Brief note on high-multipole Kerr tails
BBrief note on high-multipole Kerr tails
Tyler Spilhaus and Gaurav Khanna ∗ Physics Department and Center for Scientific Computing and Visualization Research,University of Massachusetts Dartmouth, North Dartmouth, MA 02747 (Dated: October 9, 2018)In this note we reconsider the late-time, power-law decay rate of scalar fields in a Kerr space-time background. We implement a number of mathematical and computational enhancements toour time-domain, (2+1)D Teukolsky evolution code and are able to obtain reliable decay rates formultipoles as high as (cid:96) = 16. Our numerical results suggest full agreement with the proposed decayexpressions in recent work [6, 9] for both the finite distance (including horizon), and null infinitycases. We also study the same in the context of extremal Kerr space-time and find that the sameresults hold, except that the horizon tails follow the null infinity expressions instead.
I. BACKGROUND & SUMMARY
The asymptotic late-time, power-law decay ( t n ) ratesof matter fields in Kerr black hole space-time has been amatter of some debate for several decades. However, sig-nificant progress has been made on this question over thepast few years through the development of sophisticatedtechniques [1–8] to tackle the computational challengesinherent to this problem. Recent work [6, 9] proposes thefollowing late-time decay rate expressions for the scalarfield case: n = (cid:26) − ( (cid:96) (cid:48) + (cid:96) + 3) for (cid:96) (cid:48) = 0 , − ( (cid:96) (cid:48) + (cid:96) + 1) otherwise (1)and n I + = (cid:26) − (cid:96) (cid:48) for (cid:96) ≤ (cid:96) (cid:48) − − ( (cid:96) + 2) for (cid:96) ≥ (cid:96) (cid:48) (2)by carefully studying the “inter-mode coupling” effectsthat are present in Kerr space-time due to frame-dragging. Note that these expressions above are for theaxisymmetric multipoles. (cid:96) (cid:48) refers to the initial field mul-tipole and (cid:96) is the multipole of interest under study.In this note, we borrow and implement the “best-practice” lessons from all previous numerical work onKerr tails (especially [5, 7, 10]) and perform very high-accuracy computations for the axisymmetric multipolesup to (cid:96) = (cid:96) (cid:48) = 16. Our results are in full agreement withthe expressions 1, 2 above. It should be noted that weare not presenting any new physical results in this workor developing a deeper understanding of known expres-sions; we are simply demonstrating a proposed result inthe literature [6, 9] to be accurate for a large range ofparameters. II. METHODOLOGY
We begin this section by briefly commenting on whynumerical computations are so challenging in the con- ∗ Electronic address: [email protected] text of Kerr tails: (i) These simulations are required tobe rather long duration – this is because typically the ob-served field exhibits an exponentially decaying oscillatorybehavior in the initial part of the evolution i.e. quasi-normal ringing, and only much later does this transitionover to a clean power-law decay. Therefore, one needs toevolve past the point that the initial oscillations decayaway. Moreover, as clearly shown in Refs. [6, 9] thereis an “intermediate” tail regime, wherein one observestails with various decay rates that are not necessarilythe late-time asymptotic rates that we are after. Indeed,this intermediate regime is largely responsible for muchof the confusion on this topic, in the literature. Theseintermediate tails decay faster than the asymptotic rate,but typically have dominant amplitudes for a period oftime. We must evolve past this regime as well, in order toobtain the true asymptotic rate; (ii) Because each mul-tipole has its own decay rate (which increases with anincrease in (cid:96) ) at late times one obtains numerical datain which different multipoles have widely different am-plitudes (often 30 – 40 orders of magnitude apart!). Forthis reason, the numerical solution scheme is required tohave high-order convergence (to reduce the discretizationerrors to low enough levels, to be able to track the fastdecaying multipoles); (iii) Moreover, these computationsalso require high-precision floating-point numerics, dueto the very large range of amplitudes involved, and alsoto reduce round-off error which can easily overwhelm thefast decaying modes.To address the challenges mentioned above, we per-form the following mathematical and computational ad-vancements to our time-domain, (2+1)D Teukolsky equa-tion evolution code:
A. Mathematical Enhancements
The main advance we make in this context is to re-cast the problem using the technique of hyperboloidalcompactification [11] for the Teukolsky equation in Kerrspace-time. This allows one to include null infinity I + on the computational grid by mapping the entire space-time onto a compact domain. This technique also allowsus to use a rather modest sized grid to sample the entire a r X i v : . [ g r- q c ] J a n . . –67 –59 –51 –43 –35 –27 –19 Time S ca l a r F i e l d M u l t i po l es L=1 , n= X L=
3, n=- , n=-1 L= , n=-1 L= , n=-1 L= , n=-1 L= , n= -9L= , n= -7 FIG. 1: Kerr tails for a range of (cid:96) multipoles (1 – 15) startingwith a pure (cid:96) (cid:48) = 5 multipole. These tails obey the proposedtail formula n = − ( (cid:96) + (cid:96) (cid:48) + 1). It is clear that such numericalsimulations require octal-precision floating-point arithmetic. domain, thus delivering tremendous savings towards thetotal computational cost. In short, hyperboloidal com-pactification enables us to perform long duration simula-tions using relatively modest computational resources. Inthis work, we use the compactification approach similarto Ref. [7, 8, 12] that utilizes a single horizon-penetratinghyperboloidal foliation with conformal compactification.More specifically, we utilize the compactified radial coor-dinate ρ given by r = ρ/ Ω, and Ω = 1 − ρ/S , where S isthe location of the outer boundary. For all our numericalsimulations, we choose a Kerr hole with a/M = 0 . S/M = 38 .
4, with the inner boundary at the (outer)horizon.
B. Computational Enhancements
We make three additional major enhancements to ourTeukolsky code in order to obtain the high-accuracy nu-merical results we report in the following section. First,a pseudo-spectral collocation numerical scheme is im-plemented in order to accurately and efficiently handlethe spatial discretization. Because the tail solutions aresmooth, this method converges exponentially , thus allow-ing us to significantly lower the discretization error asneeded, even with a modest grid size. We utilize a sin-gle spectral domain in this work, with the angular andradial directions expanded in Legendre and Chebyshevpolynomials, respectively, using the Gauss-Lobatto col-location points. We use 245 collocation points in theradial direction and 16 in the angular, for all the simula-tions in this work. We use the method-of-lines approachto evolve forward in time, using a fourth-order Runge- Kutta method. The time-step we use in all our compu-tations is ∆ t/M = 0 .
05. Secondly, we implement high-precision, floating-point arithmetic throughout our code.In particular, this enhancement includes full support for octal-precision numerics (256-bit or ∼
60 decimal digits).This is required to keep the round-off error in our simu-lations at acceptable levels. Finally, we also implementalgorithmic parallelism in order to speed up the compu-tations, so they complete in a reasonable amount of time.In particular, we use a small cluster of 16 Sony PlaySta-tion 3 gaming consoles (PS3) to perform all the simula-tions in this work. Each PS3 works independently on aninitial multipole (cid:96) (cid:48) case (an “embarrassing” coarse-grainparallelism). In addition, we also implement a fine-grainparallelism at the level of the high-precision computa-tions themselves (see Ref. [5, 10] for details) utilizing theparallel architecture of the PS3’s Cell Broadband Engine.Overall, we obtain over two orders-of-magnitude speed-up via this parallel computing approach.In summary, pseudo-spectral collocation method en-ables us to drastically reduce the discretization error,high-precision numerics helps us accurately track ampli-tudes down to the 10 − scale, and the parallelism andcompactification help to keep the total runtime to stayreasonable (and also obtain rates at null infinity directly). (cid:96) (cid:48) \ (cid:96) (cid:96) (cid:48) \ (cid:96) http://gravity.phy.umassd.edu/ps3.html III. NUMERICAL RESULTS
In this section, we present the outcome of the enhance-ments implemented in the previous section.
A. Non-Extremal Kerr
The initial data is a smooth Gaussian wave-packet cen-tered at ρ/M = 3 . σ/M = 2 . t/M = 2000. The results are de-tailed in four tables: two for the finite-distance rates (oddand even multipole cases separately) and another two ta-bles for the null infinity rates. It is worth pointing outthe tails on the horizon follow the same behavior as theones at finite-distance. Except for a few cases (close to (cid:96) = 16) where even octal-precision numerics prove to beinsufficient to produce reliable tail solutions, all resultsagree precisely with expressions 1, 2 above. The tablesare self-explanatory. All depicted numerical results areaccurate within a few percent or less. Figure 1 depictsthe actual simulation data for a sample (cid:96) (cid:48) = 5 case. (cid:96) (cid:48) \ (cid:96) (cid:96) (cid:48) \ (cid:96) It took approximately two days to perform all the com-putations we have presented in this section, using the 16PS3 cluster we mentioned before.
B. Extremal Kerr
For obtaining reliable results for the extremal case( a/M = 1) we use the same settings as in the previ-ous subsection, however with twice as many collocationpoints. The finite-distance and null infinity rates ob-tained in this case are identical to the results presentedin the last subsection, so we will not repeat those here.However, the horizon rates differ completely and followthe null infinity expressions instead. Due to computa-tional resource limitations we are only able to accuratelyobtain data for a small subset of the ( (cid:96) (cid:48) , (cid:96) ) cases con-sidered thus far. Those results are presented in the twotables below. (cid:96) (cid:48) \ (cid:96) (cid:96) \ (cid:96) (cid:48) (cid:96) (cid:48) \ (cid:96) (cid:96) \ (cid:96) (cid:48) The tails at the event horizon matching those at nullinfinity may seem surprising at first, but, it should benoted that this feature has been appreciated before inthe literature, at least for extremal Reissner-Nordstromblack holes [13]. This happens as a result of a discreteconformal symmetry between the horizon and null infin-ity in these space-times. Our numerical results are simplyan interesting and independent verification of this sym-metry in the context of extremal Kerr space-time.
IV. ACKNOWLEDGEMENTS