We present a method for extracting gravitational radiation from a three-dimensional numerical relativity simulation and, using the extracted data, to provide outer boundary conditions. The method treats dynamical gravitational variables as nonspherical perturbations of Schwarzschild geometry. We discuss a code which implements this method and present results of tests which have been performed with a three dimensional numerical relativity code

Abrahams, A. M.

Anderson, A.

Anninos, P.

Baumgarte, T. W.

Bishop, N. T.

Brandt, S. R.

Browne, J. C.

Camarda, K.

Choptuik, M. W.

Cook, G. B.

Correll, R. R.

Evans, C. R.

Finn, L. S.

Fox, G. C.

Gómez, R.

Haupt, T.

Huq, M. F.

Kidder, L. E.

Klasky, S. A.

Laguna, P.

Landry, W.

Lehner, L.

Lenaghan, J.

Marsa, R. L.

Masso, J.

Matzner, R. A.

Mitra, S.

Papadopoulos, P.

Parashar, M.

Rezzolla, L.

Rupright, M. E.

Saied, F.

Saylor, P. E.

Scheel, M. A.

Seidel, E.

Shapiro, S. L.

Shoemaker, D.

Smarr, L.

Szilágyi, B.

Teukolsky, S. A.

van Putten, M. H. P. M.

Walker, P.

Winicour, J.

York, J. W.

We present a method for extracting gravitational radiation from a three-dimensional numerical relativity simulation and, using the extracted data, to provide outer boundary conditions. The method treats dynamical gravitational variables as nonspherical perturbations of Schwarzschild geometry. We discuss a code which implements this method and present results of tests which have been performed with a three-dimensional numerical relativity code

Caltech Authors - Main

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Gravitational wave extraction and outer boundary conditions by perturbative
matching
The Binary Black Hole Grand Challenge Alliance:
A. M. Abrahamsa,b,c, L. Rezzollaa, M. E. Ruprightc, A. Andersonc, P. Anninosa, T. W. Baumgartea,
N. T. Bishopd,l, S. R. Brandta, J. C. Brownee, K. Camardaf , M. W. Choptuike, G. B. Cookg, C. R. Evansc,
L. S. Finnh, G. Foxi, R. Go´mezj, T. Haupti, M. F. Huqe, L. E. Kidderh, S. Klaskyi, P. Lagunaf , W. Landryg,
L. Lehnerj, J. Lenaghanc, R. L. Marsae, J. Massoa, R. A. Matznere,k, S. Mitrae, P. Papadopoulosf, M. Parashare,
F. Saieda, P. E. Saylora, M. A. Scheelc, E. Seidela, S. L. Shapiroa, D. Shoemakere, L. Smarra, B. Szila´gyij,
S. A. Teukolskyg, M. H. P. M. van Putteng, P. Walkera, J. Winicourj, J. W. York Jrc.
aUniversity of Illinois at Urbana-Champaign, Urbana, IL 61801
bJ. P. Morgan, 23 Wall St., New York, NY 10260
cUniversity of North Carolina, Chapel Hill, NC 27599
dUniversity of South Africa, P.O. Box 392, Pretoria 0001, South Africa
eThe University of Texas at Austin, Austin, Texas 78712
fPenn State University, University Park, PA 16802
gCornell University, Ithaca, NY 14853
hNorthwestern University, Evanston, IL 60208
iSyracuse University, Syracuse, NY 13244-4100
jUniversity of Pittsburgh, Pittsburgh, PA 15260
kLead Principal Investigator
lAssociate
We present a method for extracting gravitational radiation from a three-dimensional numerical
relativity simulation and, using the extracted data, to provide outer boundary conditions. The
method treats dynamical gravitational variables as nonspherical perturbations of Schwarzschild ge-
ometry. We discuss a code which implements this method and present results of tests which have
been performed with a three dimensional numerical relativity code.
PACS numbers: 04.70.Bw, 04.25.Dm, 04.25.Nx, 04.30.Db
Numerical relativity represents the only currently vi-
able method for obtaining solutions to Einstein equations
for highly dynamical and strong field sources of gravita-
tional radiation. Using these techniques to study coa-
lescing black hole binaries is the purpose of the multi-
institutional Binary Black Hole “Grand Challenge” Al-
liance effort [1] which is presently underway in the United
States. This effort is also motivated by the prospect of
observations with the next generation of gravitational
wave detectors.
In addition to tremendous demands on computational
resources, implementing the standard 3+1 [2,3] formula-
tion of Einstein theory as a Cauchy problem [4] is compli-
cated considerably by the necessity of imposing bound-
ary conditions which maintain numerical accuracy and
the physical correctness of the solution. Both inner and
outer boundary conditions have received considerable at-
tention. Recent efforts on interior boundaries have fo-
cused on the excision of the interior of the black hole from
the computational domain (see, for example, [5]). This
paper will concentrate on the problem of outer boundary
conditions applied at a finite radius around a source of
gravitational waves.
Proper boundary conditions on spacelike slices of
asymptotically flat spacetimes are essential for the ac-
curate computation of the gravitational waveforms pro-
duced in the strong field region that represent the obser-
vationally relevant aspect of the computation. Since it is
not feasible to simulate on spacelike slices out to arbitrar-
ily large distances from the source, it is necessary to ex-
tract gravitational waves comparatively near the strong
field region and to have boundary conditions that allow
radiation to pass cleanly off the mesh. If poor outgo-
ing boundary conditions are imposed, spurious radiation
is produced which can contaminate the computed grav-
itational waveform. Additionally, the outer boundary is
usually close enough to the isolated source that backscat-
ter of radiation from curvature is significant. This source
of incoming radiation needs to be built into the outer
boundary conditions. An approach to the extraction of
gravitational wave information and the computation of
outer boundary conditions that exploits the matching of
the interior numerical solution with an exterior perturba-
tive solution on spacelike slices has been developed dur-
1
ing the past decade and applied to a number of different
physical scenarios [6–8]. Extension of these techniques
to three dimensional (3D) simulations has been one of
the efforts of the Alliance. A parallel development is also
underway in the Alliance that matches interior Cauchy
solutions to exterior solutions on characteristic hypersur-
faces [9].
In this Letter we report the first successful application
of the perturbative matching approach to provide outer
boundary conditions for a 3D numerical relativity code.
In this context, the perturbative matching method should
be viewed as a general “module” that can be coupled to
any nonlinear “interior” 3D code. While the latter solves
the Einstein equations in the highly dynamical strong
field region, the former extracts gravitational wave data
and imposes outer boundary conditions. In the follow-
ing we give an overview of the method and present test
results obtained from the evolution of linear waves.
Nonspherical perturbations of Schwarzschild geome-
try have been studied for many years in the context of
black hole perturbations [10,11] and as a way to extract
gauge-invariant information about gravitational radia-
tion [6]. More recently Schwarzschild perturbation the-
ory has been found to be useful in studying the late-time
behavior of the coalescence of compact binaries in a nu-
merical simulation after the horizon has formed [12,13,8].
We base our treatment of Schwarzschild perturbation
theory on a recent hyperbolic formulation of Einstein
field equations [14]. A principal advantage of this ap-
proach is that we can easily derive perturbative wave
equations in terms of the standard 3 + 1 variables, mak-
ing it straightforward to match exterior perturbative so-
lutions to interior numerical ones. This method leads to
spatially gauge-invariant radial wave equations for each
angular mode [15] which we have used: a) to extract grav-
itational radiation from a 3D numerical relativity simula-
tion, b) to evolve this information to large radius yielding
an approximate asymptotic waveform and c) to provide
outer boundary conditions for such a simulation. One of
the major advantages of the perturbative method is that
we have replaced the computationally expensive 3D evo-
lution of the gravitational waves via the nonlinear Ein-
stein equations with a set of 1D linear equations we can
integrate to high accuracy and minimal computational
costs.
We split the gravitational quantities of interest into
background and perturbed parts: the three-metric gij =
g˜ij + hij , the extrinsic curvature Kij = K˜ij + κij , the
lapse function N = N˜ + α and the shift vector βi =
β˜i + vi, where the tilde denotes background quantities.
We assume a Schwarzschild background,
g˜ijdx
idxj = N˜−2dr2 + r2(dθ2 + sin2 θdφ2) , (1)
N˜ =
(
1−
2M
r
)1/2
. (2)
(It follows that K˜ij = 0 = β˜
i). The perturbed parts have
arbitrary angular dependence.
We use this approximation to linearize the hyperbolic
equations and, in particular, we find that the wave equa-
tion for Kij reduces to a linear wave equation for κij in-
volving also the background lapse [15]. We separate the
angular dependence in this equation by expanding κij in
terms of tensor spherical harmonics (eˆ1)ij , . . . , (fˆ4)ij . In
the notation of [11], we have
κij = a×(t, r)(eˆ1)ij + rb×(t, r)(eˆ2)ij +
N˜−2a+(t, r)(fˆ2)ij + rb+(t, r)(fˆ1)ij +
r2c+(t, r)(fˆ3)ij + r
2d+(t, r)(fˆ4)ij , (3)
where a×, b× are the odd-parity multipoles, while a+,
b+, c+, d+ are the even-parity ones. Note that we have
suppressed the angular indices (ℓ,m) of (eˆ1)ij , . . . , (fˆ4)ij
and of a×, . . . , d+ over which there is an implicit sum.
In odd-parity, we take the wave equation for κrθ
and use the linearized momentum constraints, ∇˜k(κ
k
i −
δkiκ
j
j) = 0, to eliminate the odd-parity amplitude (b×).
In even-parity, we use the wave equation for κrr to-
gether with the wave equation obtained for the trace
κ = κjj = h(t, r)Yℓm and the linearized momentum con-
straints. In this way we eliminate b+ and c+ and obtain
two coupled equations for a+ and h. For each (ℓ,m)
mode, we therefore have one odd-parity equation{
∂2t − N˜
4∂2r −
2
r
N˜2∂r −
2M
r3
(
1−
3M
2r
)
+
N˜2
[
ℓ(ℓ+ 1)
r2
−
6M
r3
]}
(a×)ℓm = 0 , (4)
and two coupled even-parity equations,[
∂2t − N˜
4∂2r −
6
r
N˜4∂r + N˜
2 ℓ(ℓ+ 1)
r2
−
6
r2
+
14M
r3
−
3M2
r4
]
(a+)ℓm +
[
4
r
N˜2
(
1−
3M
r
)
∂r +
2
r2
(
1−
M
r
−
3M2
r2
)]
(h)
ℓm
= 0 , (5)
[
∂2t − N˜
4∂2r −
2
r
N˜2∂r + N˜
2 ℓ(ℓ+ 1)
r2
+
2M
r3
−
7M2
r4
]
(h)
ℓm
−
2M
r3
(
3−
7M
r
)
(a+)ℓm = 0 . (6)
(The usual Regge-Wheeler and Zerilli equations can be
obtained by a more complete analysis [15].)
We now turn to implementation and tests of this tech-
nique. Consider a numerical relativity simulation which
2
0t
1t
Boundary
2-sphere
E
Extraction
1D Outer
r
Exterior
3D Outer
Boundary
Interior
t
FIG. 1. Location of the different outer boundaries and
of the extraction 2-sphere for two successive timeslices. The
dark shaded region shows the spatial domain over which the
3D nonlinear equation are solved.
evolves Einstein equations in either the standard 3 + 1
(ADM) or hyperbolic form on a “interior” 3D grid. (No
restrictions need be put on the choice of the 3D coordi-
nate system.) In the tests described here, we have used
the ADM 3D interior code of the Alliance [16] to evolve
linear, ℓ = 2, m = 0, Teukolsky waves [17]. The interior
3D grid uses (topologically) Cartesian coordinates and
up to (129)3 gridpoints. In the results shown, the ini-
tial data consists of a metric formed from the sum of the
background Cartesian metric and a Gaussian envelope of
amplitude A = 10−6, width b = 1 and of a vanishing ex-
trinsic curvature. The ADM evolution is undertaken on
a cubical grid of extent {x, y, z} = ± 4, with unit lapse
and zero shift.
During each time step, the procedure for extracting
radiation and imposing outer boundary conditions pro-
ceeds in three steps: (1) extraction of the independent
amplitudes and their time derivatives on a 2-sphere of
radius r
E
inscribed in the grid, (2) evolution of the ra-
dial wave equations (4)–(6) out to large radii using the
extracted amplitudes to construct inner boundary val-
ues on the “exterior” 1D grid and (3) reconstruction and
“injection” of Kij and ∂tKij at specified gridpoints at or
near the boundary of the interior grid. The last step uses
the momentum constraints and the inverse of the trans-
formation employed in step (1). In Figure 1, we show
the schematic location of the different outer boundaries
and of the extraction 2-sphere on two successive spacelike
time slices at t = t0 and t = t1.
The first step involves interpolating Kij and ∂tKij
onto the 2-sphere and evaluating their projections onto
the appropriate conjugate tensor spherical harmonics to
eliminate the angular dependence and obtain the cor-
FIG. 2. Convergence to the analytic solution of the ex-
tracted (r = 1) and evolved (r = 8) multipole (a+)20. The
amplitude is scaled by r3 to compensate for the radial fall-off.
responding multipoles. The precise location of the 2-
sphere within the interior 3D grid depends on the specific
case under investigation. In general, for the perturbative
matching to be justified, we require that the extraction
surface be located in a region where the gravitational
field is close to Schwarzschild. One must be careful that
the numerical errors introduced by the interior evolution
do not dominate the exterior solution. (Further details
will be presented elsewhere [18].)
The second step entails evolving equations (4)–(6). In
the test presented here, we consider a flat background
with M = 0. The initial data for the 1D exterior grid
is set consistently with the interior 3D initial data. Dur-
ing each time integration of the 1D exterior grid, the
extracted multipoles are used as inner boundary values
and standard Sommerfeld outgoing wave conditions are
imposed at the outer boundary (e.g., at r = 30). The
top diagram of Fig. 2 shows the only relevant multipole
for this test [19], extracted from the interior 3D grid at a
2-sphere of radius r
E
= 1 [20]; the lower diagram shows
its evolved waveform at a radius r = 8. Different curves
refer to different resolutions of the interior 3D grid and
show the convergence to the analytic solution.
The third and final step consists of computing outer
boundary values for the interior 3D grid, and this step
can proceed in one of two ways. In the first method, the
values of Kij reconstructed from the exterior 1D data
are injected as Dirichlet data at all the boundary points
of the interior 3D grid. (No restrictions are put on the
shape of the 2-surface where the data are injected.) For
a code using a hyperbolic formulation, ∂tKij can also
3
FIG. 3. Convergence to zero of the L2 norm of the error
of Kzz integrated over the 3D outer boundary surface.
be provided at the boundary. Moreover, the evolution
equation of the three-metric can be integrated using Kij
at the boundary so Dirichlet data for gij need not be
provided there. The second method relates, at the 3D
outer boundary, the null derivatives of the extrinsic cur-
vature obtained from the interior grid (Kij) and from
the perturbative module (κij , since background extrinsic
curvature is assumed to be zero):
∂
∂t
(Kij − κij) +
∂
∂r
(Kij − κij) +
2
r
(Kij − κij) = 0 (7)
This method resembles a Sommerfeld condition but is
more general since it can be used in regions where the
radiation is not dominated by the asymptotic outgoing
behavior. Moreover it takes into account arbitrary angu-
lar dependence, as well as the effects of a Schwarzschild
black hole background. Experimentation with the two
methods has shown that this “perturbative Sommerfeld”
approach is very accurate and generally more stable than
the direct injection of Dirichlet data. Figure 3 shows
convergence to zero of the L2 norm of the difference be-
tween one component of Kij (namely Kzz) and its ana-
lytic value, integrated over the entire outer boundary of
the interior 3D grid.
An important property of any outer boundary imple-
mentation is that it should allow the outgoing radiation
to escape freely to infinity. In practice, however, a fi-
nite discretization always produces a certain amount of
reflection at the outer boundary and this could drive in-
stabilities which grow exponentially. We have compared
the long-term stability obtained with the perturbative
outer boundary module with that of the best “alterna-
tive” outer boundary conditions we have tried: namely a
FIG. 4. Long-term evolution of the L2 norm of the error
in Kzz at the 3D outer boundary surface. The inset shows
Kzz at the outer boundary along the x−axis. The interior
grid has (33)3 gridpoints and the code is stopped when the
norm is unity.
standard outgoing Sommerfeld condition. Fig. 4 shows
the results of this comparison both on a long time scale
and on a shorter time scale. (The calculations were per-
formed using a very coarse resolution.) As shown in the
inset (first echoes at t ∼ 7), the amount of reflection
produced with the perturbative method is smaller than
that produced by the Sommerfeld condition. Moreover,
the use of perturbative boundary conditions delays the
onset of exponential error growth and allows for a much
longer evolution (see main figure). Increasing the interior
resolution and the number of (ℓ,m) modes used we can
further prolong the running time. Using (49)3 interior
gridpoints, we are able to evolve the code for more than
50 crossing times. In strong contrast, when a standard
Sommerfeld condition is used, increased interior resolu-
tion results in a shorter running time [18].
This work was supported by the NSF Binary Black
Hole Grand Challenge Grant Nos. NSF PHY 93–18152,
NSF PHY 93–10083, ASC 93–18152 (ARPA supple-
mented). Computations were performed at NPAC (Syra-
cuse University) and at NCSA (University of Illinois at
Urbana-Champaign).
[1] Information about the Binary Black Hole Grand Chal-
lenge Alliance, its goals and present status of research can
be found at the web page: http://www.npac.syr.edu/
4
projects/bh/.
[2] R. Arnowitt, S. Deser and C. W. Misner, in Gravitation,
edited by L. Witten, Wiley, New York (1962).
[3] J. W. York in Sources of Gravitational Radiation, edited
by L. Smarr, Cambridge Univ. Press, Cambridge (1979).
[4] Y. Choquet-Bruhat and J. W. York in General Relativity
and Gravitation I, edited by A. Held, Plenum, New York
(1980).
[5] E. Seidel and W.-M. Suen, Phys. Rev. Lett., 69, 1845
(1992).
[6] A. M. Abrahams and C. R. Evans, Phys. Rev. D, 37, 317
(1988); Phys. Rev. D, 42, 2585 (1990).
[7] A. M. Abrahams et al., Phys. Rev. D 45, 3544 (1992);
P. Anninos et al., Phys. Rev. Lett. 71, 2851 (1993).
[8] A. M. Abrahams, S. L. Shapiro and S. A. Teukolsky,
Phys. Rev. D 51, 4295 (1995); A. M. Abrahams and R. H.
Price, Phys. Rev. D 53, 1963, Ibidem, 1972 (1996).
[9] N. T. Bishop, R. Gomez, P. R. Holvorcem, R. A.
Matzner, P. Papadopoulos and J. Winicour Phys. Rev.
Lett. 76, 4303 (1996); N. T. Bishop, R. Gomez, L.
Lehner, J. Winicour Phys. Rev. D. 54, 6153 (1996).
[10] T. Regge and J.A. Wheeler, Phys. Rev. 108, 1063 (1957);
F. Zerilli, Phys. Rev. Lett. 24 737 (1970).
[11] V. Moncrief, Annals of Physics 88, 323 (1974).
[12] R. H. Price and J. Pullin, Phys. Rev. Lett. 72, 3297
(1994).
[13] A. M. Abrahams and G. Cook, Phys. Rev. D , (1994).
[14] Y. Choquet-Bruhat and J. W. York, C. R. Acad. Sci.
Paris, 321, 1089 (1995); A. Abrahams, A. Anderson,
Y. Choquet-Bruhat and J. W. York, Phys. Rev. Lett.
75, 3377 (1995).
[15] A. M. Abrahams, A. Anderson, C. R. Evans, C. Lea,
J. Lenaghan, M. Rupright and J. W. York, In prepara-
tion.
[16] The Binary Black Hole Grand Challenge Alliance, In
preparation.
[17] S. Teukolsky, Phys. Rev. D 26, 745 (1982).
[18] A. M. Abrahams, L. Rezzolla and M. Rupright, In prepa-
ration.
[19] Other (ℓ,m) multipoles are also computed and used in
the decomposition; but, given the initial data chosen,
their amplitudes are several orders of magnitude smaller
[18].
[20] On a flat background spacetime, the position of the ex-
traction 2-sphere is arbitrary. Similar results have been
obtained also for r
E
= 0.5, 1.5, 2, 2.5, 3, 3.5 [18].
5


Gravitational Wave Extraction and Outer Boundary Conditions by Perturbative Matching

Carolina Digital Repository

arXiv:gr-qc/9709082v1  30 Sep 1997Gravitational wave extraction and outer boundary conditions by perturbativematchingThe Binary Black Hole Grand Challenge Alliance:A. M. Abrahamsa,b,c, L. Rezzollaa, M. E. Ruprightc, A. Andersonc, P. Anninosa, T. W. Baumgartea,N. T. Bishopd,l, S. R. Brandta, J. C. Brownee, K. Camardaf , M. W. Choptuike, G. B. Cookg, C. R. Evansc,L. S. Finnh, G. Foxi, R. Gómezj, T. Haupti, M. F. Huqe, L. E. Kidderh, S. Klaskyi, P. Lagunaf , W. Landryg,L. Lehnerj, J. Lenaghanc, R. L. Marsae, J. Massoa, R. A. Matznere,k, S. Mitrae, P. Papadopoulosf, M. Parashare,F. Saieda, P. E. Saylora, M. A. Scheelc, E. Seidela, S. L. Shapiroa, D. Shoemakere, L. Smarra, B. Szilágyij,S. A. Teukolskyg, M. H. P. M. van Putteng, P. Walkera, J. Winicourj, J. W. York Jrc.aUniversity of Illinois at Urbana-Champaign, Urbana, IL 61801bJ. P. Morgan, 23 Wall St., New York, NY 10260cUniversity of North Carolina, Chapel Hill, NC 27599dUniversity of South Africa, P.O. Box 392, Pretoria 0001, South AfricaeThe University of Texas at Austin, Austin, Texas 78712fPenn State University, University Park, PA 16802gCornell University, Ithaca, NY 14853hNorthwestern University, Evanston, IL 60208iSyracuse University, Syracuse, NY 13244-4100jUniversity of Pittsburgh, Pittsburgh, PA 15260kLead Principal InvestigatorlAssociateWe present a method for extracting gravitational radiation from a three-dimensional numericalrelativity simulation and, using the extracted data, to provide outer boundary conditions. Themethod treats dynamical gravitational variables as nonspherical perturbations of Schwarzschild ge-ometry. We discuss a code which implements this method and present results of tests which havebeen performed with a three dimensional numerical relativity code.PACS numbers: 04.70.Bw, 04.25.Dm, 04.25.Nx, 04.30.DbNumerical relativity represents the only currently vi-able method for obtaining solutions to Einstein equationsfor highly dynamical and strong field sources of gravita-tional radiation. Using these techniques to study coa-lescing black hole binaries is the purpose of the multi-institutional Binary Black Hole “Grand Challenge” Al-liance effort [1] which is presently underway in the UnitedStates. This effort is also motivated by the prospect ofobservations with the next generation of gravitationalwave detectors.In addition to tremendous demands on computationalresources, implementing the standard 3+1 [2,3] formula-tion of Einstein theory as a Cauchy problem [4] is compli-cated considerably by the necessity of imposing bound-ary conditions which maintain numerical accuracy andthe physical correctness of the solution. Both inner andouter boundary conditions have received considerable at-tention. Recent efforts on interior boundaries have fo-cused on the excision of the interior of the black hole fromthe computational domain (see, for example, [5]). Thispaper will concentrate on the problem of outer boundaryconditions applied at a finite radius around a source ofgravitational waves.Proper boundary conditions on spacelike slices ofasymptotically flat spacetimes are essential for the ac-curate computation of the gravitational waveforms pro-duced in the strong field region that represent the obser-vationally relevant aspect of the computation. Since it isnot feasible to simulate on spacelike slices out to arbitrar-ily large distances from the source, it is necessary to ex-tract gravitational waves comparatively near the strongfield region and to have boundary conditions that allowradiation to pass cleanly off the mesh. If poor outgo-ing boundary conditions are imposed, spurious radiationis produced which can contaminate the computed grav-itational waveform. Additionally, the outer boundary isusually close enough to the isolated source that backscat-ter of radiation from curvature is significant. This sourceof incoming radiation needs to be built into the outerboundary conditions. An approach to the extraction ofgravitational wave information and the computation ofouter boundary conditions that exploits the matching ofthe interior numerical solution with an exterior perturba-tive solution on spacelike slices has been developed dur-1ing the past decade and applied to a number of differentphysical scenarios [6–8]. Extension of these techniquesto three dimensional (3D) simulations has been one ofthe efforts of the Alliance. A parallel development is alsounderway in the Alliance that matches interior Cauchysolutions to exterior solutions on characteristic hypersur-faces [9].In this Letter we report the first successful applicationof the perturbative matching approach to provide outerboundary conditions for a 3D numerical relativity code.In this context, the perturbative matching method shouldbe viewed as a general “module” that can be coupled toany nonlinear “interior” 3D code. While the latter solvesthe Einstein equations in the highly dynamical strongfield region, the former extracts gravitational wave dataand imposes outer boundary conditions. In the follow-ing we give an overview of the method and present testresults obtained from the evolution of linear waves.Nonspherical perturbations of Schwarzschild geome-try have been studied for many years in the context ofblack hole perturbations [10,11] and as a way to extractgauge-invariant information about gravitational radia-tion [6]. More recently Schwarzschild perturbation the-ory has been found to be useful in studying the late-timebehavior of the coalescence of compact binaries in a nu-merical simulation after the horizon has formed [12,13,8].We base our treatment of Schwarzschild perturbationtheory on a recent hyperbolic formulation of Einsteinfield equations [14]. A principal advantage of this ap-proach is that we can easily derive perturbative waveequations in terms of the standard 3 + 1 variables, mak-ing it straightforward to match exterior perturbative so-lutions to interior numerical ones. This method leads tospatially gauge-invariant radial wave equations for eachangular mode [15] which we have used: a) to extract grav-itational radiation from a 3D numerical relativity simula-tion, b) to evolve this information to large radius yieldingan approximate asymptotic waveform and c) to provideouter boundary conditions for such a simulation. One ofthe major advantages of the perturbative method is thatwe have replaced the computationally expensive 3D evo-lution of the gravitational waves via the nonlinear Ein-stein equations with a set of 1D linear equations we canintegrate to high accuracy and minimal computationalcosts.We split the gravitational quantities of interest intobackground and perturbed parts: the three-metric gij =g̃ij + hij , the extrinsic curvature Kij = K̃ij + κij , thelapse function N = Ñ + α and the shift vector βi =β̃i + vi, where the tilde denotes background quantities.We assume a Schwarzschild background,g̃ijdxidxj = Ñ−2dr2 + r2(dθ2 + sin2 θdφ2) , (1)Ñ =(1−2Mr)1/2. (2)(It follows that K̃ij = 0 = β̃i). The perturbed parts havearbitrary angular dependence.We use this approximation to linearize the hyperbolicequations and, in particular, we find that the wave equa-tion for Kij reduces to a linear wave equation for κij in-volving also the background lapse [15]. We separate theangular dependence in this equation by expanding κij interms of tensor spherical harmonics (ê1)ij , . . . , (f̂4)ij . Inthe notation of [11], we haveκij = a×(t, r)(ê1)ij + rb×(t, r)(ê2)ij +Ñ−2a+(t, r)(f̂2)ij + rb+(t, r)(f̂1)ij +r2c+(t, r)(f̂3)ij + r2d+(t, r)(f̂4)ij , (3)where a×, b× are the odd-parity multipoles, while a+,b+, c+, d+ are the even-parity ones. Note that we havesuppressed the angular indices (ℓ,m) of (ê1)ij , . . . , (f̂4)ijand of a×, . . . , d+ over which there is an implicit sum.In odd-parity, we take the wave equation for κrθand use the linearized momentum constraints, ∇̃k(κki −δkiκjj) = 0, to eliminate the odd-parity amplitude (b×).In even-parity, we use the wave equation for κrr to-gether with the wave equation obtained for the traceκ = κjj = h(t, r)Yℓm and the linearized momentum con-straints. In this way we eliminate b+ and c+ and obtaintwo coupled equations for a+ and h. For each (ℓ,m)mode, we therefore have one odd-parity equation{∂2t − Ñ4∂2r −2rÑ2∂r −2Mr3(1−3M2r)+Ñ2[ℓ(ℓ+ 1)r2−6Mr3]}(a×)ℓm = 0 , (4)and two coupled even-parity equations,[∂2t − Ñ4∂2r −6rÑ4∂r + Ñ2 ℓ(ℓ+ 1)r2−6r2+14Mr3−3M2r4](a+)ℓm +[4rÑ2(1−3Mr)∂r +2r2(1−Mr−3M2r2)](h)ℓm= 0 , (5)[∂2t − Ñ4∂2r −2rÑ2∂r + Ñ2 ℓ(ℓ+ 1)r2+2Mr3−7M2r4](h)ℓm−2Mr3(3−7Mr)(a+)ℓm = 0 . (6)(The usual Regge-Wheeler and Zerilli equations can beobtained by a more complete analysis [15].)We now turn to implementation and tests of this tech-nique. Consider a numerical relativity simulation which20t1tBoundary2-sphereEExtraction1D OuterrExterior3D OuterBoundaryInteriortFIG. 1. Location of the different outer boundaries andof the extraction 2-sphere for two successive timeslices. Thedark shaded region shows the spatial domain over which the3D nonlinear equation are solved.evolves Einstein equations in either the standard 3 + 1(ADM) or hyperbolic form on a “interior” 3D grid. (Norestrictions need be put on the choice of the 3D coordi-nate system.) In the tests described here, we have usedthe ADM 3D interior code of the Alliance [16] to evolvelinear, ℓ = 2, m = 0, Teukolsky waves [17]. The interior3D grid uses (topologically) Cartesian coordinates andup to (129)3 gridpoints. In the results shown, the ini-tial data consists of a metric formed from the sum of thebackground Cartesian metric and a Gaussian envelope ofamplitude A = 10−6, width b = 1 and of a vanishing ex-trinsic curvature. The ADM evolution is undertaken ona cubical grid of extent {x, y, z} = ± 4, with unit lapseand zero shift.During each time step, the procedure for extractingradiation and imposing outer boundary conditions pro-ceeds in three steps: (1) extraction of the independentamplitudes and their time derivatives on a 2-sphere ofradius rEinscribed in the grid, (2) evolution of the ra-dial wave equations (4)–(6) out to large radii using theextracted amplitudes to construct inner boundary val-ues on the “exterior” 1D grid and (3) reconstruction and“injection” of Kij and ∂tKij at specified gridpoints at ornear the boundary of the interior grid. The last step usesthe momentum constraints and the inverse of the trans-formation employed in step (1). In Figure 1, we showthe schematic location of the different outer boundariesand of the extraction 2-sphere on two successive spaceliketime slices at t = t0 and t = t1.The first step involves interpolating Kij and ∂tKijonto the 2-sphere and evaluating their projections ontothe appropriate conjugate tensor spherical harmonics toeliminate the angular dependence and obtain the cor-FIG. 2. Convergence to the analytic solution of the ex-tracted (r = 1) and evolved (r = 8) multipole (a+)20. Theamplitude is scaled by r3 to compensate for the radial fall-off.responding multipoles. The precise location of the 2-sphere within the interior 3D grid depends on the specificcase under investigation. In general, for the perturbativematching to be justified, we require that the extractionsurface be located in a region where the gravitationalfield is close to Schwarzschild. One must be careful thatthe numerical errors introduced by the interior evolutiondo not dominate the exterior solution. (Further detailswill be presented elsewhere [18].)The second step entails evolving equations (4)–(6). Inthe test presented here, we consider a flat backgroundwith M = 0. The initial data for the 1D exterior gridis set consistently with the interior 3D initial data. Dur-ing each time integration of the 1D exterior grid, theextracted multipoles are used as inner boundary valuesand standard Sommerfeld outgoing wave conditions areimposed at the outer boundary (e.g., at r = 30). Thetop diagram of Fig. 2 shows the only relevant multipolefor this test [19], extracted from the interior 3D grid at a2-sphere of radius rE= 1 [20]; the lower diagram showsits evolved waveform at a radius r = 8. Different curvesrefer to different resolutions of the interior 3D grid andshow the convergence to the analytic solution.The third and final step consists of computing outerboundary values for the interior 3D grid, and this stepcan proceed in one of two ways. In the first method, thevalues of Kij reconstructed from the exterior 1D dataare injected as Dirichlet data at all the boundary pointsof the interior 3D grid. (No restrictions are put on theshape of the 2-surface where the data are injected.) Fora code using a hyperbolic formulation, ∂tKij can also3FIG. 3. Convergence to zero of the L2 norm of the errorof Kzz integrated over the 3D outer boundary surface.be provided at the boundary. Moreover, the evolutionequation of the three-metric can be integrated using Kijat the boundary so Dirichlet data for gij need not beprovided there. The second method relates, at the 3Douter boundary, the null derivatives of the extrinsic cur-vature obtained from the interior grid (Kij) and fromthe perturbative module (κij , since background extrinsiccurvature is assumed to be zero):∂∂t(Kij − κij) +∂∂r(Kij − κij) +2r(Kij − κij) = 0 (7)This method resembles a Sommerfeld condition but ismore general since it can be used in regions where theradiation is not dominated by the asymptotic outgoingbehavior. Moreover it takes into account arbitrary angu-lar dependence, as well as the effects of a Schwarzschildblack hole background. Experimentation with the twomethods has shown that this “perturbative Sommerfeld”approach is very accurate and generally more stable thanthe direct injection of Dirichlet data. Figure 3 showsconvergence to zero of the L2 norm of the difference be-tween one component of Kij (namely Kzz) and its ana-lytic value, integrated over the entire outer boundary ofthe interior 3D grid.An important property of any outer boundary imple-mentation is that it should allow the outgoing radiationto escape freely to infinity. In practice, however, a fi-nite discretization always produces a certain amount ofreflection at the outer boundary and this could drive in-stabilities which grow exponentially. We have comparedthe long-term stability obtained with the perturbativeouter boundary module with that of the best “alterna-tive” outer boundary conditions we have tried: namely aFIG. 4. Long-term evolution of the L2 norm of the errorin Kzz at the 3D outer boundary surface. The inset showsKzz at the outer boundary along the x−axis. The interiorgrid has (33)3 gridpoints and the code is stopped when thenorm is unity.standard outgoing Sommerfeld condition. Fig. 4 showsthe results of this comparison both on a long time scaleand on a shorter time scale. (The calculations were per-formed using a very coarse resolution.) As shown in theinset (first echoes at t ∼ 7), the amount of reflectionproduced with the perturbative method is smaller thanthat produced by the Sommerfeld condition. Moreover,the use of perturbative boundary conditions delays theonset of exponential error growth and allows for a muchlonger evolution (see main figure). Increasing the interiorresolution and the number of (ℓ,m) modes used we canfurther prolong the running time. Using (49)3 interiorgridpoints, we are able to evolve the code for more than50 crossing times. In strong contrast, when a standardSommerfeld condition is used, increased interior resolu-tion results in a shorter running time [18].This work was supported by the NSF Binary BlackHole Grand Challenge Grant Nos. NSF PHY 93–18152,NSF PHY 93–10083, ASC 93–18152 (ARPA supple-mented). Computations were performed at NPAC (Syra-cuse University) and at NCSA (University of Illinois atUrbana-Champaign).[1] Information about the Binary Black Hole Grand Chal-lenge Alliance, its goals and present status of research canbe found at the web page: http://www.npac.syr.edu/4projects/bh/.[2] R. Arnowitt, S. Deser and C. W. Misner, in Gravitation,edited by L. Witten, Wiley, New York (1962).[3] J. W. York in Sources of Gravitational Radiation, editedby L. Smarr, Cambridge Univ. Press, Cambridge (1979).[4] Y. Choquet-Bruhat and J. W. York in General Relativityand Gravitation I, edited by A. Held, Plenum, New York(1980).[5] E. Seidel and W.-M. Suen, Phys. Rev. Lett., 69, 1845(1992).[6] A. M. Abrahams and C. R. Evans, Phys. Rev. D, 37, 317(1988); Phys. Rev. D, 42, 2585 (1990).[7] A. M. Abrahams et al., Phys. Rev. D 45, 3544 (1992);P. Anninos et al., Phys. Rev. Lett. 71, 2851 (1993).[8] A. M. Abrahams, S. L. Shapiro and S. A. Teukolsky,Phys. Rev. D 51, 4295 (1995); A. M. Abrahams and R. H.Price, Phys. Rev. D 53, 1963, Ibidem, 1972 (1996).[9] N. T. Bishop, R. Gomez, P. R. Holvorcem, R. A.Matzner, P. Papadopoulos and J. Winicour Phys. Rev.Lett. 76, 4303 (1996); N. T. Bishop, R. Gomez, L.Lehner, J. Winicour Phys. Rev. D. 54, 6153 (1996).[10] T. Regge and J.A. Wheeler, Phys. Rev. 108, 1063 (1957);F. Zerilli, Phys. Rev. Lett. 24 737 (1970).[11] V. Moncrief, Annals of Physics 88, 323 (1974).[12] R. H. Price and J. Pullin, Phys. Rev. Lett. 72, 3297(1994).[13] A. M. Abrahams and G. Cook, Phys. Rev. D , (1994).[14] Y. Choquet-Bruhat and J. W. York, C. R. Acad. Sci.Paris, 321, 1089 (1995); A. Abrahams, A. Anderson,Y. Choquet-Bruhat and J. W. York, Phys. Rev. Lett.75, 3377 (1995).[15] A. M. Abrahams, A. Anderson, C. R. Evans, C. Lea,J. Lenaghan, M. Rupright and J. W. York, In prepara-tion.[16] The Binary Black Hole Grand Challenge Alliance, Inpreparation.[17] S. Teukolsky, Phys. Rev. D 26, 745 (1982).[18] A. M. Abrahams, L. Rezzolla and M. Rupright, In prepa-ration.[19] Other (ℓ,m) multipoles are also computed and used inthe decomposition; but, given the initial data chosen,their amplitudes are several orders of magnitude smaller[18].[20] On a flat background spacetime, the position of the ex-traction 2-sphere is arbitrary. Similar results have beenobtained also for rE= 0.5, 1.5, 2, 2.5, 3, 3.5 [18].5

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Gravitational wave extraction and outer boundary conditions by perturbative matching

Gravitational wave extraction and outer boundary conditions by perturbative matching

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