WaveExtractCPM

Seth Hopper, Barry Wardell

September 2, 2018

1 Introduction

The WaveExtractCPM thorn uses the Cunningham-Price-Moncrief formalism [1213] to calculate first order gauge invariant waveforms from a numerical spacetime. It relies on the basic assumption that the region of the spacetime where the extraction spheres are located is well-modelled as a linear perturbation to a Schwarzschild black hole. In addition to waveforms, the thorn can also compute other quantities such as mass, angular momentum and spin.

This thorn should not be used blindly, it will always return some waveform, however it is up to the user to determine whether this is the appropriate expected first order gauge invariant waveform.

2 Physical System

Consider a known, background solution to the Einstein equations gμν. A first-order perturbation to that metric, pμν yields

gμν = gμν + pμν|pμν||gμν|. (1)

We denote covariant derivatives with respect to the background metric gμν with μ or |μ. Standard textbook analysis yields the first-order vacuum field equations in an unchosen gauge (defining |αα and p pαα)

pμν p|μν + pα ν|μα + pμ|ναα = 0. (2)

Or, in Lorenz gauge (p̄μν|ν = 0)

p̄μν + 2Rαμβνp̄αβ = 0, (3)

where an overbear indicates trace-reversal: p̄μν = pμν 1 2gμνp.

2.1 The 2 ×𝒮2 decomposition in a spherically symmetric spacetime

Now we specialize to a spherically symmetric background. In this section we introduce formalism from [?] for doing a harmonic decomposition of scalar, vectors, and tensors in such a spacetime. We specialize to Schwarzschild spacetime with Schwarzschild coordinates and decompose its metric gμν on two submanifolds, yielding gab and gAB = r2ΩAB. Here a,b, {0,1} and A,B, {𝜃,ϕ}. The xa coordinates span the “(t,r) plane” while xA are the standard two-sphere polar and azimuthal coordinates. In matrix form we have

gμν g00g01 0 0 g10 g11 0 0 0 0 r2Ω𝜃𝜃r2Ω𝜃ϕ 0 0 r2Ωϕ𝜃r2Ωϕϕ = g00g01 0 0 g10 g11 0 0 0 0 r2 0 0 0 0 r2 sin2𝜃 . (4)

Specifically, we are interested in an expression of the Schwarzschild metric that is covariant under two-dimensional transformations: xa xa. The line element can be written as

ds2 = g abdxadxb + r2Ω ABdxAdxB. (5)

In Schwarzschild coordinates, the submanifold 2 has a metric and inverse

gab f 0 0 1f ,gab 1f0 0 f ,f 12M r , (6)

where M is the mass of the system. The unit two-sphere has a metric and inverse

ΩAB 1 0 0sin2𝜃 ,ΩAB 1 0 01sin2𝜃 . (7)

Note that in general (off the unit two-sphere) we use the metric gAB r2ΩAB.

2.2 Even parity

Of the ten MP amplitudes, seven are in the even-parity sector. Using the decomposition of Martel and Poisson [?], they are

pab xμ = ,mhabmY m, p aB xμ = ,mjamY Bm, p AB xμ = r2 ,mKmΩ ABY m + GmY ABm. (8)

The even-parity scalar (Y m), vector (Y Am), and tensor (Y ABm and ΩABY m) spherical harmonics are defined in [?]. Note that Y ABm is the trace-free tensor spherical harmonic, which differs from what Regge and Wheeler used in their original work [?].

If we have the metric perturbation, we can compute the amplitudes by using the completeness of the spherical harmonics. First in the 2 sector,

habm =pabȲmdΩ (9)
jam = r2 ( + 1)paBȲmBdΩ. (10)

Km = 1 2pABgABȲmdΩ. (11) Lastly pABȲmABdΩ = r2 ,mKmΩ ABY m + GmY ABmȲ mABdΩ (12) = r2 ,mGmY ABmȲ mABdΩ (13) = r2 ,mGm 1 2r4( 1) + 1( + 2)δδmm, (14)

and thus

Gm = 2r2( 2)! ( + 2)!pABȲmABdΩ (15)

We can now expand out the sums. See Appendix ?? for the components of the vector and tensor spherical harmonics. In going from the expressions above to these, note the inverse metric gAB, which provides factors of 1r2 and 1sin2𝜃.

httm =pttȲmdΩ,htrm =ptrȲmdΩ,hrrm =prrȲmdΩ, (16)

jtm = r2 ( + 1)pt𝜃Ȳm𝜃 + p tϕȲmϕ dΩ = 1 ( + 1)pt𝜃Ȳ𝜃m + 1 sin2𝜃ptϕȲϕm dΩ (17) jrm = r2 ( + 1)pr𝜃Ȳm𝜃 + p rϕȲmϕ dΩ = 1 ( + 1)pr𝜃Ȳ𝜃m + 1 sin2𝜃prϕȲϕm dΩ (18) Km = 1 2p𝜃𝜃g𝜃𝜃 + p ϕϕgϕϕ ȲmdΩ = 1 2r2p𝜃𝜃 + 1 sin2𝜃pϕϕ ȲmdΩ (19)

Gm = 2r2( 2)! ( + 2)!p𝜃𝜃Ȳm𝜃𝜃 + 2p 𝜃ϕȲm𝜃ϕ + p ϕϕȲmϕϕ dΩ (20) = 2 r2 ( 2)! ( + 2)!p𝜃𝜃Ȳ𝜃𝜃m + 2 sin2𝜃p𝜃ϕȲ𝜃ϕm + 1 sin4𝜃pϕϕȲϕϕm dΩ (21)

For the remainder of this section, we drop and m indices for the sake of brevity.

In Schwarzschild coordinates, the amplitudes defined here are related to Regge and Wheeler’s original quantities. In the “t,r sector,” htt = fH0, htr = H1, and hrr = H2f. For the off-diagonal elements, jt = h0 and jr = h1. Finally, on the two-sphere Ghere = GRW, while Khere = KRW ( + 1)G2.

In the even-parity sector there are four gauge-invariant fields, formed from linear combinations of the metric perturbation amplitudes and their first derivatives [?]

h̃tt = htt 2tjt + 2Mf r2 jr + r2 t2G Mf rG h̃tr = htr rjt tjr + 2M fr2 jt + r2 trG + r 3M f tG h̃rr = hrr 2rjr 2M fr2 jr + r2 r2G + 2r 3M f rG K̃ = K 2f r jr + rfrG + (λ + 1)G. (22)

Note that in RW gauge G = jt = jr = 0. Examining the gauge invariant quantities, we find

h̃tt = htt,h̃tr = htr,h̃rr = hrr,K̃ = K. (23)

Written in terms of the gauge-invariant fields, the seven vacuum field equations (for our purposes, we are deep in the wave zone and are not concerned with the isolated source) are

r2K̃ 3r 5M r2f rK̃ + f rrh̃rr + λ + 2r + 2M r3 h̃rr + λ r2fK̃ = 0, (24) trK̃ + r 3M r2f tK̃ f rth̃rr λ + 1 r2 h̃tr = 0, (25) t2K̃ + (r M)f r2 rK̃ + 2f r th̃tr f rrh̃tt + (λ + 1)r + 2M r3 h̃tt f2 r2 h̃rr λf r2 K̃ = 0, (26) th̃rr rh̃tr + 1 ftK̃ 2M r2fh̃tr = 0, (27) th̃tr + rh̃tt frK̃ r M r2f h̃tt + (r M)f r2 h̃rr = 0, (28) t2h̃ rr + 2trh̃tr r2h̃ tt 1 ft2K̃ + f r2K̃ + 2(r M) r2f th̃tr r 3M r2f rh̃tt (r M)f r2 rh̃rr + 2(r M) r2 rK̃ + (λ + 1)r2 2(λ + 2)Mr + 2M2 r4f2 h̃tt (λ + 1)r2 2λMr 2M2 r4 h̃rr = 0, (29) 1 fh̃tt fh̃rr = 0, (30)

where we have introduced

Λ(r) λ + 3M r ,λ + 2 1 2 . (31)

We use the gauge invariant Zerilli-Moncrief master function (see [??], modifying the approach of [?]), which is

Ψeven(t,r) r λ + 1 K̃ + f Λ fh̃rr rrK̃, (32)

in Schwarzschild coordinates. Plugging in the gauge invariant fields from above, one finds

Ψeven(t,r) = rG 2f Λ jr + r λ + 1 K + f Λ fhrr rrK (33)

Conveniently, all the second-order derivatives cancel. We are also interested in the time derivative of the master function, which is used for computing energy and angular momentum fluxes. We differentiate Eq. (32) with respect to time, using Eq. (25) to remove the trK̃ terms. Then, substituting in the gauge invariant fields we find

tΨeven(t,r) = rtG + 1 Λ fhtr 2M r2 jt + frjt + rtK ftjr . (34)

Again, simplification happens and we are left with at most first-order derivatives of the MP amplitudes.

2.3 Odd parity

The remaining three MP amplitudes belong to the odd-parity sector,

pab xμ = 0,p aB xμ = ,mhamX Bm,p AB xμ = ,mh2mX ABm. (35)

The vector (XBm) and tensor (XABm) spherical harmonics are those defined in [?]. Note that the tensor spherical harmonics differ from those used by Regge and Wheeler by a minus sign.

If we have the metric perturbation, we can compute the amplitudes by using the completeness of the spherical harmonics

ham = r2 ( + 1)paBX̄mBdΩ. (36)
h2m = 2r4( 2)! ( + 2)!pABX̄mABdΩ (37)

We can now expand out the sums. See Appendix ?? for the components of the vector and tensor spherical harmonics. In going from the expressions above to these, note the inverse metric gAB, which provides factors of 1r2 and 1sin2𝜃.

htm = r2 ( + 1)pt𝜃X̄m𝜃 + p tϕX̄mϕ dΩ = 1 ( + 1) 1 sin𝜃 pt𝜃Ȳ,ϕm + p tϕȲ,𝜃m dΩ (38) hrm = r2 ( + 1)pr𝜃X̄m𝜃 + p rϕX̄mϕ dΩ = 1 ( + 1) 1 sin𝜃 pr𝜃Ȳ,ϕm + p rϕȲ,𝜃m dΩ (39)

h2m = 2r4( 2)! ( + 2)!p𝜃𝜃X̄m𝜃𝜃 + 2p 𝜃ϕX̄m𝜃ϕ + p ϕϕX̄mϕϕ dΩ (40) = 2( 2)! ( + 2)! 1 sin𝜃p𝜃𝜃 cos𝜃 sin𝜃Ȳ,ϕm Ȳ ,𝜃ϕm p 𝜃ϕ 1 sin2𝜃Ȳ,ϕϕm + cos𝜃 sin𝜃Ȳ,𝜃m Ȳ ,𝜃𝜃m + pϕϕ 1 sin2𝜃Ȳ,ϕ𝜃m cos𝜃 sin3𝜃Ȳ,ϕm dΩ (41)

For the remainder of this section, we again drop and m indices.

These MP amplitudes are related to Regge and Wheeler’s quantities through ht = h0, hr = h1, and h2here = h2RW.

In the odd-parity sector there are two gauge-invariant fields, formed from linear combinations of the metric perturbation amplitudes and their first derivatives [?]

h̃t ht 1 2 h2 t ,h̃r hr 1 2 h2 r + h2 r . (42)

Note that in Regge-Wheeler gauge h2 = 0 and then

h̃t ht,h̃r hr. (43)

Written in terms of the gauge-invariant fields, the three vacuum field equations (for our purposes, we are deep in the wave zone and are not concerned with the isolated source) are

trh̃r + r2h̃ t 2 rth̃r 2(λ + 1)r 4M r3f h̃t = 0, (44) t2h̃ r trh̃t + 2 rth̃t + 2λf r2 h̃r = 0, (45) 1 fth̃t + frh̃r + 2M r2 h̃r = 0, (46)

where, recall that we have defined

λ + 2 1 2 . (47)

In the odd-parity sector, we use the gauge-invariant Cunningham-Price-Moncrief master function [?], which in Schwarzschild coordinates is

Ψodd(t,r) r λ rh̃t th̃r 2 rh̃t . (48)

Plugging in the gauge invariant fields from above, we find that all the h2 terms cancel and the result is simply

Ψodd(t,r) r λ rht thr 2 rht . (49)

We are also interested in the time derivative of the master function, which is used for computing energy and angular momentum fluxes. We differentiate Eq. (48) with respect to time, using Eq. (45) to remove the trh̃t and t2h̃r terms. Then, substituting in the gauge invariant fields we find

tΨodd = f r 2hr + 2 rh2 rh2 . (50)

In RW gauge where h2 vanishes it becomes clear that the CPM function Ψodd is just twice the time integral of the original RW function ΨRW = fhrr.

2.4 Wave Forms

The energy and angular momentum fluxes, for each ,m mode, can be written as [?]

Ėm = 1 64π ( + 2)! ( 2)! Ψ̇evenm 2 + Ψ̇ oddm 2 , (51) L̇m = im 64π ( + 2)! ( 2)! Ψ̇evenmΨ m± + Ψ̇ oddmΨ m±. (52)

Here, an asterisk signifies complex conjugation. Assume a spacetime gαβ can be written as a Schwarzschild background gαβ(0) with perturbation hαβ,

gαβ = gαβ(0) + h αβ. (53)

In spherical coordinates, (t,r,𝜃,ϕ), the background metric is given by

g(0) = f 0 0 0 0 f1 0 0 0 0 r2 0 0 0 0 r2 sin2𝜃 ,f(r) = 12M r . (54)

The 3-metric perturbations γij can be decomposed using tensor spherical harmonics to obtain a set of metric perturbation amplitudes γijlm(t,r), where

γij(t,r,𝜃,ϕ) = l=0 m=llγ ijlm(t,r)

and

γij(t,r,𝜃,ϕ) = k=06p k(t,r)V k(𝜃,ϕ),

with V k being a basis for tensors on a 2-sphere in 3-D Euclidean space.

Working with the Regge-Wheeler basis (see Appendix B) the 3-metric is then expanded in terms of the (six) standard Regge-Wheeler functions c1×lm, c2×lm, h1+lm, H2+lm, K+lm, G+lm [1916]. Each of these functions is either odd (×) or even (+) parity. The decomposition is then written

γijlm = c 1×lm(ê 1)ijlm + c 2×lm(ê 2)ijlm + h1+lm(f̂ 1)ijlm + A2H 2+lm(f̂ 2)ijlm + R2K+lm(f̂ 3)ijlm + R2G+lm(f̂ 4)ijlm, (55)

which we can write in an expanded form as

γrrlm = A2H 2+lmY lm, (56) γr𝜃lm = c 1×lm 1 sin𝜃Y lm,ϕ + h1+lmY lm,𝜃, (57) γrϕlm = c 1×lm sin𝜃Y lm,𝜃 + h1+lmY lm,ϕ, (58) γ𝜃𝜃lm = c 2×lm 1 sin𝜃(Y lm,𝜃ϕ cot𝜃Y lm,ϕ) + R2K+lmY lm + R2G+lmY lm,𝜃𝜃, (59) γ𝜃ϕlm = c 2×lm sin𝜃1 2 Y lm,𝜃𝜃 cot𝜃Y lm,𝜃 1 sin2𝜃Y lm + R2G+lm(Y lm,𝜃ϕ cot𝜃Y lm,ϕ), (60) γϕϕlm = sin𝜃c 2×lm(Y lm,𝜃ϕ cot𝜃Y lm,ϕ) + R2K+lm sin2𝜃Y lm + R2G+lm(Y lm,ϕϕ + sin𝜃cos𝜃Y lm,𝜃). (61)

A similar decomposition allows the four gauge components of the 4-metric to be written in terms of three even-parity variables H0, H1 and h0, and one odd-parity variable c0,

gttlm = N2H 0+lmY lm, (62) gtrlm = H 1+lmY lm, (63) gt𝜃lm = h 0+lmY lm,𝜃 c0×lm 1 sin𝜃Y lm,ϕ, (64) gtϕlm = h 0+lmY lm,ϕ + c0×lm sin𝜃Y lm,𝜃. (65)

Also, from gtt = α2 + βiβi,we have

αlm = 1 2NH0+lmY lm. (66)

It is useful to also write this with the perturbation split into even and odd parity parts:

gαβ = gαβ(0) + l,mhαβlm,odd + l,mhαβlm,even

where (dropping some superscripts)

hαβodd = 00 c0 1 sin 𝜃Y lm,ϕ c0 sin𝜃Y lm,𝜃 . 0 c1 1 sin 𝜃Y lm,ϕ c1 sin𝜃Y lm,𝜃 . .c2 1 sin 𝜃(Y lm,𝜃ϕ cot𝜃Y lm,ϕ)c21 2 1 sin 𝜃Y lm,ϕϕ + cos𝜃Y lm,𝜃 sin𝜃Y lm,𝜃𝜃 . . . c2(sin𝜃Y lm,𝜃ϕ + cos𝜃Y lm,ϕ) hαβeven = N2H0Y lm H1Y lm h0Y lm,𝜃 h0Y lm,ϕ . A2H2Y lm h1Y lm,𝜃 h1Y lm,ϕ . . R2KY lm + r2GY lm,𝜃𝜃 R2(Y lm,𝜃ϕ cot𝜃Y lm,ϕ) . . . R2Ksin2𝜃Y lm + R2G(Y lm,ϕϕ + sin𝜃cos𝜃Y lm,𝜃)

Now, for such a Schwarzschild background we can define two (and only two) unconstrained gauge invariant quantities Qlm× = Qlm×(c1×lm,c2×lm) and Qlm+ = Qlm+(K+lm,G+lm,H2+lm,h1+lm), which from [3] are

Qlm× = 2(l + 2)! (l 2)! c1×lm + 1 2 rc2×lm 2 rc2×lm S r (67) Qlm+ = 1 Λ 2(l 1)(l + 2) l(l + 1) (4rS2k 2 + l(l + 1)rk1) (68) 1 Λ 2(l 1)(l + 2) l(l + 1) l(l + 1)S(r2 rG+lm 2h 1+lm) + 2rS(H 2+lm r rK+lm) + ΛrK+lm (69)

where

k1 = K+lm + S r (r2 rG+lm 2h 1+lm) (70) k2 = 1 2S H2+lm r rk1 1 M rSk1 + S12 r(r2S12 rG+lm 2S12h 1+lm) (71) 1 2S H2 rK,r r 3M r 2MK (72)

NOTE: These quantities compare with those in Moncrief [16] by

Moncriefs odd parity Q: Qlm× = 2(l + 2)! (l 2)! Q Moncriefs even parity Q: Qlm+ = 2(l 1)(l + 2) l(l + 1) Q

Note that these quantities only depend on the purely spatial Regge-Wheeler functions, and not the gauge parts. (In the Regge-Wheeler and Zerilli gauges, these are just respectively (up to a rescaling) the Regge-Wheeler and Zerilli functions). These quantities satisfy the wave equations

(t2 r2)Q lm× + S l(l + 1) r2 6M r3 Qlm× = 0 (t2 r2)Q lm+ + S 1 Λ2 72M3 r5 12M r3 (l 1)(l + 2) 1 3M r + l(l 1)(l + 1)(l + 2) r2Λ Qlm+ = 0

where

Λ = (l 1)(l + 2) + 6Mr r = r + 2Mln(r2M 1)

3 Numerical Implementation

The implementation assumes that the numerical solution, on a Cartesian grid, is approximately Schwarzshild on the spheres of constant r = (x2 + y2 + z2) where the waveforms are extracted. The general procedure is then:

3.1 Project onto Spheres of Constant Radius

This is performed by interpolating the metric components, and if needed the conformal factor, onto the spheres. Although 2-spheres are hardcoded, the source code could easily be changed here to project onto e.g. 2-ellipsoids.

3.2 Calculate Radial Transformation

The areal coordinate r̂ of each sphere is calculated by

r̂ = r̂(r) = 1 4πγ𝜃𝜃 γϕϕd𝜃dϕ12 (73)

from which

dr̂ dη = 1 16πr̂γ𝜃𝜃,ηγϕϕ + γ𝜃𝜃γϕϕ,η γ𝜃𝜃 γϕϕ d𝜃dϕ (74)

Note that this is not the only way to combine metric components to get the areal radius, but this one was used because it gave better values for extracting close to the event horizon for perturbations of black holes.

3.3 Calculate S factor and Mass Estimate

S(r̂) = r̂ r 2γrrd𝜃dϕ (75)
M(r̂) = r̂1 S 2 (76)

3.4 Calculate Regge-Wheeler Variables

c1×lm = 1 l(l + 1) γr̂ϕY lm,𝜃 γr̂𝜃Y lm,ϕ sin𝜃 dΩ c2×lm = 2 l(l + 1)(l 1)(l + 2) 1 sin2𝜃γ𝜃𝜃 + 1 sin4𝜃γϕϕ (sin𝜃Y lm,𝜃ϕ cos𝜃Y lm,ϕ) + 1 sin𝜃γ𝜃ϕ(Y lm,𝜃𝜃 cot𝜃Y lm,𝜃 1 sin2𝜃Y lm,ϕϕ)dΩ h1+lm = 1 l(l + 1) γr̂𝜃Y lm,𝜃 + 1 sin2𝜃γr̂ϕY lm,ϕdΩ H2+lm = Sγr̂r̂Y lmdΩ K+lm = 1 2r̂2 γ𝜃𝜃 + 1 sin2𝜃γϕϕ Y lmdΩ + 1 2r̂2(l 1)(l + 2)γ𝜃𝜃 γϕϕ sin2𝜃 Y lm,𝜃𝜃 cot𝜃Y lm,𝜃 1 sin2𝜃Y lm,ϕϕ + 4 sin2𝜃γ𝜃ϕ(Y lm,𝜃ϕ cot𝜃Y lm,ϕ)dΩ G+lm = 1 r̂2l(l + 1)(l 1)(l + 2) γ𝜃𝜃 γϕϕ sin2𝜃 Y lm,𝜃𝜃 cot𝜃Y lm,𝜃 1 sin2𝜃Y lm,ϕϕ + 4 sin2𝜃γ𝜃ϕ(Y lm,𝜃ϕ cot𝜃Y lm,ϕ)dΩ

where

γr̂r̂ = r r̂ r r̂γrr (77) γr̂𝜃 = r r̂γr𝜃 (78) γr̂ϕ = r r̂γrϕ (79)

3.5 Calculate Gauge Invariant Quantities

Qlm× = 2(l + 2)! (l 2)! c1×lm + 1 2 r̂c2×lm 2 r̂c2×lm S r̂ (80) Qlm+ = 1 (l 1)(l + 2) + 6Mr̂ 2(l 1)(l + 2) l(l + 1) (4r̂S2k 2 + l(l + 1)r̂k1) (81)

where

k1 = K+lm + S r̂(r̂2 r̂G+lm 2h 1+lm) (82) k2 = 1 2S[H2+lm r̂ r̂k1 (1 M r̂S)k1 + S12 r̂(r̂2S12 r̂G+lm 2S12h 1+lm (83)

4 Using This Thorn

Use this thorn very carefully. Check the validity of the waveforms by running tests with different resolutions, different outer boundary conditions, etc to check that the waveforms are consistent.

4.1 Output Files

Although Extract is really an ANALYSIS thorn, at the moment it is scheduled at POSTSTEP, with the iterations at which output is performed determined by the parameter itout. Output files from Extract are always placed in the main output directory defined by CactusBase/IOUtil.

Output files are generated for each detector (2-sphere) used, and these detectors are identified in the name of each output file by R1, R2, ….

The extension denotes whether coordinate time (ṫl) or proper time (u̇l) is used for the first column.

5 History

This document (and the WaveExtractCPM thorn itself) are based on the Extract thorn writen by Gabrielle Allen. Much of the source code for Extract comes from a code written outside of Cactus for extracting waveforms from data generated by the NCSA G-Code for comparison with linear evolutions of waveforms extracted from the Cauchy initial data. This work was carried out in collaboration with Karen Camarda and Ed Seidel.

A Spherical harmonics

Now, consider spherical harmonics, starting with the scalar case. They are eigenfunctions, satisfying the equation

1 sin𝜃𝜃 sin𝜃 𝜃 + 1 sin2𝜃ϕ2 + + 1Y m(𝜃,ϕ) = 0. (84)

Acting on a test scalar function f we have

ΩABD ADBf = ΩAB AB ΓC ABC f (85) = 𝜃2f ΓC 𝜃𝜃Cf + 1 sin2𝜃ϕ2f 1 sin2𝜃ΓC ϕϕCf (86) = 1 sin𝜃𝜃 sin𝜃 𝜃 + 1 sin2𝜃ϕ2 f. (87)

So, we can write Eq. (84) in the compact form

ΩABD ADB + + 1Y m(𝜃,ϕ) = 0. (88)

The solution to this equation with standard normalization [?] is

Y m = 2 + 1 4π ( m)! ( + m)!Pm(cos𝜃)eimϕ (89)

where Pm are the associated Legendre functions. These are an orthonormal set of functions,

Y m(𝜃,ϕ)Ȳm(𝜃,ϕ)dΩ = δδmm. (90)

Here dΩ = sin𝜃d𝜃dϕ and the overbar represents complex conjugation.

We can use the covariant derivative DA to take derivatives of this scalar function to define vector and tensor spherical harmonics. There are even- and odd-parity vector spherical harmonics. We define the even ones as the covariant derivative of the scalar harmonics:

Y Am 𝜃,ϕ D AY m 𝜃,ϕ 𝜃Y m ϕY m . (91)

In order to create the odd-parity vectorial harmonics we need to define the Levi-Civita tensor on the two-sphere:

𝜀AB 0 sin𝜃 sin 𝜃 0 . (92)

Using this, the odd-parity harmonics are

XAm 𝜃,ϕ 𝜀 ABD BY m 𝜃,ϕ = ΩCB𝜀 ACY Bm 𝜃,ϕ. (93)

Switching to matrices we can calculate the components:

XAm 𝜃,ϕ 0 sin𝜃 sin 𝜃 0 1 0 01sin2𝜃 𝜃Y m ϕY m (94) ϕY msin𝜃 sin𝜃𝜃Y m . (95)

The tensor spherical harmonics also are either even- and odd-parity. There are two even-parity ones,

Y mΩAB Y m 0 0 sin2𝜃Y m (96)

and the more complicated

Y ABm D ADB + 1 2 + 1ΩAB Y m (97) = ABY m ΓC ABCY m + 1 2 + 1ΩABY m. (98)

We’ve already calculated the connection coefficients, so evaluating this is straightforward, leaving us with the components

Y ABm 𝜃2 + (+1) 2 Y m 𝜃ϕ cot𝜃ϕ Y m 𝜃ϕ cot𝜃ϕ Y m ϕ2 + sin𝜃cos𝜃𝜃 + (+1) 2 sin2𝜃Y m . (99)

The odd-parity tensor harmonics are

XABm = 1 2 𝜀ACD B + 𝜀BCD A DCY m (100) = 1 2 𝜀A𝜃D BD𝜃 + 𝜀AϕD BDϕ + 𝜀BϕD AD𝜃 + 𝜀BϕD ADϕ Y m. (101)

In matrix form we have

XABm 1 sin 𝜃𝜃ϕ + cos 𝜃 sin 2𝜃ϕ Y m 1 2 ϕ2 sin 𝜃 + cos𝜃𝜃 sin𝜃𝜃2 Y m 1 2 ϕ2 sin 𝜃 + cos𝜃𝜃 sin𝜃𝜃2 Y m sin𝜃ϕ𝜃 cos𝜃ϕ Y m . (102)

Now we look at some identities involving these spherical harmonics. We have already seen in Eq. (90) that the scalar spherical harmonics are orthonormal. Now consider

Y mAȲ Am dΩ = 1 r2ΩABD AY mDBȲmdΩ (103) = 1 r2𝜃Y m𝜃Ȳm + 1 sin2𝜃ϕY mϕȲmsin𝜃d𝜃dϕ. (104)

We integrate by parts (note that surface terms vanish by periodicity as we integrate of the full 4π steradians) and find

Y mAȲ Am dΩ = 1 r2 1 sin𝜃𝜃 sin𝜃𝜃Y m Ȳm 1 sin2𝜃ϕ2Y mȲmsin𝜃d𝜃dϕ (105) = 1 r2( + 1)δδmm. (106)

The odd-parity equivalent is

XmAX̄ Am dΩ =𝜀A CY mC𝜀 ABȲ Bm dΩ. (107)

This 2D contraction of the Levi-Civita tensor gives the negative of the Kronecker delta, and therefore

XmAX̄ Am dΩ =δB CY mCȲ Bm dΩ =Y mAȲ Am dΩ = 1 r2( + 1)δδmm. (108)

Now, when we contract the even and odd-parity vector harmonics we get

Y mAX̄ Am dΩ = DAY m𝜀ABD BȲm dΩ. (109)

By parts integration we have

Y mAX̄ Am dΩ =𝜀ABD ADBY mȲm dΩ = 0 =ȲmAX Am dΩ, (110)

because of the derivatives commute while the Levi-Civita tensor is antisymmetric. Consider now ΩABDADBY Cm = ΩABDADBDCY m. The two closest covariant derivatives commute, but we have to use the rule

DA,DB V C = RC DABV D D A,DB V C = RCABDV D (111)

to commute the outer two, and therefore

ΩABD ADBY Cm = ΩABD ADCDBY m (112) = ΩAB D CDADB + RBACDD D Y m. (113)

Using the differential equation for the scalar harmonics, we get

ΩABD ADBY Cm = ( + 1)Y Cm + ΩAB 1 r2ΩDE Ω BAΩEC ΩBCΩEA Y Dm (114) = 1 ( + 1)Y Cm. (115)

Additionally, we have

ΩABD ADBXCm = 𝜀 CDΩABD ADBY Dm = 1 ( + 1)X Cm. (116)

Taking the divergence Y mA and XmA gives

DAY mA = 1 r2ΩABD ADBY m = ( + 1) r2 Y m (117) DAXmA = 1 r2ΩABD A𝜀BCD CY m = 1 r2𝜀ACD ADCY m = 0. (118)

Now we consider contractions of the tensor harmonics. First of all, because they are each trace free, we have

ΩABY ABm = ΩABX ABm = 0. (119)

This is clear from inspecting the matrix forms of these harmonics above. Note that this implies that both Y ABm and XABm are orthogonal to ΩABY m. Now, we consider

Y mABȲ ABm dΩ =gACgBD D CDD + + 1 2 ΩDC Y m DADB + + 1 2 ΩAB Ȳm dΩ (120) = 1 r4ΩACΩBDD ADCDDY mDBȲm 1 2 + 1 + 1Y mȲm dΩ (121)

So, in order to evaluate this we need the harmonic operator (ΩABDADB) acting on Y C, which we calculated above. Using it and the completeness of the scalar harmonics gives

Y mABȲ ABm dΩ = 1 r4ΩBD1 ( + 1)D DY mDBȲm dΩ 1 2r42 + 12δ δmm (122) = 1 2r4( 1) + 1( + 2)δδmm. (123)

A similar, though slightly longer calculation for the odd-parity case gives

XmABX̄ ABm dΩ = 1 2r4( 1) + 1( + 2)δδmm. (124)

For the divergence of the tensor harmonics we first consider the even-parity case,

DBY ABm = 1 r2ΩBCD C DADBY m + 1 2 + 1ΩABY m , (125) = 1 r2ΩBC D ADCDBY m + R BCADD DY m + 1 2r2 + 1DAY m, (126) = 1 r2 1 1 2 + 1Y Am. (127)

For the odd-parity harmonics we have

DBX ABm = 1 2 1 r2ΩBDD D DBXAm + D AXBm , (128) = 1 2r21 ( + 1)XAm + 1 2r2ΩBD D ADDXBm + R BCDAXmC . (129)

The divergence of XBm vanishes, so we are left with

DBX ABm = 1 2r21 ( + 1)XAm + 1 2r2ΩBDr2 Ω BDΩCA ΩBAΩCD XmC, (130) = 1 r2 1 1 2 + 1XAm. (131)

B Regge-Wheeler Harmonics

(ê1)lm = 0 1 sin 𝜃Y lm,ϕ sin𝜃Y lm,𝜃 . 0 0 . 0 0 (ê2)lm = 0 0 0 0 1 sin 𝜃(Y lm,𝜃ϕ cot𝜃Y lm,ϕ) . 0 sin 𝜃 2 [Y lm,𝜃𝜃 cot𝜃Y lm,𝜃 1 sin 2𝜃Y lm,ϕϕ] sin𝜃[Y lm,𝜃ϕ cot𝜃Y lm,ϕ] (f̂1)lm = 0Y lm,𝜃Y lm,ϕ . 0 0 . 0 0 (f̂2)lm = Y lm00 0 00 0 0 0 (f̂3)lm = 0 0 0 0 Y lm 0 0 0 sin2𝜃Y lm (f̂4)lm = 0 0 0 0 Y lm,𝜃𝜃 . 0Y lm,𝜃ϕ cot𝜃Y lm,ϕY lm,ϕϕ + sin𝜃cos𝜃Y lm,𝜃

C Transformation Between Cartesian and Spherical Coordinates

First, the transformations between metric components in (x,y,z) and (r,𝜃,ϕ) coordinates. Here, ρ = x2 + y2 = rsin𝜃,

x r = sin𝜃cosϕ = x r y r = sin𝜃sinϕ = y r z r = cos𝜃 = z r x 𝜃 = rcos𝜃cosϕ = xz ρ y 𝜃 = rcos𝜃sinϕ = yz ρ z 𝜃 = rsin𝜃 = ρ x ϕ = rsin𝜃sinϕ = y y ϕ = rsin𝜃cosϕ = x z ϕ = 0

γrr = 1 r2(x2γ xx + y2γ yy + z2γ zz + 2xyγxy + 2xzγxz + 2yzγyz) γr𝜃 = 1 rρ(x2zγ xx + y2zγ yy zρ2γ zz + 2xyzγxy + x(z2 ρ2)γ xz + y(z2 ρ2)γ yz) γrϕ = 1 r(xyγxx + xyγyy + (x2 y2)γ xy yzγxz + xzγyz) γ𝜃𝜃 = 1 ρ2(x2z2γ xx + 2xyz2γ xy 2xzρ2γ xz + y2z2γ yy 2yzρ2γ yz + ρ4γ zz) γ𝜃ϕ = 1 ρ(xyzγxx + (x2 y2)zγ xy + ρ2yγ xz + xyzγyy ρ2xγ yz) γϕϕ = y2γ xx 2xyγxy + x2γ yy

or,

γrr = sin2𝜃cos2ϕγ xx + sin2𝜃sin2ϕγ yy + cos2𝜃γ zz + 2sin2𝜃cosϕsinϕγ xy + 2sin𝜃cos𝜃cosϕγxz +2sin𝜃cos𝜃sinϕγyz γr𝜃 = r(sin𝜃cos𝜃cos2ϕγ xx + 2 sin𝜃cos𝜃sinϕcosϕγxy + (cos2𝜃 sin2𝜃)cosϕγ xz + sin𝜃cos𝜃sin2ϕγ yy +(cos2𝜃 sin2𝜃)sinϕγ yz sin𝜃cos𝜃γzz) γrϕ = rsin𝜃(sin𝜃sinϕcosϕγxx sin𝜃(sin2ϕ cos2ϕ)γ xy cos𝜃sinϕγxz + sin𝜃sinϕcosϕγyy +cos𝜃cosϕγyz) γ𝜃𝜃 = r2(cos2𝜃cos2ϕγ xx + 2cos2𝜃sinϕcosϕγ xy 2sin𝜃cos𝜃cosϕγxz + cos2𝜃sin2ϕγ yy 2sin𝜃cos𝜃sinϕγyz + sin2𝜃γ zz) γ𝜃ϕ = r2 sin𝜃(cos𝜃sinϕcosϕγ xx cos𝜃(sin2ϕ cos2ϕ)γ xy + sin𝜃sinϕγxz + cos𝜃sinϕcosϕγyy sin𝜃cosϕγyz) γϕϕ = r2 sin2𝜃(sin2ϕγ xx 2sinϕcosϕγxy + cos2ϕγ yy)

We also need the transformation for the radial derivative of the metric components:

γrr,η = sin2𝜃cos2ϕγ xx,η + sin2𝜃sin2ϕγ yy,η + cos2𝜃γ zz,η + 2sin2𝜃cosϕsinϕγ xy,η +2sin𝜃cos𝜃cosϕγxz,η + 2sin𝜃cos𝜃sinϕγyz,η γr𝜃,η = 1 rγr𝜃 + r(sin𝜃cos𝜃cos2ϕγ xx,η + sin𝜃cos𝜃sinϕcosϕγxy,η + (cos2𝜃 sin2𝜃)cosϕγ xz,η +sin𝜃cos𝜃sin2ϕγ yy,η + (cos2𝜃 sin2𝜃)sinϕγ yz,η sin𝜃cos𝜃γzz,η) γrϕ,η = 1 rγrϕ + rsin𝜃(sin𝜃sinϕcosϕγxx,η sin𝜃(sin2ϕ cos2ϕ)γ xy,η cos𝜃sinϕγxz,η +sin𝜃sinϕcosϕγyy,η + cos𝜃cosϕγyz,η) γ𝜃𝜃,η = 2 rγ𝜃𝜃 + r2(cos2𝜃cos2ϕγ xx,η + 2cos2𝜃sinϕcosϕγ xy,η 2sin𝜃cos𝜃cosϕγxz,η +cos2𝜃sin2ϕγ yy,η 2sin𝜃cos𝜃sinϕγyz,η + sin2𝜃γ zz,η) γ𝜃ϕ,η = 2 rγ𝜃ϕ + r2 sin𝜃(cos𝜃sinϕcosϕγ xx,η cos𝜃(sin2ϕ cos2ϕ)γ xy,η + sin𝜃sinϕγxz,η +cos𝜃sinϕcosϕγyy,η sin𝜃cosϕγyz,η) γϕϕ,η = 2 rγϕϕ + r2 sin2𝜃(sin2ϕγ xx,η 2sinϕcosϕγxy,η + cos2ϕγ yy,η)

D Integrations Over the 2-Spheres

This is done by using Simpson’s rule twice. Once in each coordinate direction. Simpson’s rule is

x1x2 f(x)dx = h 3[f1 + 4f2 + 2f3 + 4f4 + + 2fN2 + 4fN1 + fN] + O(1N4) (132)

N must be an odd number.

References

[1]   Abrahams A.M. & Cook G.B. “Collisions of boosted black holes: Perturbation theory predictions of gravitational radiation” Phys. Rev. D 50 R2364-R2367 (1994).

[2]   Abrahams A.M., Shapiro S.L. & Teukolsky S.A. “Calculation of gravitational wave forms from black hole collisions and disk collapse: Applying perturbation theory to numerical spacetimes” Phys. Rev. D. 51 4295 (1995).

[3]   Abrahams A.M. & Price R.H. “Applying black hole perturbation theory to numerically generated spacetimes” Phys. Rev. D. 53 1963 (1996).

[4]   Abrahams A.M. & Price R.H. “Black-hole collisions from Brill-Lindquist initial data: Predictions of perturbation theory” Phys. Rev. D. 53 1972 (1996).

[5]   Abramowitz, M. & Stegun A. “Pocket Book of Mathematical Functions (Abridged Handbook of Mathematical Functions”, Verlag Harri Deutsch (1984).

[6]   Andrade Z., & Price R.H. “Head-on collisions of unequal mass black holes: Close-limit predictions”, preprint (1996).

[7]   Anninos P., Price R.H., Pullin J., Seidel E., and Suen W-M. “Head-on collision of two black holes: Comparison of different approaches” Phys. Rev. D. 52 4462 (1995).

[8]   Arfken, G. “Mathematical Methods for Physicists”, Academic Press (1985).

[9]   Baker J., Abrahams A., Anninos P., Brant S., Price R., Pullin J. & Seidel E. “The collision of boosted black holes” (preprint) (1996).

[10]   Baker J. & Li C.B. “The two-phase approximation for black hole collisions: Is it robust” preprint (gr-qc/9701035), (1997).

[11]   Brandt S.R. & Seidel E. “The evolution of distorted rotating black holes III: Initial data” (preprint) (1996).

[12]   Cunningham C.T., Price R.H., Moncrief V., “Radiation from collapsing relativistic stars. I. Linearized Odd-Parity Radiation” Ap. J. 224 543-667 (1978).

[13]   Cunningham C.T., Price R.H., Moncrief V., “Radiation from collapsing relativistic stars. I. Linearized Even-Parity Radiation” Ap. J. 230 870-892 (1979).

[14]   Landau L.D. & Lifschitz E.M., “The Classical Theory of Fields” (4th Edition), Pergamon Press (1980).

[15]   Mathews J. “”, J. Soc. Ind. Appl. Math. 10 768 (1962).

[16]   Moncrief V. “Gravitational perturbations of spherically symmetric systems. I. The exterior problem” Annals of Physics 88 323-342 (1974).

[17]   Press W.H., Flannery B.P., Teukolsky S.A., & Vetterling W.T., “Numerical Recipes, The Art of Scientific Computing” Cambridge University Press (1989).

[18]   Price R.H. & Pullin J. “Colliding black holes: The close limit”, Phys. Rev. Lett. 72 3297-3300 (1994).

[19]   Regge T., & Wheeler J.A. “Stability of a Schwarzschild Singularity”, Phys. Rev. D 108 1063 (1957).