Towards a wave-extraction method for numerical relativity. V. Extracting the Weyl scalars in the quasi-Kinnersley tetrad from spatial data

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by  Lior M. Burko

Towards a wave–extraction method for numerical relativity. V. Extracting the Weyl
scalars in the quasi-Kinnersley tetrad from spatial data
Lior M. Burko
Department of Physics, University of Alabama in Huntsville, Huntsville, Alabama 35899
(Dated: January 17, 2007)
We extract the Weyl scalars Ψ0 and Ψ4 in the quasi-Kinnersley tetrad by finding initially the
(gauge–, tetrad–, and background–independent) transverse quasi-Kinnersley frame. This step still
leavestwoundetermineddegreesoffreedom: theratio|Ψ0|/|Ψ4|,andoneofthephases(theproduct
|Ψ0|·|Ψ4| and the sum of the phases are determined by the so-called BB radiation scalar). The
residual symmetry (“spin/boost”) can be removed by gauge fixingof spin coefficients in two steps:
First, we break the boost symmetry by requiring that ρ corresponds to a global constant mass
parameterthatequalstheADMmass(or,equivalentlyinperturbationtheory,thatρorµequaltheir
values in the no-radiation limits), thus determining the two moduli of the Weyl scalars |Ψ0|,|Ψ4|,
7 while leaving their phases as yet undetermined. Second, we break the spin symmetry by requiring
0 thattheratioπ/τ givestheexpectedpolarizationstateforthegravitationalwaves,thusdetermining
0 the phases. Our method of gauge fixing—specifically its second step—is appropriate for cases for
2 which the Weyl curvature is purely electric. Applying this method to Misner and Brill–Lindquist
data, we explicitly findthe Weylscalars Ψ0 and Ψ4 perturbatively in thequasi-Kinnersley tetrad.
n
a
PACSnumbers: 04.25.Dm,04.30.Nk,04.70.Bw
J

8
1
I. INTRODUCTION AND SUMMARY

1
v Theoretical understanding of sources for gravitational waves and their emitted waveforms is an important step
1 for their detection and for gravitational-wave astronomy. Such theoretical studies are typically done as numerical
0
simulations. Anumericalrelativitysimulationcomprisesofthreemajorsteps: First,oneneedstoconstructtheinitial
1
data for the physical system of interest; Second, these initial data are evolved in time [1], and third, the evolved
1
data are to be interpreted, and the outgoing waveforms are to be extracted. This paper addresses the last step,
0
7 specifically the problem of wave extraction. It is, however, also closely related to the first step, namely the problem
0 of initial-data determination. Indeed, although our method is applied in this paper to initial data sets (as they are
/ available analytically—for the specific problem of interest here), it is equally applicable also for evolution data. This
c
q paper is a continuation of its prequels [2, 3, 4, 5, 6], and is based on the method developed therein.
- Thelong-termgoalofnumericalrelativityistoevolveabinarysystemovermanyorbits,startingwithslow,adiabatic
r
g inspiral, going through a plunge, and culminating in a ringdown of the produced black hole. Despite dramatic recent
: advances [7], it is unlikely that current numerical relativity codes will be able to to evolve hundreds of orbits in the
v
very near future. The alternative is to start the simulation with the binary being at small orbital separation, so
i
X that only few orbits are left before the plunge. This alternative presents the construction of initial-data sets with a
r fundamental problem. Specifically, it is unclear how to encode in the initial data the gravitational waves that would
a emerge from inspiral fromlarge orbital separationsin the past of the initial time for the simulation. In fact, different
constructions of initial data for the same physical system lead to distinct physical solutions, with different future
evolutions and different gravitational waveforms [1, 8]. The difference between such initial-data sets is basically the
amount of spurious gravitational waves that are present at the initial time. Lack of control over how much “junk”
gravitational waves are present at the initial time implies that even if the numerical evolution were done sufficiently
accurately,itwouldtypicallybeunknownwhichphysicalsystemthatsolutionrepresents,ifthe“junk”dataarestrong
enough.
Thelaststepofthenumericalrelativitysimulationrequirestheextractionoftheoutgoingwaveform. Thelatter—to
coincidewiththewavesthatdetectorssuchasLIGOorLISAwouldmeasure—needstobeextractedintheKinnersley
tetrad. IftheKinnersleyframeisnotidentified,theextractedWeylscalarΨ wouldgenerallybealinearcombination
4
of all five Weyl scalars of the Kinnersley frame, and therefore include not just the information about the outgoing
transverseradiativedegreesoffreedom,butalsoinformationaboutincomingmodes,Coulombicdata,andpuregauge
(longitudinal) information. Indeed, it was shown recently that the extracted waveform changes markedly when the
tetrad basis vectors are transformed [6] (see also [9]), so that if the Weyl scalars are extracted in a tetrad that is not
sufficiently close to the Kinnersley one, the predicted waveform would deviate significantly from the waves detected
experimentally. Identifying—fromnumericalrelativitydata—the physicallyrelevanttetradinwhichthe Weylscalars
are to be extracted is therefore an important problem, and this papers—like its prequels—addresses it.
An important element in the proposed method for wave extraction is the quasi-Kinnersely (qK) frame. A frame
heremeansanequivalenceclassoftetrads(“particularsetoffourbasisvectors”)thatarerelatedtoeachotherbytype
2
III (“spin/boost”) rotations (and exchange) transformations [4]. The quasi-Kinnerley frame is a frame that evolves
continuously into the asymptotic Kinnersley frame, as the simulated (generically Petrov type–I) spacetime settles
down to quiescence (generically Petrov type–D spacetime). A particular subset of all quasi-Kinnersley frames is the
transverse quasi-Kinnersleyframe, defined by the requirementthat it is a quasi-Kinnersleyframe for which the Weyl
scalars Ψ and Ψ vanish. Transverseframes come in threefold in type–I spacetimes, the transversequasi-Kinnersley
1 3
frame being one of these three. The transverse quasi-Kinnersley frame is unique for any generic (type–I) spacetime,
and hence lies its importance. In Ref. [3] we showed how the transverse quasi-Kinnersley frame can be found from
spatial data. The analysis of Ref. [3] still leaves an unbroken residualsymmetry, namely the (continuous) spin/boost
rotations (and the (discrete) exchange operation). In this paper we address the problem of breaking this residual
symmetry. We do not develop a general method in this paper, that would parallel the approach of Ref. [3]. Instead,
we take the approach of addressing a particular (simple, but not too simple) spacetime, and attempting to ad hoc
break the residual symmetry. We believe that this paper sheds light on the general symmetry breaking problem, as
we discuss below.
Specifically, in this paper we consider two initial data sets for the time-symmetric (momentarily stationary) head-
on collision of two non-rotating black holes, namely Misner [10] and Brill–Lindquist [11] data. These initial-data
sets differ in the topological properties of the geometry extended through the black holes’ throats. For an external
observer,thetwoinitial-datasetsdifferintheamountofdistortionthateachblackholesuffersbecauseofthepresence
of the other black hole. Time symmetric black hole binaries were popular in the early days of numerical relativity,
and their gravitational-wave contents have been compared using other methods [12, 13]. In this paper, we revisit
this question using a new method, that we believe to be robust: some questions get answers that are background–,
coordinate–, and tetrad–independent, in addition to being fully calculable from pure spatial data. Other questions
requires gauge fixing in order to be answered, but this gauge fixing can be done in a robust and generic way for at
least some problems.
Thebenefitoftime-symmetricblackholebinarydataisthattheyallowforanalytic(perturbative)solutions,atleast
for small black hole separations. Specifically, the magnetic part of the Weyl curvature vanishes for time-symmetric
data. Not only does this property simplify the derivation of the solution considerably, it also facilitates the gauge
fixing that is required in order to find the quasi-Kinnersley tetrad. Unlike earlier works, we can answer the question
of which initial data set has more “junk waves” without actually evolving the data: our method allows us to find the
energy flux in gravitationalwaves using pure spatial data.
For either initial data set, we obtain the solution for the Weyl scalars Ψ and Ψ using a combination of gauge
0 4
invariant and gauge fixing techniques. Specifically, we first find the transverse quasi-Kinnersley frame, which is a
family of (null) transversetetrads connected by a spin/boost rotation. The transversequasi-Kinnersleyframe can be
found in a completely gauge-, tetrad-, and background-independent method from pure spatial data as are commonly
used in numerical relativity. The residual spin/boost symmetry implies that there are four degrees of freedom to be
determined: the two moduli and the two phases of the Weyl scalars Ψ and Ψ . However, already at this gauge-
0 4
invariant level two of these four degrees of freedom are determined: we determine the product of the two moduli and
the sum of the two phases by finding the so-called Beetle–Burko (BB) radiation scalar ξ [2], which is invariantunder
spin/boost rotations. Recall, that in a transverse frame ξ = Ψ Ψ , so that the phase of the BB scalar ξ equals the
0 4
sum ofthe phases of the two Weyl scalars,andits modulus equals the product ofthe moduli of Ψ andΨ , when the
0 4
latter are in a transverse (quasi–Kinnersley) frame.
We break the remaining spin/boost symmetry in two steps: First, we fix the boost degree of freedom by requiring
that the expansion (or spin coefficient ρ) corresponds to a global mass parameter that equals the system’s ADM
mass [14]. This method is equivalent for the specific problem of interest to the requirement that ρ (or, equivalently
µ) coincides with the no radiation limit (in which the two black holes degenerate into one). While this gauge fixing
is problem adapted, we believe it—or an adaptation thereof—is general enough for most applications in numerical
relativity simulations. Specifically, we require that the spin coefficient ρ (or, equivalently, the expansion θ) coincides
with that of a Schwarzschildblack hole when the proper distance along the spacelike geodesic connecting the throats
of the two black holes vanishes. This method is applicable also to spacetimes that include perturbed rotating black
holes. This gauge fixing determines the two moduli Ψ and Ψ completely.
0 4
| | | |
Theremainingdegreeoffreedomcanalsobedeterminedbygaugefixing,atleastforthespecificproblemofinterest
here, although not without additional information. Specifically, we argue that the spin rotation can be fixed by
choosing the ratio of the spin coefficients π (which below we denote by ̟) and τ. In principle, each of these spin
coefficients has a non-zero spin weight and vanishing boost weight, and could be used to fix the remaining degree
of freedom. However, both have zeros, where spin rotations leave them vanishing. Consequently, gauge fixing is
not unambiguous. However, their ratio exists everywhere (for the initial data sets of interest in this paper), and
has vanishing boost weight and non-zero spin weight, so that its use is vary natural, and completely determines the
Weyl scalars Ψ and Ψ . Specifically, we extend results from point particle motion and expect that only the h
0 4 +
polarization state is excited significantly. Indeed, most studies of point particle motion near black holes show that
3
the h polarization state is extremely hard to excite [15]. As the motion of the two black holes in the problem of
intere×st is radial, indeed no excitation of the h polarization state is expected. We therefore choose the gauge fixing
so that the generated waves include the h pol×arization state only.
+
Our proposed gauge-fixing requires knowledge of the polarization state of the emitted gravitational waves. While
the total emitted energy in gravitational waves is independent of this second step of gauge fixing (and therefore the
flux of energy can be found without first breaking it), its distribution between the h and h polarization states is
+
not. We argue that a physical assumption on the polarization of the waves needs to be made×. Our method of gauge
fixing, specifically its second step, is not specific to the head-on collision case. We expect it to be appropriate for
cases for which the Weyl curvature is purely electric. When the magnetic part does not vanish, π and τ may have
non-coincidingzeros,thushinderingtheuseoftheirratio. ThebreakingoftheresidualspinsymmetrywhentheWeyl
curvature has a non-vanishing magnetic part awaits further consideration.
TheorganizationofthisPaperisasfollows: InSectionIIwereviewtheconstructionofthe quasi-Kinnersleyframe.
We then construct this frame—and break first the boost and then the spin symmetry to find the quasi-Kinnersley
tetrad—for Misner (Section III) and Brill–Lindquist (Section IV) initial data sets. Finally, we compare our results
for Misner and Brill–Lindquist initial data in Section V.
II. CONSTRUCTION OF THE QUASI-KINNERSLEY FRAME
In this Section we outline the calculation of the quasi-Kinnersley frame from spatial data. See Ref. [3] for more
detail. Ina3+1decomposition,thegravitationalfieldisexpressedintermsofthespatialmetricγ andtheextrinsic
ij
curvature K . From these, the electric and magnetic components E and B of the Weyl tensor can be computed
ij ij ij
from
= R +KK K K k 4πS (1)
Eij ij ij − ik j − ij
1
E = γ γkl (2)
ij ij ij kl
E − 3 E
where S =γ aγ bT is the spatial projection of the stress energy tensor T , and
ij i j ab ab
= ǫ kl K (3)
Bij − i ∇k lj
B = . (4)
ij (ij)
B
Here, ǫ := τmǫ is the intrinsic spatial volume element, where τm is the unit future-directed normal to the
ijk mijk
hypersurface, and ǫ is the spacetime volume element.
mijk
By construction, E and B are both symmetric and traceless. We then define the complex tensor
ij ij
Ci =Ei +iBi , (5)
j j j
from which the scalar curvature invariants I and J can be computed as
1
I = Ci Cj (6)
2 j i
and
1
J = Ci CjCl . (7)
−6 j l i
In terms of the dimensionless Baker–Campanellispeciality index (“the S invariant”) [16]
J2
S =27 (8)
I3
the Coulomb scalar can be written as
3J W (S)1/3+W (S) 1/3
χ0, = χ χ − , (9)
±
−2I √S
where W (S)=√S √S 1, and the BB scalar is
χ
− −
1
ξ0, = I 2 W (S)1/3 W (S) 1/3 , (10)
± ξ ξ −
4 − −
h i
4
whereW (S)=2S 1+2 S(S 1). Ingeneral,boththeCoulombandthe BBscalarsadmitthreedistinctcomplex
ξ
− −
roots,sothatξ andχ aremultiply defined. We denotethese threerootswiththe superscripts“0, ”. Thisambiguity
p ±
is related to the fact that generic (i.e., algebraically general) spacetimes admit three distinct transverse frames, in
which the Weyl scalars take different values. The value associated with the principal branch will be denoted with
a superscript 0. The other two branches with superscripts , respectively. The quasi-Kinnersley [4] value will have
±
no superscript. For the cases studied here, there is a clear limit in which spacetime is algebraically special (more
specifically, is Petrov type-D). One can then find the eigenvector σˆa corresponding to the quasi-Kinnersley frame
0
and the BB scalar in the quasi-Kinnersley frame using the methods of Refs. [3, 5]. Specifically, the quasi-Kinnersley
framecanbe identifiedby choosingthe eigenvectorσˆα ofCa correspondingto the eigenvalueofgreatestmodulus [4].
0 b
Notably,theBBscalarisnotneededexplicitlyinordertofindthequasi-Kinnesleyframe. Theonlyrequiredquantity
is the eigenvectorσˆα. We emphasize that while the BB scalaris a promising toolto (partially) describe the radiative
0
degrees of freedom, the wave extraction method is not based on finding it. Rather, it is an auxiliary quantity, which
need not be found for determining the quasi-Kinnersley frame.
Once the eigenvectorσˆa has been found, one might like to find the elements ofsome tetrad inthe quasi-Kinnersley
0
frame. This, too, can be done using only spatial, rather than space-time, data. The real null vectors of an arbitrary
tetradprojectinto a space-likehypersurfaceto define a pair ofrealspatialvectors. While the normalizationsofthese
spatial projections naturally vary if one performs a spin/boost on the tetrad, they define a pair of invariant rays in
the tangent space at a point. These rays correspond to a pair of real unit vectors λˆa and νˆa which are generally
completely independent ofoneanother. In particular,theypointin exactlyopposite directionsonly whenthe normal
τˆa to the spatial slice lies in the space-time tangent 2-plane spanned by ℓa and na. This degenerate case, which
happens when the magnetic part of the Weyl tensor vanishes, is treated separately in Ref. [3], and is applied below
for the moment of time symmetry of Misner and Brill–Lindquist initial data sets. The fiducial time foliation used in
numerical relativity is overwhelmingly likely be “boosted” relative to the 2-plane defined by a given null tetrad, and
only parallel λˆa and νˆa are actually disallowed.
Since λˆa and νˆa determine the directions of the space-time vectors ℓa and na, they also suffice to determine the
frame associated to σˆa, and thus σˆa itself. The explicit relation is
σˆa =(1 λˆ νˆ)−1(νˆa λˆa+iǫabcλˆbνˆc). (11)
− · −
Note that when λˆa and νˆa are interchanged, σˆa simply changes sign, as it should since the bivector Σ changes sign
ab
under an exchange operation in the corresponding frame. (See Ref. [3] for definitions and detail.)
To invert Eq. (11) and solve for λˆa and νˆa given σˆa, we separate σˆa into real and imaginary parts:
σˆa =xa+iya. (12)
In general,neither xa nor ya is unit, but the normalizationcondition for σˆa demands they be orthogonalwith norms
satisfying x 2 y 2 =1. Taking the electric part of Eq. (10) of Ref. [3] then yields
k k −k k
ǫabcx y xa ǫabcx y +xa
λˆa = b c− and νˆa = b c . (13)
x 2 x 2
k k k k
Finally, the elements of the most general null tetrad (ℓa,na,ma,m¯a) in the frame associated to σˆa take the form
ℓa = |c|−1 (τˆa+λˆa)
√1 λˆνˆ
− ·
na = |c| (τˆa+νˆa) (14)
√1 λˆνˆ
− ·
ma = eiϑ √1+λˆ·νˆ(τˆa+µˆa),
√2 √1 λˆνˆ
− ·
where the complex unit projection of ma is
µˆa =(1+λˆ νˆ) 1(λˆa+νˆa+iǫabcλˆ νˆ ). (15)
− b c
·
These vectors are the result of a spin/boost with parameter c = ceiϑ, acting on a preferred tetrad fixed by setting
| |
ℓaτˆ =naτˆ <0andmaτˆ =m¯aτˆ 0. Notethatwhenλˆa andνˆa areindeedanti-parallel,the expressionforma in
a a a a
≤
Eq. (14) is ill-defined. However, the limit as νˆa λa does exist, though it does depend on how the limit is taken.
→−
This is naturalsince the fixedtetrad is determinedonly up to spin transformationsin this case. This degeneratecase
is treated in detail in Ref. [3], and is used below for the head-on collision case of two nonrotating black holes as an
illustration. NoticethatEqs.(14)aredetermineduptoacontinuouscomplexfunctionc,thatis,thebasisvectorsare
a two (real) parameter family (“equivalence class”) of tetrads. This family of tetrads is the quasi-Kinnersley frame,
and the desired quasi-Kinnersley tetrad is a single member of this family of tetrads.
5
III. MISNER DATA
Misner data can be written in spherical isotropic coordinates as
dσ2 =Φ4(R,θ;µ ) dR2+R2 dθ2+ sin2θdφ2 (16)
0
with unit lapse and vanishing shift, where the confor(cid:2)mal factor i(cid:0)s given by (cid:1)(cid:3)
ℓ+1
M
Φ (R,θ;µ )=1+2 κ (µ ) P (cosθ), (17)
Misner 0 ℓ 0 ℓ
R
ℓ=0,2,4, (cid:18) (cid:19)
X ···
where P (cosθ) are the Legendre functions. The parameter µ is related to the separation of the two holes (in a
ℓ 0
manner to be made clear below) and M is the ADM mass of the system. The coefficients κ (µ ) are given by
ℓ 0
1 ∞ (cothnµ0)ℓ ∞ 1
κ (µ ) := and Σ := . (18)
ℓ 0 (4Σ )ℓ+1 sinhnµ 1 sinhnµ
1 0 0
n=1 n=1
X X
As pointed outby Anninoset al. [12], Misner datacanrepresentperturbationsovera singleSchwarzschildblack hole
for µ 1. Repeating the approximations in Ref. [12], for ℓ 1, we assume that (cothnµ )ℓ/ sinhnµ (nµ )ℓ
0 0 0 0
(which≪strictlyspeakingiscorrectfornµ0 ≪1)forsmallµ0,soth≥atweonlykeeptermsinsummationsuptoN≈∼µ−01.
We therefore have
∞ (cothnµ0)ℓ ζ(ℓ+1) lnµ0 ζ(ℓ+1)
, Σ | | ,and κ (µ ) . (19)
n=1 sinhnµ0 ≈ µℓ0+1 1 ≈ µ0 ℓ 0 ≈ |4lnµ0|ℓ+1
X
Here, ζ is the Riemann zeta function. Note that Σ =M/2.
1
The proper distance L along the spacelike geodesic connecting the throats of the two black holes is
∞ n
L=2 1+2µ , (20)
0
sinh(nµ )
" 0 #
n=1
X
which in the limit of small separation(µ 1) is given by
0
≪
π2
L M, (21)
≈ 4 lnµ
0
| |
so that the initial data, to hexadecapole order, can be written in this approximationas
3 5
M ζ(3)P (cosθ) L ζ(5)P (cosθ) L
2 4
Φ 1+ + + , (22)
Misner ≈ 2R 2π6 R 2π10 R
(cid:18) (cid:19) (cid:18) (cid:19)
A. The quasi-Kinnersley frame
We first find the elements of the tensor Ci to O(L3). They are
j
MR3 26ζ(3) 24R2+2MR+M2
CR = 27 P (cosθ) L3+O(L5) (23)
R − (2R+M)6 − π6 2 (2R+M)7
MR3 25ζ(3) 2P (cosθ)(14R2+3MR+M2) (2R+M)2
Cθ = 26 + 2 − L3+O(L5) (24)
θ (2R+M)6 π6 (2R+M)7
MR3 25ζ(3) 2P (cosθ)(10R2 MR)+(2R+M)2
Cφ = 26 + 2 − L3+O(L5) (25)
φ (2R+M)6 π6 (2R+M)7
1
Cθ = CR
R R2 θ
ζ(3) 8R+M
= 3 25 sinθ cosθL3+O(L5). (26)
− · π6 R(2R+M)6
6
The eigenvalues of Ci are
j
λ =CR +O(L6)
1 R
λ =Cθ +O(L6)
2 θ
λ =Cφ
3 φ
so that the eigenvalue that corresponds to the quasi-Kinnersley frame is λ , and the corresponding eigenvector is
1
vα =a(δα +X δα), where
1 R M θ
ζ(3) 8R+M
X = sinθ cosθL3+O(L5) (27)
M 2π6 MR4
and
4R2 8ζ(3)
a=Φ−2(1+R2XM2 )−1/2 = (2R+M)2 − π6(2R+M)3 P2(cosθ)L3+O(L5). (28)
(Notice, that X affects a only at O(L6).)
M
Following the method outlined in Refs. [3, 5], we find the BB scalar to equal
28 32ζ2(3) 5 29 3ζ(3)ζ(5)
ξ = · sin4θL6 + · · [7P (cosθ)+2] sin4θL8 + O(L9) (29)
M π12(M +2R)10 π16R2(M +2R)10 2
and the Coulomb scalar to equal
26MR3 25ζ(3)(24R2+2MR+M2) 3ζ(5)(10R2+5MR+M2)
χ = P (cosθ)L3 P (cosθ)L5+O(L6).
M −(M +2R)6 − π6(M +2R)7 2 − π10R2(M +2R)7 4
(30)
Becausethisisatimesymmetricconfiguration,themagneticpartoftheWeyltensorvanishes,fromwhichitfollows
that the eigenvectorof Weyl is real,as is evident from Eq.(27). This implies that the two realprojections of the two
real null vectors of the quasi-Kinnersley frame onto the hypersurface are anti-parallel. As vˆα is already normalized,
1
we take λˆα = νˆα =vˆα, and then the two null spacetime vector are
− 1
c 1 1
ℓα = | |− δα+ (δα +X δα) (31)
√2 " t Φ2 1+R2X2 R M θ #
M
p
c 1
nα = | | δα (δα +X δα) (32)
√2 " t − Φ2 1+R2X2 R M θ #
M
and the complex null vector is p
eiϑ 1 i
mα = δa+ δa R2X δa (33)
√2Φ2R"sinθ φ 1+R2X2 θ − M R #
M
(cid:0) (cid:1)
Here, c:= c exp(iϑ) is a complex valued function on sppacetime with modulus c and argument ϑ.
| | | |
The Weyl scalars in the basis (31)–(33) are given by
3 24ζ(3) sin2θ 5 24ζ(5) L5
Ψ0 = |c|−2e2iϑ · π6 (2R+M)5 L3+|c|−2e2iϑ ·π10 sin2θ[7P2(cosθ)+2]R2(2R+M)5
23ζ2(3) 2P (cosθ)(64R2+49MR+M2)+(2R+M)(32R+M)
c−2e2iϑ sin2θ 2 L6+O(L7) (34)
− | | π12 MR3(2R+M)6
26MR3 25ζ(3)(24R2+2MR+M2)
Ψ = P (cosθ)L3+O(L5) (35)
2 −(M +2R)6 − π6(M +2R)7 2
3 24ζ(3) sin2θ 5 24ζ(5) L5
Ψ4 = |c|2e−2iϑ · π6 (2R+M)5 L3+|c|2e−2iϑ ·π10 sin2θ[7P2(cosθ)+2]R2(2R+M)5
23ζ2(3) 2P (cosθ)(64R2+49MR+M2)+(2R+M)(32R+M)
+ c2e−2iϑ sin2θ 2 L6+O(L7). (36)
| | π12 MR3(2R+M)6
7
Because this is an approximatesolution,the longitudinal WeylscalarsΨ andΨ are notexactly zero,but atO(L5).
1 3
Specifically, as we expressed X to O(L3), the coefficients of Ψ and Ψ at O(L3) vanishes, and the next term, at
M 1 3
O(L5) remains. We can nullify these (and successive) terms by keeping terms to sufficiently high order on Φ and in
X .
M
B. The qK tetrad: Breaking the boost symmetry
Webreakthespin/boostsymmetryintwosteps: first,webreaktheboostsymmetrywhileleavingthespinsymmetry
unbroken. At the next step we address the breaking of the spin symmetry. To break the boost symmetry we are
looking for a spin coefficient with non-zero boost weight but vanishing spin weight. The two candidates are the spin
coefficients ρ and µ. The spin coefficient
ρ := mαm¯β ℓ
β α
− ∇
R(2R M)
= 2√2 c−1 − +O(L3). (37)
− | | (2R+M)3
Under Type III rotations ρ transforms like ℓα, i.e., ρ c 1ρ. We can therefore scale the null basis vectors ℓα and
−
nα by demanding that ρ equals, in the limit L = 0 wh→en|n|o radiation exists, the Schwarzschild black hole value ρ .
S
As ρ is real, the twist vanishes, and ρ equals the expansion θ. Recall that for L = 0, we have a single non-rotating
black hole with ADM mass M, for which the spin coefficient ρ = R 1(1+M/2R) 2. Requiring that for L=0 we
S − −
−
get ρ=ρ , we find that
S
1 2R M
c = − . (38)
| | √2 2R+M
Alternatively, we can recall that for L = 0 (also for a Kerr black hole), Ψ /ρ3 = M. Under a type III rotation the
2
combinationΨ /ρ3hasboostweight3andzerospinweight. Wecanthereforefindaboostrotationfunction c sothat
2
| |
this combination equals a global constant. To fix the remaining multiplicative constant factor in c we require that
| |
the global constant equals M. (In practice we apply this method here in the L 0 limit.) Putting these expressions
together,wesolvefor c,andrecoverthe previousresult. Onecanalsousethe s→pincoefficientµ:=m¯αmβ n ,and
β α
| | ∇
demand that it would equal the no-radiation limit as L 0, or µ = 2R(2R M)2/(2R+M)4. This calculation
S
→ − −
recovers the value of c.
| |
We can now find Ψ and Ψ in the quasi-Kinnersley tetrad
0 4
| | | |
3 25ζ(3) L3 5 25ζ(5) 1
Ψ = · sin2θ + · sin2θ[7P (cosθ)+2]L5
| 0| π6 (2R M)2(2R+M)3 π10 R2(2R M)2(2R+M)3 2
(cid:12) − −
(cid:12)(cid:12)(cid:12)24ζ2(3) sin2θ 2P2(cosθ)(64R2+49MR+M2)+(2R+M)(32R+M)L6+O(L7) (39)
− π12 MR3(2R+M)4(2R M)2
− (cid:12)
(cid:12)
(cid:12)
(cid:12)
3 23ζ(3) (2R M)2L3 5 23ζ(5) (2R M)2
Ψ = · sin2θ − + · − sin2θ[7P (cosθ)+2]L5
| 4| π6 (2R+M)7 π10 R2(2R+M)7 2
(cid:12)
(cid:12)(cid:12)(cid:12)22ζ2(3) sin2θ 2P2(cosθ)(64R2+49MR+M2)+(2R+M)(32R+M)(2R M)2L6+O(L7) (40)
− π12 MR3(2R+M)8 −
(cid:12)
(cid:12)
At great distances R M, Ψ4 L3/R5 and Ψ0 L3/R5. (cid:12)(cid:12)
≫ | |∼ | |∼
The energy flux in gravitational waves depends on the (time domain) Weyl scalars Ψ and Ψ in a non-local way,
0 4
as is manifest from the integral over preceding times in the expression for the energy flux in gravitational waves (at
infinity)
dE 1 t 2
(t)= lim dω dt ψ(t,r,θ,φ) , (41)
′ ′
dt r→∞"4πr6 Zω (cid:12)(cid:12)Z−∞ (cid:12)(cid:12) #
(cid:12) (cid:12)
where ψ is the Teukolsky function, r is the geometrical ra(cid:12)dial coordinate, and(cid:12)dω is the integration measure over
the 2-sphere. The contribution to the energy flux (at finite distances) in gravitational waves at the moment of time
8
symmetry (“t=0”) from that moment in time (“t =0”) is given by
′
:= R2 Ψ 2dΩ
0 0
F | |
Z
3 211ζ2(3) L6 3 211ζ3(3) 64R2+22MR+M2
= · · L9
5π11 R2(2R+M)2(2R M)4 − 5 7π17 MR5(2R+M)3(2R M)4
− · −
5 211ζ2(5) L10 212ζ2(3)ζ(5) 160R+173M
+ · L11+O(L12) (42)
π19 R6(2R+M)2(2R M)4 − 5 7π21 MR6(2R+M)3(2R M)4
− · −
:= R2 Ψ 2dΩ
4 4
F | |
Z
3 27ζ2(3) (2R M)4 3 27ζ3(3) (64R2+22MR+M2)(2R M)4
= · − L6 · − L9
5π11 R2(2R+M)10 − 5 7π17 MR5(2R+M)11
·
5 27ζ2(5) (2R M)4 28ζ2(3)ζ(5) (160R+173M)(2R M)4
+ · − L10 − L11+O(L12), (43)
π19 R6(2R+M)10 − 5 7π21 MR6(2R+M)11
·
where is the flux in Ψ .
i i
F
C. The qK tetrad: Breaking the spin symmetry
Tobreakthespinsymmetrywelookforspincoefficientsthathavenon-zerospinweightandvanishingboostweight.
The two candidates are the spin coefficients ̟1 and τ. The spin coefficient
̟ := m¯aℓb n
b a
∇
ζ(3) e iϑ
= 2√2i sinθ cosθ − L3+O(L5) (44)
− π6 MR2(2R+M)
and
τ := manb ℓ
b a
− ∇
ζ(3) eiϑ
= 2√2i sinθ cosθ L3+O(L5). (45)
− π6 MR2(2R+M)
We cannot just consider a spin rotation (i.e., a type III rotation with c =1) directly on ̟ and τ, because ̟ and
| |
τ have zeros on the equatorial plane and at the poles. At the zeros (and numerically also in neighborhoods thereof)
the rotation parameter is arbitrary. However, because both ̟ and τ have zeros at the same locations, the limit of
their ratio exists everywhere. We therefore consider the ratio
̟
=e 2iϑ. (46)
−
τ
Notably, Eq. (46) is satisfied nonperturbatively. Under type III rotation by angle ϕ, ̟/τ e 2iϕ̟/τ (i.e., the ratio
−
→
̟/τ has spin weight 2), so that by choosing a spin rotation function ϕ we can determine the ratio ̟/τ uniquely.
−
Choosingthe function ϕ we in fact determine the localpolarizationstate of the gravitationalwavesin the wavezone,
where Ψ and Ψ have a clear physical interpretation in term of outgoing and incoming waves, respectively. Notice,
4 0
that once the boost degree of freedom is fixed, the total energy flux in gravitational waves is determined. The spin
degreeoffreedomwill neverthelessmovepartofthis flux betweenthe two polarizationstates. This propertysuggests
to us that the spin degree of freedom can only be determined based on the understanding of the polarization of the
wavesin the wave zone. As already discussed in the Introduction, studies on radialmotion in the point particle limit
have shown that the h polarization state is extremely hard to excite. Following Berti et al [15] we may presumably
lean on the point part×icle results, and assume that only the h polarization state is generated. In our case, indeed
+
this conclusion is guaranteed by the diagonality of the metric (16).
1 Conventionally,m¯aℓb∇bna isdenotedbyπ. Here,toavoidconfusionwiththegeometricalπ,wechangetheusualnotation,anddenote
itwith̟.
9
Our ability to break the spin symmetry depends on the observation that the ratio ̟/τ is free of zeros, which may
be attributed to the vanishing of Bi .2 We assert that this method for breaking the spin symmetry by gauge fixing
j
̟/τ according to the polarization state is applicable to purely electric cases for which Bi =0. The question of how
j
to break the spin symmetry for the magnetic case is not answered by the above discussion, and we hope to address
this question elsewhere.
Inpractice,bothΨ andΨ wouldberealforthiscase. Choosingϕ= ϑwecanset̟/τ =1. Underthisrotation,
0 4
Ψ e2iϕΨ and Ψ e 2iϕΨ , so that the new Weyl scalars equal th−eir moduli, i.e.,
0 0 4 − 4
→ →
3 25ζ(3) L3 5 25ζ(5) 1
Ψ = · sin2θ + · sin2θ[7P (cosθ)+2]L5
0 π6 (2R M)2(2R+M)3 π10 R2(2R M)2(2R+M)3 2
− −
24ζ2(3) 2P (cosθ)(64R2+49MR+M2)+(2R+M)(32R+M)
sin2θ 2 L6+O(L7) (47)
− π12 MR3(2R+M)4(2R M)2
−
3 23ζ(3) (2R M)2L3 5 23ζ(5) (2R M)2
Ψ = · sin2θ − + · − sin2θ[7P (cosθ)+2]L5
4 π6 (2R+M)7 π10 R2(2R+M)7 2
22ζ2(3) 2P (cosθ)(64R2+49MR+M2)+(2R+M)(32R+M)
sin2θ 2 (2R M)2L6+O(L7). (48)
− π12 MR3(2R+M)8 −
We emphasize that this breaking of the spin symmetry relies on physical intuition based on extension of the point
particleresultstotheequalmasscase. Anyotherchoiceforϕwoulddojustasfineintheabsenceofknowledgeofthe
local polarizationstate of the gravitationalwaves. Notice, in contrast,that breaking the boost symmetry is a unique
gauge fixing by virtue of the radiation free limit, and that getting the quasi-Kinnnersley frame is gauge invariant.
Finally, we can write explicitly the basis vectors for the quasi-Kinnersley tetrad:
2R+M 1
ℓα = δα+ (δα +X δα) (49)
2R M " t Φ2 1+R2X2 R M θ #
− M
p
2R M 1
nα = − δα (δα +X δα) (50)
2(2R+M) " t − Φ2 1+R2X2 R M θ #
M
p
1 1 i
mα = δa+ δa R2X δa . (51)
√2Φ2R"sinθ φ 1+R2X2 θ − M R #
M
(cid:0) (cid:1)
p
1 1 i
m¯α = δa δa R2X δa . (52)
√2Φ2R"sinθ φ− 1+R2X2 θ − M R #
M
(cid:0) (cid:1)
p
IV. BRILL–LINDQUIST DATA
Next, we write Brill–Lindquistinitial data. The metric hasthe form(16), but now the conformalfactor is givenby
m 1 1
Φ (R,θ;z )=1+ + , (53)
BL 0
2  R2 sin2θ+(R cosθ z )2 R2 sin2θ+(R cosθ+z )2
0 0
−
q q 
where the two black holes are at positions z along the z axis. The parameter m is related to the mass of the black
0
±
holes.
2 WhereBi 6=0therealunitvectors λˆa andνˆa arenotco-linear,sothatthelimit̟/τ maynotexisteverywhere.
j
10
A. The quasi-Kinnersley frame
In this Section we find the quasi-Kinnersleyframe for B–L data. Our calculation parallels its Misner data counter-
part, so we describe it briefly.
mR3 6R2+mR+m2
CR = 4 4m P (cosθ)z2+O(z4) (54)
R − (R+m)6 − (R+m)7 2 0 0
mR3 (7R2+3mR+2m2) P (cosθ) (R+m)2
Cθ = 2 +2m 2 − z2+O(z4) (55)
θ (R+m)6 (R+m)7 0 0
mR3 R(5R m) P (cosθ)+(R+m)2
Cφ = 2 +2m − 2 z2+O(z4) (56)
φ (R+m)6 (R+m)7 0 0
1
Cθ = CR
R R2 θ
m+4R
= 6m sinθ cosθz2+O(z4) (57)
− R(R+m)6 0 0
The eigenvalues of Ci are
j
λ =CR +O(z4)
1 R 0
λ =Cθ +O(z4)
2 θ 0
λ =Cφ
3 φ
so that the eigenvalue that corresponds to the quasi-Kinnersley frame is λ , and the corresponding eigenvector is
1
vα =a(δα +X δα), where
1 R BL θ
4R+m
X = sinθ cosθz2+O(z4) (58)
BL R4 0 0
and
R2 m
a=Φ−2(1+R2XB2L)−1/2 = (R+m)2 −2(R+m)3 P2(cosθ)z02+O(z04). (59)
(Notice, that X affects a only at O(z4).)
BL 0
We repeat the calculation for Brill–Lindquist initial data, and find that the BB scalar equals
9m2 6m2
ξ = sin4θz4+ (3R2 14mR 2m2)P (cosθ) (6R2+7mR+m2) sin4θz6+O(z8)
BL (R+m)10 0 R3(R+m)11 − − 2 − 0 0
(cid:2) (cid:3) (60)
and the Coulomb scalar is given by
2mR3 2m(6R2+Rm+m2)
χ = P (cosθ)z2+O(z4). (61)
BL −(R+m)6 − (R+m)7 2 0 0
The Weyl scalars in the quasi-Kinnersley frame are
3m sin2θ m sin2θ
Ψ0 = |c|−2e2iϑ (R+m)5 z02+|c|−2e2iϑ R3(R+m)6 [(3R2−14mR−2m2)P2(cosθ)+(6R2+7mR+m2)]z04+O((z6062))
mR3 6R2+mR+m2
Ψ = 2 2m P (cosθ)z2+O(z4) (63)
2 − (R+m)6 − (R+m)7 2 0 0
3m sin2θ m sin2θ
Ψ4 = |c|2e−2iϑ (R+m)5 z02+|c|2e−2iϑR3(R+m)6 [(3R2−14mR−2m2)P2(cosθ)+(6R2+7mR+m2)]z04+O(z(6064)).