Scalar and gravitational hair for extreme Kerr black holes
aa r X i v : . [ g r- q c ] M a y Scalar and gravitational hair for extreme Kerr black holes
Lior M. Burko , Gaurav Khanna , and Subir Sabharwal Theiss Research, La Jolla, California 92037, USA Department of Physics, University of Massachusetts, Dartmouth, Massachusetts 02747, USA (Dated: May 13, 2020)For scalar perturbations of an extreme Reissner-Nordstr¨om black hole we show numerically thatthe Ori pre-factor equals the Aretakis conserved charge. We demonstrate a linear relation of ageneralized Ori pre-factor – a certain expression obtained from the late-time expansion or the per-turbation field at finite distances – and the Aretakis conserved charge for a family of scalar orgravitational perturbations of an extreme Kerr black hole, whose members vary only in the radiallocation of the center of the initial packet. We infer that it can be established that there is an Are-takis conserved charge for scalar or gravitational perturbations of extreme Kerr black holes. Thisconclusion, in addition to the calculation of the Aretakis charge, can be made from measurementsat a finite distance: Extreme Kerr black holes have gravitational hair that can be measured at finitedistances. This gravitational hair can in principle be detected by gravitational-wave detectors.
Introduction and summary.
Extreme sphericallysymmetric and charged black holes [extreme Reissner-Nordstr¨om black holes (BHs), hereafter ERN] have beenshown to carry massless scalar hair that can be mea-sured at future null infinity ( I + ) [1]. This scalar hairis a certain quantity s [ ψ ] which is evaluated at I + andwhich equals the Aretakis charge, a non-vanishing quan-tity H [ ψ ] which is calculated on the BH’s event horizon(EH, H + ) but vanishes if the BH is non-extreme.Since the scalar hair at I + is intimately related tothe Aretakis conserved charge on H + , one may suspectthat corresponding conserved charges for other fields oneither ERN or extreme Kerr (EK) BHs may also be re-lated to observable hair at I + , or be measurable at finitedistances. Specifically, conserved Aretakis charges werefound in ERN, in addition for massless scalar fields [2]also for massive scalar fields, for coupled linearized grav-itational and electromagnetic fields [3], for charged scalarperturbations [4], and in EK for scalar [2], electromag-netic, and gravitational perturbations [5–7].Ori showed that the Aretakis charge can also be usedin order to determine a certain pre-factor e [ ψ ] in the latetime expansion of scalar field perturbation fields in ERNas measured at a finite distance [8]. Here, we first shownumerically that for scalar perturbations of ERN the Oripre-factor e [ ψ ] equals H [ ψ ], and therefore can be usedin order to measure the Aretakis conserved charge at afinite distance. It follows that e [ ψ ] can be interpreted asscalar hair measured outside the BH.We then go beyond the framework of scalar perturba-tions of ERN to EK, and show numerically that anal-ogous pre-factors can be formulated also for scalar andgravitational perturbations of EK. Since the value of theAretakis charge depends on the initial data of the pertur-bation field, it follows that information on the prepara-tion of the perturbation field can be inferred at great dis-tances from the BH measurements, in apparent contra-diction of the established no-hair theorem [9–11]. Thatis, we bring evidence that in addition to the three ex-ternally observable classical parameters, specifically theBH’s mass M , charge q , and spin angular momentum a , it is in principle possible to also detect with a gravitational-wave detector the gravitational Aretakis charge of EK. Setting up the problem.
Following Ori [8] we write thelate time expansion of a field ψ I s,ℓ,m as ψ I s,ℓ,m ( t, r, θ ) = e I s,ℓ,m r ( r − M ) − p I s,ℓ,m t − n I s,ℓ,m Θ I s,ℓ,m ( θ )+ O ( t − n I s,ℓ,m − k I s,ℓ,m ) (1)in Boyer-Lindquist coordinates, where s is the field’s spin, ℓ, m are the spherical harmonic numbers, and the index Icorresponds to the BH type, i.e., I = { ERN , EK } . Here, e I s,ℓ,m is a generalized Ori pre-factor. The case studiedin [8] corresponds to ψ ERN0 ,ℓ, , for which it was found that e I0 ,ℓ, = ( − ℓ +1 M ℓ +2 e [8], where e [ ψ ] is a certain pre-factor that depends on the initial data (and which is givenexplicitly in [8]), and p ERN0 ,ℓ, = ℓ + 1, n ERN0 ,ℓ, = 2 ℓ + 2, andΘ ERN0 , , ( θ ) = 1. The late-time expansion (1) is expected tobe valid for t ≫ r ∗ , where r ∗ is the tortoise coordinate.Specifically, we may expect r -dependent correction termswhen this condition is not satisfied. Comparing [1] and[8] we expect that e ERN0 , , [ ψ ] = − M H [ ψ ]. Numerical approach.
To test this prediction, and toset up the framework for generalization to EK and togravitational perturbations, we write the 2+1 Teukol-sky equation in ERN or EK backgrounds for azimuthal( m = 0) modes in compactified hyperboloidal coordi-nates ( τ, ρ, θ, ϕ ), such that I + is included in the compu-tational domain at a finite radial (in ρ ) coordinate [12].We re-write the second-order hyperbolic partial differen-tial equation as a coupled system of two first-order hy-perbolic equations. We solve this system for the scalarfield case by implementing a second-order Richtmeyer-Lax-Wendroff iterative evolution scheme [13, 14]. Forthe gravitational case we implement a sixth-order (in ρ ) WENO (Weighted Essentially Non-Oscillatory) finite-difference scheme with explicit time-stepping [6]. Thesecodes converge with second-order temporally and angu-larly.The initial data are a compactly supported “trun-cated” gaussian with non-zero initial field values on H + ,but similar results are expected also for other forms ofinitial data. Specifically, in hyperboloidal coordinates( ρ, τ ) (see [13] for definitions), the initially spherical( ℓ = 0) Gaussian pulse is non-vanishing in the range ρ/M ∈ [0 . , . M and centered closeto the BH (at ρ/M = 1 .
0, 1 .
1, 1 .
2, 1 .
3, 1 . . H + , is at ρ = 0 . M for ERNand EK in these coordinates.) The outer boundary islocated at S = ρ ( I + ) = 19 . M .The computations were performed on IBM 32-corePower9 servers accelerated by Nvidia V100 GPGPUs.Our resolution for each production run was ∆ ρ = M/ , τ = M/ , θ = π/
64, which we runin quadrupole precision (128-bit, i.e., to ∼
30 decimaldigits). The combination of quadruple-precision float-ing point numerics and the extremely high-resolution re-sulted in computationally intensive simulations, whichtook two weeks for each run to get to t/M ∼ , Scalar perturbations of ERN.
We calculate e ERN0 , , [ ψ ] di-rectly from Eq. (1), and H ERN0 , , [ ψ ] from H I0 , , [ ψ ] = − M π Z H + ∂ r ( rψ ) d Ω , (2)where I = ERN. To determine e ERN0 , , [ ψ ] we calculate itfor a set of finite values of the time. Figure 1(a) shows e ERN0 , , [ ψ ] at a number of time values as a function of theSchwarzschild coordinate r , for the initial data set forwhich the gaussian is centered at ρ/M = 1 .
0. Noticethat the numerical constancy of ( t/M ) (1 − M/r ) ψ ERN0 , , for small values of r/M suggests that p ERN0 , , = 1 and n ERN0 , , = 2, as expected from [8]. For larger values of r/M the constant value starts to vary, as expected from theexpansion of [8]. Equation (1) suggests that e ERN0 , , [ ψ ]( t )is time dependent, and that when ( t/M ) (1 − M/r ) ψ ERN0 , , is plotted as a function of inverse time, the value of k canbe determined. We see in Fig. 1 that there is indeed timedependence as expected.The time dependence of e ERN , , [ ψ ]( t ) is shown in greaterdetail in Fig 2, which displays for each initial data setthe values of e ERN0 , , [ ψ ]( t ). We then extrapolate the valuesto M/t → e ERN0 , , [ ψ ]. Thelinearity suggests that k ERN0 , , = 1, in agreement with [8].The values of e ERN0 , , [ ψ ] depend on the choice of the ini-tial data set. In Fig. 3(a) we show ( t/M ) (1 − M/r ) ψ ERN0 , , for each initial data set as functions of r/M . As the cen-ter of the initial gaussian packet moves outward (to larger ρ values) the value of ( t/M ) (1 − M/r ) ψ ERN0 , , decreases.Finally, Fig. 4(a) shows the values of e ERN0 , , [ ψ ] as afunction of the corresponding H ERN0 , , [ ψ ] for the differ-ent data sets. Fitting our numerical data to e ERN0 , , [ ψ ] = α H ERN0 , , [ ψ ] + β we find that α = − . ± . β = (1 . ± . × − , consistently with our expectation.The Ori pre-factor e equals the Aretakis charge H . Scalar perturbations of EK.
We next extend the anal-ysis from the case of a scalar field in ERN to scalar
Figure 1: The values of e I s,ℓ, [ ψ ]( t ) as functions of r/M .These values are shown for the data set for which at thegaussian’s center ρ/M = 1 .
0. Top panel (a): ERN with s = 0 , ℓ = 0. Middle panel (b): EK with s = 0 , ℓ = 0.Bottom panel (c): EK with s = − , ℓ = 2. The valuesare plotted for t/M = 1100, 1200, 1300, 1400, 1500, and1600. [For panel (c) the time value was replaced with 1553.]The function f ( t, r ) = ( t/M ) (1 − M/r ) and the function g ( t, r ) = M ( t/M ) ( r/M ) (1 − M/r ) . -3 Figure 2: The values of e ERN0 , , [ ψ ]( t ), normalized by their valuesas t → ∞ , as functions of M/t . These values are shown foreach initial data set, parametrized by the ρ/M value at thecenter of the gaussian packet. and gravitational perturbations of EK. First, we set upthe initial value problem for scalar field perturbationssimilarly as for ERN. We use the expansion 1 as anAnsatz. The results for the scalar case in EK are shownin Figs. 1(b), 3(b), and 4(b). These result suggest thatEq. (1) describes well also the field for this case. Fittingthe parameters to this Ansatz, we find that p EK0 , , = 1and n EK0 , , = 2. We also find that Θ EK0 , , ( θ ) = 1. To find H EK0 , , [ ψ ] we again use Eq. (2) with I = EK. Seeking alinear relation of the form e EK0 , , [ ψ ] = α H EK0 , , [ ψ ] + β we Figure 3: The values of e I s,ℓ, [ ψ ]( t/M = 1500) as functionsof r/M , shown for each initial data set, parametrized by the ρ/M value at the center of the gaussian packet. Left panel(a): ERN with s = 0 , ℓ = 0. Center panel (b): EK with s = 0 , ℓ = 0. Right panel (c): EK with s = − , ℓ = 2. -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 00246-1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 001020-0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 00200400600 Figure 4: The pre-factor e I s,ℓ, [ ψ ] shown as a function of theAretakis charge H I s,ℓ, [ ψ ] for the different initial data sets(parametrized with ρ/m at the center of the gaussian ini-tial packet). Top panel (a): ERN with s = 0 , ℓ = 0. Middlepanel (b): EK with s = 0 , ℓ = 0. Bottom panel (c): EK with s = − , ℓ = 2. find that α = − . ± .
03 and β = − . ± . e EK0 , , [ ψ ] and H EK0 , , [ ψ ] suggest thatalso in this case the Aretakis conserved charge can bemeasured at a finite distance, and that a generalized Oripre-factor can be used in order to measure it. Gravitational perturbations of EK.
Finally, we considerEK gravitational perturbations with s = − ℓ = 2.We write the Teukolsky equation for a Kerr BH withparameters M, a for the variable Φ − , which is relatedto the Teukolsky function Ψ K − in the Kinnersley tetradand Boyer-Lindquist coordinates via Φ − = ( r/ ∆ ) Ψ K − , -3 t / M = 1,350 t / M = 1,250 t / M = 1,150 t / M = 1,050 Figure 5: The relative difference of the Weyl scalar ψ (nor-malized by its maximal value) and Θ( θ ) = sin θ as a functionon the polar angle θ at a fixed value of ρ/M = 2 for four differ-ent time values, t/M = 1 ,
050 (dotted), 1,150 (dash-dotted),1,250 (dashed), and 1,350 (solid). On the scale shown theseplots cannot be resolved. where ∆ = r − M r + a . Since the Weyl scalar ψ HH4 in the Hartle-Hawking tetrad is related to its Kinnersleytertrad counterpart, ψ K4 , via a type-III transformation,or ψ HH4 = 4( r + a ) ∆ − ψ K4 [15] and that Ψ K − = ( r − ia cos θ ) ψ K4 [16] we find thatΦ − = r ( r − ia cos θ ) r + a ) ψ HH4 , (3)and use Φ − with ℓ = 2 , m = 0 and a = M for ψ EK − , , . Note that at great distances, as r ≫ M , ψ EK − , , ∼ ( r/ ψ HH4 ∼ rψ K4 . Therefore, determinationof ψ EK − , , at great distances allows us to measure di-rectly the Weyl scalar ψ K4 in the Kinnersley tetrad. Con-versely, measurement with a gravitational wave detectorat a great distance of ψ K4 allows us to calculate ψ EK − , , ifthe distance to the source is known.We plot Φ − for a fixed ρ as a function of θ for a setof τ values in Fig. 5. Since our angular resolution is∆ θ = π/
64 and our code converges angularly with sec-ond order, we would expect our angular numerical errorto be (a few) × − . We find that the angular functionΘ( θ ) deviates from sin θ by no more than (a few) × − .Therefore, we could not distinguish numerically betweenour numerical function Θ( θ ) and sin θ .We calculate e EK − , , [ ψ ] directly from Eq. (1), and mo-tivated by [3], we calculate H EK − , , [ ψ ] by H EK − , , [ ψ ] = − π M Z H + ∂ r Φ − d Ω . (4)(Note that ψ decays to 0 at late times on H + .) Weonly calculate here the real part of ψ : Because of thelinearity of the Teukolsky equation we can always per- I s ℓ p n Θ( θ ) α β ERN 0 0 1 2 1 − . ± . . ± . × − EK 0 0 1 2 1 − . ± . − . ± . θ − . ± . − . ± . α, β in the linear relation e I s,ℓ, [ ψ ] = α H I s,ℓ, [ ψ ] + β . form a Wick rotation, and obtain commensurate resultsfor the imaginary part.The results for the Weyl scalar ψ are shown inFigs. 1(c), 3(c), and 4(c). Again, we find that the Ansatz(1) describes the field behavior well. Fitting the parame-ters to this Ansatz, we find that p EK − , , = 5 and n EK − , , =6. Seeking a linear relation of the form e EK − , , [ ψ ] = α H EK − , , [ ψ ] + β we find that α = − . ± . β = − . ± .
3. The linear relation of e EK − , , [ ψ ] and H EK − , , [ ψ ] suggest that also in this case the Aretakis con-served charge can be measured at a finite distance, andthat a generalized Ori pre-factor can be used in order tomeasure it. We summarize our results in Table I. Discussion.
The values for the Ori pre-factor, andtherefore also for the Aretakis charge – when comparedbetween members of the same initial data family whichdiffer from each other just by the distance of the centerof the initial packet – are suggested by our results to beuniversal, i.e., they depend only weakly on the spin ofthe field and on whether the BH is ERN or EK (Fig. 3).The linear relation of the Ori pre-factor and the Are-takis conserved charge for either scalar or gravitationalperturbations of EK suggests that we could make mea-surements at a finite distance and conclude that the BHhas a conserved charge, and therefore establish also thatit is an extreme BH. Moreover, by using the (numericallydetermined) value of the parameter α (or, in the case of scalar perturbations of ERN, its analytical value) we cancalculate the value of the Aretakis charge. If the mea-sured quantity appears to behave as for an ERN or EKfor some time, and then decays as for a non-extreme BH(i.e., it is a transient behavior), we can establish that itis a nearly extreme BH (see also [17]). Since the valueof the Aretakis charge depends on the perturbation field(cf. Fig. 3), and this value can be found from observationsat a finite distance, this is a procedure for detecting grav-itational hair of EK.Extreme Kerr BHs that are perturbed gravitationallyhave hair, and this determination and also the calcula-tion of the strength of the hair can be made at finitedistances by measuring the Weyl scalar ψ directly fromthe gravitational wave strain. Specifically, gravitationalwave detectors can be used to measure this gravitational-field hair of extreme black holes.This apparent contradiction of the no-hair theorempertains to extreme BHs, which require fine tuning ofthe astrophysical processes that created them. RealisticBHs are more likely to be nearly extreme, and thereforewould present transient hair that could in principle bedetected by gravitational-wave detectors.Work on higher- ℓ modes and non-azimuthal ( m = 0)modes is currently underway. Measurement of gravita-tional hair of EK at I + awaits further work. Acknowledgements.
The authors thank Shahar Hadarand Achilleas Porfyriadis for discussions. S.S. thanksthe University of Massachusetts, Dartmouth for hospi-tality duration the performance of this work. Manyof the computations were performed on the MIT/IBMSatori GPU supercomputer supported by the Mas-sachusetts Green High Performance Computing Center(MGHPCC). G.K. acknowledges research support fromNSF Grants No. PHY-1701284 and No. DMS-1912716and Office of Naval Research/Defense University Re-search Instrumentation Program (ONR/DURIP) GrantNo. N00014181255. [1] Y. Angelopoulos, S. Aretakis, and D. Gajic,Phys. Rev. Lett. , 131102 (2018)[2] S. Aretakis, arXiv:1206.6598 [gr-qc] (2012)[3] J. Lucietti, K. Murata, H.S. Reall, and N. Tanahashi, J.High Energy Phys. , 35 (2013)[4] P. Zimmerman, Phys. Rev. D , 124032 (2017)[5] J. Lucietti and H.S. Reall, Phys. Rev. D , 104030(2012)[6] L.M. Burko and G. Khanna, Phys. Rev. D , 061502(R)(2018)[7] S.E. Gralla and P. Zimmerman, Class. QuantumGrav. , 095002 (2018)[8] A. Ori, arXiv:1305.1564 (2013)[9] J. D. Bekenstein, Phys. Rev. Lett. , 452 (1972) [10] J. D. Bekenstein, Phys. Rev. D , 1239 (1972)[11] J. D. Bekenstein, Phys. Rev. D , 2403 (1972)[12] A. Zengino˘glu, Class. Quantum Grav. P , 021017(2011)[14] L.M. Burko, G. Khanna, and A. Zengino˘glu,Phys. Rev. D , 041501(R) (2016), [Erratum:Phys. Rev. D , 129903(E) (2017)][15] E. Poisson, Phys. Rev. D , 084044 (2004)[16] S.A. Teukolsky, Astrophys. J. , 635-647 (1973)[17] L.M. Burko, G. Khanna, and S. Sabharwal,Phys. Rev. Research1