Extracting Gravitational Waves and Other Quantities from Numerical Spacetimes

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1 Extracting Gravitational Waves and Other Quantities from Numerical Spacetimes Gabrielle Allen Abstract 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. Physical System. Wave Forms Assume a spacetime g αβ which can be written as a Schwarzschild background gαβ Schwarz with perturbations h αβ : g αβ = gαβ Schwarz + h αβ () with {gαβ Schwarz }(t, r, θ, φ) = S S r r sin θ S(r) = M r The 3-metric perturbations γ ij can be decomposed using tensor harmonics into γij lm (t, r) where and γ ij (t, r, θ, φ) = γ ij (t, r, θ, φ) = l l=0 m= l γ lm ij (t, r) 6 p k (t, r)v k (θ, φ) k=0 where {V k } is some basis for tensors on a -sphere in 3-D Euclidean space. Working with the Regge- Wheeler basis (see Section 6) the 3-metric is then expanded in terms of the (six) standard Regge-Wheeler functions {c lm, c lm, h +lm, H +lm, K +lm, G +lm } [9], [6]. Where each of the functions is either odd ( ) or even (+) parity. The decomposition is then written γij lm = c lm (ê ) lm ij + c lm (ê ) lm ij + h +lm ( ˆf ) lm ij + A H +lm ( ˆf ) lm ij + R K +lm ( ˆf 3 ) lm ij + R G +lm ( ˆf 4 ) lm ij (3) which we can write in an expanded form as () γ lm rr = A H +lm Y lm (4)

2 γrθ lm = c lm sin θ Y lm,φ + h +lm Y lm,θ (5) γrφ lm = c lm sin θy lm,θ + h +lm Y lm,φ (6) γθθ lm = c lm sin θ (Y lm,θφ cot θy lm,φ ) + R K +lm Y lm + R G +lm Y lm,θθ (7) γθφ lm = c lm sin θ ( Y lm,θθ cot θy lm,θ ) sin θ Y lm + R G +lm (Y lm,θφ cot θy lm,φ ) (8) γ lm φφ = sin θc lm (Y lm,θφ cot θy lm,φ ) + R K +lm sin θy lm + R G +lm (Y lm,φφ + sin θ cos θy lm,θ )(9) A similar decomposition allows the four gauge components of the 4-metric to be written in terms of three even-parity variables {H 0, H, h 0 } and the one odd-parity variable {c 0 } Also from g tt = α + β i β i we have gtt lm = N H 0 +lm Y lm (0) gtr lm = H +lm Y lm () 0 sin θ Y lm,φ () gtφ lm = h +lm 0 Y lm,φ + c lm 0 sin θy lm,θ (3) gtθ lm = h +lm 0 Y lm,θ c lm α lm = NH+lm 0 Y lm (4) It is useful to also write this with the perturbation split into even and odd parity parts: g αβ = g background αβ + l,m g lm,odd αβ + l,m g lm,even αβ where (dropping some superscripts) 0 0 c 0 sin θ Y lm,φ c 0 sin θy lm,θ {gαβ odd } =. 0 c sin θ Y lm,φ c sin θy lm,θ.. c sin θ (Y ( lm,θφ cot θy lm,φ ) c sin θ Y ) lm,φφ + cos θy lm,θ sin θy lm,θθ... c ( sin θy lm,θφ + cos θy lm,φ ) N H 0 Y lm H Y lm h 0 Y lm,θ h 0 Y lm,φ {gαβ even } =. A H Y lm h Y lm,θ h Y lm,φ.. R KY lm + r GY lm,θθ R (Y lm,θφ cot θy lm,φ )... R K sin θy lm + R G(Y lm,φφ + sin θ cos θy lm,θ ) Now, for such a Schwarzschild background we can define two (and only two) unconstrained gauge invariant quantities Q lm = Q lm (c lm, c lm ) and Q + lm = Q+ lm (K+lm, G +lm, H +lm, h +lm ), which from [3] are Q lm = (l + )! (l )! [ c lm + ( r c lm )] S r c lm r Q + lm = (l )(l + ) (4rS k + l(l + )rk ) (6) Λ l(l + ) (l )(l + ) ( l(l + )S(r r G +lm h +lm ) + rs(h +lm r r K +lm ) + ΛrK +lm) (7) Λ l(l + ) where k = K +lm + S r (r r G +lm h +lm ) (8) k = [ ( H +lm r r k M ) ] k + S / r (r S / r G +lm S / h +lm ) (9) S rs [ H rk,r r 3M ] S r M K (0) (5)

3 3 NOTE: These quantities compare with those in Moncrief [6] by Moncriefs odd parity Q: Q lm = (l + )! (l )! Q Moncriefs even parity Q: Q + lm = (l )(l + ) Q l(l + ) 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 where ( t r )Q lm + S [ l(l + ) r 6M r 3 ( t r )Q+ lm + S [ Λ ( 7M 3 r 5 ] Q lm = 0 M (l )(l + ) r3 ( 3M r Λ = (l )(l + ) + 6M/r r = r + M ln(r/m ) )) + ] l(l )(l + )(l + ) r Q + lm Λ = 0 3 Numerical Implementation The implementation assumes that the numerical solution, on a Cartesian grid, is approximately Schwarzshild on the spheres of constant r = (x +y +z ) where the waveforms are extracted. The general procedure is then: Project the required metric components, and radial derivatives of metric components, onto spheres of constant coordinate radius (these spheres are chosen via parameters). Transform the metric components and there derivatives on the -spheres from Cartesian coordinates into a spherical coordinate system. Calculate the physical metric on these spheres if a conformal factor is being used. Calculate the transformation from the coordinate radius to an areal radius for each sphere. Calculate the S factor on each sphere. estimate of the mass. Combined with the areal radius This also produces an Calculate the six Regge-Wheeler variables, and required radial derivatives, on these spheres by integration of combinations of the metric components over each sphere. Contruct the gauge invariant quantities from these Regge-Wheeler variables. 3. Project onto Spheres of Constant Radius This is performed by interpolating the metric components, and if needed the conformal factor, onto the spheres. Although -spheres are hardcoded, the source code could easily be changed here to project onto e.g. -ellipsoids. 3. Calculate Radial Transformation The areal coordinate ˆr of each sphere is calculated by ˆr = ˆr(r) = [ ] / γθθγ φφdθdφ () 4π

4 4 from which dˆr dη = γθθ,η γ φφ + γ θθ γ φφ,η dθdφ () 6πˆr γθθ γ φφ 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 ( ) ˆr S(ˆr) = γ rr dθdφ (3) r M(ˆr) = ˆr S (4) 3.4 Calculate Regge-Wheeler Variables where c lm = c lm = h +lm = H +lm = S K +lm = G +lm = γˆrφ Ylm,θ γˆrθy lm,φ dω l(l + ) sin θ {( l(l + )(l )(l + ) sin θ γ θθ + sin 4 θ γ φφ + sin θ γ θφ(ylm,θθ cot θylm,θ } sin θ Y lm,φφ) dω { l(l + ) γˆrˆr YlmdΩ ˆr ( γ θθ + sin θ γ φφ + ˆr (l )(l + ) γˆrθ Y lm,θ + sin θ γˆrφy lm,φ ) Y lmdω {( γ θθ γ φφ sin θ } + 4 sin θ γ θφ(y lm,θφ cot θy ˆr l(l + )(l )(l + ) } dω lm,φ) dω {( γ θθ γ φφ sin θ } + 4 sin θ γ θφ(y lm,θφ cot θy lm,φ) 3.5 Calculate Gauge Invariant Quantities dω [ Q lm = (l + )! c lm + ( ˆr c lm (l )! Q + lm = (l )(l + ) + 6M/ˆr ) (sin θy lm,θφ cos θy lm,φ) ) ( Ylm,θθ cot θylm,θ ) sin θ Y lm,φφ ) ( Ylm,θθ cot θylm,θ ) sin θ Y lm,φφ γˆrˆr = r r ˆr ˆr γ rr (5) γˆrθ = r ˆr γ rθ (6) γˆrφ = r ˆr γ rφ (7) ˆr c lm )] S ˆr (8) (l )(l + ) (4ˆrS k + l(l + )ˆrk ) (9) l(l + )

5 5 where k = K +lm + Sˆr (ˆr ˆr G +lm h +lm ) (30) k = S [H+lm ˆr ˆr k ( MˆrS )k + S / ˆr (ˆr S / ˆr G +lm S / h +lm (3) 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. Basic Usage 4. 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 (-sphere) used, and these detectors are identified in the name of each output file by R, R,.... The extension denotes whether coordinate time (ṫl) or proper time ( ul) is used for the first column. rsch R?.[tu]l The extracted areal radius on each -sphere. mass R?.[tu]l Mass estimate calculated from g rr on each -sphere. Qeven R???.[tu]l The even parity gauge invariate variable (waveform) on each -sphere. This is a complex quantity, the nd column is the real part, and the third column the imaginary part. Qodd R???.[tu]l The odd parity gauge invariate variable (waveform) on each -sphere. This is a complex quantity, the nd column is the real part, and the third column the imaginary part. ADMmass R?.[tu]l Estimate of ADM mass enclosed within each -sphere. (To produce this set doadmmass = yes ). momentum [xyz] R?.[tu]l Estimate of momentum at each -sphere. (To produce this set do momentum = yes ). spin [xyz] R?.[tu]l Estimate of momentum at each -sphere. (To produce this set do spin = yes ). 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 6 6 Appendix: Regge-Wheeler Harmonics (ê ) lm = (ê ) lm = ( ˆf ) lm = ( ˆf ) lm = ( ˆf 3 ) lm = ( ˆf 4 ) lm = 0 sin θ Y lm,φ sin θy lm,θ sin θ (Y lm,θφ cot θy lm,φ ). 0 sin θ [Y lm,θθ cot θy lm,θ sin θ Y lm,φφ] sin θ[y lm,θφ cot θy lm,φ ] 0 Y lm,θ Y lm,φ Y lm Y lm sin θy lm Y lm,θθ. 0 Y lm,θφ cot θy lm,φ Y lm,φφ + sin θ cos θy lm,θ 7 Appendix: Transformation Between Cartesian and Spherical Coordinates First, the transformations between metric components in (x, y, z) and (r, θ, φ) coordinates. Here, ρ = x + y = r sin θ, x r = sin θ cos φ = x r y r = sin θ sin φ = y r z r = cos θ = z r x = r cos θ cos φ = xz θ ρ y = r cos θ sin φ = yz θ ρ z θ = r sin θ = ρ x φ = r sin θ sin φ = y y φ = r sin θ cos φ = x z φ = 0 γ rr = r (x γ xx + y γ yy + z γ zz + xyγ xy + xzγ xz + yzγ yz ) γ rθ = rρ (x zγ xx + y zγ yy zρ γ zz + xyzγ xy + x(z ρ )γ xz + y(z ρ )γ yz ) γ rφ = r ( xyγ xx + xyγ yy + (x y )γ xy yzγ xz + xzγ yz )

7 7 γ θθ = ρ (x z γ xx + xyz γ xy xzρ γ xz + y z γ yy yzρ γ yz + ρ 4 γ zz ) γ θφ = ρ ( xyzγ xx + (x y )zγ xy + ρ yγ xz + xyzγ yy ρ xγ yz ) γ φφ = y γ xx xyγ xy + x γ yy or, γ rr = sin θ cos φγ xx + sin θ sin φγ yy + cos θγ zz + sin θ cos φ sin φγ xy + sin θ cos θ cos φγ xz + sin θ cos θ sin φγ yz γ rθ = r(sin θ cos θ cos φγ xx + sin θ cos θ sin φ cos φγ xy + (cos θ sin θ) cos φγ xz + sin θ cos θ sin φγ yy +(cos θ sin θ) sin φγ yz sin θ cos θγ zz ) γ rφ = r sin θ( sin θ sin φ cos φγ xx sin θ(sin φ cos φ)γ xy cos θ sin φγ xz + sin θ sin φ cos φγ yy + cos θ cos φγ yz ) γ θθ = r (cos θ cos φγ xx + cos θ sin φ cos φγ xy sin θ cos θ cos φγ xz + cos θ sin φγ yy sin θ cos θ sin φγ yz + sin θγ zz ) γ θφ = r sin θ( cos θ sin φ cos φγ xx cos θ(sin φ cos φ)γ xy + sin θ sin φγ xz + cos θ sin φ cos φγ yy sin θ cos φγ yz ) γ φφ = r sin θ(sin φγ xx sin φ cos φγ xy + cos φγ yy ) We also need the transformation for the radial derivative of the metric components: γ rr,η = sin θ cos φγ xx,η + sin θ sin φγ yy,η + cos θγ zz,η + sin θ cos φ sin φγ xy,η + sin θ cos θ cos φγ xz,η + sin θ cos θ sin φγ yz,η γ rθ,η = r γ rθ + r(sin θ cos θ cos φγ xx,η + sin θ cos θ sin φ cos φγ xy,η + (cos θ sin θ) cos φγ xz,η + sin θ cos θ sin φγ yy,η + (cos θ sin θ) sin φγ yz,η sin θ cos θγ zz,η ) γ rφ,η = r γ rφ + r sin θ( sin θ sin φ cos φγ xx,η sin θ(sin φ cos φ)γ xy,η cos θ sin φγ xz,η + sin θ sin φ cos φγ yy,η + cos θ cos φγ yz,η ) γ θθ,η = r γ θθ + r (cos θ cos φγ xx,η + cos θ sin φ cos φγ xy,η sin θ cos θ cos φγ xz,η + cos θ sin φγ yy,η sin θ cos θ sin φγ yz,η + sin θγ zz,η ) γ θφ,η = r γ θφ + r sin θ( cos θ sin φ cos φγ xx,η cos θ(sin φ cos φ)γ xy,η + sin θ sin φγ xz,η + cos θ sin φ cos φγ yy,η sin θ cos φγ yz,η ) γ φφ,η = r γ φφ + r sin θ(sin φγ xx,η sin φ cos φγ xy,η + cos φγ yy,η ) 8 Appendix: Integrations Over the -Spheres This is done by using Simpson s rule twice. Once in each coordinate direction. Simpson s rule is x x f(x)dx = h 3 [f + 4f + f 3 + 4f f N + 4f N + f N ] + O(/N 4 ) (3) N must be an odd number. References [] Abrahams A.M. & Cook G.B. Collisions of boosted black holes: Perturbation theory predictions of gravitational radiation Phys. Rev. D 50 R364-R367 (994).

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