Extracting Gravitational Waves and Other Quantities from Numerical Spacetimes

Gabrielle Allen

September 2, 2018

Abstract

NB: This documentation is taken from the Extract thorn on which WaveExtractL is based. There may be some di๏ฌ€erences between WaveExtractL and Extract, which are not documented here.

1 Introduction

Thorn Extract calculates ๏ฌrst order gauge invariant waveforms from a numerical spacetime, under the basic assumption that, at the spheres of extract the spacetime is approximately Schwarzschild. In addition, other quantities such as mass, angular momentum and spin can be determined.

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 ๏ฌrst order gauge invariant waveform.

2 Physical System

2.1 Wave Forms

Assume a spacetime gαβ which can be written as a Schwarzschild background gαβSchwarz with perturbations hαβ:

gαβ = gαβSchwarz + h αβ (1)

with

{gαβSchwarz}(t,r,๐œƒ,ฯ•) = S 0 0 0 0 S1 0 0 0 0 r2 0 0 0 0 r2 sin2๐œƒ S(r) = 12M r (2)

The 3-metric perturbations γij can be decomposed using tensor harmonics into γijlm(t,r) where

γij(t,r,๐œƒ,ฯ•) = l=0 m=llγ ijlm(t,r)

and

γij(t,r,๐œƒ,ฯ•) = k=06p k(t,r)V k(๐œƒ,ฯ•)

where {V k} is some basis for tensors on a 2-sphere in 3-D Euclidean space. Working with the Regge-Wheeler basis (see Sectionย 6) the 3-metric is then expanded in terms of the (six) standard Regge-Wheeler functions {c1ร—lm,c2ร—lm,h1+lm,H2+lm,K+lm,G+lm}ย [19],ย [16]. Where each of the functions is either odd (ร—) or even (+) parity. The decomposition is then written

γijlm = c 1ร—lm(รช 1)ijlm + c 2ร—lm(รช 2)ijlm + h1+lm(fฬ‚ 1)ijlm + A2H 2+lm(fฬ‚ 2)ijlm + R2K+lm(fฬ‚ 3)ijlm + R2G+lm(fฬ‚ 4)ijlm (3)

which we can write in an expanded form as

γrrlm = A2H 2+lmY lm (4) γr๐œƒlm = c 1ร—lm 1 sin๐œƒY lm,ฯ• + h1+lmY lm,๐œƒ (5) γrฯ•lm = c 1ร—lm sin๐œƒY lm,๐œƒ + h1+lmY lm,ฯ• (6) γ๐œƒ๐œƒlm = c 2ร—lm 1 sin๐œƒ(Y lm,๐œƒฯ• cot๐œƒY lm,ฯ•) + R2K+lmY lm + R2G+lmY lm,๐œƒ๐œƒ (7) γ๐œƒฯ•lm = c 2ร—lm sin๐œƒ1 2 Y lm,๐œƒ๐œƒ cot๐œƒY lm,๐œƒ 1 sin2๐œƒY lm + R2G+lm(Y lm,๐œƒฯ• cot๐œƒY lm,ฯ•) (8) γฯ•ฯ•lm = sin๐œƒc 2ร—lm(Y lm,๐œƒฯ• cot๐œƒY lm,ฯ•) + R2K+lm sin2๐œƒY lm + R2G+lm(Y lm,ฯ•ฯ• + sin๐œƒcos๐œƒY lm,๐œƒ) (9)

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

gttlm = N2H 0+lmY lm (10) gtrlm = H 1+lmY lm (11) gt๐œƒlm = h 0+lmY lm,๐œƒ c0ร—lm 1 sin๐œƒY lm,ฯ• (12) gtฯ•lm = h 0+lmY lm,ฯ• + c0ร—lm sin๐œƒY lm,๐œƒ (13)

Also from gtt = α2 + βiβi we have

αlm = 1 2NH0+lmY lm (14)

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

gαβ = gαβbackground + l,mgαβlm,odd + l,mgαβlm,even

where (dropping some superscripts)

{gαβodd} = 00 c0 1 sin ๐œƒY lm,ฯ• c0 sin๐œƒY lm,๐œƒ . 0 c1 1 sin ๐œƒY lm,ฯ• c1 sin๐œƒY lm,๐œƒ . .c2 1 sin ๐œƒ(Y lm,๐œƒฯ• cot๐œƒY lm,ฯ•)c21 2 1 sin ๐œƒY lm,ฯ•ฯ• + cos๐œƒY lm,๐œƒ sin๐œƒY lm,๐œƒ๐œƒ . . . c2(sin๐œƒY lm,๐œƒฯ• + cos๐œƒY lm,ฯ•) {gαβeven} = N2H0Y lm H1Y lm h0Y lm,๐œƒ h0Y lm,ฯ• . A2H2Y lm h1Y lm,๐œƒ h1Y lm,ฯ• . . R2KY lm + r2GY lm,๐œƒ๐œƒ R2(Y lm,๐œƒฯ• cot๐œƒY lm,ฯ•) . . . R2Ksin2๐œƒY lm + R2G(Y lm,ฯ•ฯ• + sin๐œƒcos๐œƒY lm,๐œƒ)

Now, for such a Schwarzschild background we can de๏ฌne two (and only two) unconstrained gauge invariant quantities Qlmร— = Qlmร—(c1ร—lm,c2ร—lm) and Qlm+ = Qlm+(K+lm,G+lm,H2+lm,h1+lm), which from [3] are

Qlmร— = 2(l + 2)! (l 2)! c1ร—lm + 1 2 rc2ร—lm 2 rc2ร—lm S r (15) Qlm+ = 1 Λ 2(l 1)(l + 2) l(l + 1) (4rS2k 2 + l(l + 1)rk1) (16) 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 (17)

where

k1 = K+lm + S r (r2 rG+lm 2h 1+lm) (18) k2 = 1 2S H2+lm r rk1 1 M rSk1 + S1โˆ•2 r(r2S1โˆ•2 rG+lm 2S1โˆ•2h 1+lm) (19) 1 2S H2 rK,r r 3M r 2MK (20)

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

Moncriefsย oddย parityย Q:ย Qlmร— = 2(l + 2)! (l 2)! Q Moncriefsย evenย parityย Q:ย 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 lmร— + S l(l + 1) r2 6M r3 Qlmร— = 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) + 6Mโˆ•r r = r + 2Mln(rโˆ•2M 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 rฬ‚ of each sphere is calculated by

rฬ‚ = rฬ‚(r) = 1 4πγ๐œƒ๐œƒ γฯ•ฯ•d๐œƒdฯ•1โˆ•2 (21)

from which

drฬ‚ dη = 1 16πrฬ‚γ๐œƒ๐œƒ,ηγฯ•ฯ• + γ๐œƒ๐œƒγฯ•ฯ•,η γ๐œƒ๐œƒ γฯ•ฯ• d๐œƒdฯ• (22)

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(rฬ‚) = rฬ‚ r 2γrrd๐œƒdฯ• (23)
M(rฬ‚) = rฬ‚1 S 2 (24)

3.4 Calculate Regge-Wheeler Variables

c1ร—lm = 1 l(l + 1) γrฬ‚ฯ•Y lm,๐œƒ γrฬ‚๐œƒY lm,ฯ• sin๐œƒ dΩ c2ร—lm = 2 l(l + 1)(l 1)(l + 2) 1 sin2๐œƒγ๐œƒ๐œƒ + 1 sin4๐œƒγฯ•ฯ• (sin๐œƒY lm,๐œƒฯ• cos๐œƒY lm,ฯ•) + 1 sin๐œƒγ๐œƒฯ•(Y lm,๐œƒ๐œƒ cot๐œƒY lm,๐œƒ 1 sin2๐œƒY lm,ฯ•ฯ•)dΩ h1+lm = 1 l(l + 1) γrฬ‚๐œƒY lm,๐œƒ + 1 sin2๐œƒγrฬ‚ฯ•Y lm,ฯ•dΩ H2+lm = Sγrฬ‚rฬ‚Y lmdΩ K+lm = 1 2rฬ‚2 γ๐œƒ๐œƒ + 1 sin2๐œƒγฯ•ฯ• Y lmdΩ + 1 2rฬ‚2(l 1)(l + 2)γ๐œƒ๐œƒ γฯ•ฯ• sin2๐œƒ Y lm,๐œƒ๐œƒ cot๐œƒY lm,๐œƒ 1 sin2๐œƒY lm,ฯ•ฯ• + 4 sin2๐œƒγ๐œƒฯ•(Y lm,๐œƒฯ• cot๐œƒY lm,ฯ•)dΩ G+lm = 1 rฬ‚2l(l + 1)(l 1)(l + 2) γ๐œƒ๐œƒ γฯ•ฯ• sin2๐œƒ Y lm,๐œƒ๐œƒ cot๐œƒY lm,๐œƒ 1 sin2๐œƒY lm,ฯ•ฯ• + 4 sin2๐œƒγ๐œƒฯ•(Y lm,๐œƒฯ• cot๐œƒY lm,ฯ•)dΩ

where

γrฬ‚rฬ‚ = r rฬ‚ r rฬ‚γrr (25) γrฬ‚๐œƒ = r rฬ‚γr๐œƒ (26) γrฬ‚ฯ• = r rฬ‚γrฯ• (27)

3.5 Calculate Gauge Invariant Quantities

Qlmร— = 2(l + 2)! (l 2)! c1ร—lm + 1 2 rฬ‚c2ร—lm 2 rฬ‚c2ร—lm S rฬ‚ (28) Qlm+ = 1 (l 1)(l + 2) + 6Mโˆ•rฬ‚ 2(l 1)(l + 2) l(l + 1) (4rฬ‚S2k 2 + l(l + 1)rฬ‚k1) (29)

where

k1 = K+lm + S rฬ‚(rฬ‚2 rฬ‚G+lm 2h 1+lm) (30) k2 = 1 2S[H2+lm rฬ‚ rฬ‚k1 (1 M rฬ‚S)k1 + S1โˆ•2 rฬ‚(rฬ‚2S1โˆ•2 rฬ‚G+lm 2S1โˆ•2h 1+lm (31)

4 Using This Thorn

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

4.1 Basic Usage

4.2 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 ๏ฌles from Extract are always placed in the main output directory de๏ฌned by CactusBase/IOUtil.

Output ๏ฌles are generated for each detector (2-sphere) used, and these detectors are identi๏ฌed in the name of each output ๏ฌle by R1, R2, ….

The extension denotes whether coordinate time (แนซl) or proper time (uฬ‡l) is used for the ๏ฌrst column.

5 History

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 compare with linear evolutions of waveforms extracted from the Cauchy initial data. This work was carried out in collaboration with Karen Camarda and Ed Seidel.

6 Appendix: Regge-Wheeler Harmonics

(รช1)lm = 0 1 sin ๐œƒY lm,ฯ• sin๐œƒY lm,๐œƒ . 0 0 . 0 0 (รช2)lm = 0 0 0 0 1 sin ๐œƒ(Y lm,๐œƒฯ• cot๐œƒY lm,ฯ•) . 0 sin ๐œƒ 2 [Y lm,๐œƒ๐œƒ cot๐œƒY lm,๐œƒ 1 sin 2๐œƒY lm,ฯ•ฯ•] sin๐œƒ[Y lm,๐œƒฯ• cot๐œƒY lm,ฯ•] (fฬ‚1)lm = 0Y lm,๐œƒY lm,ฯ• . 0 0 . 0 0 (fฬ‚2)lm = Y lm00 0 00 0 0 0 (fฬ‚3)lm = 0 0 0 0 Y lm 0 0 0 sin2๐œƒY lm (fฬ‚4)lm = 0 0 0 0 Y lm,๐œƒ๐œƒ . 0Y lm,๐œƒฯ• cot๐œƒY lm,ฯ•Y lm,ฯ•ฯ• + sin๐œƒcos๐œƒY lm,๐œƒ

7 Appendix: Transformation Between Cartesian and Spherical Coordinates

First, the transformations between metric components in (x,y,z) and (r,๐œƒ,ฯ•) coordinates. Here, ρ = x2 + y2 = rsin๐œƒ,

x r = sin๐œƒcosฯ• = x r y r = sin๐œƒsinฯ• = y r z r = cos๐œƒ = z r x ๐œƒ = rcos๐œƒcosฯ• = xz ρ y ๐œƒ = rcos๐œƒsinฯ• = yz ρ z ๐œƒ = rsin๐œƒ = ρ x ฯ• = rsin๐œƒsinฯ• = y y ฯ• = rsin๐œƒcosฯ• = x z ฯ• = 0

γrr = 1 r2(x2γ xx + y2γ yy + z2γ zz + 2xyγxy + 2xzγxz + 2yzγyz) γr๐œƒ = 1 rρ(x2zγ xx + y2zγ yy zρ2γ zz + 2xyzγxy + x(z2 ρ2)γ xz + y(z2 ρ2)γ yz) γrฯ• = 1 r(xyγxx + xyγyy + (x2 y2)γ xy yzγxz + xzγyz) γ๐œƒ๐œƒ = 1 ρ2(x2z2γ xx + 2xyz2γ xy 2xzρ2γ xz + y2z2γ yy 2yzρ2γ yz + ρ4γ zz) γ๐œƒฯ• = 1 ρ(xyzγxx + (x2 y2)zγ xy + ρ2yγ xz + xyzγyy ρ2xγ yz) γฯ•ฯ• = y2γ xx 2xyγxy + x2γ yy

or,

γrr = sin2๐œƒcos2ฯ•γ xx + sin2๐œƒsin2ฯ•γ yy + cos2๐œƒγ zz + 2sin2๐œƒcosฯ•sinฯ•γ xy + 2sin๐œƒcos๐œƒcosฯ•γxz +2sin๐œƒcos๐œƒsinฯ•γyz γr๐œƒ = r(sin๐œƒcos๐œƒcos2ฯ•γ xx + 2 sin๐œƒcos๐œƒsinฯ•cosฯ•γxy + (cos2๐œƒ sin2๐œƒ)cosฯ•γ xz + sin๐œƒcos๐œƒsin2ฯ•γ yy +(cos2๐œƒ sin2๐œƒ)sinฯ•γ yz sin๐œƒcos๐œƒγzz) γrฯ• = rsin๐œƒ(sin๐œƒsinฯ•cosฯ•γxx sin๐œƒ(sin2ฯ• cos2ฯ•)γ xy cos๐œƒsinฯ•γxz + sin๐œƒsinฯ•cosฯ•γyy +cos๐œƒcosฯ•γyz) γ๐œƒ๐œƒ = r2(cos2๐œƒcos2ฯ•γ xx + 2cos2๐œƒsinฯ•cosฯ•γ xy 2sin๐œƒcos๐œƒcosฯ•γxz + cos2๐œƒsin2ฯ•γ yy 2sin๐œƒcos๐œƒsinฯ•γyz + sin2๐œƒγ zz) γ๐œƒฯ• = r2 sin๐œƒ(cos๐œƒsinฯ•cosฯ•γ xx cos๐œƒ(sin2ฯ• cos2ฯ•)γ xy + sin๐œƒsinฯ•γxz + cos๐œƒsinฯ•cosฯ•γyy sin๐œƒcosฯ•γyz) γฯ•ฯ• = r2 sin2๐œƒ(sin2ฯ•γ xx 2sinฯ•cosฯ•γxy + cos2ฯ•γ yy)

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

γrr,η = sin2๐œƒcos2ฯ•γ xx,η + sin2๐œƒsin2ฯ•γ yy,η + cos2๐œƒγ zz,η + 2sin2๐œƒcosฯ•sinฯ•γ xy,η +2sin๐œƒcos๐œƒcosฯ•γxz,η + 2sin๐œƒcos๐œƒsinฯ•γyz,η γr๐œƒ,η = 1 rγr๐œƒ + r(sin๐œƒcos๐œƒcos2ฯ•γ xx,η + sin๐œƒcos๐œƒsinฯ•cosฯ•γxy,η + (cos2๐œƒ sin2๐œƒ)cosฯ•γ xz,η +sin๐œƒcos๐œƒsin2ฯ•γ yy,η + (cos2๐œƒ sin2๐œƒ)sinฯ•γ yz,η sin๐œƒcos๐œƒγzz,η) γrฯ•,η = 1 rγrฯ• + rsin๐œƒ(sin๐œƒsinฯ•cosฯ•γxx,η sin๐œƒ(sin2ฯ• cos2ฯ•)γ xy,η cos๐œƒsinฯ•γxz,η +sin๐œƒsinฯ•cosฯ•γyy,η + cos๐œƒcosฯ•γyz,η) γ๐œƒ๐œƒ,η = 2 rγ๐œƒ๐œƒ + r2(cos2๐œƒcos2ฯ•γ xx,η + 2cos2๐œƒsinฯ•cosฯ•γ xy,η 2sin๐œƒcos๐œƒcosฯ•γxz,η +cos2๐œƒsin2ฯ•γ yy,η 2sin๐œƒcos๐œƒsinฯ•γyz,η + sin2๐œƒγ zz,η) γ๐œƒฯ•,η = 2 rγ๐œƒฯ• + r2 sin๐œƒ(cos๐œƒsinฯ•cosฯ•γ xx,η cos๐œƒ(sin2ฯ• cos2ฯ•)γ xy,η + sin๐œƒsinฯ•γxz,η +cos๐œƒsinฯ•cosฯ•γyy,η sin๐œƒcosฯ•γyz,η) γฯ•ฯ•,η = 2 rγฯ•ฯ• + r2 sin2๐œƒ(sin2ฯ•γ xx,η 2sinฯ•cosฯ•γxy,η + cos2ฯ•γ yy,η)

8 Appendix: 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(1โˆ•N4) (32)

N must be an odd number.

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