Superradiant Instability and Backreaction of Massive Vector Fields around Kerr Black Holes
SSuperradiant Instability and Backreaction of Massive Vector Fields around KerrBlack Holes
William E. East and Frans Pretorius Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada and Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
We study the growth and saturation of the superradiant instability of a complex, massive vector(Proca) field as it extracts energy and angular momentum from a spinning black hole, using numer-ical solutions of the full Einstein-Proca equations. We concentrate on a rapidly spinning black hole( a = 0 .
99) and the dominant m = 1 azimuthal mode of the Proca field, with real and imaginarycomponents of the field chosen to yield an axisymmetric stress-energy tensor and, hence, spacetime.We find that in excess of 9% of the black hole’s mass can be transferred into the field. In all casesstudied, the superradiant instability smoothly saturates when the black hole’s horizon frequencydecreases to match the frequency of the Proca cloud that spontaneously forms around the blackhole. Introduction. —A remarkable feature of spinningblack holes (BHs) is that a portion of their mass—up to 29% for extremal spin—can, in principle, beextracted. One way to realize this liberation of ro-tational energy is through the interaction of the BHwith an impinging wave—be it scalar, electromag-netic, or gravitational—with frequency ω < m Ω BH ,where Ω BH is the BH horizon frequency and m is theazimuthal number of the wave. Waves satisfying thiscriterion exhibit superradiance and carry away energyand angular momentum from the BH. An analogousphenomenon can occur for charged BHs, where theelectromagnetic energy of the BH is superradiantlytransferred to an interacting charged matter field in-teracting with the BH.Going back to Ref. [1], there has been speculationof how superradiance could be combined with a con-fining mechanism to force the wave to continuouslyinteract with the BH and hence undergo exponen-tial growth—a so called “black hole bomb.” The firstnonlinear studies of this process were recently under-taken for a charged scalar field around a charged BHin spherical symmetry, both in a reflective cavity inasymptotically flat space [2], and in the naturally con-fining environment of an asymptotically anti-de Sitterdomain [3].However, there is an exciting possibility that a vari-ation of this scenario could, in fact, be realized aroundastrophysical spinning BHs. Massive bosonic fieldswith a Compton wavelength comparable to, or largerthan, the horizon radius of a BH can form boundstates around the BH, and if the latter is spinning,the bound states can grow from a seed perturbationthrough superradiance [4–6]. This implies that stellarmass BHs can probe the existence of ultralight bosons with masses (cid:46) − eV that are weakly coupled toordinary matter and thus difficult to detect by othermeans. Theoretical scenarios where this might occurinclude the string axiverse [7, 8], the QCD axion [9],and dark photons [10, 11]. Such particles could formlarge clouds, spinning down the BH in the process.This is of particular interest now that LIGO has be-gun observing gravitational waves (GWs) [12], sincemeasurements of BH masses and spins from binarymergers can be used to rule out or provide evidencefor such particles, in addition to direct searches forthe GW signatures of boson clouds [13, 14]. See [15]for a review.Though details of the nonlinear growth and satu-ration of the rotational superradiant instability willbe important to help observe or rule out such mas-sive fields, there are presently few results of relevanceto this regime where the backreaction on the BH issignificant. In Ref. [16], it was found that, for suffi-ciently large GWs superradiantly scattering off a KerrBH, backreaction effects decrease the efficiency of en-ergy extraction (for the analogous case of the scatter-ing of a charged scalar field by a Reissner-Nordstr¨omBH, see [17]). The nonlinear behavior of the superra-diant instability of massive bosons has not been ad-dressed before. This is because of the computationalcost of solving the equations, in part due to the dis-parate time scales between the oscillation of the fieldand the growth rate of the instability and the lackof symmetries to reduce it to a (1 + 1)-dimensionalproblem (unlike the charged case). Important ques-tions include what the efficiency of energy and angu-lar momentum extraction is, how explosive the nonlin-ear phase of growth is (e.g., can the energy extractionovershoot limits implied by the parameters of the field a r X i v : . [ g r- q c ] J un and BH [2]), and what the final state is after a non-negligible amount of energy has been transferred tothe Proca field (e.g., does a stable cloud form aroundthe BH, or could there be something akin to a bosen-ova where the entire field is rapidly expelled from thevicinity of the BH).In this Letter we begin to address these questionsrelated to the nonlinear behavior of the superradiantinstability of massive bosonic fields around a spin-ning BH. We focus on the case of a vector field, asit exhibits faster growth than a scalar field. The lin-ear regime of the instability for Proca fields has beenstudied before in various limits [11, 14, 18–20]. Herewe find numerical solutions of the full Einstein-Procafield equations. To make the problem computation-ally tractable, we use a complex field with prescribed m = 1 azimuthal dependence to give an axisymmetricstress-energy tensor and, hence, spacetime geometry.Beginning with a seed field about a rapidly rotatingBH, we find that the instability efficiently grows intothe nonlinear regime and smoothly saturates when theBH horizon frequency decreases to match that of theProca cloud. This frequency depends on the massparameter of the field, and for a value near wherewe expect maximal energy extraction, we find thatwhen the instability saturates a large Proca cloud hasformed, containing 9% of the initial BH mass (and38% its initial angular momentum). We use units with G = c = 1 throughout. Methodology. —We consider a Kerr BH with initialmass M and dimensionless spin a = 0 .
99 in thepresence of a complex Proca field X a with constantmass parameter µ , and numerically evolve the coupledEinstein-Proca equations. The Proca field equation ofmotion is ∇ a F ab = µ X b , where F ab = ∇ a X b −∇ b X a ,and its corresponding stress-energy tensor is T ab = 12 ( F ac ¯ F bd + ¯ F ac F bd ) g cd − g ab F cd ¯ F cd + µ X a ¯ X b + ¯ X a X b − g ab X c ¯ X c ) , (1)where the overbar indicates complex conjugation.We evolve the Proca equations in a form similar toRef. [21] (which also evolved the Einstein-Proca equa-tions, though without symmetry restrictions and fo-cusing on nonlinear interactions between the field anda nonspinning BH). We restrict to cases where theProca field has an m = 1 azimuthal dependence andthe resulting stress-energy tensor and spacetime areaxisymmetric—i.e., in terms of the Lie derivative withrespect to the axisymmetric Killing vector ( ∂/∂φ ) b , L φ X a = iX a . This allows us to use a two-dimensional numerical domain for the spatial discretization, whichis essential in making evolutions on time scales of ∼ M computationally feasible. Here we studycases with ˜ µ := M µ =0.25, 0.3, 0.4, and 0.5. Aswe discuss below, ˜ µ = 0 .
25 is near the value thatmaximizes the energy extracted from the BH, while˜ µ = 0 . ∼ − M arounda Kerr BH (ignoring the effect of this small field onthe initial spacetime geometry) and study the subse-quent evolution. We have verified that using differentor lower amplitude perturbations gives similar results.We evolve the Einstein equations using the gener-alized harmonic formulation, with the gauge degreesof freedom set by fixing the source functions to thevalues of the initial BH solution in Kerr-Schild coor-dinates, as in Ref. [16]. To mitigate the accumulationof truncation error during the long period it takes forthe Proca field to grow large enough to significantlybackreact on the spacetime, we use the background er-ror subtraction technique described in Ref. [23], withthe initial isolated spinning BH as the background so-lution. As the spacetime evolves, we keep track ofthe BH apparent horizon and measure its area A andangular momentum J BH , from which we can derive amass using the Christodoulou formula M BH = (cid:18) M + J M (cid:19) / , (2)where M irr = (cid:112) A/ π is the irreducible mass. Wealso measure the flux of Proca field energy ˙ E H andangular momentum ˙ J H through the BH horizon. Inaddition, for the Proca field we keep track of the en-ergy density ρ E = − αT tt and angular momentum den-sity ρ J = − αT tφ (where α = [ − g tt ] − / is the lapse),the volume integrals of which give a measure of the to-tal energy E and angular momentum J outside of theBH horizon. Details on the numerical resolution andconvergence are given in the appendix; more informa-tion on how we evolve the Proca equations, as well asresults on the instability in the test-field regime, areprovided in Ref. [22]. Results. —All the cases studied here are susceptibleto a linear superradiant instability, and after a brieftransient period the energy and angular momentum inthe Proca field enter a period of exponential growthas shown in Fig. 1. As also shown there, the corre-sponding loss of mass and angular momentum by theBH, as measured from its horizon properties, closelytracks this. The cases with larger µ have larger growth M )0.000.020.040.060.080.10 M / M ˜µ = 0. 25˜µ = 0. 3˜µ = 0. 4˜µ = 0. 5 E−∆M BH M )0.000.050.100.150.200.250.300.350.40 J / M ˜µ = 0. 25˜µ = 0. 3˜µ = 0. 4˜µ = 0. 5 J−∆J BH FIG. 1. The energy (top) and angular momentum (bot-tom) in the Proca field as a function of time (solid lines),along with the loss in mass (top) and angular momentum(bottom) of the BH (dashed lines). rates for the instability and also saturate with smallerenergy and angular momentum. Though the mass ofthe BH is decreasing in each case, as required by BHthermodynamics the irreducible mass M irr is alwaysincreasing, and smaller µ cases saturate with a largeroverall increase in M irr .The reason for the saturation of the superradiantinstability is illustrated in Fig. 2, where we plot boththe horizon frequency of the BH Ω BH and the ratio ofProca field energy to angular momentum flux throughthe horizon ˙ E H / ˙ J H . When Ω BH > ˙ E H / ˙ J H , the su-perradiant condition is met and the Proca cloud willextract rotational energy from the BH. However, asshown in Fig. 2, eventually the BH’s horizon frequencydecreases to the point where Ω BH ≈ ˙ E H / ˙ J H , and the M )0.250.300.350.400.45 Ω H M ˜µ = 0. 25˜µ = 0. 3˜µ = 0. 4˜µ = 0. 5 Ω BH ˙E H / ˙J H FIG. 2. The BH horizon frequency Ω BH , as calculatedfrom the BH’s mass and angular momentum, and the ratioof the flux of Proca field energy and angular momentum˙ E H / ˙ J H through the BH horizon, as a function of time. instability saturates.We can obtain simple estimates of the final stateproperties of the black hole if we assume, as roughlyconsistent with the simulations, that the instabilitywill extract energy and angular momentum in somefixed proportion ω ( µ ) = ˙ E H / ˙ J H [where ω ( µ ) ≈ µ (1 − ˜ µ /
2) in the linear/small ˜ µ limit [24, 25]] un-til ω ( µ ) = Ω BH . We plot the results in Fig. 3, alongwith the four end-state points from the full nonlinearsimulations, showing excellent agreement with the ap-proximation. This indicates an efficient extraction ofenergy and angular momentum, with a negligible ad-ditional increase in irreducible mass (equivalently, BHentropy). This is likely due to the relatively slow evo-lution of the instability compared to the light-crossingtime of the BH, even approaching saturation (similarconclusions were reached using a “quasiadiabatic” ap-proximation for the massive scalar field instability inRefs. [26, 27]). We see that the energy lost by the BHshould be maximized at − ∆ M BH /M ≈ . − ∆ M BH /M ≈ .
092 found for ˜ µ = 0 . µ , less energy, but moreangular momentum will be extracted, with the insta-bility just converting the Kerr BH into a nonspinningBH of the same mass in the µ → ω0.50.60.70.80.91.0 F i n a l B H p a r a m e t e r M irr /M M BH /M J BH /M FIG. 3. The final BH irreducible mass, total mass, andangular momentum after saturation of the m = 1 super-radiant instability for a BH with a = 0 .
99 ( M irr , /M ≈ .
76) initially. The lines show the prediction obtained withthe assumption that the BH will lose energy and angularmomentum in fixed proportion ω , until Ω BH = ω , while thepoints show the measured values from the simulations. sulting clouds are illustrated in Fig. 4 for two cases.Away from the BH, the Proca clouds have a roughlyspherical energy density, falling off exponentially withdistance from the BH. For the larger µ cases, the cloudis concentrated on much smaller scales near the BH.As expected, given the close match between the en-ergy and angular momentum lost by the BH and thatgained by the Proca cloud, the radiation from bothGWs and the Proca field is negligible (the dominantcontribution of which comes from the other modesin the seed perturbation leading to initial radiation).The fact that we are considering a complex Proca fieldand restricting our study to axisymmetric spacetimeswill suppress the gravitational radiation. We can es-timate what the gravitational radiation would be fora single real Proca field by using the GW luminos-ity results from the test field limit [22] and scalingthem using P GW ∝ E . This gives P GW ∼ × − ,2 × − , 6 × − , and 6 × − for the Proca fieldclouds at the end of the ˜ µ = 0 .
25, 0.3, 0.4, and 0.5simulations, respectively. This means that once theBH has spun down to below the superradiant regime,the Proca cloud will decay via GW emission on timescales of ∼ –10 M .Our Proca field ansatz only allows exploration ofthe m = 1 mode instability. Higher m modes arealso unstable, even after the BH has spun down tothe point where the m = 1 becomes stable (ignoring FIG. 4. The energy (left) and angular momentum density(right) of the Proca field in the final state with ˜ µ = 0 . µ = 0 . z axis) and perpendicular to the equatorialplane ( z = 0); note the different scales for the two cases. the Proca cloud). However, the grow rates becomesignificantly longer with increasing m . For example,for ˜ µ = 0 . m = 2 instability has a growth rate ∼ m = 1 mode [22], andthe disparity is worse for smaller µ (with the relativegrowth rates scaling as µ ). Conclusion. —We have studied the growth and sat-uration of the superradiant instability of a massivevector field around a Kerr BH. We find that in allcases the instability efficiently extracts energy fromthe BH and then smoothly shuts off as the BH hori-zon frequency decreases to the threshold of instability.This contrasts with Ref. [2], where the energy extrac-tion of a charged BH in a reflecting cavity was seen toovershoot in some cases; this could be due to the pres-ence of multiple unstable modes [3]. We further findthat at saturation essentially all the energy and angu-lar momentum extracted from the BH has gone intoforming a cloud of complex Proca “hair” with station-ary energy density surrounding the BH. A family ofstationary hairy BH solutions with this property andthe same matter model was constructed in Ref. [28]which is plausibly the same as our end states; it wouldbe interesting to investigate in detail how close thesesolutions are to what we find at saturation. In ourcase, the Proca clouds persist for the relatively shorttimes we have extended the runs beyond saturation,though this is not adequate to comment on their long-term stability.
Acknowledgements. — We thank Asimina Arvan-itaki, Masha Baryakhtar, Stephen Green, RobertLasenby, Luis Lehner, Vasilis Paschalidis, Justin Rip-ley, Kent Yagi, and Huan Yang for stimulating discus-sions. This research was supported in part by Perime-ter Institute for Theoretical Physics (WE), the Na-tional Science Foundation through Grant No. PHY-1607449 (FP), and the Simons Foundation (FP). Re-search at Perimeter Institute is supported by the Gov-ernment of Canada through the Department of Inno-vation, Science and Economic Development Canadaand by the Province of Ontario through the Ministryof Research, Innovation and Science. Simulations wererun on the Perseus Cluster at Princeton Universityand the Sherlock Cluster at Stanford University. [1] W. H. Press and S. A. Teukolsky, Nature (Lon-don) , 211 (1972).[2] N. Sanchis-Gual, J. C. Degollado, P. J. Montero, J. A.Font, and C. Herdeiro, Phys. Rev. Lett. , 141101(2016), 1512.05358.[3] P. Bosch, S. R. Green, and L. Lehner, Phys. Rev.Lett. , 141102 (2016), 1601.01384.[4] T. Damour, N. Deruelle, and R. Ruffini, Lettere AlNuovo Cimento Series 2 , 257 (1976).[5] S. L. Detweiler, Phys.Rev. D22 , 2323 (1980).[6] T. Zouros and D. Eardley, Annals Phys. , 139(1979).[7] A. Arvanitaki, S. Dimopoulos, S. Dubovsky,N. Kaloper, and J. March-Russell, Phys. Rev.
D81 ,123530 (2010), 0905.4720.[8] A. Arvanitaki and S. Dubovsky, Phys. Rev.
D83 ,044026 (2011), 1004.3558.[9] A. Arvanitaki, M. Baryakhtar, and X. Huang, Phys.Rev.
D91 , 084011 (2015), 1411.2263.[10] B. Holdom, Phys. Lett.
B166 , 196 (1986).[11] P. Pani, V. Cardoso, L. Gualtieri, E. Berti, andA. Ishibashi, Phys. Rev.
D86 , 104017 (2012), 1209.0773.[12] Virgo, LIGO Scientific, B. P. Abbott et al. , Phys.Rev. Lett. , 061102 (2016), 1602.03837.[13] A. Arvanitaki, M. Baryakhtar, S. Dimopoulos,S. Dubovsky, and R. Lasenby, Phys. Rev.
D95 ,043001 (2017), 1604.03958.[14] M. Baryakhtar, R. Lasenby, and M. Teo, (2017),1704.05081.[15] R. Brito, V. Cardoso, and P. Pani, editors,
Super-radiance , , Lecture Notes in Physics, Berlin SpringerVerlag Vol. 906, 2015, 1501.06570.[16] W. E. East, F. M. Ramazano˘glu, and F. Pretorius,Phys. Rev.
D89 , 061503 (2014), 1312.4529.[17] O. Baake and O. Rinne, Phys. Rev.
D94 , 124016(2016), 1610.08352.[18] P. Pani, V. Cardoso, L. Gualtieri, E. Berti, andA. Ishibashi, Phys. Rev. Lett. , 131102 (2012),1209.0465.[19] S. Endlich and R. Penco, JHEP , 052 (2017),1609.06723.[20] H. Witek, V. Cardoso, A. Ishibashi, and U. Sperhake,Phys. Rev. D87 , 043513 (2013), 1212.0551.[21] M. Zilh˜ao, H. Witek, and V. Cardoso, Class. Quant.Grav. , 234003 (2015), 1505.00797.[22] W. E. East, (2017), 1705.01544.[23] W. E. East and F. Pretorius, Phys. Rev. D87 , 101502(2013), 1303.1540.[24] S. R. Dolan, Phys. Rev.
D76 , 084001 (2007),0705.2880.[25] J. G. Rosa and S. R. Dolan, Phys. Rev.
D85 , 044043(2012), 1110.4494.[26] S. R. Dolan, Phys. Rev.
D87 , 124026 (2013),1212.1477.[27] R. Brito, V. Cardoso, and P. Pani, Classical andQuantum Gravity , 134001 (2015), 1411.0686.[28] C. Herdeiro, E. Radu, and H. Runarsson, Class.Quant. Grav. , 154001 (2016), 1603.02687. Details of numerical methods
We employ fourth-order accurate finite differencemethods to integrate the equations. For all of the sim-ulations presented here we used a grid hierarchy withseven levels of mesh refinement, with 2:1 refinementratio, centered on the BH. For the cases with ˜ µ = 0 . ×
192 points on eachrefinement level and a resolution of dx/M ≈ . µ = 0 . / µ = 0 . / / E / M ˜µ = 0. 4 Low Res.Med. Res.High Res. M )−1. 2−1. 0−0. 8−0. 6−0. 4−0. 20. 00. 20. 4 ∆ E / M ×10 −3 (Med. - Low)×1. 46(High - Med.)×3. 94 E / M ×10 −2 ˜µ = 0. 5 Low Res.Med. Res.High Res. M )−0. 20. 00. 20. 40. 60. 81. 01. 21. 4 ∆ E / M ×10 −3 (Med. - Low)×1. 46(High - Med.)×3. 37 FIG. 5. The energy in the Proca field for simulationsperformed at three different resolutions, and the differencein this quantity with resolution, scaled assuming fourth-order convergence for ˜ µ = 0 . µ = 0 . of the energy of the field are a few percent at thelowest resolution, and are less than 1% for the highresolution runs throughout the simulation run time.We did not perform convergence studies for the twolower µ cases, as the much slower growth rates make such studies prohibitively expensive. However, basedon the convergence studies for the higher µ cases, wesuspect errors in the lower µ runs — which have largercharacteristic length scales — are still relatively small,at a few percent at most. In the main text all resultsare taken from the highest resolution data available.Though we do not include the effect of the pertur-bation of the initial Proca field configuration on theinitial BH spacetime, we have verified that it so smallas to introduce negligible error. This is illustrated in −5 0 5 10(t − t )/(10 M )0.000.010.020.030.040.050.060.07 − ∆ M B H / M ˜µ = 0. 4 E(t = 0)/M = 1. 5 × 10 −3 E(t = 0)/M = 3. 8 × 10 −4 FIG. 6. The change in BH mass as a function of timefor ˜ µ = 0 . E ( t = 0) /M = 1 . × − , and onewhere the initial Proca field energy is four times smaller,but otherwise identical. A time shift has been applied sothat the two curves align when − ∆ M BH /M = 0 . Fig. 6 where we show a comparison of the change inBH mass as a function of time for a simulation wherewe use an initial Proca field perturbation that is halfthe amplitude of the standard case, for ˜ µ = 0 .
4. Afterapplying a relative time shift (of ≈ M0