Extracting Gravitational Waves and Other Quantities from
Numerical Spacetimes
Gabrielle Allen
September 2, 2018
NB: This documentation is taken from the Extract thorn on which WaveExtractL is based. There
may be some di๏ฌerences between WaveExtractL and Extract, which are not documented
here.
1 Introduction
Thorn Extract calculates ๏ฌrst 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 ๏ฌrst order gauge invariant waveform.
2 Physical System
2.1 Wave Forms
Assume a spacetime which can be
written as a Schwarzschild background
with perturbations :
| (1) |
with
| (2) |
The 3-metric perturbations can be
decomposed using tensor harmonics into
where
and
where
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
ย [19],ย [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
| (14) |
It is useful to also write this with the perturbation split into even and odd parity parts:
where (dropping some superscripts)
Now, for such a Schwarzschild background we can de๏ฌne 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
| (21) |
from which
| (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
factor and Mass Estimate
| (23) |
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 di๏ฌerent resolutions,
di๏ฌerent 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 ๏ฌles from Extract are always placed in
the main output directory de๏ฌned by CactusBase/IOUtil.
Output ๏ฌles are generated for each detector (2-sphere) used, and these detectors are identi๏ฌed in the name of
each output ๏ฌle by R1, R2, ….
The extension denotes whether coordinate time (แนซl) or proper time (uฬl) is used for the ๏ฌrst column.
- rsch_R?.[tu]l
The extracted areal radius on each 2-sphere.
- mass_R?.[tu]l
Mass estimate calculated from
on each 2-sphere.
- Qeven_R?_??.[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.
- Qodd_R?_??.[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.
- ADMmass_R?.[tu]l
Estimate of ADM mass enclosed within each 2-sphere. (To produce this set doADMmass = โโyes’’).
- momentum_[xyz]_R?.[tu]l
Estimate of momentum at each 2-sphere. (To produce this set do_momentum = โโyes’’).
- spin_[xyz]_R?.[tu]l
Estimate of momentum at each 2-sphere. (To produce this set do_spin = โโyes’’).
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
| (32) |
must be
an odd number.
References
[1]ย ย ย Abrahams A.M. & Cook G.B. โCollisions of boosted black holes: Perturbation theory predictions
of gravitational radiationโ Phys. Rev. D 50 R2364-R2367 (1994).
[2]ย ย ย Abrahams A.M., Shapiro S.L. & Teukolsky S.A. โCalculation of gravitational wave forms from
black hole collisions and disk collapse: Applying perturbation theory to numerical spacetimesโ Phys.
Rev. D. 51 4295 (1995).
[3]ย ย ย Abrahams A.M. & Price R.H. โApplying black hole perturbation theory to numerically generated
spacetimesโ Phys. Rev. D. 53 1963 (1996).
[4]ย ย ย Abrahams A.M. & Price R.H. โBlack-hole collisions from Brill-Lindquist initial data: Predictions
of perturbation theoryโ Phys. Rev. D. 53 1972 (1996).
[5]ย ย ย Abramowitz, M. & Stegun A. โPocket Book of Mathematical Functions (Abridged Handbook of
Mathematical Functionsโ, Verlag Harri Deutsch (1984).
[6]ย ย ย Andrade Z., & Price R.H. โHead-on collisions of unequal mass black holes: Close-limit predictionsโ,
preprint (1996).
[7]ย ย ย Anninos P., Price R.H., Pullin J., Seidel E., and Suen W-M. โHead-on collision of two black holes:
Comparison of di๏ฌerent approachesโ Phys. Rev. D. 52 4462 (1995).
[8]ย ย ย Arfken, G. โMathematical Methods for Physicistsโ, Academic Press (1985).
[9]ย ย ย Baker J., Abrahams A., Anninos P., Brant S., Price R., Pullin J. & Seidel E. โThe collision of
boosted black holesโ (preprint) (1996).
[10]ย ย ย Baker J. & Li C.B. โThe two-phase approximation for black hole collisions: Is it robustโ preprint
(gr-qc/9701035), (1997).
[11]ย ย ย Brandt S.R. & Seidel E. โThe evolution of distorted rotating black holes III: Initial dataโ
(preprint) (1996).
[12]ย ย ย Cunningham C.T., Price R.H., Moncrief V., โRadiation from collapsing relativistic stars. I.
Linearized Odd-Parity Radiationโ Ap. J. 224 543-667 (1978).
[13]ย ย ย Cunningham C.T., Price R.H., Moncrief V., โRadiation from collapsing relativistic stars. I.
Linearized Even-Parity Radiationโ Ap. J. 230 870-892 (1979).
[14]ย ย ย Landau L.D. & Lifschitz E.M., โThe Classical Theory of Fieldsโ (4th Edition), Pergamon Press
(1980).
[15]ย ย ย Mathews J. โโ, J. Soc. Ind. Appl. Math. 10 768 (1962).
[16]ย ย ย Moncrief V. โGravitational perturbations of spherically symmetric systems. I. The exterior
problemโ Annals of Physics 88 323-342 (1974).
[17]ย ย ย Press W.H., Flannery B.P., Teukolsky S.A., & Vetterling W.T., โNumerical Recipes, The Art of
Scienti๏ฌc Computingโ Cambridge University Press (1989).
[18]ย ย ย Price R.H. & Pullin J. โColliding black holes: The close limitโ, Phys. Rev. Lett. 72 3297-3300
(1994).
[19]ย ย ย Regge T., & Wheeler J.A. โStability of a Schwarzschild Singularityโ, Phys. Rev. D 108 1063
(1957).