waves are reflected in-wards. Via holographic duality [14], this system has adual interpretation as a 2+1 dimensional strongly cou-pled quantum field theory living on the boundary.

For two decades it has been known that sufficientlyrapidly spinning Kerr-AdS black holes are susceptibleto superradiant instabilities [15, 16]. The spectrum ofunstable modes in the Kerr-AdS spacetime was studiedin [17]. It was subsequently shown that any black holein asymptotically AdS spacetime with an ergoregion —meaning a region where a Killing vector becomes space-like — must be unstable [18]. It has been suggested thatthe AdS4 instability may have no end-state and can result

in violations of cosmic censorship [19] . Despite the longhistory of the AdS4 superradiant instability, the questionof what the final state is has thus far remained elusive.Our goal in this paper is to study this long standing prob-lem with numerical relativity simulations.

Suppose the dominant unstable mode in the Kerr-AdS spacetime has (complex) frequency ω and azimuthalquantum number m. At a linear level the mode’s timeand azimuthal angle dependence is then given by the ex-ponential e−iωt+imϕ. Hence the associated gravitationalwave rotates in ϕ at angular velocity Re ωm while slowly

growing in amplitude like eImω t. If the superradiant in-stability was a one stage process, with this mode and itsharmonics merely plateauing after some time, the finalstate geometry would have a single Killing vector,

K = ∂t + Ω ∂ϕ, (1)

with Ω ≈ Re ωm . Black holes with single Killing vector— coined black resonators — were first constructed inRef. [20]. Their geometry rigidly rotates with angularvelocity Ω, meaning they oscillate with a single funda-mental frequency, and they are thermodynamically pre-ferred over the Kerr-AdS solution with the same massand angular momentum. However, they are also unsta-ble [18]. Their angular velocities satisfy

Ω >1

L, (2)

which means K is always spacelike near the AdS bound-ary, signaling the presence of an ergoregion. It is there-fore reasonable to expect that additional instabilities oc-cur at some stage of the evolution. A natural guessis that the superradiant instability results in the Kerr-AdS geometry transitioning to a black resonator, whichthen experiences its own distinct superradiant instabili-ties, thereby transitioning to another state.

Previously we numerically simulated the AdS4 super-radiant instability and found evolution consistent withthe transition of the Kerr-AdS geometry to an approx-imate black resonator geometry [21]. While secondaryinstabilities were also observed, numerical evolution was

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not carried out long enough to ascertain their end state,leaving the final fate of the system uncertain. Never-theless, Ref. [21] found several unstable modes duringthe black resonator epoch. These modes oscillated atdifferent fundamental frequencies and rotated with dif-ferent angular velocities. This suggests that the finalstate has no continuous symmetries. In other words, itis reasonable to expect that the final state of the su-perradiant instability is a black hole geometry oscillat-ing with several fundamental frequencies, and withouta Killing vector or ergoregion. Following the nomen-clature of Ref. [22], we will refer to black holes oscil-lating with multiple fundamental frequencies as multi-oscillating black holes. Multi-oscillating black hole solu-tions to an Einstein-scalar system in AdS5 were recentlystudied in Ref. [23].

One of the primary challenges in numerically simulat-ing superradiant instabilities is the slow growth rates as-sociated with unstable modes. Compounding this, theAdS4 superradiant instability has no symmetries to ex-ploit, meaning one must numerically solve the full 3+1dimensional Einstein equations. This is a common fea-ture of superradiant instabilities with real fields, and con-sequently there are few simulations which reach the finalstate [9]. In contrast, with complex fields it is possibleto study superradiant instabilities in cylindrical symme-try [8] or charged superradiance with spherical symmetry[24–26], thereby allowing much faster simulations.

To study the AdS4 superradiant instability one mustemploy fast and stable numerical algorithms. To thisend, we use a characteristic evolution scheme we previ-ously developed for asymptotically AdS spacetimes. Thisscheme — which has seen use in a wide variety of prob-lems [21, 27–41] and is reviewed in detail in Ref. [42]— is remarkably efficient, allowing even five dimensionalproblems to be solved on a single CPU [38]. This effi-ciency is largely due to the fact that within the scheme,the apparent horizon can naturally be chosen to lie atsome fixed radial coordinate. That is, the entire compu-tation domain exterior to the apparent horizon and in-terior to the AdS boundary can be chosen to be a staticspherical shell. This means the angular and radial di-rections can be discretized with tensor product grids, al-lowing efficient numerical integration and differentiationoperations. Additionally, with tensor product grids it iseasy to employ spectral and pseudo-spectral discretiza-tions, which converge much faster than finite differenceschemes [43]. This allows far coarser grids to be used andameliorates CFL instabilities, meaning one can also usea larger time step compared to a finite difference scheme.We also judiciously choose our initial conditions so thatthe instabilities are reasonably fast, but that the Kerr-AdS black hole is not too close to extremality. We do thisbecause near-extremal black holes can develop structurenear the horizon, requiring finer grids there and longerrun times. For reference, each of our simulations runs inapproximately five weeks on a 2020 MacBook Air.

We simulate the Kerr-AdS superradiant instability

with two sets of seed perturbations, both with the samemass and spin. In both cases we find that the AdS4 su-perradiant instability results in a two stage process withdistinct initial and secondary instabilities. At the end ofthe secondary instability the geometry is that of a multi-oscillating black hole with several distinct fundamentalfrequencies. The multi-oscillating black hole is remark-ably close to a black resonator geometry, albeit with abit of gravitational hair localized far from the horizon.We see no obvious signs of additional instabilities in themulti-oscillating epoch. Indeed, at late times the appar-ent horizon area is consistent with the exponential ap-proach to a constant. Multi-oscillating black holes aretherefore a plausible candidate for the endpoint of thesuperradiant instability. We also demonstrate that hairyblack resonators are not unique and depend on initialconditions.

An outline of our paper is as follows. In Sec. II weoutline the setup of our problem, including initial condi-tions and our numerical evolution scheme. In Sec. III wepresents the results of our numerical simulations. Finally,in Sec. IV we discuss our results.

II. SETUP

We numerically solve the vacuum Einstein’s equationswith negative cosmological constant Λ = −3/L2. We setthe AdS radius L to unity. Our characteristic evolutionscheme is reviewed in detail in Ref. [42]. Here we out-line the details salient for characteristic evolution withspherical coordinates and an appropriate choice of basisfunctions.

Our metric ansatz takes the form,

ds2 = λ2gµν(xα, λ)dxµdxν + 2dvdλ, (3)

with Greek indices (µ, ν) running over the AdS boundaryspacetime coordinates xµ = v, θ, ϕ, where v is timeand θ and ϕ are the usual polar and azimuthal anglesin spherical coordinates. The coordinate λ is the AdSradial coordinate, with the AdS boundary located at λ =∞. Note lines of constant xµ are infalling radial nullgeodesics affinely parameterized by λ. Correspondingly,the metric ansatz (3) is invariant under shifts in λ,

λ→ λ+ ξ(xµ), (4)

for any function ξ(xµ). We exploit this residual diffeo-morphism invariance to fix the location of the apparenthorizon to be at λ = 1. This means horizon excisionis performed by restricting the computational domain toλ ≥ 1.

Near the AdS boundary Einstein’s equations can besolved with the power series expansion,

gµν(xα, λ) = g(0)µν (xα)+ · · ·+g(3)µν (xα)/λ3 +O(1/λ4). (5)

The expansion coefficient g(0)µν is the AdS boundary met-

ric (i.e. the metric the dual quantum field theory lives

3

in). As a boundary condition we fix

g(0)µν = ηµν , (6)

where

ηµν = diag(−1, 1, sin2 θ), (7)

is the metric on the unit sphere. This boundary condi-tion means gravitational waves are reflected off the AdSboundary.

A convenient diffeomorphism invariant observable isthe stress tensor Tµν in the dual quantum field theory,

which is determined by g(3)µν via [44],

Tµν = g(3)µν + 13ηµνg

(3)00 . (8)

Note that Einstein’s equations imply Tµν is traceless,ηµνTµν = 0, and covariantly conserved,

∇µTµν = 0, (9)

where ∇µ is the covariant derivative under the boundarymetric ηµν .

Within our characteristic evolution scheme, evolutionvariables consist of the conserved densities T 0µ, a gaugeparameter ξ used to shift the horizon to be at λ = 1 viaEq. (4), and the rescaled angular metric,

gab ≡√

det ηcddet gcd

gab, (10)

with lower case latin indices running over the angulardirections θ, ϕ. All other components of the metricare determined by non-dynamical equations, which aresolved on each slice of constant v. The rescaled metricsatisfies

det gab = det ηab = sin2 θ, (11)

which means gab contains two independent degrees offreedom. These two degrees of freedom encode the twopropagating degrees of freedom in gravitational waves. Inour numerical simulations we decompose gab as follows,

gab =[1 + 1

2habhab

]1/2ηab + hab, (12)

where indices are raised with ηab and hab is traceless,meaning haa = ηabhab = 0. Note the expansion (5) andboundary condition (6) imply that near the boundaryhab ∼ 1/λ3.

A. Discretization

We expand the angular dependence of all functions ina basis of scalar, vector and tensor harmonics. Theseare eigenfunctions of the covariant Laplacian on the unit

sphere. The scalar eigenfunctions are just spherical har-monics y`m.1 There are two vector harmonics, Vs`ma ,which read [45]

V1`ma = 1√

`(`+1)∇ay`m, (13a)

V2`ma = 1√

`(`+1)ε ba ∇by`m, (13b)

where ε ba has non-zero components ε ϕθ = csc θ and

ε θϕ = − sin θ. V1`ma is longitudinal, meaning it points

in the direction of ∇a, and V2`ma is transverse, meaning

∇aV2`ma = 0 . There are three symmetric tensor harmon-

ics T s`mab . However, since hab is traceless, we only needthe two traceless tensor harmonics, which read [45]

T 1`mab = 1√

`(`+1)(`(`+1)

2 −1)ε c(a ∇b)∇cy`m, (14a)

T 2`mab = 1√

`(`+1)( `(`+1)2 −1)

[∇a∇b + `(`+1)2 ηab]y

`m. (14b)

The scalar, vector and tensor harmonics are orthonor-mal and complete. Upon expanding SO(3) scalar, vec-tor, and tensor components of the metric and boundarystress in terms of these basis functions, angular deriva-tives can be computed by differentiating the basis func-tions themselves. In order to efficiently transform be-tween real space and mode space, we employ a Gauss-Legendre grid in θ with `max + 1 points and a Fouriergrid in ϕ with 2`max + 1 points. For details see Ref. [43].The grid points θi are the zeros of the `max + 1 orderLegendre polynomial,

P`max+1(cos θi) = 0, (15)

and the associated integration weights read

wθi =2

(dP`max+1/dθ)2

∣∣∣∣θ=θi

. (16)

Likewise, the ϕ grid points are

ϕj =2πj

2`max + 1, j = 0, 1, . . . , 2`max, (17)

with associated integration weights wϕj = 2π2`max+1 . The

transformation between real space and mode space canthen be performed using Gaussian quadrature.2 UsingGaussian quadrature, the orthonormality of the scalar,vector and tensor harmonics is exact up to angular mo-mentum ` = `max. We choose `max = 40.

We note that the mode expansions contain approxi-mately half as many degrees of freedom as that which

1 Note we employ the convention y`−m = y`m∗.2 Note that integration over ϕ can also be performed via FastFourier Transforms. However, with the small values of `max weemploy in this paper, we have found Gaussian quadrature to befaster.

4

live on the real space grid. Because of this, we choose touse mode amplitudes as our evolution variables. Whencomputing nonlinear products, we simply transform themode amplitudes to real space first.

For the radial dependence of the metric, we employ aninverse radial coordinate u ≡ 1

λ ∈ (0, 1) and expand theu dependence in a pseudo-spectral basis of Chebyshevpolynomials. We employ domain decomposition in the udirection with seven domains. The first domain interfacelies at u = 0.05 and the remaining interfaces are equallyspaced between u = 0.05 and u = 1. We use seven pointsin the domain closest to the boundary and 14 points inall other domains. Note that using fewer points in thedomain closest to the boundary helps ameliorate CFLinstabilities, thereby allowing larger time steps.

We evolve forward in time 3060 units using a 4th or-der Runge–Kutta method with constant time step dv =0.008. To test convergence of our numerics we have alsoran simulations with approximately 15% more grid pointsin each direction, and with 20% smaller time step. Like-wise, we also increased the filter parameters `0 and σ0(discussed below) by approximately 15%. Reassuringly,these simulations, which were ran until time v = 1500,which encompasses the times during which the moststructure exists, produced results indistinguishable fromthose presented in this paper. In particular, the differ-ences in Fig. 4 were far smaller than the widths of eachline.

0 10 20 30 400

0.2

0.4

0.6

0.8

1

FIG. 1: The angular filter function, Eq. (18). Modes with` . 20 are not appreciably modified by the filter.

B. Filtering

An important practical matter is filtering short wave-length excitations artificially generated during numericalevolution. We apply a short wavelength filter directly tothe time derivatives of the fields. Following [42], to filterthe radial direction, we simply interpolate the real spaceradial grid to a courser grid, and then reinterpolate back

to the original grid. For the course grid we use the samedomain interface locations, but just with fewer points.Specifically, we use one fewer point in the domain clos-est to the boundary, and two fewer points in all otherdomains.

To filter in the angular directions, at each time stepwe multiply the time derivatives of the mode amplitudesby the filter function

F (`) ≡ 1

2

[1 + erf

(− (`−`0)√

2σ

)]. (18)

where

`0 = 28, σ = 2.5. (19)

This function, plotted in Fig. 1, is ostensibly a regularizedstep function. Note that at ` = 15 we have 1−F ≈ 10−7

and at ` = 20 we have 1− F ≈ 7× 10−4. Hence this fil-ter does not appreciably modify low angular momentummodes (e.g. those with ` . 20).

We note that our decomposition of the angular metric(12) and use of only traceless tensor harmonics is differ-ent from our previous study of the AdS4 superradiantinstability [21]. In particular, in Ref. [21] we expandedgab in a basis of three tensor harmonics, with the addi-tional one being proportional to ηab, and imposed theconstraint (11) numerically during each time step. How-ever, imposing the constraint (11) involves nonlinear op-erations which can themselves numerically excite highangular momentum modes. While this can be amelio-rated by employing a stronger filter than that used inthis paper, the cost of doing so is reducing the effectiveangular resolution for a given value of `max. Hence, whileour value of `max is essentially the same as that used inRef. [21], our effective angular resolution is superior.

C. Initial data

The Kerr-AdS solution is parameterized by mass andspin parameters M and a, and the AdS radius L, whichwe have set to unity. For initial data we choose

hab = hKerrab , (20a)

T0µ = TKerr0µ + ∆T0µ, (20b)

where the superscript “Kerr” refers to the metric andboundary stress of the Kerr-AdS solution. If ∆T 0µ = 0,then the resulting geometry is exactly that of the Kerr-AdS solution. Hence, with these initial conditions the su-perradiant instability is seeded by non-vanishing ∆T 0µ.We choose

∆T00 = −4

3Re

4∑

`=2

α` y``, (21a)

∆T0a = −Re

4∑

`=2

α` (V1``a + V2``

a ). (21b)

5

IC α2 α3 α4

A 2.745× 10−2 −6.863× 10−3 0

B 5.490× 10−2eiπ/3 −1.373× 10−2e−iπ/3 3.768× 10−4e2πi/3

TABLE I: Mode amplitudes for our two sets of initial conditions.

'

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FIG. 2: The boundary energy density T 00 at eight different times for I.C. A. By v = 110 a small amplitude m = 2 mode isexcited. At subsequent times more structure develops via the excitation of higher m modes. These modes rotate at differentangular velocities. However, by time v = 1650 much of this structure has relaxed. At late times the energy density isapproximately that of a black resonator, rigidly rotating in ϕ at constant angular velocity. Indeed, the energy density atv = 1650 and v = 3000 mostly differs by a rotation in ϕ. Additionally, at late times there are also small amplitude m = 2, 4and 6 modes rotating at different angular velocities. These modes are responsible for the small differences seen in the structureof the energy density peaks at v = 1650 and v = 3000. Note the appearance of negative energy density v ≥ 1650.

Therefore, our initial conditions are determined by themass parameter M , spin parameter a, and the mode am-plitudes α`. Note that our choice of ∆T 0µ yields vanish-ing contribution to the total mass and angular momen-tum of the system. Hence the mass and angular momen-tum are identical to that of the Kerr-AdS solution withmass parameter M and spin parameter a.

We employ mass and spin parameters

M = 0.2375, a = 0.2177, (22)

and two sets of mode amplitudes α` given in Table I,which we will refer to as initial condition (I.C.) A and B.Note our chosen masses and spins differ from those usedin our previous work [21] by ∼ 5%.

III. RESULTS

We begin by showcasing results for I.C. A. In the toppanel of Fig. 2 we plot snapshots of boundary energy den-sity T 00 at eight times between v = 110 and v = 3000.

Note that the color scaling is different at each time. Byv = 110 a small amplitude m = 2 excitation is visi-ble. At subsequent times more structure develops viathe excitation of higher m modes, with different modesrotating in ϕ at different angular velocities. Neverthe-less, by v = 1650 most of this structure has relaxed andthe energy density approximately rotates rigidly at con-stant angular velocity. As we elaborate on further below,there are also tiny excitations with m = 2, 4, 6 rotating atdifferent angular velocities. These tiny excitations are re-sponsible for small differences seen in the energy densitypeaks at times v = 1650 and v = 3000.

To quantify the growth of different modes, includingtheir frequency content, we define the mode amplitudes

Fs`m(v, ω) ≡∫d2xdv′eiωv

′V∗s`ma (x)W (v−v′)T 0a(v′,x),

(23)where W (v) is a Gaussian window function with width15, and the vector spherical harmonics Vs`ma are givenby Eq. (13). F1`m and F2`m are essentially the Fourierspace amplitudes of the longitudinal and transverse com-ponents of the momentum density, respectively. More

6

m = 2

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m = 4

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m = 6

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v

!/m

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transv

erse

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longi

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FIG. 3: Spectrograms showing |Fs`m| as a function of v and ω/m for I.C. A. All plots are on the same scale. At sufficientlylate times the mode amplitudes |Fs`m| decay everywhere except in the vicinity of a few fundamental frequencies, which aredenoted by the yellow dashed lines. The dashed red line shows a fundamental frequency of the intermediate state. This modeis excited by the initial instability, which occurs in the (s`m) = (222) channel.

precisely, Fs`m is the short-time Fourier transform of thevector spherical harmonic transform of the momentumdensity.

In all of our simulations we see no significant growth inFs`m with m = 0 or odd m. We therefore focus on evenm. Plotted in Fig. 3 are spectrograms showing |Fs`m|as a function of v and ω/m for m = 2, 4, 6 and ` =m,m + 1,m + 2. All plots are on the same scale. Noteω/m is the angular velocity of a mode with frequencyω and azimuthal quantum number m. At sufficientlylate times the mode amplitudes |Fs`m| decay everywhereexcept in the vicinity of a few fundamental frequencies.The four yellow dashed lines in Fig. 3 lie at frequenciesΩs`m, where |Fs`m(v,Ωs`m)| is approximately constantat late times. These angular velocities are

Ω122 = 1.29, Ω176 = 1.41. Ω154 = 1.47, Ω242 = 1.65.(24)

Additionally, the red dashed line shows

Ω222 = 1.68. (25)

The mode amplitudes |Fs`m(v,Ωs`m)| evaluated alongthe dashed curves are shown below in Fig. 4. Note thatthe left and right columns in Fig. 4 show identical data,with the left column employing a logarithmic scale whilethe right column uses a linear scale.

From Figs. 3 and 4 it is evident there are two distinctepochs of growth in |Fs`m|, which we shall refer to asinitial and secondary instabilities [21]. The initial insta-bility is due to superradiant instabilities in the Kerr-AdSgeometry. Our numerics are consistent with an initialinstability in the (s`m) = (222) channel with complexfrequency

ω222 = 3.41 + 0.020i. (26)

7

0 500 1000 1500 2000 2500 300010-5

10-3

10-1

0 500 1000 1500 2000 2500 30000

0.2

0.4

0 500 1000 1500 2000 2500 300010-5

10-4

10-3

10-2

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1

2

10-3

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0.25

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0.75

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10-1

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0.025

0.05

0 500 1000 1500 2000 2500 300010-8

10-6

10-4

10-2

0 500 1000 1500 2000 2500 30000

1.5

3 10-3

|e Fs`m

|

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v

e0.02v

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e0.02v

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e0.00223v

<latexit sha1_base64="/8Qs3+FVYH5htME60+01QSwmcSM=">AAAB9HicbVDLTgJBEOzFF+IL9ehlIjHxRHYRo0eiF4+YyCOBlcwODUyYfTgzS0I2fIcXDxrj1Y/x5t84C3tQsJJOaqq6M93lRYIrbdvfVm5tfWNzK79d2Nnd2z8oHh41VRhLhg0WilC2PapQ8AAbmmuB7Ugi9T2BLW98m/qtCUrFw+BBTyN0fToM+IAzqo3k4mNil227Urkgk1mvWEofKcgqcTJSggz1XvGr2w9Z7GOgmaBKdRw70m5CpeZM4KzQjRVGlI3pEDuGBtRH5SbzpWfkzCh9MgilqUCTufp7IqG+UlPfM50+1SO17KXif14n1oNrN+FBFGsM2OKjQSyIDkmaAOlziUyLqSGUSW52JWxEJWXa5FQwITjLJ6+SZqXsVMuX99VS7SaLIw8ncArn4MAV1OAO6tAABk/wDK/wZk2sF+vd+li05qxs5hj+wPr8AYxbkKo=</latexit>

e0.00205v

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e0.00437v

<latexit sha1_base64="L+gmLMWm0ToPu1cbMJ1xk3iHMnk=">AAAB9HicbVDLTgJBEOzFF+IL9ehlIjHxRHYVg0eiF4+YyCOBlcwODUyYfTgzS0I2fIcXDxrj1Y/x5t84C3tQsJJOaqq6M93lRYIrbdvfVm5tfWNzK79d2Nnd2z8oHh41VRhLhg0WilC2PapQ8AAbmmuB7Ugi9T2BLW98m/qtCUrFw+BBTyN0fToM+IAzqo3k4mNil227clklk1mvWEofKcgqcTJSggz1XvGr2w9Z7GOgmaBKdRw70m5CpeZM4KzQjRVGlI3pEDuGBtRH5SbzpWfkzCh9MgilqUCTufp7IqG+UlPfM50+1SO17KXif14n1oNrN+FBFGsM2OKjQSyIDkmaAOlziUyLqSGUSW52JWxEJWXa5FQwITjLJ6+S5kXZqZSv7iul2k0WRx5O4BTOwYEq1OAO6tAABk/wDK/wZk2sF+vd+li05qxs5hj+wPr8AZcRkLE=</latexit>

FIG. 4: The mode amplitudes |Fs`m(v,Ωs`m)| evaluated along the dashed yellow and red lines shown in Fig. 3. The bluecurves correspond to I.C. A while the red curves correspond to I.C. B. The left and right columns show identical data, with theleft column employing a logarithmic scale and the right column using a linear scale. Evolution arising from both sets of initialconditions exhibits both initial and secondary instabilities. The initial instability has |F222| ∼ e0.02v. After |F222| plateaus,the secondary instability kicks in with |F122| ∼ e0.0022v, |F154| ∼ e0.0021v and |F176| ∼ e0.0041v. After the secondary instabilityterminates, the mode amplitudes approach constants, which are the same for both sets of initial conditions.

8

FIG. 5: R as a function of inverse radius 1/λ and frequency ω for I.C. A. The four yellow lines superimposed on the figurecorrespond to the four angular velocities ω/m = Ωs`m shown in Fig. 3. The black lines show harmonics of the black resonatorfundamental frequency, ω = 2Ω122 = 2.59. Note that the color scaling is on a logarithmic scale. R is peaked at several discretefrequencies, with the largest peaks at ω = 0 and the black resonator’s fundamental frequency. Modes corresponding to theblack resonator’s hair are largest at large radii.

The growth of |F222| begins to slow at v ∼ 300, whilealso slightly shifting in frequency, and reaches its peakamplitude around v ∼ 500. During the subsequent inter-val, 500 . v . 1200, there is a plateau-like structure in|F222(v,Ω222)|. It is also during this time interval thatthe secondary instability kicks in, with

|F122| ∼ e0.0022v, |F154| ∼ e0.0021v, |F176| ∼ e0.0041v.(27)

Note that the exponential growth of |F154| and |F176| isnot as crisp as that of |F122|.

The fact that |F222| is the largest amplitude modeat the early stages of the secondary instability suggeststhat the intermediate state is an excited black resonator.However, it is noteworthy that near v ∼ 500 there areadditional modes excited with amplitudes comparable to|F222|, but with fundamental frequencies different fromΩ222. See Fig. 3. We comment on this further below inthe Discussion section.

As seen in both Figs. 3 and 4, after v ∼ 1200, |F222|begins to precipitously decay while |F242|, |F122|, |F154|and |F176| approach constants shortly thereafter. Ac-tually, all three of these modes are decaying extremelyslowly at late times, with ∂v log |Fs`m| ∼ −1.9 × 10−5,although this is nearly impossible to see in Fig. 4. Weare unsure if this tiny decay rate is real or a numericalartifact. We comment on this below in the Discussionsection. At v = 3000 we have,

|F122| ≈ 0.70, |F154| ≈ 0.024. (28a)

|F172| ≈ 1.9× 10−3, |F242| ≈ 4× 10−4. (28b)

It is noteworthy that |F122| is nearly 30 times larger thanthe next largest mode, |F154|. If Ω122 was the only angu-lar velocity excited, then the geometry would be a blackresonator. Evidently, the final state of our numericalevolution is a black resonator with a small amount ofgravitational hair.

A natural question is where in the bulk is the blackresonator’s hair localized? Is it localized near the bound-ary or is it spread throughout the bulk, including nearthe horizon? To answer this question we consider theKretschmann scalar K ≡ RµναβRµναβ . Near the bound-ary K → 24, indicating that K itself is not a good probeof near-boundary excitations. We wish to construct aquantity from K which has a nontrivial limit near theboundary and which can discriminate contributions fromdifferent frequencies, including those with small modeamplitudes. To this end we take the short-time Fouriertransform of the difference ∆K ≡ K − 24,3

∆K(v, ω, λ,x) ≡∫dv′eiωv

′W (v−v′)∆K(v, λ,x), (29)

and then average |∆K| over the angular directions,

〈|∆K(v, ω, λ)|〉 ≡∫d2x |∆K(v, ω, λ,x)|. (30)

〈|∆K(v, ω, λ)|〉 vanishes near the boundary and is insen-sitive to the precise value of v (at sufficiently late times).The dimensionless ratio

R ≡ 〈|∆K|〉〈|∆K|〉|ω=0

, (31)

has a nontrivial boundary limit and measures the ampli-tude of a bulk mode relative to the zero frequency limit.

In Fig. 5 we plot R at time v = 2750 as a function ofthe radial coordinate λ and frequency ω. The four yellowlines superimposed on the figure correspond to the fourangular velocities ω/m = Ωs`m shown in Fig. 3 and listed

3 Due to memory constraints relevant for the calculation of K,here we use a Gaussian window function W (v) with width 10.

9

0 1000 2000 30000.95

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1.15

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10-9

10-6

10-3

100

e20.02v

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e20.00223v

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FIG. 6: The apparent horizon area A (left) and the rate of area growth A ≡ ∂vA (right) for I.C. A (blue) and I.C. B (red).The areas are normalized by the Kerr-AdS horizon area. The dashed black lines in the right panel are the squares of thetwo exponential curves seen in the (s`m) = (222) and (s`m) = (122) panels of Fig. 4. For both sets of initial conditions,

two distinct epochs of horizon area growth are clearly visible. At late times A decreases exponentially. At v = 3000 the twoapparent horizon areas agree at order 1 part in 106.

in Eq. 24. The black lines show harmonics of the blackresonator fundamental frequency, ω = 2Ω122 = 2.59.Note that the color scaling is on a logarithmic scale. Asis evident from the figure, R is peaked at several dis-crete frequencies, with the largest peaks at ω = 0 andthe black resonator’s fundamental frequency. The peakscorresponding to the black resonator’s hair are largest atlarge radii and smallest at the horizon (λ = 1). In par-ticular, at the horizon R < 0.4× 10−4 everywhere exceptnear ω = 0 and the black resonator’s fundamental fre-quency and harmonics. In contrast, at λ = 5, the peakat ω = 4Ω154 has amplitude 0.02. This signals that theblack resonator’s hair is concentrated away from the hori-zon. Similar behavior was seen in Ref. [25], where thefinal state of the (spherically symmetric) charged AdSsuperradiant instability was studied.

Plotted in Fig. 6 is the apparent horizon area A (left

panel) and its time derivative A ≡ ∂vA (right panel),both normalized by the area of the associated Kerr-AdSblack hole. To increase the fidelity of the plots of A, aswell as smooth out short time structure in A due to in-dividual superradiant amplification cycles, we computeA by convolving A with the time derivative of a normal-ized Gaussian with width 8. The two dashed lines inthe right panel are simply the squares of the exponen-tial curves seen in the (s`m) = (222) and (s`m) = (122)panels of Fig. 4. Evidently, during both the initial andsecondary instabilities, the horizon area grows roughlyas the square of the these modes. Note that A decreasesexponentially at late times. At v = 3000 the horizon areais 11% larger than that of the Kerr-AdS black hole withthe same mass and spin.

Finally, we turn to evolution generated by our secondinitial condition. I.C. B has the same energy and angularmomentum as I.C. A, but different initial seed perturba-tions. It turns out that many of the features of the result-ing solutions are identical, both qualitatively and quan-titatively. The resulting spectrogram for I.C. B looks

strikingly similar to that shown in Fig. 3, with angularvelocities Ωs`m identical to those found with I.C. A. Alsoincluded in Fig. 4 are the amplitudes |Fs`m(v,Ωs`m)|obtained with I.C. B. During the initial and secondaryinstabilities the exponential growth rates are the samefor both sets of initial conditions. Likewise, as shownin Fig. 6, A and A also have similar structure for bothsets of initial conditions. Remarkably, the final values of|F122|, |F242|, |F154| and |F176| are also nearly identi-cal for both set of initial conditions. Moreover, the finalapparent horizon areas agree at order 1 part in 106.

Despite these similarities, the final state obtained withI.C. B is in fact distinct from that obtained with I.C.A. One difference lies in relative phase shifts betweendifferent modes. Define the phases

δs`m = arg(Fs`m

), with δs`m ∈ [−π, π]. (32)

In order the exclude the possibility that the phases δs`mof our two solutions are merely related by a shift in vand/or ϕ, we define the weighted phase differences

∆1 = 8(Ω154 − Ω242)δ122 + 4(Ω242 − Ω122)δ154 (33)

+ 8(Ω122 − Ω154)δ242,

∆2 = 24(Ω154 − Ω176)δ122 + 12(Ω176 − Ω122)δ154 (34)

+ 8(Ω122 − Ω154)δ176.

It is easy to check that ∆1 and ∆2 are invariant undershifts in v and ϕ. If the late-time solutions generated bydifferent initial conditions are identical, then they musthave identical ∆1 and ∆2 at late times.

In Fig. 7 we plot ∆1 and ∆2. The saw-tooth like struc-ture seen in both plots is due to the fact that the phasesδs`m are not continuous functions of time. For both setsof initial conditions ∆1 and ∆2 approach constants at latetimes. However, these constants are different for each ini-tial condition. This demonstrates that the gravitationalhair obtained with I.C. A is distinct from that generated

10

0 1000 2000 3000-5

0

5

0 1000 2000 3000-20

-10

0

10

20

FIG. 7: Weighted phase differences ∆1 and ∆2 for both sets of initial conditions. The saw-tooth like structure seen in bothplots is due to the fact that the phases δs`m are not continuous functions of time. For both sets of initial conditions ∆1 and∆2 approach constants at late times. However, these constants are different for I.C. A and I.C. B. This demonstrates that thegravitational hair obtained with I.C. A is distinct from that generated with I.C. B.

with I.C. B. Simply put, the final state of our evolution issensitive to seed perturbation in the Kerr-AdS geometry.

IV. DISCUSSION

It is remarkable that after starting in the Kerr-AdS ge-ometry, and then having many large m modes excited —as is evident from the plots of the energy density seen inFig. 2 — that the geometry finds its way to a hairy blackresonator state. Is a hairy black resonator, or more gener-ally a multi-oscillating black hole, the final state of the su-perradiant instability? While our numerics are certainlyconsistent with this — mode amplitudes and the hori-zon area appear to plateau — we cannot exclude otherpossibilities. For example, it is always possible there areadditional instabilities which are too small to be observedwith our current numerical simulations. Likewise, its al-ways possible there are instabilities in other channels notprobed by our limited set of initial conditions.

Even if the final state is a hairy black resonator, itis possible the hair could undergo nontrivial slow dy-namics. For example, the hair could decay or differ-ent modes could interact via nonlinear couplings, re-sulting in slow changes to the hairy mode amplitudes.We do see some evidence of slow mode amplitude dy-namics in our simulations. The hairy modes seen inFig. 4 all are decaying very slowly at late times, with∂v log |Fs`m| ∼ −1.9 × 10−5. Additionally, we do see atiny amount of late time growth in |F133| ∼ 3 × 10−3,which roughly grows like ∂v log |F133| ∼ +7×10−6. Thismode has angular velocity Ω133 = 1.27, which only differsfrom Ω122 by 1.5%. However, while these tiny rates couldindicate interesting hairy dynamics (or even instabilitiesof the black resonator itself), they lie at the thresholdof fidelity of our current numerical simulations. Indeed,we have also ran short duration simulations of our finalstate at 15% reduced resolution and found these rates tochange significantly. To help assess whether the time de-

pendence of these mode amplitudes is physical, it wouldbe useful run simulations with different discretization andfiltering schemes (e.g. cubing the sphere as opposed tousing spherical harmonics), as well as long duration highresolution simulations.

We have also ran short duration simulations of a va-riety of perturbations of our final state. In doing so wefound that after initial transients decay, the hairy modeamplitudes become approximately constant, just as seenin Fig. 4. However, we have found that the associatedconstants are sensitive to the perturbations and can dif-fer by O(1) factors from those seen in Fig. 4. Notablyhowever, these solutions have virtually identical horizonareas: all simulations resulted in the same horizon areaat order 1 part in 106. This presumably reflects the factthat, as highlighted in Fig. 5, the hair is localized nearthe boundary, meaning it does not affect the near-horizongeometry.

It would clearly be of considerable value to study themode spectrum of black resonators. First, it would beinteresting to see how instabilities and decay rates of theblack resonator associated with our final state comparewith our numerical simulations. Second, it would be in-teresting to compare our intermediate state to a blackresonator. While the spectrograms in Fig. 3 demonstratethat the intermediate state is initially dominated by asingle angular velocity, Ω222, there are other modes ap-preciably excited with different angular velocities. Thisobservation alone does not exclude the possibility thatthe intermediate state is an excited black resonator. In-deed, far-from-equilibrium black branes in AdS5 are well-approximated by large amplitude linear perturbations ontop of a stationary background [30]. To solidify the in-tuition that the the intermediate state is an approxi-mate black resonator, it would be useful to compare theblack resonator spectrum to the spectrum of decayingand growing modes seen during the intermediate stageof our evolution. Finally, it would also be interesting tounderstand why instabilities associated with the inter-

11

mediate state are strong, while none are visible in ourfinal state. At the boundary our intermediate state isdominated by transverse modes whereas our final stateis dominated by longitudinal modes. It is possible thatinstabilities associated with these two different types ofblack resonators could be qualitatively different.

We have only studied evolution with a single set ofmasses and spins. A natural question then is how genericare our results? Is the resulting multi-oscillating blackhole always close to a black resonator? Its noteworthythat our previous work, Ref. [21], which studied evolutionwith a different mass and spin, resulted in |F154| ∼ |F122|during the intermediate stage of the evolution. This sug-gests it is possible that the final state can be a multi-oscillating black hole which is significantly different froma black resonator. Simulating such a scenario likely re-quires higher resolution and longer runtimes than thesimulations presented in this paper.

Finally, via AdS/CFT duality our results also have in-teresting consequences for quantum field theory. Firstly,the existence of the AdS superradiant instability impliesthat rotation invariance is spontaneously broken in thedual field theory. This behavior evidently only occurs

in small systems, where there is are dual superradiantinstabilities. From field theory arguments alone, we donot know why one should have expected this. Second,both sets initial conditions studied in this paper yieldevolution with negative energy density. This was alsofound in Ref. [21]. This indicates that the dual quan-tum field theory state has exotic properties. Regions ofnegative energy density have also been seen in the holo-graphic models studied in [34, 39, 46, 47]. Finally, it isremarkable that the system does not appear to approacha stationary configuration despite having a large entropy.It would be interesting to explore to what extent, if any,hairy black resonators correspond to thermal states inthe dual quantum field theory. This should be possibleusing the methods developed in Refs. [48, 49]. We leavethis and many other interesting question for future work.

Acknowledgments

I thank Luis Lehner and Larry Yaffe for helpful com-ments and discussions.

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