Towards a more robust algorithm for computing the Kerr quasinormal mode frequencies

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Towards a more robust algorithm for computing the Kerr quasinormal mode

frequencies

Sashwat Tanay

1, ∗

Department of Physics and Astronomy, The University of Mississippi, University, MS 38677, USA

Leaver’s method has been the standard for computing the quasinormal mode (QNM) frequencies

for a Kerr black hole (BH) for a few decades. We start with a spectral variant of Leaver’s method

introduced by Cook and Zalutskiy [Phys. Rev. D 90, 124021 (2014)], and propose improvements

in the form of computing the necessary derivatives analytically, rather than by numerical ﬁnite

diﬀerencing. We also incorporate this derivative information into

qnm

, a Python package which

ﬁnds the QNM frequencies via the spectral variant of Leaver’s method. We conﬁne ourselves to ﬁrst

derivatives only.

I. INTRODUCTION

When the two component black holes (BHs) of a binary

black hole (BBH) system merge together to form a single

BH, this BH oscillates in the so-called quasinormal modes

(QNMs). This ﬁnal state of the system is referred to

as the ringdown state, in contrast to the initial inspiral

state wherein the two component BHs of the system

slowly rotate around a common center. All this while, the

BBH system and and resulting single BH keep on emitting

gravitational waves (GWs). The construction of templates

for accurate parameter estimation using GWs requires

the modeling on the ringdown state as well. The QNM

frequencies of a Kerr BH are functions of its parameters

like mass and spin. All this makes the determination of

the QNM frequencies becomes a matter of importance.

The QNMs have a ﬁfty-plus years long history; see

Refs. [

] for reviews on the QNM literature. We will

conﬁne ourselves to the literature immediately useful for

this paper. Leaver’s method has become the standard

way to hunt for the Kerr QNM frequencies [

], although

modiﬁcations have been suggested over the years [

]. The

method starts with Teukolsky equations which describe

the perturbations to the Kerr geometry. One then feeds a

Frobenius series ansatz (representing these perturbations)

to the Teukolsky equations, both in the radial and angular

sectors. This leads to recurrence relations which then are

turned into two inﬁnite continued fraction (CF) equations.

Complex

(the QNM frequency) and a separation con-

stant

are the two roots of these two CF equations which

are to be numerically found. All this is done for a ﬁxed

dimensionless BH spin parameter a, where 0<a<1.

Cook and Zalutskiy in Ref. [

], presented a variant of

Leaver’s method where the CF approach was retained

while dealing with the radial sector but spectral decom-

position was introduced in the angular sector, thereby

recasting the problem as a CF equation and an eigenvalue

equation, both coupled together; also see Appendix A of

Ref. [

]. The advantage of the spectral approach is that it

leads to rapid convergence towards the solution. Recently,

∗stanay@go.olemiss.edu

qnm

, a Python package was released with Ref. [

] which

implemented the spectral approach of Ref. [4].

This paper aims to establish some theoretical founda-

tions on which improvements to the spectral variant of

Leaver’s method of Ref. [

] can be brought about. This is

mainly done by introducing ways to replace the computa-

tion of numerical derivatives (via ﬁnite diﬀerencing) with

those of analytical ones. This is so because in general,

analytical derivatives are more accurate and reliable than

their numerical counterparts. In this paper, we conﬁne

ourselves to computing only the ﬁrst derivatives, while

leaving second derivatives for future work. We also in-

corporate these improvements in the Python package

qnm.

The organization of the paper is as follows. The prob-

lem of ﬁnding the QNM frequencies of a Kerr BH is

introduced in Sec. II as a root-ﬁnding and an eigenvalue

problem, coupled together. In Sec. III, we highlight some

aspects of the present-day methods of computing these

frequencies where the inclusion of analytical derivative

information could be useful. Then in Sec. IV, we show

how to compute the necessary derivatives analytically,

while leaving some calculational details to Sec. V. We

devote some time to discussing the implementation of

the derivative information in the Python package

qnm

in Sec. VI, before summarizing in Sec. VII.

II. QNMS AS ROOTS OF A CONTINUED

FRACTION EQUATION

We will try to isolate the reader from some general

relativistic and other mathematical considerations while

casting the problem of ﬁnding the QNM frequencies of

a Kerr BH as a root-ﬁnding problem and an eigenvalue

problem, coupled together. The interested reader is re-

ferred to Ref. [

] and the references therein for details.

Our starting points will be Eqs. (44) and (56) of Ref. [

which we reproduce here (with

c≡aω

, where

is the di-

mensionless BH spin parameter and

is the sought-after

arXiv:2210.03657v1 [gr-qc] 7 Oct 2022

QNM frequency).

C(ω, sA`m, a, s;n, N)

≡βn−αn−1γn

βn−1−

αn−2γn−1

βn−2−. . . α0γ1

β0

−αnγn+1

βn+1−

αn+1γn+2

βn+2−. . . αN−1γN

βN+αNrN

= 0 (1)

M·~

C`m(c) = sA`m(c)~

C`m(c)(2)

A few clarifying comments are warranted at this point.

First of all, unlike Ref. [

], we will use

instead of

to de-

note the CF or any of its inversions.

(

a, sA`m, ω, s

;

n, N

)

denotes the nth inversion of the truncated CF

C(a, sA`m, ω, s;N)

≡ C(a, sA`m, ω, s; 0, N)

≡β0−α0γ1

β1−

α1γ2

β2−

α2γ3

β3−. . . αN−1γN

βN+αNrN

.(3)

Next,

αn

’s,

βn

’s and

γn

’s are functions of

sA`m

(

)(the

eigenvalue in Eq.

(2)

, and spin of the gravitational

ﬁeld

. For ﬁxed values of

m, s

and

c≡aω

, we have the

matrix

whose eigenvalues

sA`m

(

)’s are labeled by

Altogether, this means that for ﬁxed values of

s, `, m

(dis-

crete parameters) and

(continuous parameter), Eqs.

(1)

and

(2)

are two equations to be solved for the eigenvalue

sA`m

and root

. This point of view gives meaning to

expressions like

(

)or

sA`m

(

). Stated diﬀerently, for

ﬁxed values of

s, `, m

and

, Eq.

(1)

is a root-ﬁnding prob-

lem in

(if

sA`m

is ﬁxed), and Eq.

(2)

is an eigenvalue

problem for the eigenvalue sA`m (if ωis ﬁxed).

For ﬁxed and physically relevant values of (

s, `, m, a, ω

(which implies a ﬁxed

sA`m

), there are an inﬁnite number

of QNM frequencies

’s (along with the

sA`m

) that sat-

isfy Eq.

(1)

. These

’s are generally labeled with another

(overtone) index

= 0

· · ·

. In general, the

th over-

tone QNM frequency

is the most stable root of the

inversion of the CF

(

p, N

). We will sometimes denote

sA`m

as simply

, thus suppressing the dependence on

s, `

and

. This concludes our brief review of casting the

problem of ﬁnding QNM frequencies as that of a coupled

root-ﬁnding problem (the root being

) and an eigenvalue

problem (the eigenvalue being

), involving Eqs.

(1)

and

(2).

III. SPECTRAL VARIANT OF LEAVER’S

METHOD

We now brieﬂy highlight some aspects of computations

of QNM frequencies adopted in Refs. [

] which require

computation of certain derivatives. In the context of

Eq.

(2)

, we may write

(

)as

(

a, ω

)since

c≡aω

. With

this, we can display the functional dependencies involved

in Eq. (1) as1

C(ω, A(c(a, ω)) , a) = C(a, ω)=0.(4)

The main point is that Eq.

(4)

expresses a parameterized

(by

) root-ﬁnding problem in a single complex variable

ω.

A few key aspects of the standard approach to comput-

ing QNM frequencies are as follows

The derivative

∂C/∂ω

needed to ﬁnd the slope of

(

a, ω

)for the Newton-Raphson procedure, is com-

puted numerically via ﬁnite diﬀerencing.

After having found the solutions

ω0

(

)and

(

)

at BH spin

, we move to the next value of the

BH spin

+∆

. The initial guesses

ωguess

and

Aguess

for

+ ∆

is usually constructed by a

quadratic extrapolation using solutions at three pre-

vious values of

. Note that this quadratic extrapo-

lation is the numerical ﬁnite-diﬀerencing equivalent

of taking the second derivatives of

(

)and

(

)

with respect to a.

The point of this paper is to suggest analytical replace-

ments to the above-mentioned instances where ﬁnite-

diﬀerencing is employed to compute the derivatives in the

standard approaches. This is so because ﬁnite-diﬀerencing

is considered as an unreliable way to compute the deriva-

tives2.

There is an interesting aspect about supplying the ini-

tial guesses for (

ω, A

)to the root-ﬁnding routine that

warrants some discussion. If we use the previous solu-

tion for (

ω0, A0

)at

as a guess for the solution at

+ ∆

, then we found from numerical inspection

with a handful of cases that the maximum step size in

that we can take is ∆

a0∼

02. A larger step-size results

in the numerical procedure not being able to ﬁnd the root

or ending up on a diﬀerent root than what it is supposed

to. When the ﬁrst derivative information (

dω/da

and

dA/da

) is supplied to determine

ωguess

and

Aguess

for the

next value of

(via linear extrapolation), we found that

we can safely take a step size of ∆

a0∼

25. When the

(numerical) second derivative information is supplied (via

quadratic extrapolation), a step size of even ∆

a0∼

looks possible. The above ﬁndings indicate that including

the derivative information to generate

ωguess

and

Aguess

at the next value of

can allow us to take much larger

steps ∆

than would be possible otherwise. One should

note that although the derivative information lets us take

large step size in

, but accurate numerical computation

We have suppressed the display of (

n, N

)dependence of

in the

above equation because it is irrelevant to the ongoing discussion.

The reader is referred to Secs. 5.7 and 9.4 of Ref. [

] for a detailed

discussion on why ﬁnite-diﬀerencing is not the preferred way to

compute the derivatives in general and particularly so in the context

of the Newton-Raphson method, respectively.

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TowardsamorerobustalgorithmforcomputingtheKerrquasinormalmodefrequenciesSashwatTanay1,*1DepartmentofPhysicsandAstronomy,TheUniversityofMississippi,University,MS38677,USALeaver'smethodhasbeenthestandardforcomputingthequasinormalmode(QNM)frequenciesforaKerrblackhole(BH)forafewdecades.Westartwithaspect...

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Towards a more robust algorithm for computing the Kerr quasinormal mode frequencies

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