The first task in analyzing data from a numerical simulation is simply extract- ing the relevant waveform. There are two common methods used to achieve this
6More specifically, we need|∂t/∂R| ¿ |∂r
∗/∂R|. This condition needs to be checked for all radii used, at all times in the simulation. For the data presented below, we have checked this, and shown it to be a valid assumption, at the radii used for extrapolation.
3.3. Waveform extraction 73 aim. One uses Regge–Wheeler–Zerilli functions to extract the metric perturbation directly. This method explicitly decomposes the full spacetime into a spherically symmetric background and a perturbation. Particular combinations of the pertur- bations are then taken, making the final result invariant to first-orderunder gauge perturbations. The second method uses Newman–Penrose scalars to decompose the Weyl tensor, directly reading off the curvature. We now examine these two techniques in more detail.
3.3.1
Regge–Wheeler–Zerilli functions
Both methods of extracting gravitational-wave information have their origins in treatments of perturbations of Schwarzschild and Kerr black holes. While these analyses are valid right down to—or even within—the black hole’s horizon, we only need them at some distance from the center of our domain. The physical situation of interest to simulations of binary black holes is an asymptotically flat spacetime. In this case, the spacetime in the outer parts of a simulation can be thought of as a perturbation of Schwarzschild.
Regge and Wheeler [222] first introduced a formalism for studying odd-parity perturbations of a Schwarzschild black hole in Schwarzschild coordinates, while investigating the stability of black holes. Zerilli [255] extended their formalism to allow for even-parity perturbations. The Regge–Wheeler–Zerilli (RWZ) meth- ods are gauge dependent, however, requiring the gauge to be fixed before any interpretation can be made of the physical content of their results. Moncrief [196] suggested an improvement to the combined formalism, making it linearly invari- ant to gauge perturbations. Sarbach and Tiglio [228] have extended this further,
with a geometric treatment that can be used to study perturbations on an ar- bitrary slicing of a spherically symmetric spacetime.7 These generalized RWZ functions obey wave equations, and can be used to extract the (linearly) gauge- invariant metric perturbation. Second-order methods have also been developed [146, 83, 203, 147].
We begin by assuming the existence of a spherically symmetric background metric
_gµν that can be decomposed into a time–radius sector and a spherical sector:
_gµνdx
µdxν=gˇ
abdxadxb+r2Ω˚ABdxAdxB . (3.15)
Here, lowercase Latin indices refer to thet–r sector, as does the check (suggesting the null-cone) over the metric. Similarly, uppercase Latin indices refer to theϑ–ϕ sector, as does the ring (suggesting the sphere) over the metric. The metricΩ˚AB is assumed to be the standard unit-sphere metric. We will also need the derivative operator∇˚ and antisymmetric tensor²˚appropriate to the spheroidal submanifold spanned by ϑ
,
ϕ.Next, we define the metric perturbation δgµν as the full metric minus the background:
δgµν
B
gµν−_gµν . (3.16)We then expand the perturbation in (tensor, vector, and scalar) spherical harmon- 7Useful reviews of RWZ methods can be found in Refs. [198, 71, 226].
3.3. Waveform extraction 75 ics: δgab=X l,m ˜ Pabl,m Yl,m
,
(3.17a) δgAb= X l,m h ˜ Qbl,m ∇˚AYl,m+R˜bl,m ²˚CA∇˚CYl,m i,
(3.17b) δgAB= X l,m h ˜ Sl,m r2Ω˚ABYl,m+T˜l,m ∇˚(A∇˚B)Yl,m+U˜l,m ²˚C(A∇˚B)∇˚CYl,m i . (3.17c) The components P˜abl,m,
...,
U˜l,m will be gauge dependent. Even in flat spacetime, we could introduce a gauge perturbation which mimics a gravitational wave in terms of its effect on some of the components. For example, if we only look at T˜l,m and ignore other components, we may easily be deceived by a gauge perturbation. Moncrief, however, pointed out that it is possible to take certain combinations of the components to obtain components that are gauge invariant to first order.For example, take the magnetic-parity part of the metric perturbation:
δgAb=R˜bl,m ²˚CA∇˚CYl,m
,
δgAB=U˜l,m ²˚C(A∇˚B)∇˚CYl,m . (3.18) A general magnetic-parity gauge transformation looks likeξa=0 ξA=ς(t
,
r) ˚²CA∇˚CYl,m,
(3.19) for some function ς(t,
r). If this is an infinitesimal gauge transformation, the components of the metric perturbation transform as˜
Rlb,m→R˜bl,m+ ∇bς
,
U˜l,m→U˜l,m+ς . (3.20) Looking at these expressions for a moment suggests a way to cancel the gauge dependence:The new component Rlb,m is invariant to linear order under gauge transforma- tions. Analogous—though more complicated—expressions may be obtained for the other components of the metric perturbation [228, 224].
These invariant components are then used to reconstruct the metric, which can be decomposed in the standard way [42] by defining the polarization tensors εαβ+
B
ϑαϑβ−ϕαϕβ and εαβ×B
ϑαϕβ+ϕαϑβ,
(3.22)and using these to define
h+
B
12εαβ+ δgαβ and h×B
12εαβ× δgαβ . (3.23) Here, ϑα and ϕα are the standard coordinate vectors, which do not necessarily have unit length. These two quantities are commonly combined into a single complex field:h
B
h+−ih× . (3.24)Details of the procedure used for the data presented here are given in [224]. In the notation of that paper, the metric perturbation is given by [231]
h=1 r ∞ X l=2 l X m=−l p (l+2)(l+1)l(l−1)³Φl(+,m) +iΦl(−,m)´−2Yl,m
,
(3.25) where−2Yl,m are the spin-weight −2 spherical harmonics discussed in Sec. 3.3.3.3.3.2
Newman–Penrose scalars
Newman and Penrose introduced a useful decomposition of the Weyl tensor, among other geometric quantities [201]. At each point in space, they define a complex null tetrad (lα
,
nα,
mα,
m¯α), where m¯α is simply the complex conjugate3.3. Waveform extraction 77 of mα. The tetrad is assumed to obey the orthonormality conditions lαnα= −1, mαm¯
α=1, with all other products being zero, and the vector mα assumed to be a complex combination of spacelike vectors. This tetrad is then used to define five complex functions of space and time:8
Ψ0
B
Cαβγδlαmβlγmδ;
(3.26a)Ψ1
B
Cαβγδlαmβlγnδ;
(3.26b)Ψ2
B
Cαβγδlαmβm¯γnδ;
(3.26c)Ψ3
B
Cαβγδlαnβm¯γnδ;
(3.26d)Ψ4
B
Cαβγδnαm¯βnγm¯δ . (3.26e) These ten real degrees of freedom correspond to the ten degrees of freedom in the Weyl tensor itself, effectively encoding the curvature of a vacuum spacetime. Indeed, it is not difficult to re-express the Weyl tensor in terms of these five functions and the tetrad—essentially inverting the equations above. (See, e.g., Chandrasehkar’s Eq. (1.298) [92].) Because we will always be working in vacuum, we note thatCαβγδ=Rαβγδ.A general spacetime, naturally, has nonzero values for each of these quanti- ties. Moreover, because the choice of tetrad is not unique, we can generally mix components by rotating the tetrad. However, it can be shown [227] that—in vac- uum, asymptotically flat spacetimes—there exists a tetrad for whichΨ4 isO(r−1) along an outgoing null ray, and all other Newman–Penrose scalars fall off more quickly. This is essentially the so-called peeling theorem [235]. It shows that, if 8Note that Newman and Penrose use the opposite metric-signature convention and convention for the Riemann tensor, as compared to the one used here (see Sec. A.2). The equations they use to defineΨn also have opposite signs, so overall, the signs of theΨn used here should agree with those of Newman and Penrose.
we hope to find radiation from an isolated source using the Newman–Penrose formalism, Ψ4 is—at least—a crucial element.
In fact, there exist many tetrads such thatΨ4isO(r−1), and all other Newman– Penrose scalars fall off more quickly. This variety—and the ambiguity it brings about—makes the issue of choosing a particular such tetrad in a general spacetime a delicate one. Here, we simply define the tetrad components by the coordinate basis used in an evolution. Explicitly, we define the tetrad components by
lα
B
p1 2 ¡ tα+rα¢,
(3.27a) nαB
p1 2 ¡ tα−rα¢,
(3.27b) mαB
p1 2 ¡ ϑα+iϕα¢ . (3.27c)(When evaluation on thez axis is necessary, we always take the limit withϕ=0.) The vector tα is a unit-magnitude vector, normal to the spacelike hypersurface of constant time in the simulation. The vector rα is a unit vector within that spacelike hypersurface which is normal to the extraction sphere at that time. Both of the latter are normalized by the full spacetime metric. Again, the vectors ϑα and ϕα are the standard coordinate vectors. This choice actually leaves the orthonormality conditions for mα and m¯α unsatisfied in general; they do not have unit magnitude, and are not orthogonal. Nonetheless, when the metric in these coordinates asymptotically approaches the standard Minkowski metric, we can expect that the orthonormality conditions will be satisfied asymptotically. In turn, we can also expect that this will suffice because the physically relevant part of the resulting Ψ4 will be selected by extrapolation.
3.3. Waveform extraction 79
Relation of Ψ4 to h
In the linear theory, we can construct a Minkowskian background, and small perturbations on top of it. In this case, we define the tetrad above with respect to the background, orthonormalizing the tetrad with respect to the background metric. Then, for plane waves propagating along the nα vector, it is not difficult to show that9
Ψ4= −h¨= −¡h¨+−i ¨h×¢ in the linear approximation, (3.28)
where double dots denote double time derivatives. Note thath, h+, and h× were
defined in Eqs. (3.23) and (3.24). This gives us further confidence that Ψ4 is a physically relevant quantity to extract. Unfortunately, it also shows us that we need to integrate Ψ4 twice, if we wish compute h from Ψ4. Various integration techniques have been developed, the most successful of which will be discussed in later chapters.
It is worth emphasizing that this relation is true only in the linear approxima- tion, and only with a particular choice of tetrad. In highly nonlinear spacetimes, there is no reason to expect this relation to hold. Indeed, even in the linear approximation, this need not hold for general tetrads satisfying the conditions set out by Newman and Penrose; this simple expression is only obtained when the wave is propagating along the nα vector and when (ε+−iε×)αβ=2 ¯mαm¯β.
(Recall the definitions of ε+ and ε× from Eq. (3.22).) For example, if we were to
rotate the mα vector by an angle θ, we would have Ψ
4= −e−2iθh¨.
9There are many sign choices that need to be made before deriving this relation: the sign of Riemann; the sign ofΨ4; the metric signature; the sign of metric perturbations; etc. Equation (3.28) is correct for the signs chosen here. For a review of those signs, see Appendix A.
3.3.3
Spin-weighted spherical harmonics
Newman and Penrose [202] introduced a useful set of functions similar to the standard spherical harmonics, designed to decompose functions like Ψ4—the spin-weighted spherical harmonics (SWSHs), sYl,m.10 In particular,Ψ4and h can be decomposed in terms of SWSHs of spin weight s= −2:
Ψl4,m(t