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 differences between WaveExtractL and Extract, which are not documented here.

1 Introduction

Thorn Extract calculates first 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 first 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,πœƒ,Ο•) = S 0 0 0 0 S1 0 0 0 0 r2 0 0 0 0 r2 sin2π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,πœƒ,Ο•) = l=0 m=llγ ijlm(t,r)

and

γij(t,r,πœƒ,Ο•) = k=06p k(t,r)V k(πœƒ,Ο•)

where {V k} is some basis for tensors on a 2-sphere in 3-D Euclidean space. Working with the Regge-Wheeler basis (see SectionΒ 6) the 3-metric is then expanded in terms of the (six) standard Regge-Wheeler functions {c1Γ—lm,c2Γ—lm,h1+lm,H2+lm,K+lm,G+lm}Β [19],Β [16]. Where each of the functions is either odd (Γ—) or even (+) parity. The decomposition is then written

γijlm = c 1Γ—lm(Γͺ 1)ijlm + c 2Γ—lm(Γͺ 2)ijlm + h1+lm(fΜ‚ 1)ijlm + A2H 2+lm(fΜ‚ 2)ijlm + R2K+lm(fΜ‚ 3)ijlm + R2G+lm(fΜ‚ 4)ijlm (3)

which we can write in an expanded form as

γrrlm = A2H 2+lmY lm (4) γrπœƒlm = c 1Γ—lm 1 sinπœƒY lm,Ο• + h1+lmY lm,πœƒ (5) γrΟ•lm = c 1Γ—lm sinπœƒY lm,πœƒ + h1+lmY lm,Ο• (6) γπœƒπœƒlm = c 2Γ—lm 1 sinπœƒ(Y lm,πœƒΟ• cotπœƒY lm,Ο•) + R2K+lmY lm + R2G+lmY lm,πœƒπœƒ (7) γπœƒΟ•lm = c 2Γ—lm sinπœƒ1 2 Y lm,πœƒπœƒ cotπœƒY lm,πœƒ 1 sin2πœƒY lm + R2G+lm(Y lm,πœƒΟ• cotπœƒY lm,Ο•) (8) γϕϕlm = sinπœƒc 2Γ—lm(Y lm,πœƒΟ• cotπœƒY lm,Ο•) + R2K+lm sin2πœƒY lm + R2G+lm(Y lm,ϕϕ + sinπœƒcosπœƒY lm,π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) gtπœƒlm = h 0+lmY lm,πœƒ c0Γ—lm 1 sinπœƒY lm,Ο• (12) gtΟ•lm = h 0+lmY lm,Ο• + c0Γ—lm sinπœƒY lm,π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 πœƒY lm,Ο• c0 sinπœƒY lm,πœƒ . 0 c1 1 sin πœƒY lm,Ο• c1 sinπœƒY lm,πœƒ . .c2 1 sin πœƒ(Y lm,πœƒΟ• cotπœƒY lm,Ο•)c21 2 1 sin πœƒY lm,ϕϕ + cosπœƒY lm,πœƒ sinπœƒY lm,πœƒπœƒ . . . c2(sinπœƒY lm,πœƒΟ• + cosπœƒY lm,Ο•) {gαβeven} = N2H0Y lm H1Y lm h0Y lm,πœƒ h0Y lm,Ο• . A2H2Y lm h1Y lm,πœƒ h1Y lm,Ο• . . R2KY lm + r2GY lm,πœƒπœƒ R2(Y lm,πœƒΟ• cotπœƒY lm,Ο•) . . . R2Ksin2πœƒY lm + R2G(Y lm,ϕϕ + sinπœƒcosπœƒY lm,πœƒ)

Now, for such a Schwarzschild background we can define two (and only two) unconstrained gauge invariant quantities QlmΓ— = QlmΓ—(c1Γ—lm,c2Γ—lm) and Qlm+ = Qlm+(K+lm,G+lm,H2+lm,h1+lm), which from [3] are

QlmΓ— = 2(l + 2)! (l 2)! c1Γ—lm + 1 2 rc2Γ—lm 2 rc2Γ—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 + S1βˆ•2 r(r2S1βˆ•2 rG+lm 2S1βˆ•2h 1+lm) (19) 1 2S H2 rK,r r 3M r 2MK (20)

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

MoncriefsΒ oddΒ parityΒ Q:Β QlmΓ— = 2(l + 2)! (l 2)! Q MoncriefsΒ evenΒ parityΒ Q:Β 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 lmΓ— + S l(l + 1) r2 6M r3 QlmΓ— = 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) + 6Mβˆ•r r = r + 2Mln(rβˆ•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 rΜ‚ of each sphere is calculated by

rΜ‚ = rΜ‚(r) = 1 4πγπœƒπœƒ γϕϕdπœƒdΟ•1βˆ•2 (21)

from which

drΜ‚ dη = 1 16πrΜ‚γπœƒπœƒ,ηγϕϕ + γπœƒπœƒγϕϕ,η γπœƒπœƒ γϕϕ dπœƒdΟ• (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(rΜ‚) = rΜ‚ r 2γrrdπœƒdΟ• (23)
M(rΜ‚) = rΜ‚1 S 2 (24)

3.4 Calculate Regge-Wheeler Variables

c1Γ—lm = 1 l(l + 1) γrΜ‚Ο•Y lm,πœƒ γrΜ‚πœƒY lm,Ο• sinπœƒ dΩ c2Γ—lm = 2 l(l + 1)(l 1)(l + 2) 1 sin2πœƒγπœƒπœƒ + 1 sin4πœƒγϕϕ (sinπœƒY lm,πœƒΟ• cosπœƒY lm,Ο•) + 1 sinπœƒγπœƒΟ•(Y lm,πœƒπœƒ cotπœƒY lm,πœƒ 1 sin2πœƒY lm,ϕϕ)dΩ h1+lm = 1 l(l + 1) γrΜ‚πœƒY lm,πœƒ + 1 sin2πœƒγrΜ‚Ο•Y lm,Ο•dΩ H2+lm = SγrΜ‚rΜ‚Y lmdΩ K+lm = 1 2rΜ‚2 γπœƒπœƒ + 1 sin2πœƒγϕϕ Y lmdΩ + 1 2rΜ‚2(l 1)(l + 2)γπœƒπœƒ γϕϕ sin2πœƒ Y lm,πœƒπœƒ cotπœƒY lm,πœƒ 1 sin2πœƒY lm,ϕϕ + 4 sin2πœƒγπœƒΟ•(Y lm,πœƒΟ• cotπœƒY lm,Ο•)dΩ G+lm = 1 rΜ‚2l(l + 1)(l 1)(l + 2) γπœƒπœƒ γϕϕ sin2πœƒ Y lm,πœƒπœƒ cotπœƒY lm,πœƒ 1 sin2πœƒY lm,ϕϕ + 4 sin2πœƒγπœƒΟ•(Y lm,πœƒΟ• cotπœƒY lm,Ο•)dΩ

where

γrΜ‚rΜ‚ = r rΜ‚ r rΜ‚γrr (25) γrΜ‚πœƒ = r rΜ‚γrπœƒ (26) γrΜ‚Ο• = r rΜ‚γrΟ• (27)

3.5 Calculate Gauge Invariant Quantities

QlmΓ— = 2(l + 2)! (l 2)! c1Γ—lm + 1 2 rΜ‚c2Γ—lm 2 rΜ‚c2Γ—lm S rΜ‚ (28) Qlm+ = 1 (l 1)(l + 2) + 6Mβˆ•rΜ‚ 2(l 1)(l + 2) l(l + 1) (4rΜ‚S2k 2 + l(l + 1)rΜ‚k1) (29)

where

k1 = K+lm + S rΜ‚(rΜ‚2 rΜ‚G+lm 2h 1+lm) (30) k2 = 1 2S[H2+lm rΜ‚ rΜ‚k1 (1 M rΜ‚S)k1 + S1βˆ•2 rΜ‚(rΜ‚2S1βˆ•2 rΜ‚G+lm 2S1βˆ•2h 1+lm (31)

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 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 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

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

(Γͺ1)lm = 0 1 sin πœƒY lm,Ο• sinπœƒY lm,πœƒ . 0 0 . 0 0 (Γͺ2)lm = 0 0 0 0 1 sin πœƒ(Y lm,πœƒΟ• cotπœƒY lm,Ο•) . 0 sin πœƒ 2 [Y lm,πœƒπœƒ cotπœƒY lm,πœƒ 1 sin 2πœƒY lm,ϕϕ] sinπœƒ[Y lm,πœƒΟ• cotπœƒY lm,Ο•] (fΜ‚1)lm = 0Y lm,πœƒY lm,Ο• . 0 0 . 0 0 (fΜ‚2)lm = Y lm00 0 00 0 0 0 (fΜ‚3)lm = 0 0 0 0 Y lm 0 0 0 sin2πœƒY lm (fΜ‚4)lm = 0 0 0 0 Y lm,πœƒπœƒ . 0Y lm,πœƒΟ• cotπœƒY lm,Ο•Y lm,ϕϕ + sinπœƒcosπœƒY lm,πœƒ

7 Appendix: Transformation Between Cartesian and Spherical Coordinates

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

x r = sinπœƒcosΟ• = x r y r = sinπœƒsinΟ• = y r z r = cosπœƒ = z r x πœƒ = rcosπœƒcosΟ• = xz ρ y πœƒ = rcosπœƒsinΟ• = yz ρ z πœƒ = rsinπœƒ = ρ x Ο• = rsinπœƒsinΟ• = y y Ο• = rsinπœƒcosΟ• = x z Ο• = 0

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

or,

γrr = sin2πœƒcos2Ο•γ xx + sin2πœƒsin2Ο•γ yy + cos2πœƒγ zz + 2sin2πœƒcosΟ•sinΟ•γ xy + 2sinπœƒcosπœƒcosΟ•γxz +2sinπœƒcosπœƒsinΟ•γyz γrπœƒ = r(sinπœƒcosπœƒcos2Ο•γ xx + 2 sinπœƒcosπœƒsinΟ•cosΟ•γxy + (cos2πœƒ sin2πœƒ)cosΟ•γ xz + sinπœƒcosπœƒsin2Ο•γ yy +(cos2πœƒ sin2πœƒ)sinΟ•γ yz sinπœƒcosπœƒγzz) γrΟ• = rsinπœƒ(sinπœƒsinΟ•cosΟ•γxx sinπœƒ(sin2Ο• cos2Ο•)γ xy cosπœƒsinΟ•γxz + sinπœƒsinΟ•cosΟ•γyy +cosπœƒcosΟ•γyz) γπœƒπœƒ = r2(cos2πœƒcos2Ο•γ xx + 2cos2πœƒsinΟ•cosΟ•γ xy 2sinπœƒcosπœƒcosΟ•γxz + cos2πœƒsin2Ο•γ yy 2sinπœƒcosπœƒsinΟ•γyz + sin2πœƒγ zz) γπœƒΟ• = r2 sinπœƒ(cosπœƒsinΟ•cosΟ•γ xx cosπœƒ(sin2Ο• cos2Ο•)γ xy + sinπœƒsinΟ•γxz + cosπœƒsinΟ•cosΟ•γyy sinπœƒcosΟ•γyz) γϕϕ = r2 sin2πœƒ(sin2Ο•γ xx 2sinΟ•cosΟ•γxy + cos2Ο•γ yy)

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

γrr,η = sin2πœƒcos2Ο•γ xx,η + sin2πœƒsin2Ο•γ yy,η + cos2πœƒγ zz,η + 2sin2πœƒcosΟ•sinΟ•γ xy,η +2sinπœƒcosπœƒcosΟ•γxz,η + 2sinπœƒcosπœƒsinΟ•γyz,η γrπœƒ,η = 1 rγrπœƒ + r(sinπœƒcosπœƒcos2Ο•γ xx,η + sinπœƒcosπœƒsinΟ•cosΟ•γxy,η + (cos2πœƒ sin2πœƒ)cosΟ•γ xz,η +sinπœƒcosπœƒsin2Ο•γ yy,η + (cos2πœƒ sin2πœƒ)sinΟ•γ yz,η sinπœƒcosπœƒγzz,η) γrΟ•,η = 1 rγrΟ• + rsinπœƒ(sinπœƒsinΟ•cosΟ•γxx,η sinπœƒ(sin2Ο• cos2Ο•)γ xy,η cosπœƒsinΟ•γxz,η +sinπœƒsinΟ•cosΟ•γyy,η + cosπœƒcosΟ•γyz,η) γπœƒπœƒ,η = 2 rγπœƒπœƒ + r2(cos2πœƒcos2Ο•γ xx,η + 2cos2πœƒsinΟ•cosΟ•γ xy,η 2sinπœƒcosπœƒcosΟ•γxz,η +cos2πœƒsin2Ο•γ yy,η 2sinπœƒcosπœƒsinΟ•γyz,η + sin2πœƒγ zz,η) γπœƒΟ•,η = 2 rγπœƒΟ• + r2 sinπœƒ(cosπœƒsinΟ•cosΟ•γ xx,η cosπœƒ(sin2Ο• cos2Ο•)γ xy,η + sinπœƒsinΟ•γxz,η +cosπœƒsinΟ•cosΟ•γyy,η sinπœƒcosΟ•γyz,η) γϕϕ,η = 2 rγϕϕ + r2 sin2πœƒ(sin2Ο•γ xx,η 2sinΟ•cosΟ•γxy,η + cos2Ο•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(1βˆ•N4) (32)

N must be an odd number.

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