Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
You have accessResearch articles

High-order asymptotics for the spin-weighted spheroidal equation at large real frequency

Marc Casals

Marc Casals

Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, CEP 22290-180, Brazil

School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland

Google Scholar

Find this author on PubMed

,
Adrian C. Ottewill

Adrian C. Ottewill

School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland

Google Scholar

Find this author on PubMed

and
Niels Warburton

Niels Warburton

School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland

[email protected]

Google Scholar

Find this author on PubMed

Published:https://doi.org/10.1098/rspa.2018.0701

    Abstract

    The spin-weighted spheroidal eigenvalues and eigenfunctions arise in the separation by variables of spin-field perturbations of Kerr black holes. We derive a large, real-frequency asymptotic expansion of the spin-weighted spheroidal eigenvalues and eigenfunctions to high order. This expansion corrects and extends existing results in the literature and we validate it via a high-precision numerical calculation.

    1. Introduction

    Teukolsky [1,2] derived a single ‘master’ equation for spin-field perturbations of rotating (Kerr) black holes. This (3 + 1)-dimensional master equation separates by variables, with the polar-angular factor in the solution being the so-called spin-weighted spheroidal eigenfunction. The corresponding eigenvalue also appears in the equation satisfied by the radial factor of the Teukolsky master solution. Thus, both the eigenvalues and the eigenfunctions are important for studying perturbations of astrophysical black holes.

    Neither the spin-weighted spheroidal eigenfunctions nor the eigenvalues are known in closed form but they can be calculated using numerical and analytical techniques (see [3,4] for a review). As for the analytical techniques, for example, expansions have been obtained for small-frequency [58] and asymptotic analyses have been carried out for large, purely imaginary frequency [3,4,911].

    In this paper, we are instead interested in the asymptotics for large, real frequency. These asymptotics are interesting for various reasons, such as for synchrotron radiation [9,12], for the study of divergences in either the quantum or classical field theories (e.g. [13,14] for Wentzel–Kramers–Brillouin method in the case of spherically symmetric space–times and [15,16] for expressions for expectation values involving (spin-weighted) spheroidal harmonics in Kerr) and gravitational waves from rapidly rotating black holes [17,18]. Analytic approximations are also extremely valuable as checks on numerical calculation schemes. The large, real-frequency behaviour of the scalar spheroidal eigenfunctions and eigenvalues was studied in [1921]. The first large, real-frequency study in the non-zero spin case was carried out in [9]. However, this work contained an error which was corrected by Breuer, Ryan and Waller [22] (BRW). BRW provided an asymptotic expansion for the eigenvalue up to six leading orders, which depended crucially on a parameter sqm that was left undetermined for the case of non-zero spin. Furthermore, their analysis for non-zero spin had an error in the asymptotic behaviour of the eigenfunctions which was later corrected in [23]. This correction further allowed [23] to analytically obtain the parameter sqm as well as the correct first term in a large, real-frequency series expansion for the eigenfunctions.

    As it turns out, however, the last three orders in the asymptotic expansion for the eigenvalue provided in BRW formally in terms of sqm were also incorrect. In this paper, we correct these third to sixth leading orders and extend the expansion up to four higher orders, thus providing the correct 10 leading orders of the eigenvalue for large, real frequency. We also provide the first few coefficients in the large, real-frequency expansion of the eigenfunctions, thus going, for the first time, beyond leading order. We compare our asymptotic expansions for both the eigenvalues and eigenfunctions with high-precision numerical calculations and find excellent agreement. The results of this paper together with those in [23] thus provide a correct, high-order asymptotic expansion of the eigenvalues and eigenfunctions for large, real frequency.

    The layout of the rest of the paper is as follows. In §2, we introduce the spin-weighted spheroidal equation and its symmetries. In §3, we perform the large-frequency asymptotic analysis of the spin-weighted spheroidal eigenfunctions and eigenvalues. We compare our asymptotic analysis and our numerical results in §4. In Appendix A we give explicit expressions for the coefficients in the series for the eigenfunctions, and in Appendix B we describe the implementation of the asymptotics for the eigenvalue in a Mathematica toolkit.

    2. Spin-weighted spheroidal equation

    Teukolsky [1,2] managed to decouple and separate by variables the linear spin-field perturbations of Kerr black holes. He achieved this for the radiative components of the massless fields of spin1 s = 0 (scalar), ± 1/2 (neutrino), ± 1 (electromagnetic) and ± 2 (gravitational). The polar-angular factors of the perturbations are the so-called spin-weighted spheroidal harmonics sSmc. These functions satisfy the following linear, second-order ordinary differential equation (ODE):

    (ddx((1x2)ddx)+c2x22scx(m+sx)21x2+sEmcs2)sSmc(x)=0,2.1
    where x≡cosθ∈[ − 1, + 1] is the physical region of interest and θ∈[0, π] is the (Boyer–Lindquist) polar angle. Here, c, where aR is the angular momentum per unit mass of the black hole and ωC is the frequency of the field mode. The multipole number =|s|,|s|+1,|s|+2, serves to label the eigenvalue sEmc and m = − ℓ, − ℓ + 1, ℓ + 2, …, + ℓ is the azimuthal number. The ODE (2.1) has two regular singular points at x = ± 1 and an irregular singular point at x = ∞. The eigenvalue sEmc is chosen so that the corresponding solution sSmc is regular over x∈[ − 1, + 1]. The case s = 0 yields the (scalar) spheroidal equation [1921], whereas the case c = 0 yields the spin-weighted spherical equation [24] (in which case,2 sEm0 = ℓ(ℓ + 1)).

    Other common parameterizations of the eigenvalue are

    sλmcsEmcs(s+1)+c22mc2.2a
    and
    sAmcsEmcs(s+1).2.2b
    The manifest symmetries of the spin-weighted spheroidal equation imply that
    sSmc(x)=(1)+msSmc(x)andsS(m)(c)(x)=(1)+ssSmc(x),2.3
    where the choice of signs ensures consistency with the so-called Teukolsky–Starobinsky identities [25,26], and
    sEmc=sEmcandsEm(c)=sE(m)c.2.4
    While sλmc is most common in the current literature, we shall use sAmc in the following sections to ease comparison with BRW.

    3. Large, real-frequency asymptotics

    In this paper, we are interested in the large, real-‘frequency’ (by which we really mean c →  ± ∞) behaviour of the eigenvalues and eigenfunctions. In addition, by equations (2.3) and (2.4) we may assume c positive and deduce the c negative behaviour from changing m to −m. We, therefore, restrict ourselves to c > 0 from now on.

    We here generalize to arbitrary spin Flammer's [20] approach in the scalar case—this is essentially BRW's path, although they obtained some incorrect results which we specify and correct below. We start by writing solutions of the spin-weighted spheroidal equation (2.1) as

    sSmc±(x)=(1x)|m+s|/2(1+x)|ms|/2ec(1x)g±(x),3.1
    where g±(x) are regular functions. The powers of (1 − x) and (1 + x) are dictated by the Frobenius method, so that the solution is regular at both boundary points x = + 1 and −1. The exponential factor ec(1±x) is included for convenience when looking for an asymptotic solution ‘near’ x = ± 1.

    From equation (2.3), it follows that the solution g(x) is obtained from g+(x) under the transformation {s →  − s, x →  − x}, modulo an overall sign of the solution. Hence, from now on we focus on g+.

    Looking for the asymptotic solution valid near x = + 1, we introduce u≡2c(1 − x). (For a full discussion see [23], where this procedure defines an asymptotic solution3 Sinn,+1(x) for 01xO(cδ) with −1 < δ < 0.) Inserting the expression (3.1) into the ODE (2.1), we find that g+(u) satisfies the following equation:

    ug++(|m+s|+1u)g++12(sqm|m+s|1s)g++sA¯mcg+14c(u(ug++(|m+s|+|ms|+2u)g+)+12((|m+s|+1)(|ms|+1)(m+1)2+s2(|m+s|+|ms|+2+2s)u)g+)=0,3.2
    where primes denote derivatives with respect to u. Here, we have defined
    sA¯mc14c(s(s+1)m(m+1)+c22sqmc+sAmc),3.3
    introducing the parameter sqm, discussed in the Introduction, which is chosen so that
    sA¯mc=o(1),asc.3.4

    BRW left the parameter sqm undetermined for non-zero spin. Its value may be determined by requiring that the number of zeros of our asymptotic expansion4 is equal to the number of zeros of the spin-weighted spheroidal harmonics (which is given in eqn. (4.1) in [23]). Casals & Ottewill [23] determined this value to be5

    sqm=+1z0,if|s|m,3.5
    and
    sqm=2+1|s|m,if<|s|m,3.6
    where sm≡|m + s| + s and
    z0{0if+meven,1if+modd.3.7
    The value of z0 indicates whether sSmc(x) has a zero ‘near’ x = 0 for large c: it is z0 = 1 if sSmc(x) has a zero ‘near’ x = 0 and it is z0 = 0 if it does not. Regarding the values of sqm, we note, in particular, that: (i) sqm is an integer if s is an integer, whereas sqm is a half-integer if s is a half-integer; (ii) sqm = sqm.

    In the limit of infinitely large c, only the first line in equation (3.2) survives and the solution of the resulting ODE which is regular at u = 0 (x = 1) is

    1F1(spm+,|m+s|+1,u),3.8
    where we have introduced spm+12(sqm|m+s|s1) and 1F1 is the regular confluent hypergeometric function [19]. We note that spm+Z for 2sZ.

    Equation (3.2) then suggests that we express the function g+ as

    g+(u)=n=an1F1(spm+n,|m+s|+1,u),3.9
    where without loss of generality we assume that a0 = 1. The series coefficients an satisfy a three-term recurrence relation
    (2n+sqm|ms|+s+1)(2n+sqm|m+s|s+1)an+1+2(8cn(2n+sqm)2+2s2(m+1)28csA¯mc)an+(2n+sqm+|ms|+s1)(2n+sqm+|m+s|s1)an1=0.3.10
    These recurrence relations are obtained by inserting the series representation (3.9) into (3.2) and using the following recurrence relations satisfied by the hypergeometric functions [19]:
    u1F1(α,β,u)+(γu)1F1(α,β,u)=(γβ)1F1(α,β,u)+α1F1(α,β,u),3.11a
    u1F1(α,β,u)=α(1F1(α+1,β,u)1F1(α,β,u))3.11b
    andu1F1(α,β,u)=α1F1(α+1,β,u)(2αβ)1F1(α,β,u)+(αβ)1F1(α1,β,u),3.11c
    for constant α, β and γ. Because of the analyticity of the coefficients of the spin-weighted spheroidal differential equation in parameter c, we can assume a series expansion in powers of 1/c for sĀmc (see, for example, theorems 2.9 and 4.9 in ch. 8 in [27]),
    sA¯mck=1A¯kckasc.3.12
    Correspondingly, we now expand the coefficients for large c,
    ank=|n|an,kckasc,3.13
    where the structure of this coefficient expansion follows from the dominant first term in the coefficient of an in the recurrence relation. We give explicit expressions for the series coefficients an, k for n: − 3 → 3 and k:|n| → 3 in appendix A.

    Inserting the equations (3.12) and (3.13) into the recurrence relation (3.10) and requiring it to be satisfied order by order determines the expansion coefficients. Specifically, we find sAmc takes the form

    sAmc=c2+2sqmc12(sqm2m2+2s+1)+k=17Akck+O(1c8).3.14
    Other common definitions of the eigenvalue are easily computed from this using equations (2.2a) and (2.2b). Dropping the subscripts on sqm for compactness, the Ak's are given by
    A1=18(q3m2q+q2s2(q+m)),3.15
    A2=164(m4+6m2q2+2m25q410q21+4s2(m2+4mq+3q2+1)),3.16
    A3=1512(q(37+13m4+114q2+33q42m2(25+23q2))+4(13mm3+25q+9m2q+33mq2+23q3)s28(m+q)s4),3.17
    A4=11024(14+2m6239q2340q463q63m4(6+13q2)+10m2(3+23q2+10q4)+4(m49m3q+5m2(2+3q2)+mq(93+73q2)+5(3+23q2+10q4))s28(2+3m2+9mq+6q2)s4),3.18
    A5=18192(q(100953m6+5221q2+4139q4+527q6+5m4(127+93q2)m2(1591+3750q2+939q4))+2(14m545m4q+130m2q(3+q2)20m3(7+18q2)+2m(303+1820q2+685q4)+q(1591+3750q2+939q4))s280(7m+m3+11q+9m2q+18mq2+10q3)s4+16(m+q)s6),3.19
    A6=1131072(374751m886940q2205898q4101836q69387q8+12m6(85+167q2)6m4(939+5078q2+1855q4)+12m2(701+8657q2+9575q4+1547q6)+8(19m6+270m5qm4(191+309q2)4m3q(919+725q2)+m2(949+1482q2255q4)+2mq(8135+15310q2+3363q4)+3(701+8657q2+9575q4+1547q6))s2+16(467+17m4236m3q3438q21455q44mq(919+725q2)2m2(407+849q2))s4+128(4+7m2+19mq+12q2)s6)3.20
    andA7=12097152(q(822221+4093m8+5771940q2+7568470q4+2520820q6+175045q81540m6(65+43q2)+42m4(16371+29350q2+6375q4)4m2(353449+1345421q2+847819q4+95167q6))8(257m71253m6q+35m4q(379+169q2)35m5(181+381q2)+7m2q(6821+6070q2+3567q4)+7m3(5389+32190q2+12045q4)q(353449+1345421q2+847819q4+95167q6)m(112285+1057707q2+953715q4+136773q6))s2+112(31m5+363m4q6m3(107+131q2)10m2q(1135+749q2)q(10573+23530q2+5673q4)m(5389+32190q2+12045q4))s4+896(73m+19m3+105q+113m2q+189mq2+95q3)s6640(m+q)s8).3.21

    While for compactness we have given just the first 10 orders (to order 1/c7) for sAmc in equation (3.14), the process is easy to automate as it is for the an's. We have implemented code into the SpinWeightedSpheroidalHarmonicS package of the Black Hole Perturbation Toolkit to compute the high-frequency expansion of the eigenvalue (see appendix B). We also provide additional code to compute the an,k's and Ak's to arbitrary order.

    We note that BRW gave an expansion for sAmc to the first six orders (i.e. to order 1/c3) but, while their first three orders were as in equation (3.14), our values of A1, A2 and A3 correct the corresponding last three orders in eqn. (4.12) in BRW.6 We also note that for s = 0 our results for the Ak's agree with [28]. Finally, we note that it could also be interesting to consider the limit where both c and m with a fixed m/c ratio. This has been analysed in the s = 0 case [29] but has not, to the best of our knowledge, been analysed in the s≠0 case. We leave such an analysis for future work.

    4. Comparison with numerical calculation

    We validate our high-frequency asymptotic expansions by comparing them against a numerical calculation. For the numerical results, we use the SpinWeightedSpheroidalHarmonics Mathematica package, which is part of the Black Hole Perturbation Toolkit (http://bhptoolkit.org/). This package employs both a spectral method [30] and Leaver's method [31,32], combining them in a similar fashion to the method used by Falloon et al. [33] for the s = 0 case, to rapidly compute high-precision values for the spin-weighted spheroidal harmonics and their eigenvalues.

    For the eigenvalue calculation, we use the SpinWeightedSpheroidalEigenvalue to compare against equation (3.14). Note that equation (3.14) gives the expansion for sAmc, whereas the SpinWeightedSpheroidalEigenvalue command returns sλmc so we use equations (2.2a) and (2.2b) to convert between them. The results of the comparison are shown in figure 1, which shows that our high-frequency expansion agrees extremely well with the numerical results for large c. The comparison is further discussed in the figure's caption.

    Figure 1.

    Figure 1. Difference between the numerically computed eigenvalue, λnum (computed to 40 digits of accuracy), and its high-frequency expansion, λHF, for the case {s, ℓ, m} = {2, 2, 2}. We present the same results on both a log–log scale (a) and a log scale (b). The different curves are computed using successively higher orders in the high-frequency expansion. On the right of the graph, the top curve plots the numerical value of the eigenvalue. The subsequent lower curves are computed by subtracting the high-frequency series truncated at O(c1),O(c0),O(c1)O(c27), respectively. For c ≳1 including additional terms in the high-frequency series improves the comparison with the numerical results up to a point. After this, adding more terms does not improve the agreement. The shape of the curve beyond which adding terms not does improve the agreement is clearest on a log scale (b). This suggests that in addition to admitting a series expansion in c−1 there is an exponential term which the power law expansion cannot capture. Finally, as one would expect of a high-frequency expansion, for c≲1 adding terms acts to worsen the agreement with the numerical results. The eigenvalue is better approximated by small-frequency expansions around c = 0 in this region. (Online version in colour.)

    For the numerical calculation of the eigenfunctions, we use the SpinWeightedSpheroidal-Harmonics command, which computes sSmc ei in our notation. These harmonics are normalized such that

    02π0πsS1m1c(θ)eim1φsS2m2c(θ)eim2φsinθdθdφ=δ12δm1m2,4.1
    where δ is the Kronecker delta function. On the other hand, we do not know the normalization of the sS±mc in equation (3.1) with the g± given by (3.9) and its {s →  − s, x →  − x} counterpart. To make a meaningful comparison with the numerical calculation of the harmonics we numerically integrate sS+mc over x∈[1, 0] and sSmc over x∈[0, − 1] to obtain their normalization. With this information we can ensure that the numerical and asymptotic approximate solutions are normalized the same. Figure 2 presents an example of the excellent agreement we find between the numerical calculation and the high-frequency approximation of the eigenfunctions. The convergence of the expansion in equation (3.9) becomes slower the further the point is from x = + 1; similarly for g from x = − 1. This means that the combined asymptotic expansion of sS+mc and sSmc converges more slowly near x = 0, as reflected in figure 2. The convergence near x = 0 could be improved by incorporating the ‘outer’ solution of eqn. (3.26) in [23] in the manner done in §3c of that paper.
    Figure 2.

    Figure 2. Example of the high-frequency approximation to the spheroidal-harmonic eigenfunction for parameters {s, ℓ, m, c} = {2, 7, 3, 20}. (a) The (red) solid curve shows the numerically computed value of 2S7,3,20. The leading-order approximation is shown with the (blue) dotted curve. (b) Including higher order terms in the expansion improves the agreement with the numerical results. In this figure, the top curve is the difference between the leading-order expansion and the numerical data. Successive lower curves are the difference between the numerical expansion and successively higher order expansions. (Online version in colour.)

    For the eigenvalues and the eigenfunctions, the excellent agreement we observe between the high-frequency asymptotics and the numerical results give us confidence in both.

    Data accessibility

    Code to compute the expansions in this paper to arbitrary order has been integrated into the open source Black Hole Perturbation Toolkit (bhptoolkit.org)—see appendix B for more details.

    Author's contributions

    M.C. and A.C.O. calculated the high-order large-frequency expansions presented in this work. N.W. made detailed comparisons of these expansions with high-precision numerical calculations and integrated all three authors' codes into the Black Hole Perturbation Toolkit.

    Competing interests

    We have no competing interests.

    Funding

    M.C. acknowledges partial financial support by CNPq (Brazil), process no. 310200/2017-2. N.W. gratefully acknowledges support from a Royal Society—Science Foundation Ireland University Research Fellowship.

    Acknowledgments

    This work makes use of the Black Hole Perturbation Toolkit.

    Appendix A. Eigenfunction coefficients

    For completeness we here give the first three orders for the coefficients in the eigenfunction asymptotic expansion—see equations (3.9) and (3.13). The coefficients an, k may conveniently be expressed in terms of spm±12(sqm|m±s|s1), again dropping the subscripts on spm ±  and sqm for compactness:

    a1,1=14pp+,A 1a
    a1,1=14(pqs)(p+q+s),A 1b
    a2,2=132p(p1)p+(p+1),A 1c
    a1,2=18pp+(q1),A 1d
    a1,2=18(pqs)(p+q+s)(q+1),A 1e
    a2,2=132(pqs)(pqs1)(p+q+s)(p+q+s1),A 1f
    a3,3=1384p(p1)(p2)p+(p+1)(p+2),A 1g
    a2,3=164p(p1)p+(p+1)(2q3),A 1h
    a1,3=1128pp+((p(pqs+1)qs2)×(p+(p+q+s+1)q+s2)+2(q2)(5q2)2s2),A 1i
    a1,3=1128(pqs)(p+q+s)×((p(pqs+1)2)(p+(p+q+s+1)2)+2(q+2)(5q+2)2s2),A 1j
    a2,3=164(pqs)(pqs1)(p+q+s)(p+q+s1)(2q+3)A 1k
    anda3,3=1384(pqs)(pqs1)(pqs2)(p+q+s)×(p+q+s1)(p+q+s2).A 1l

    These series coefficients were not given in BRW or, to the best of our knowledge, anywhere else in the literature.

    As noted in the body of the paper, it is spm+Z for 2sZ, and it is straightforward to show that spm≥0 for s≥0 and spm+≥0 for s ≤ 0. The structure of the expanded recursion relations then shows that the functional expansion (3.9) terminates with finite lower limit −spm for s≥0, reflected in the vanishing of the coefficients an, k for n < − spm. Corresponding comments hold for s ≤ 0 with spm replaced by spm+. For s = 0, 0pm = 0pm+ and our observation agrees with eqn. (8.2.9) of Flammer [20].

    Appendix B. Implementation in the Black Hole Perturbation Toolkit

    We have implemented the calculation of the high-frequency expansion of the spin-weighted spheroidal eigenvalue and eigenfunction into the Mathematica SpinWeightedSpheroidal Harmonics package, which is part of the open-source Black Hole Perturbation Toolkit. This package allows for the numerical and (where possible) analytic calculation of the eigenvalue and eigenfunction of the spin-weighted spheroidal equation. It also allows the user to compute small-frequency expansions of these functions using the standard Mathematica Series[..] function. Following this work, we have implemented the high (real) frequency expansion of the eigenfunction as well.

    As an example, the high-frequency expansion of the eigenvalue, sλmc, for {s, ℓ, m} = {2, 7, 3} about c = ∞ can be computed via

    Series[SpinWeightedSpheroidalEigenvalue[2,7,3,c],{c,,5}]=10c3045c4052c298558c3176852c4226111532c5+O[c6].B 1
    The expansion can also be computed around c = − ∞; for example,
    Series[SpinWeightedSpheroidalEigenvalue[2,7,3,c],{c,,5}]=22c3051c5012c2130178c3246032c4328314932c5+O[c6].B 2

    We have also included an example notebook in the Toolkit which demonstrates the use of this function and provides code to calculate the an,k and Ak coefficients that appear in equations (3.13) and (3.14), respectively.

    Footnotes

    1 The symbol s really corresponds to the helicity of the spin field, although, in keeping with general convention, we refer to it as the spin.

    2 This corrects a typographical error in eqn. (1.3) [23].

    3 The function Sinn,+1(x) in [23] corresponds to the leading-order term in the expansion for g+ which we provide in this paper.

    4 Specifically, this number is obtained by adding the number of zeros of the leading-order expression equation (3.8) near x = + 1, the number of zeros of its counterpart near x = − 1 and the number of zeros z0 in equation (3.7) near x = 0.

    5 It can be checked that our expressions for sqm and z0 here are equivalent to—but simpler than—those given in eqns. (4.5) and (4.6) in [23].

    6 Eqn. (4.12) in BRW was merely reproduced in [23] and in [3] without previously checking it, and so containing the last three erroneous terms of the original BRW version.

    Published by the Royal Society. All rights reserved.