Gravitational wave extraction based on Cauchy-characteristic extraction and characteristic evolution

Abstract

We implement a code to find the gravitational news at future null infinity by using data from a Cauchy code as boundary data for a characteristic code. This technique of Cauchy-characteristic Extraction (CCE) allows for the unambiguous extraction of gravitational waves from numerical simulations. Comment: 11 pages, 5 figures; corrections after referee's comments; add section on Teukolsky wave

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arXiv:gr-qc/0501008v2 27 Sep 2005
AEI-2005-004
Gravitational wave extraction based on Cauchy-characteristic extraction and characteristic
evolution
Maria Babiuc,1B´ ela Szil´ agyi,1,2Ian Hawke,2,3and Yosef Zlochower4
1Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 15260, US
2Max-Planck-Institut f¨ ur Gravitationsphysik, Albert-Einstein-Institut, Am M¨ uhlenberg 1, D-14476 Golm, Germany
3School of Mathematics, University of Southampton, Southampton SO17 1BJ, UK
4Department of Physics and Astronomy, and Center for Gravitational Wave Astronomy,
The University of Texas at Brownsville, Brownsville, TX 78520, US
(Dated: Date: 2005/09/20 14:30:02 )
We implement a code to find the gravitational news at future null infinity by using data from a Cauchy code
as boundary data for a characteristic code. This technique of Cauchy-characteristic Extraction (CCE) allows
for the unambiguous extraction of gravitational waves from numerical simulations. We first test the technique
on non-radiative spacetimes: Minkowski spacetime, perturbations of Minkowski spacetime and static black
hole spacetimes in various gauges. We show the convergence and limitations of the algorithm and illustrate
its success in cases where other wave extraction methods fail. We further apply our techniques to a standard
radiative test case for wave extraction: a linearized Teukolsky wave, presenting our results in comparison to the
Zerilli technique and we argue for the advantages of our method of extraction.
PACS numbers: 04.25.Dm, 95.30.Sf, 97.60.Lf
I.INTRODUCTION
The importanceofgravitationalwaveformtemplates forgravitationalwave detectorsimplies a needforaccurate3D numerical
simulations of isolated sources such as binary black hole mergers. These simulations are often done with Cauchy codes based
on a “3+1” slicing of spacetime. With the slicing conditions most commonly used in numerical simulations, two problems arise:
artificial boundary conditions must be placed on the computational domain, and information such as the gravitational news
cannot be extracted at future null infinity I+.
To avoid these problems two possible approaches have been suggested. One way is to evolve the entire spacetime. For
example, a hyperboloidal slicing of the spacetime [1, 2, 3] allows information to propagate to I+in finite time. Conformal
compactifications, such as those suggested by the conformal field equations [4, 5] or employed in [6] allow I+to be located
at a finite computational coordinate in a regular way. Another example is characteristic evolution based on a Bondi-Sachs line
element , which gives a natural description of the radiation-regionof spacetime extendingto I+. Characteristic numerical codes
have been used to study tail decay [7], critical phenomena [8, 9, 10, 11, 12], singularity structure [13, 14, 15, 16] and fluid
collapse [17, 18, 19], to list just a few examples.
Unfortunately, characteristic methods suffer from the appearance of caustics in the inner strong field region. The problem of
causticscanbeavoidedbyevolvingthestrongfieldregionwithastandardCauchyslicingwhilstusingthecharacteristicapproach
for the exterior. This technique of Cauchy-characteristic matching (CCE) has proved successful in numerical evolution of the
spherically symmetric Klein-Gordon-Einstein field equations [20], for 3D non-linear wave equations [21] and for the linearized
harmonic Einstein system [22].
The second way of avoiding problems with standard Cauchy codes is a perturbative approach. The standard way of extracting
gravitational waves from a numerical simulation is based on perturbation theory, either using the quadrupole formula or first
order gauge invariant formalisms based on the work of Zerilli and Moncrief [23, 24]. The vast majority of waveform templates
are currently based upon these approaches. Numerical codes able to solve a generalization of the Zerilli equation to a three
dimensional Cartesian coordinate system and to extract the gravitational signal are reported in [25, 26, 27, 28]. Extraction of
gravitationalwavesbasedontheZerilli-Moncriefformalisminfullythree-dimensionalsimulationsarepresentedalsoin [29, 30].
Perturbative methods have been also used to provide boundary conditions at the outer boundary of the Cauchy grid. This
approach of Cauchy-perturbative matching has been implemented numerically in [31, 32, 33]. All those works are impressive,
but discrepancies between the results of a perturbative approach and the full non-linear theory cannot be determined without
solving the fully non-linear problem, although error estimates have been made [34].
Of these approaches, CCM has many appealing properties. The characteristic description of the exterior, particularly at
I+, allows for a natural extraction of gravitational information such as the news. Matching to a standard Cauchy code in the
interior implies that the methods employed in current 3D numerical simulations may immediately be used. As both Cauchy and
characteristic approaches have been well tested and commonly used, CCM is a natural way of avoiding the potential problems
of caustics in the characteristic region and artificial boundaries in the Cauchy region.
In this work we take a step towards the full CCM method by using a Cauchy code to provide boundary data for a charac-
teristic code which propagates the solution to I+to extract the waveform. This procedure of Cauchy-characteristic extraction

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(CCE) allows the computation of gravitational waves in an unambiguous fashion. CCE was succesfully implemented in the
quasispherical approximation in [35]; here we demonstrate it in the fully non-linear case.
In this work the outer region extending to I+is numerically evolved by the Pitt Null Code [20, 35, 36, 37, 38, 39, 40, 41, 42].
The link between the Cauchy and the characteristic modules is done by a non-linear 3D CCE algorithm [35, 41, 42, 43, 44]. At
the outer edge of the characteristic grid the Bondi news is computed [37, 39, 45, 46]. We have imported the CCE code into the
Cactus computational infrastructure [47, 48, 49, 50]. Within this infrastructure we have been able to use two separate Cauchy
codes implementing the BSSN and harmonic 3 + 1 formulations of the Einstein equations. This allows us to test the robustness
of the CCE approach.
We compareCCE with the Zerilli extractiontechnique[23, 24, 51] forboth non-radiativeandradiativespacetimes. The Zerilli
formalism is based on the paper of Nagar and Rezzolla [52] which reviews and collects the relevant expressions related to the
Regge-Wheeler and Zerilli equations for the odd and even-parity perturbations of a Schwarzschild spacetime. The conventions
presented in their review are implemented in the Wave Extract code, included in the Cactus computational toolkit open source
infrastructure [49].
For non-radiative spacetimes, the comparison gives results consistent with reference [34]: the CCE algorithm is O(∆2)
accurate (where ∆ is the computational grid-step), while the accuracy of the perturbative (Zerilli) approach is O(r−2) where
r is the radius of the wave extraction sphere. Since the signal is O(r−1), this implies an O(r−1) relative error in the Zerilli
approach. As a result, it is shown in [34] that CCM is more efficient computationally in the sense that, as the desired error goes
to zero, the amount of computation required for CCM becomes negligible compared to a pure Cauchy computation.
We further apply our algorithms to the study of the propagation of a linearized Teukolsky gravitational wave [27, 53, 54, 55]
and we compare the CCE and Zerilli wavesignal. For the case of a large extraction radius, we find that both methods give very
good results and we demonstrate convergenceto the analytical waveform of the Teukolsky solution for the CCE news. We show
that the CCE waveform does not depend upon the extraction radius, which is a major advantage of the CCE method. A small
extraction radius has no effect on the CCE waveform but it introduces errors in the Zerilli waveform.
In Sec. II we outline some general background and notation. In Sec. III we detail the transformation from the Cauchy slice
to the characteristic slice. In Sec. IVA we give a brief summary of the characteristic evolution code. In Sec. IVB we describe
how the news is computed at I+. Finally, in Sec. V we give two sets of tests in full 3D numerical relativity to validate the
accuracy, convergenceand robustness of the CCE algorithm. The first set contains four non-radiativetests, with no gravitational
wave content. The second one is a radiative three-dimensional test. The results show that CCE is valid and accurate in both
non-radiative and truly radiative situations.
II.NOTATION, GEOMETRY AND METRICS
The implementationof CCE described here follows previous descriptionsof Cauchy-characteristicextractionand matching in
the literature. Much of the work has been presented earlier [35, 41, 42, 43, 44]. Here we briefly outline the notation, geometry
and metrics used.
The geometry is described by two separate foliations, neither of which cover the entire spacetime. The Cauchy foliation is
described using a standard 3 + 1 ADM type metric [56],
ds2= −(α2− βiβi)dt2+ 2βidtdxi+ γijdxidxj.
(2.1)
Many different formulations can be used, given initial and boundary data, to evolve the 3-metric γij. In what follows we
shall either consider the BSSN formulation [57, 58] as implemented in [59] or the generalized harmonic formulation [60] as
implemented in the Abigel code [61]. In both cases all interaction between the Cauchy and characteristic foliations will be
performed in terms of the ADM metric, Eq. (2.1).
The Cauchy slice does not extend to asymptotic infinity. Instead an artificial boundary is placed at |xi| = Li. Within this
artificial boundary a world-tube Γ is constructed such that its intersection with any t = const. Cauchy slice is defined as a
Cartesian sphere x2+ y2+ z2= R2, with angular coordinates labeled by ˜ y˜ A,˜A = (2,3). The world-tube Γ is then used as the
inner boundary of a characteristic foliation which uses the standard Bondi-Sachs metric [62, 63]
ds2= −
?
e2βV
r− r2hABUAUB
?
du2− 2e2βdudr − 2r2hABUBdudyA+ r2hABdyAdyB.
(2.2)
Here u labels the outgoing null hypersurfaces, yA= ˜ y˜ Athe angular coordinates (the null rays emanating from the world-tube),
and r is a radial surface area distance. The angular metric hABobeys the condition
det(hAB) = det(qAB),
(2.3)
where qABis the unit sphere metric. This coordinate system consistently covers the world-tube Γ and the exterior spacetime as
long as it is free of caustics.

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The free variables in the Bondi-Sachs metric are then V,β,UAand hAB. The physical interpretation of these variables is that
hABcontains the 2 radiative degrees of freedom, e2βmeasures the expansion of the nullcone, and V/r is the counterpart of the
Newtonian gravitational potential. The 2+1 decomposition of the intrinsic metric on the r = const. world-tube
γijdyidyj= −e2βV
rdu2+ r2hAB(dyA− UAdu)(dyB− UBdu)
(2.4)
identifies r2hABas the intrinsic 2-metric of the u foliation, −UAas the shift vector and e2βV/r as the square of the lapse.
III.THE CCE ALGORITHM
The crucial task of the CCE algorithm is to take Cauchy data given in the ADM form Eq. (2.1) in a neighborhood of the
world-tube Γ and to transform it into boundary data for the Bondi-Sachs metric Eq. (2.2). Then the Bondi code can use the
hypersurface equations to evolve the appropriate quantities out to I+so that the gravitational news may be extracted there.
Much of the present version of the CCE algorithm has been presented in earlier work [35, 41, 42, 43, 44], so in this section we
will give a brief description, highlighting a few new features.
In section II the world-tube was defined as a Cartesian sphere x2+ y2+ z2= R2, with angular coordinates labeled by
˜ y˜ A,˜A = (2,3). In addition to the angular coordinates, we set u = t on the world-tube and choose the fourth coordinate, λ to be
the affine parameter along the radial direction, with λ|Γ= 0. The characteristic cones are constructed such that λ is the future
oriented, outgoingnull direction normal to the foliation of Γ. (In order to avoid a singular Jacobian for the Cauchy-characteristic
coordinate transformation, we require that the world-tube Γ be timelike.) The affine parameter λ is used because the world-tube
is not constructed to be a surface of constant Bondi r.
The choice of the angular coordinates is determined following [64] by the use of two stereographic patches. The use of two
patches avoids numerical difficulties in taking derivatives near the poles when standard spherical coordinates are used. The use
of only two patches may not give the most accurate numerical results; various ways of discretizing the 2-sphere on multiple
patches are discussed in [65]. The two patches are centered aroundthe North and South poles with the stereographiccoordinates
related to the usual spherical coordinates (θ,φ) by
ξNorth=
?1 − cosθ
1 + cosθeiφ,ξSouth=
?1 + cosθ
1 − cosθe−iφ,
(3.1)
and ξ = ˜ y2+ i˜ y3= q + ip, where i =√−1 . We also introduce the complex vector on the sphere qA= (P/2)(δA
its co-vector qA= (2/P)(δ2
construction. The unit sphere metric corresponding to these coordinates is
2+ iδA
3) and
A+ iδ3
A), with P = 1 + ξ¯ξ = 1 + q2+ p2. The orthogonality condition ¯ qAqA= 2 is satisfied by
qAB=1
2(qA¯ qB+ ¯ qAqB) =
4
P2
?1 0
0 1
?
.
(3.2)
The angular subspace in the Bondi code is treated by use of the eth formalism and spin-weighted quantities. (See [39, 64] for
details.)
The CCE algorithm starts by interpolating the Cauchy metric γij, lapse α and shift βiand their spatial derivatives onto the
world-tube. Time derivatives are computed via backwards finite-differencing done along Γ, e.g., at t = tNwe write
(∂tF)[N]=
1
2∆t
?
3F[N]− 4F[N−1]+ F[N−2]
?
+ O?(∆t)2?.
(3.3)
Knowledge of the Cauchy metric and its 4-derivative is enough to compute the affine metric ˜ η˜ α˜βas a Taylor series expansion
around the world-tube
˜ η˜ α˜β= ˜ η˜ α˜β|Γ+ λ˜ η˜ α˜β
,λ|Γ+ O(λ2).
(3.4)
Next a second null coordinate system (the Bondi-Sachs system) is introduced
yα= (y1,yA,y4),
where
yA= ˜ y
˜ A,y1= ˜ y1= u.
(3.5)
The fourth coordinate y4= r is a surface area coordinate, defined by
r =
?det(˜ η˜
A˜ B)
det(qAB)
?1
4
=P
2det(˜ η˜
A˜ B)
1
4.
(3.6)

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The Bondi-Sachs metric on the extraction world-tube can be computed as
ηµν
|Γ=
?∂yµ
∂˜ y˜ α
∂yν
∂˜ y˜β˜ η˜ α˜β
?
|Γ
.
(3.7)
Note that the metric on the sphere is unchanged by this coordinate transformation, i.e., ηAB= ˜ η˜
to work with the Jacobian components that correspond to derivatives of r.
In terms of qA, ¯ qAthe two dimensional metric ηABcan be encoded into the metric functions
A˜ B. Therefore one only needs
J ≡1
2qAqBhAB,K ≡1
2qA¯ qBhAB.
(3.8)
The determinant condition Eq. (2.3) translates into
K2= 1 + J¯J.
(3.9)
With hABsymmetric and of fixed determinant, there are only two degrees of freedomin the angular metric that are encoded into
the complex function J.
From Eq. (2.2) we have
β = −1
2log(−ηrr) =1
2log(r,λ).
(3.10)
This quantity is a measure of the expansion of the light rays as they propagate outwards. The CCE and the Bondi codes have
been implemented with the assumption that r,λ> 0 (or that β is real).
The radial-angular components ηrAcan be represented by
U ≡ UAqA=ηrA
ηru;
(3.11)
while the radial-radial component ηrris contained in
W ≡V − r
r2
.
(3.12)
In addition to these quantities, in [40] the auxiliary variables
ν ≡¯ðJ =1
k ≡ ðK =1
B ≡ ðβ = β,AqA,
2hAB,CqAqB¯ qC,
(3.13)
2hAB,CqA¯ qBqC+ 2ξK,
(3.14)
(3.15)
have been introducedto eliminate the need to explicitlyuse second angularderivativesin the Bondi evolutioncode. The required
boundary data is J,β,U,∂rU,W,ν,k, and B. (See Sec. IVA.) Notice that once the Bondi-Sachs metric is known on the world-
tube one can only obtain J,β,U,W. In order to provide the rest of the necessary boundary data we need the radial derivative of
the Bondi-Sachs metric
(∂ληµν)|Γ=
?∂2yµ
∂λ∂˜ y˜ α
∂yν
∂˜ y˜β˜ η˜ α˜β+∂yµ
∂˜ y˜ α
∂2yν
∂λ∂˜ y˜β˜ η˜ α˜β+∂yµ
∂˜ y˜ α
∂yν
∂˜ y˜β˜ η˜ α˜β
,λ
?
|Γ
.
(3.16)
With ˜ η˜ α˜β
,λalready known, the only non-trivial parts in Eq.(3.16) are the Jacobian terms
r,λ˜ α=
∂2y1
∂λ∂˜ y˜ α.
(3.17)
These depend on the second derivatives of the Cauchy metric. In order to avoid possible numerical problems caused by interpo-
lating second derivatives onto the world-tube, we calculate r,λ˜
calculate r,λuby backwards differencing in time along the world-tube. The remaining term r,λλis calculated using the identity
Aby taking centered derivatives of r,λon the world-tube and we
β,λ= −r,λλ
2r,λ,
(3.18)

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and the characteristic equation
β,r=r
8
?J,r¯J,r− (K,r)2?.
(3.19)
(The right hand side of Eq. (3.19) can be computed in terms of J,r = J,λ/r,λ, which, in turn, can be computed in terms of
already known quantities.)
Knowledge of the radial derivative of the Bondi metric is important not only for obtaining (∂rU)|Γbut also because the grid-
structure of the Bondi code is based on the radial coordinate r, and the extraction world-tube will not, in general, coincide with
any of these radial grid-points. We need to use, therefore, Taylor series expansions to fill the Bondi gridpoints surrounding Γ
with the necessary boundary data. We write, e.g.,
β = β|Γ+ λβ,λ+ O(λ2).
(3.20)
Anotherproblemis the need to providethe auxiliaryangularvariables, B,k and ν on the world-tube. These have been defined
as the ð-derivatives of Bondi fields in the yαframe, while taking angular derivatives on the world-tube amounts to computing ð
derivatives in the ˜ y˜ αframe. The correction term between the two frames is
ðF =˜ðF −F,λ
r,λ
˜ðr.
(3.21)
The quantities J,˜ðJ and J,λare known from the interpolated Cauchy data, while J,λλis not computed in CCE. Thus one can
compute ðJ but not ðJ,λ. As a consequence we cannot use a simple Taylor series expansion to place ν on the Bondi grid to
O(λ2) accuracy. The solution to this problem is to first place J on Bondi grid-points surrounding the world-tube, then compute
ðJ on those points (i.e., compute angular derivatives in the Bondi-Sachs frame), and then calculate ∂λðJ on the world-tube by
use of the neighboring Bondi grid values of ðJ. With ν = ðJ and its radial derivative known on the world-tube, we can then
use the standard O(λ2) expansion to provide ν at Bondi gridpoints surrounding the world-tube. This way we make maximal
use of the Cauchy data as interpolated onto the world-tube and minimal use of the finite difference ð algorithm that has its own
discontinuous O(∆2) error at the edges of the stereographic patches. A similar approach is used for k = ðK and B = ðβ.
IV. THE BONDI CODE AND THE NEWS ALGORITHM
A.The Bondi code
The inner workings of the Bondi code had been described in detail elsewhere (see [20, 35, 36, 37, 38, 39, 40, 41, 42]) so
here we give only a brief overview of the algorithm. As already stated in Sec. III, the variables of the code are J,β,U,W as
well as ν,k,B. Out of these the equation for J is the only one to contain a time derivative. For this reason J is updated via an
evolution stencil that involves two time-levels. The rest of the variables are integrated radially from the world-tube out to I+.
The integration constants are set by CCE. All of these radial integration equations are first differential order in r except for the
U equation which contains U,rr. For this reason integrating U requires two constants, which explains the need to provide, at the
world-tube, U as well as U,r.
B. The News algorithm
The calculation of the Bondi news function on I+is based on the algorithm developed in [39] with the modifications intro-
duced in [45, 46] (an alternative calculation for the news was recently introduced in [66]). Here we present an overview of the
algorithm.
The Bondi-Sachs metric Eq. (2.2) in a neighborhoodof I+in (u,xA,l = 1/r) coordinates (after multiplying by a conformal
factor l2) has the form
l2ds2= O(l2)du2+ 2e2βdudl − 2hABUBdudxA+ hABdxAdxB,
where the metric variables β, UA, and hABhave the asymptotic expansions β = H + O(l2), UA= LA+ O(l), and hAB=
HAB+ lcAB+ O(l2). We can always find coordinates (uB,qB,pB,lB) (hereafter referred to as ‘inertial’ coordinates), and an
associated conformal metric dsB2= ω2ds2(ω > 0), such that (i)
of I+, (ii) lB= ω l + O(l2), (iii) the conformal metric in the subspace (uB= const.,lB= const.) is the unit sphere metric on
I+. We fix a null-tetradon I+by choosingnato be affine and to point along the null generatorsof I+and by choosingm(a¯ mb)
(4.1)
?
∂
∂uB
?
is null and affine and points along the null generators

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to be the unit-sphere covariant metric in the 2-dimensional angular subspace. In inertial coordinates (uB,qB,pB,lB) on I+the
tetrad has the form
˜ na= (1,0,0,0)
˜la= (0,0,0,1)
˜ ma= (0,˜P/2,i˜P/2,0),
(4.2)
(4.3)
(4.4)
where the tilde denotes a quantity defined with respect to inertial observers. Note that ˜ m(A¯˜ mB)= qABas required. We define
a complex vector Fa, analogous to ˜ ma(i.e. F(A¯FB)= HAB), adapted to the coordinates used in the characteristic evolution
code via
Fa= (0,FA,0),
(4.5)
where
FA= qA
?
K0+ 1
2
− J0¯ qA
?
1
2(K0+ 1),
(4.6)
where qAis the dyad defined in Sec. III, J0= qAqBHAB/2, and K0= qA¯ qBHAB/2. Faand ˜ maare related by
˜ ma= e−iδω−1Fa+ γ˜ na.
(4.7)
The γ˜ naterm will not enter the news calculation.
To calculate the Bondi news one needs to evolve two scalar quantities δ, the phase factor in Eq. (4.7), and the conformal
factor ω, as well the relations ξ(ξB) between the angular coordinates used in the characteristic evolution and inertial angular
coordinates, and uB(u,ξB) between the inertial time slicing and the time slicing of the characteristic evolution code. Then δ,
ξ(ξB), and uB(u,ξB) are evolved using the following ODE’s along the null generators of I+
?
2(K0+ 1)
dξ
du
duB
du
dδ
du
=
1
2ℑ
¯J0,uJ0
K0+ 1+J0
?U0¯ð¯J0+¯U0ð¯J0
?
+ J0¯ð¯U0+ K0¯ðU0+ 2U0¯ξ
?
(4.8)
=
1
2(1 + ξ¯ξ)U0
(4.9)
= ωe2H,
(4.10)
where U0= qALA. Also ω is evolved using the PDE
∂ulogω = −ℜ
?
¯U ðlogω +1
2ð¯U
?
(4.11)
The Bondi news function up to a phase factor of e−2iδis given by
N =1
2ω−2e−2HFAFB
?
(∂u+ LL)cAB−1
2cABDCLC+ 2ωDA[ω−2DB(ωe2H)]
?
,
(4.12)
where DAis the covariant derivative with respect to HAB. The Bondi news is calculated in three steps. First Eq. (4.12) is
evaluated, ignoring the e−2iδphase factor, as a function of the evolution coordinates (u,ξ). Then using the relation ξ(ξB),
the news is interpolated onto a fixed inertial angular grid (i.e. N(u,ξB)) and multiplied by the phase factor e−2iδ(which is
only known on the inertial grid). Finally, in post-processing, the news is interpolated in time onto fixed inertial time slices (i.e.
N(uB,ξB)). Once the news is obtained in inertial coordinates it can be decomposed into spin-weighted spherical harmonics.
V.TESTS
We apply the algorithm described above for two sets of tests: non-radiative spacetimes and radiative spacetimes. In the first,
non-radiative set of tests (section VA), we analyze Minkowsy and Schwazchild spacetimes. The Minkowski (Sec. VA1) or
small perturbation of Minkowski (Sec. VA2) tests are used to show the stability of the code and the errors due to transforming
between the different coordinate systems and sets of variables.

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The Schwazchild tests describe a static spherically symmetric black hole in a centered frame (Sec. VA3) and in an oscillating
frame (Sec. VA4). These tests indicate the accuracy of the code in non-trivial spacetimes.
The last test (Sec. VB) is a standard radiative test: namely a linearized Teukolsky wave, and indicates that CCE is a valid and
accurate method in truly radiative situations.
In the non-radiative cases, the extracted gravitational wave signal should vanish identically. As this is true uniformly for all
points at I+we simplify the computationfor the norms of the gravitational wave signal. When the results of the CCE are shown
the norm used is the simple L2norm over all grid points at I+, without any weighting by the area element. In the radiative case,
we plot the real part of the extracted news waveform (∂th+), versus time.
The Zerilli Moncriefformalism (see [23, 24, 51] for the implementationused here) approximatesthe spacetime as a linearized
perturbationofasphericallysymmetricspacetime. Suchperturbationsaredecomposedintermsofsphericalharmonicswhichare
then expressedin terms of some basis set. An exampleusing the Regge-Wheelerset is where the evenparitymetric perturbations
are decomposed as
?hlm
µν
?(e)=



e2aH0Ylm
H1Ylm
h(e)
H1Ylm
e2bH2Ylm
H(e)
A∇cYlm
A∇cYlm
1,H2and b are found from the decomposition, the even parity master function Q+
r2?KYlmγcd+ G∇d∇cYlm?


.
(5.1)
Once the basis functions such as G,K,h(e)
can be computed from
Q+=
?
2(l + 2)!
(l − 2)!
r
?
rΛ
?
K +
1
e2b
?
r∂rG −2
rh(e)
1
??
+2r
e4b
?e2bH2− eb∂r
?rebK???
Λ[r(Λ − 2) + 6M]
.
(5.2)
Equivalent definitions in terms of other basis functions, alternative conventions and expressions for the odd parity sector can be
found in [52].
Using this notation where Q×gives the odd parity master function, the asymptotic value for the wave forms is
h+− ih×=
1
√2r
?
l,m
?
Q+
lm− i
?t
−∞
Q×
lm(t′)dt′
?
−2Ylm(θ,φ) + O
?1
r2
?
.
(5.3)
For the non-radiative cases, when we compute the wavesignal from the Zerilli approximation, we use the norm
(?h?2)2=
?
2r2(Q+
1
2r2δℓℓ′δmm′(Q+
(h+− ih×)(h+− ih×)†dΩ
1
ℓm− iq×
(5.4)
=
ℓm)(Q+
ℓ′m′ − iq×
ℓ′m′)†
?
−2Yℓm−2Yℓ′m′†dΩ
=
ℓm− iq×
ℓm)(Q+
ℓ′m′− iq×
ℓ′m′)†,
(5.5)
where dΩ is the element of solid angle. The last equality Eq. (5.5) follows from the orthogonality of the spin-weighted tensor
spherical harmonics. We expect h+− ih×to vanish for all angles so this norm should be identically zero. In what follows we
shall look at the spherical harmonics up to ℓ = m = 7 although in practice there is little contribution from the higher modes.
For the Teukolsky wave used here we expect all terms except for the Q+
which leads to a simple but tedious computation of the wavesignal.
2,0term in Eq. (5.3) to be negligible asymptotically,
A.Non-radiative Spacetime Tests
1.Minkowski Flat Spacetime
Minkowski space in standard coordinates is evolved in the harmonic Abigel code, with the metric given analytically on all
Cauchy slices. The domain extends between xi∈ [−10,10], the extraction world-tube is placed at r = 7, and the simulation is
run until t = 10. The coarsest simulation used 503points for the Cauchy grid and 352× 31 points for the characteristic grid.
Simulations to test convergence scaled all grids by factors of 2. For this test, extraction using the Zerilli method gives results
that are identically zero, as expected.
This test indicates the level of numerical round off error in transforming from the Cauchy variables to the Bondi variables on
the world-tube. As shown in Fig. 1, the extracted news is small in all cases but the level of truncation error increases linearly
with the number of points on the extraction world-tube. Since the construction of the boundary data for the CCE code involves
derivatives of the Cauchy metric, this sensitivity to noise in the Cauchy data is expected. However, given the extremely low
amplitude of the noise, it is unlikely that this will cause problems in practical simulations.

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FIG. 1: The L2norm of the CCE news for Minkowski space in standard coordinates. This indicates the level of numerical noise introduced by
transforming variables on the world-tube. In this case the noise is linearly dependent on the number of points on the extraction world-tube. It
is, however, extremely small.
2.Random Perturbations around Minkowski Flat Spacetime
This test is in the spirit of the robust stability tests of [22, 67, 68]. The stable evolution and extraction of white noise initial
data with no frequency dependent growth of the wavesignal is a good indication that the combined evolution and extraction
codes are stable. The domain extends between xi∈ [−10,10], the extraction world-tube is placed at r = 7, and the simulation
is run until t = 100. The coarsest Cauchy grid has 513points and the coarsest characteristic grid 352× 31 points.
The results in Fig. 2 show that the error is independent of the resolution of both Cauchy and characteristic grids. This is
strong evidence that the CCE code is stable against small perturbations that in a practical run would be induced by numerical
error. The wavesignal from the Zerilli extraction is about an order of magnitude smaller. This is to be expected as the Zerilli
extraction approach uses the field values. The error in the CCE news has a jump at the beginning. This arises because on the
initial characteristic data is set to zero so that it takes some time for the noise to build up on the outgoing null cone.
3. Schwarzchild Black Hole in a ”centered” frame
To test a simple black hole spacetime we use a Schwarzschild black hole in ingoingEddington-Finklesteincoordinates(ˆt, ˆ xi),
with the line element
ds2= −
?
1 −2M
ˆ r
?
dˆt2+
?4M
ˆ r
?
dˆtdˆ r +
?
1 +2M
ˆ r
?
dˆ r2+ ˆ r2dΩ2.
(5.6)
This is manifestly static in these coordinates. The spacetime is evolved using the BSSN code and excision methods described
in [69]. The coarsest Cauchy grid has 293points with ∆xi= 0.4M, whilst the characteristic grid has 352× 31 points. The
world-tube is at r = 7M. In the Cauchy evolution domain octant symmetry is used.
The evolution is only performed for a short time (to t = 100M). Over this timescale we see second order convergence for
the CCE news until t ≈ 7M and second order convergence in the waveform from Zerilli extraction until t ≈ 20M, as seen in
Fig. 3. By varying the location of the world-tube or the Zerilli extraction sphere we can see that the errors come from a variety
of locations.
The difference between the times at which the two extraction methods lose convergence is probably due to the greater dif-
ferencing error seen in the Zerilli extraction. At early times (where finite differencing error dominates) the absolute value of
the error is larger for Zerilli extraction by a factor ≈ 4, as seen in Fig. 3. At late times (where outer Cauchy boundary errors
dominate) the error for Zerilli extraction is of the same order as for the CCE extraction.
The non-convergent errors are probably caused by the boundary conditions on the Cauchy grid which do not satisfy the
constraints. As in [69] we are simply applying Sommerfeld type boundary conditions to all fields. This condition does not a

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FIG. 2: The L2 norm of the CCE news for random perturbations of Minkowski where the Cauchy slice is evolved using the Abigel code.
The news displays very slow growth that is independent of the frequency of the initial data. This is a good indication that the CCE method is
stable. The right hand figure shows that the norm of the waveform extracted from the Cauchy grid using the Zerilli method is about one order
of magnitude smaller.
priori satisfy the constraints and is not known to be well-posed. Thus we might expect errors to be induced by the use of these
boundaryconditions. It is likely that constraintsatisfying boundaryconditionsor Cauchy-characteristicmatchingwouldsolve or
at least greatly reducethis problem. Due to the complicatedpattern of the reflected errors arising from a boundaryconditionon a
cubical surface, we have been unable to determine exactly how this error depends on the location of the outer Cauchy boundary.
FIG. 3: The scaled L2norm of the CCE news for a static spherically symmetric black hole where the Cauchy slice is evolved using the BSSN
formalism. At early times, when the world-tube iscausally disconnected from the outer boundary of the Cauchy slice, the CCE news converges
to zero at second order as it should. Later, when the outer boundary is causally connected to the world-tube, deviations from second order
convergence appear, indicating the effect of the Cauchy boundary conditions on the extracted wavesignal. The left panel shows the CCE news
extracted at I+, whilst the right panel shows the “norm” of the wavesignal extracted by the Zerilli method at r = 7.

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4.Schwarzchild Black Hole in an oscillating frame
We use the line element for the standard Schwarzschild black hole in ingoing Eddington-Finklestein coordinates, given in
Eq. (5.6). The moving coordinate frame (t,xi) is given by
t =ˆt, xi= ˆ xi+ Bib(t),
(5.7)
where Biare parameters specifying the velocity and b(t) is a simple periodic function turned on after some time t > t0; here
we use
b(t) =
?
0,
sin(ω(t − t0))3, t > t0.
t ≤ t0
(5.8)
For the test shown here we set Bx= 0.2,By= 0.5,Bz= 0.3 for the velocity and ω = 2π × 0.05,t0= 0.5 for the function b.
The metric is given analytically on the Cauchy grid to avoid any boundary effects. The coarsest Cauchy grid has 513points
and covers the domain xi∈ [−10M,10M]. The coarsest characteristic grid has 352× 31 points. The extraction world-tube is
at r = 7M. The simulation is evolved until t = 100M.
The CCE news convergesto zero as it should. However, Zerilli extraction does not give a signal that convergesto zero as grid
resolution is refined. Instead, an erroneous non-trivial signal is computed.
FIG. 4: The scaled L2 norm of the CCE news for a static spherically symmetric black hole in an oscillating frame. The CCE news converges
to zero at second order as it should.
B. Teukolsky Wave Test
1. Theoretical part
TheTeukolskysolution[27, 53, 55]tothelinearizedEinsteinequationis a weakgravitationalwavepropagatingthroughspace,
and represents one of the most valuable standard test cases for numerical codes that implement wave extraction techniques.
The general form of the spacetime metric is
ds2= −dt2+ (1 + Afrr)dr2+ 2Bfrθrdrdθ
+ 2Bfrφrsinθdrdφ + (1 + Cf(1)
+ 2(A − 2C)fθφr2sinθdθdφ + (1 + Cf(1)
θθ+ Af(2)
θθ)r2dθ2
φφ+ Af(2)
φφ)r2sin2θdφ2
(5.9)

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FIG. 5: The wavesignal computed using Zerilli extraction for the static spherically symmetric black hole in an oscillating frame. The “norm”
is computed as described in the text with the radial dependence removed. The left panel shows that as the resolution of both the Cartesian
grid and extraction sphere are increased the signal does not converge to zero. The right panel shows how this erroneous non-trivial wavesignal
decays as O(r−2) as the extraction radius is increased. In the main plot the standard scaling of the signal is used. In the inset the curves are
scaled by the extraction radius so as to overlay each other for the expected decay rate.
where the angular functions fij, corresponding to an l = 2, m = 0, spin − weight = 2 spherical harmonic, are
frr= 2 − 3sin2θ,frθ= −3sinθcosθ,frφ= 0,
f(1)
fθφ= 0,f(1)
θθ= 3sin2θ,f(2)
φφ= −f(1)
θθ= −1,
θθ,f(2)
φφ= 3sin2θ − 1.
(5.10)
The functions A, B and C are given in terms of a free generating function F(x) = Axe−x2/λ2/λ2, where A is the amplitude
and λ determines the width of the wave, by
A = 3?d2
B = −?d3
C =
xF
r3
xF
r2
xF
r
+3dxF
r4
+3F
r5
?
+3d2
xF
r3
xF
r2
+6dxF
r4
xF
r3
+6F
r5
?
1
4
?d4
+2d3
+9d2
+21dxF
r4
+21F
r5
?,
(5.11)
where dn
xF denotes
dn
xF =dnF(x)
dxn
.
(5.12)
Here, x = t − r corresponds to an outgoing wave. To obtain the ingoing wave solution coresponding to x = t + r, we change
the sign in front of all the terms with odd numbers of F.
We consider a superposition of ingoing t + r and outgoing t − r waves [44], centered at the origin of the coordinate system
at t = 0, which provides a moment of time symmetry. This solution, after is linearized in amplitude, is implemented in the
harmonic Abigel code.
We further give the analytical form of the Teukolsky wave signal. The real part of the Bondi news function is proportional
to the time derivative of the ”plus” polarization mode of the gravitational wave signal. The time derivative of the CCE news
satisfies
˙N = lim
r→∞rΨ4
(5.13)

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12
where Ψ4is the Newman-Penrose component of the Weyl tensor
Ψ4= −Cµρντnµ¯ mρnν¯ mτ
(5.14)
with nνan ingoing null vector, and mρa complex unit vector, oriented in the angular directions.
The connection between mρand the dyad vector qAon the unit sphere is
mA=1
rqA.
(5.15)
Moreover, nµ∂µ= ∂x. A straightforward computation starting from Eq. (5.13), gives
˙N = −1
r¯ qA¯ qB∂2
xgAB.
(5.16)
At infinity, in the linearized regime, only the O(1/r) terms in the Teukolsky functions A, B and C contribute to the news. The
term that determines the gravitational wave signal is d4
xF. We find
˙N = −3sin2θ
4
∂6
tF.
(5.17)
Alternatingly, computation of the linearized expression for the Bondi news from Eq. (4.12), gives
N = −3sin2θ
4
∂5
tF.
(5.18)
C. Numerical part
We give the metric specified by the Teukolsky solution analytically on the Cauchy grid. Then data is extracted at the world-
tubeandevolvedbytheCCE code. We carryouta series ofsimulations, varyingboththelocationofthecharacteristicworld-tube
andthe radiusof theZerilli sphere. We use 803, 1203, and1603gridpointsforthe Cartesian Cauchygridand 602×80,902×120,
and1202×160gridpointsforthe nullgrid. Thedomainextendsbetweenxi∈ [−15,15]and the simulationsare rununtil t = 30.
We study the dependence of the signal with the amplitude and conclude that CCE code resolves correctly amplitudes of
A ≥ 10−8. At smaller amplitudes, clean convergence behavior is contaminated by round-off error. We show results only for
A = 10−5, λ = 1. We study also the dependence of the wave signal with the world-tube radius and conclude that the accuracy
of the computed CCE news is preserved even for radii as small as r = 5.
Fig. 6 shows the convergenceof the CCE news to the analytical solution, for a world-tube radius of r = 10. The convergence
rate ˜ crof the CCE news to the analytical data, is given by
˜ cr= log2
?||N80− Nana||
||N160− Nana||
?,
(5.19)
The convergence rate of the computed CCE news to the analytic value Eq. (5.18) at t − r = 0, corresponding to the peak of the
radiated signal, is
˜ cr= 2.159.
(5.20)
Hence, the CCE wavesignal is second order convergentand independent of the world-tube radius, as expected.
Fig. 7 shows that for small extractionradii, the Zerilli waveformhas a slight asymmetry,which is caused by the dependenceof
the Zerilli formalismuponthe extractionradius. We haveto increasethe extractionradiusin orderto decreasethis error. Because
the error does not fall off sufficiently fast with radius in the near zone, where we can realistically carry out the simulation, we
instead analytically compute the Zerilli news at the extraction radius r = 300. Fig. 8 demonstrates that the agreement between
the computed CCE news at small radii and the analytical Zerilli news at big radii, is very good.
VI.CONCLUSIONS
We have demonstrated the accuracy of Cauchy-characteristic extraction in 3D numerical relativity. The interface at the ex-
traction world-tube does not introduce significant error with either the BSSN [59] or harmonic [22, 61] formulations.

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FIG. 6: Left panel: the convergence of CCE news to the analytical solution. In this case the grid resolution was (Cartesian / characteristic)
803/602× 80, 1203/902× 120 and 1603/1202× 160 respectively. Right panel: the absolute value of the error in CCE news for the grid
resolutions of 803/602× 80 and 1603/1202× 160.
FIG. 7: Left panel: the Zerilli news shows a dependence on the extraction radius. The asymmetry decreases as the extraction radius increases.
Right panel: the absolute value of the error in Zerilli news with radius (grid resolution of 1203/902× 120.
The CCE news is stable against small perturbations on the Cauchy grid, as shown in Sec. VA2, and the level of truncation
error caused by transforming from the Cauchy to the characteristic variables is small, as shown in Sec. VA1. In a non-trivial
black hole spacetime, CCE performs as expected. Second order convergence is found in the moving Schwarzhild black hole
test in Sec. VA4 and in the early time behavior of the BSSN test in Sec. VA3. However, as seen in the late time behavior in
Sec. VA3, the effect of improper BSSN outer boundary conditions on the Cauchy grid is clearly visible in the extracted CCE
news.
In the linearized Teukolsky wave test in Sec. VB, we demonstrate that the computed CCE news converges to the analytic
Teukolsky waveform and does not depend on the world-tube radius. The accuracy of the CCE news is preserved even for small
radii, while the Zerilli news is affected by near zone error. When the extraction radius is sufficiently large, both Zerilli and CCE
give excellent results. The advantage of CCE over the Zerilli method is that the extracted CCE news does not depend on the
extraction radius.

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FIG. 8: When the extraction radius is sufficiently large, the comparison between the Zerilli and the CCE news is in very good agreement.
Cauchy-characteristic extraction can be seen either as a first step towards a full Cauchy-characteristic matching code or as
an improved gravitational wave extraction method in its own right. The results of this paper show that as a stand alone wave
extraction method, it produces the correct results in situations where other methods, such as Zerilli extraction, may fail. Such a
situationis seen in Sec. VA4, wherethe Zerilli methodfails to convergeas a resultof the puregaugemotionofthe Schwarzchild
metric.
Finally, these tests also show the need for improvement of the current implementation of the code. Any angular dependence
in the solution whether through genuine physics, gauge effects, or due to the imposition of boundary conditions on the cubical
Cauchy boundary, leads to short wavelenght error that is poorly resolved by the CCE code. This is particularly noticeable for
features in the region where the stereographic patches overlap. Improvements in this area of the implementation will enhance
the accuracy of the code in extracting the gravitational news from astrophysical simulations requiring high resolution. To this
end, we are investigating the use of different multiple patch implementations such as [65].
Acknowledgments
We are grateful to Nigel Bishop for help at the start of this project. Also, we are thankful to Jeff Winicour for his support
throughoutthe project as well for his careful reading of the manuscript. Computer simulations were done at the PEYOTE cluster
of the Albert Einstein Institut in Golm, Germany and at the Pittsburgh SupercomputingCenter under grant PHY040015P.While
importing the extraction code into the Cactus infrastructure we have benefited from the support of the Cactus teams at AEI and
LSU, for which we are grateful.
We thank LSU for its hospitality. MB was supported by the National Science Foundation under grant PHY-0244673 to the
University of Pittsburgh. BSz was partially supported by the National Science Foundation under grant PHY-0244673 to the
University of Pittsburgh. IH was partially supported by PPARC grant PPA/G/S/2002/00531. YZ was partially supported by the
NASA Center for Gravitational Wave Astronomy at The University of Texas at Brownsville (NAG5-13396) and by NSF grants
PHY-0140326 and PHY-0354867.
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