xx Extracting Gravitational Waves and Other Quantities from Numerical Spacetimes

Extracting Gravitational Waves and Other Quantities from Numerical Spacetimes

Gabrielle Allen

Abstract

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 which can be written as a Schwarzschild background with perturbations :

with The 3-metric perturbations can be decomposed using tensor harmonics into where

. γ_{ij}(t, r, θ, ϕ) = ∑_{l=0}^{∞} ∑_{m=−l}^{l} γ_{ij}^{lm}(t, r)

and

. γ_{ij}(t, r, θ, ϕ) = ∑_{k=0}^{6} p_{k}(t, r)V_{k}(θ, ϕ)

where is some basis for tensors on a 2-sphere in 3-D Euclidean space. Working with the Regge-Wheeler basis (see Sectionx6) the 3-metric is then expanded in terms of the (six) standard Regge-Wheeler functions x[19],x[16]. Where each of the functions is either odd () or even () parity. The decomposition is then written which we can write in an expanded form as A similar decomposition allows the four gauge components of the 4-metric to be written in terms of three even-parity variables and the one odd-parity variable Also from we have It is useful to also write this with the perturbation split into even and odd parity parts:

. g_{αβ} = g_{αβ}^{background} + ∑_{l,m} g_{αβ}^{lm,odd} + ∑_{l,m} g_{αβ}^{lm,even}

where (dropping some superscripts) Now, for such a Schwarzschild background we can define two (and only two) unconstrained gauge invariant quantities and , which from [3] are where NOTE: These quantities compare with those in Moncrief [16] by 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 3 Numerical Implementation

The implementation assumes that the numerical solution, on a Cartesian grid, is approximately Schwarzshild on the spheres of constant 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 2-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 factor on each sphere. Combined with the areal radius This also produces an estimate of the mass.

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.

3.1 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 of each sphere is calculated by from which 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 factor and Mass Estimate

3.4 Calculate Regge-Wheeler Variables

where

3.5 Calculate Gauge Invariant Quantities where

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, x. The extension denotes whether coordinate time (xl) or proper time (u̇ l) is used for the first column.

     rschxR?.[tu]l

The extracted areal radius on each 2-sphere.

massxR?.[tu]l

Mass estimate calculated from on each 2-sphere.

QevenxR?x??.[tu]l

The even parity gauge invariate variable (waveform) on each 2-sphere. This is a complex quantity, the 2nd column is the real part, and the third column the imaginary part.

QoddxR?x??.[tu]l

The odd parity gauge invariate variable (waveform) on each 2-sphere. This is a complex quantity, the 2nd column is the real part, and the third column the imaginary part.

ADMmassxR?.[tu]l

Estimate of ADM mass enclosed within each 2-sphere. (To produce this set doADMmass = ``yes'').

momentumx[xyz]xR?.[tu]l

Estimate of momentum at each 2-sphere. (To produce this set doxmomentum = ``yes'').

spinx[xyz]xR?.[tu]l

Estimate of momentum at each 2-sphere. (To produce this set doxspin = ``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 Appendix: Regge-Wheeler Harmonics

7 Appendix: Transformation Between Cartesian and Spherical Coordinates First, the transformations between metric components in and coordinates. Here, ,

or,

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

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 must be an odd number.

References [1]xxxAbrahams A.M. & Cook G.B. “Collisions of boosted black holes: Perturbation theory predictions

of gravitational radiation” Phys. Rev. D 50 R2364-R2367 (1994).

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

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

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

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

[7]xxxAnninos 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]xxxArfken, G. “Mathematical Methods for Physicists”, Academic Press (1985).

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

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

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

[12]xxxCunningham C.T., Price R.H., Moncrief V., “Radiation from collapsing relativistic stars. I.

Linearized Odd-Parity Radiation” Ap. J. 224 543-667 (1978). [13]xxxCunningham C.T., Price R.H., Moncrief V., “Radiation from collapsing relativistic stars. I.

Linearized Even-Parity Radiation” Ap. J. 230 870-892 (1979). [14]xxxLandau L.D. & Lifschitz E.M., “The Classical Theory of Fields” (4th Edition), Pergamon Press

(1980). [15]xxxMathews J. “”, J. Soc. Ind. Appl. Math. 10 768 (1962). [16]xxxMoncrief V. “Gravitational perturbations of spherically symmetric systems. I. The exterior

problem” Annals of Physics 88 323-342 (1974). [17]xxxPress W.H., Flannery B.P., Teukolsky S.A., & Vetterling W.T., “Numerical Recipes, The Art of

Scientific Computing” Cambridge University Press (1989). [18]xxxPrice R.H. & Pullin J. “Colliding black holes: The close limit”, Phys. Rev. Lett. 72 3297-3300

(1994). [19]xxxRegge T., & Wheeler J.A. “Stability of a Schwarzschild Singularity”, Phys. Rev. D 108 1063

(1957). [20]xxxSeidel E. Phys Rev D. 42 1884 (1990). [21]xxxThorne K.S., “Multipole expansions of gravitational radiation”, Rev. Mod. Phys. 52 299 (1980). [22]xxxVishveshwara C.V., “Stability of the Schwarzschild metric”, Phys. Rev. D. 1 2870, (1970).

[23]xxxZerilli F.J., “Tensor harmonics in canonical form for gravitational radiation and other applications”, J. Math. Phys. 11 2203, (1970).

[24]xxxZerilli F.J., “Gravitational field of a particle falling in a Schwarzschild geometry analysed in tensor harmonics”, Phys. Rev. D. 2 2141, (1970).

9 Parameters

11allxmodes 10Scope: privatex01 BOOLEAN

11Description: Extract: all l,m modes up to l

11                  10  x01  Default: yes

11

11cauchy  10Scope: privatex01  BOOLEAN

11Description: Do Cauchy data extraction at given timestep

11                  10  x01  Default: no

11

11cauchyxdr  10Scope: privatex01  REAL

11Description: Gridspacing for Cauchy data extraction

11  Range  10  x01  Default: 0.2

11*:*  10

11

11cauchyxr1  10Scope: privatex01  REAL

11Description: First radius for Cauchy data extraction

11  Range  10  x01  Default: 1.0

11*:*  10

11

11cauchyxtimestep  10Scope: privatex01  INT

11Description: Timestep for Cauchy data extraction

11  Range  10  x01  Default: (none)

110:*  10

11

11detector1  10Scope: privatex01  REAL

11Description: Coordinate radius of detector 1

11  Range  10  x01  Default: 5.0

110:*  10

11

11detector2  10Scope: privatex01  REAL

11Description: Coordinate radius of detector 2

11  Range  10  x01  Default: 5.0

110:*  10

11

11detector3  10Scope: privatex01  REAL

11Description: Coordinate radius of detector 3

11  Range  10  x01  Default: 5.0

110:*  10

11detector4  10Scope: privatex01  REAL

11Description: Coordinate radius of detector 4

11  Range  10  x01  Default: 5.0

110:*  10

11

11detector5  10Scope: privatex01  REAL

11Description: Coordinate radius of detector 5

11  Range  10  x01  Default: 5.0

110:*  10

11

11detector6  10Scope: privatex01  REAL

11Description: Coordinate radius of detector 6

11  Range  10  x01  Default: 5.0

110:*  10

11

11detector7  10Scope: privatex01  REAL

11Description: Coordinate radius of detector 7

11  Range  10  x01  Default: 5.0

110:*  10

11

11detector8  10Scope: privatex01  REAL

11Description: Coordinate radius of detector 8

11  Range  10  x01  Default: 5.0

110:*  10

11

11detector9  10Scope: privatex01  REAL

11Description: Coordinate radius of detector 9

11  Range  10  x01  Default: 5.0

110:*  10

11

11doxmomentum  10Scope: privatex01  BOOLEAN

11Description: Calculate momentum at extraction radii

11                  10  x01  Default: no

11

11doxspin  10Scope: privatex01  BOOLEAN

11Description: Calculate spin at extraction radii

11                  10  x01  Default: no

11

11doadmmass  10Scope: privatex01  BOOLEAN

11Description: Calculate ADM mass at extraction radii

11                  10  x01  Default: no

11

11interpolationxoperator 10Scope: privatex01  STRING

11Description: Interpolation operator to use (check LocalInterp)

11  Range  10  x01  Default: uniform cartesian

11.+  10

11

11interpolationxorder 10Scope: privatex01  INT

11Description: Order for interpolation

11 Range 10 x01 Default: 1
111:4  10Choose between first and forth order interpolation

11itout  10Scope: privatex01  INT

11Description: How often to extract, in iterations

11  Range  10  x01  Default: 1

110:*  10

11

11lxmode  10Scope: privatex01  INT

11Description: l mode

11  Range  10  x01  Default: 2

110:*  10

11

11mxmode  10Scope: privatex01  INT

11Description: m mode (ignore if extracting all modes

11  Range  10  x01  Default: (none)

110:*  10

11

11np  10Scope: privatex01  INT

11Description: Number of phi divisions

11  Range  10  x01  Default: 100

110:*  10

11

11nt  10Scope: privatex01  INT

11Description: Number of theta divisions

11  Range  10  x01  Default: 100

110:*  10

11

11numxdetectors  10Scope: privatex01  INT

11Description: Number of detectors

11  Range  10  x01  Default: (none)

110:*  10

11

11originxx  10Scope: privatex01  REAL

11Description: x-origin to extract about

11  Range  10  x01  Default: 0.0

11*:*  10

11

11originxy  10Scope: privatex01  REAL

11Description: y-origin to extract about

11  Range  10  x01  Default: 0.0

11*:*  10

11

11originxz  10Scope: privatex01  REAL

11Description: z-origin to extract about

11  Range  10  x01  Default: 0.0

11*:*  10

11

11timecoord  10Scope: privatex01  KEYWORD

11Description: Which time coordinate to use

11 Range 10 x01 Default: both
11proper  10
11coordinate  10

11both  10

11

11verbose  10Scope: privatex01  BOOLEAN

11Description: Say what is happening

11                  10  x01  Default: no

11outxdir  10Scope: shared from IOx01STRING

11

10 Interfaces

General Implements: extract

Inherits: grid admbase staticconformal io

Grid Variables

10.0.1 PRIVATE GROUPS

11x Group Names x10x Variable Names xx01 Details x11x
11  temps  10  x01  compact110
11                 10  temp3d  x01 dimensions113
11                 10  g00  x01 distribution11DEFAULT
11                 10  x01 group type11GF
11                 10  x01 timelevels111
11                 10  x01variable type11REAL

11

11 Schedule This section lists all the variables which are assigned storage by thorn EinsteinAnalysis/Extract. Storage can either last for the duration of the run (Always means that if this thorn is activated storage will be assigned, Conditional means that if this thorn is activated storage will be assigned for the duration of the run if some condition is met), or can be turned on for the duration of a schedule function.

Storage NONE

Scheduled Functions

CCTKxPARAMCHECK x extractxparamcheck x check parameters x

11x11Language:11c
11 x 11Options: 11global
11 x 11Type: 11function

11

CCTKxPOSTSTEP x extract x extract waveforms x

11x11Language:11fortran
11 x 11Storage: 11temps
11 x 11Type: 11function