# Extraction of gravitational waves in numerical relativity

## Abstract

A numerical-relativity calculation yields in general a solution of the Einstein equations including also a radiative part, which is in practice computed in a region of finite extent. Since gravitational radiation is properly defined only at null infinity and in an appropriate coordinate system, the accurate estimation of the emitted gravitational waves represents an old and non-trivial problem in numerical relativity. A number of methods have been developed over the years to “extract” the radiative part of the solution from a numerical simulation and these include: quadrupole formulas, gauge-invariant metric perturbations, Weyl scalars, and characteristic extraction. We review and discuss each method, in terms of both its theoretical background as well as its implementation. Finally, we provide a brief comparison of the various methods in terms of their inherent advantages and disadvantages.

### Keywords

Gravitational wave extraction Numerical relativity Binary mergers Black holes Neutron stars## 1 Introduction

With the commissioning of the second generation of laser interferometric gravitational-wave detectors, and the recent detection of gravitational waves (Abbott 2016), there is considerable interest in gravitational-wave astronomy. This is a huge field, covering the diverse topics of: detector hardware construction and design; data analysis; astrophysical source modeling; approximate methods for gravitational-wave calculation; and, when the weak field approach is not valid, numerical relativity.

Numerical relativity is concerned with the construction of a numerical solution to the Einstein equations, so obtaining an approximate description of a spacetime, and is reviewed, for example, in the textbooks by Alcubierre (2008), Bona et al. (2009), Baumgarte and Shapiro (2010), Gourgoulhon (2012) and Rezzolla and Zanotti (2013). The physics in the simulation may be only gravity, as is the case of a binary black hole scenario, but it may also include matter fields and/or electromagnetic fields. Thus numerical relativity may be included in the modeling of a wide range of astrophysical processes. Often (but not always), an important desired outcome of the modeling process will be a prediction of the emitted gravitational waves. However, obtaining an accurate estimate of gravitational waves from the variables evolved in the simulation is normally a rather complicated process. The key difficulty is that gravitational waves are unambiguously defined only at future null infinity (\(\mathcal {J}^+\)), whereas in practice the domain of numerical simulations is a region of finite extent using a “3+1” foliation of the spacetime. This is true for most of the numerical codes, but there are also notable exceptions. Indeed, there have been attempts towards the construction of codes that include both null infinity and the central dynamic region in the domain, but they have not been successful in the general case. These attempts include the hyperboloidal method (Frauendiener 2004), Cauchy characteristic matching (Winicour 2005), and a characteristic code (Bishop et al. 1997b). The only successful application to an astrophysical problem has been to axisymmetric core collapse using a characteristic code (Siebel et al. 2003).

In the linearized approximation, where gravitational fields are weak and velocities are small, it is straightforward to derive a relationship between the matter dynamics and the emission of gravitational waves, the well-known quadrupole formula. This can be traced back to work by Einstein (1916, 1918) shortly after the publication of general relativity. The method is widely used to estimate gravitational-wave production in many astrophysical processes. However, the strongest gravitational-wave signals come from highly compact systems with large velocities, that is from processes where the linearized assumptions do not apply. And of course, it is an event producing a powerful signal that is most likely to be found in gravitational-wave detector data. Thus it is important to be able to calculate gravitational-wave emission accurately for processes such as black hole or neutron star inspiral and merger, stellar core collapse, etc. Such problems cannot be solved analytically and instead are modeled by numerical relativity, as described in the previous paragraph, to compute the gravitational field near the source. The procedure of using this data to measure the gravitational radiation far from the source is called “extraction” of gravitational waves from the numerical solution.

In addition to the quadrupole formula and full numerical relativity, there are a number of other approaches to calculating gravitational-wave emission from astrophysical sources. These techniques are not discussed here and are reviewed elsewhere. They include post-Newtonian methods (Blanchet 2014), effective one-body methods (Damour and Nagar 2016), and self-force methods (Poisson et al. 2011). Another approach, now no-longer pursued, is the so-called “Lazarus approach”, that combined analytical and numerical techniques (Baker et al. 2000b, 2002a, b).

In this article we will review a number of different extraction methods: (a) Quadrupole formula and its variations (Sect. 2.3); (b) methods using the Newman–Penrose scalar \(\psi _4\) evaluated on a worldtube (\(\varGamma \)) (Sect. 3.3); (c) Cauchy Perturbative methods, using data on \(\varGamma \) to construct an approximation to a perturbative solution on a known curved background (Sects. 4, 5; Abrahams and Evans 1988, 1990); and (d) Characteristic extraction, using data on \(\varGamma \) as inner boundary data for a characteristic code to find the waveform at \(\mathcal {J}^+\) (Sects. 6, 7). The description of the methods is fairly complete, with derivations given from first principles and in some detail. In cases (c) and (d), the theory involved is quite lengthy, so we also provide implementation summaries for the reader who is more interested in applying, rather than fully understanding, a particular method, see Sects. 5.6 and 7.8.

In addition, this review provides background material on gravitational waves (Sect. 2), on the “3+1” formalism for evolving the Einstein equations (Sect. 3), and on the characteristic formalism with particular reference to its use in estimating gravitational radiation (Sect. 6). The review concludes with a comparison of the various methods for extracting gravitational waves (Sect. 8). This review uses many different symbols, and their use and meaning is summarized in “Appendix 1”. Spin-weighted, and other, spherical harmonics are discussed in “Appendix 2”, and various computer algebra scripts and numerical codes are given in “Appendix 3”.

Throughout, we will use a spacelike signature \((-,+,+,+)\) and a system of geometrised units in which \(G = c = 1\), although when needed we will also indicate the speed of light, *c*, explicitly. We will indicate with a boldface any tensor, e.g., \(\varvec{V}\) and with the standard arrow any three-dimensional vector or operator, e.g., \(\mathbf {\varvec{v}}\) and \(\mathbf {\nabla }\). Four-dimensional covariant and partial derivatives will be indicated in general with \(\nabla _{\mu }\) and \(\partial _{\mu }\), but other symbols may be introduced for less common definitions, or when we want to aid the comparison with classical Newtonian expressions. Within the standard convention of a summation of repeated indices, Greek letters will be taken to run from 0 to 3, while Latin indices run from 1 to 3.

We note that some of the material in this review has already appeared in books or other review articles. In particular, we have abundantly used parts of the text from the book “Relativistic Hydrodynamics”, by Rezzolla and Zanotti (2013), from the review article “Gauge-invariant non-spherical metric perturbations of Schwarzschild black-hole spacetimes”, by Nagar and Rezzolla (2006), as well as adaptations of the text from the article “Cauchy-characteristic matching”, by Bishop et al. (1999a).

## 2 A quick review of gravitational waves

### 2.1 Linearized Einstein equations

^{1}Using this notation, the linearised Einstein equations (9) take the more compact form

*Dalambertian*(or wave) operator, i.e., \(\partial _{\alpha }\partial ^{\alpha }{\bar{h}}_{\mu \nu } = \square {\bar{h}}_{\mu \nu }\). At this stage, we can exploit the gauge freedom inherent in general relativity (see also below for an extended discussion) to recast Eq. (11) in a more convenient form. More specifically, we exploit this gauge freedom by choosing the metric perturbations \(h_{\mu \nu }\) so as to eliminate the terms in (11) that spoil the wave-like structure. Most notably, the coordinates can be selected so that the metric perturbations satisfy

*Lorenz*(or Hilbert)

*gauge*, the linearised field equations take the form

*amplitude tensor*. Substitution of the ansatz (15) into Eq. (14) implies that \(\kappa ^{\alpha }\kappa _{\alpha }=0\) so that \(\varvec{\kappa }\) is a null four-vector. In such a solution, the plane wave (15) travels in the spatial direction \({\varvec{k}} = (\kappa _x,\kappa _y,\kappa _z)/\kappa ^0\) with frequency \(\omega :=\kappa ^0 = (\kappa ^j\kappa _j)^{1/2}\). The next step is to substitute the ansatz (15) into the Lorenz gauge condition Eq. (12), yielding \(A_{\mu \nu }\kappa ^{\mu }=0\) so that \(\varvec{A}\) and \(\varvec{\kappa }\) are orthogonal. Consequently, the amplitude tensor \(\varvec{A}\), which in principle has \(16-6=10\) independent components, satisfies four conditions. Thus the imposition of the Lorenz gauge reduces the independent components of \(\varvec{A}\) to six. We now investigate how to reduce the number of independent components to match the number of dynamical degrees of freedom of general relativity, i.e., two.

*gauge changes*, i.e., through infinitesimal coordinate transformations that are consistent with the gauge that has been selected. The freedom to make such a transformation follows from a foundation of general relativity, the principle of general covariance. To better appreciate this matter, consider an infinitesimal coordinate transformation in terms of a small but otherwise arbitrary displacement four-vector \(\varvec{\xi }\)

*four*arbitrary constants \(C^{\alpha }\), a gauge transformation that changes arbitrarily

*four*components of \(\varvec{A}\) in addition to those coming from the condition \(\varvec{A} \cdot \varvec{\kappa }=0\). Effectively, therefore, \(A_{\mu \nu }\) has only \(10-4-4=2\) linearly independent components, corresponding to the number of degrees of freedom in general relativity (Misner et al. 1973).

Note that these considerations are not unique to general relativity and similar arguments can also be made in classical electrodynamics, where the Maxwell equations are invariant under transformations of the vector potentials of the type \(A_{\mu } \rightarrow A_{\mu '} = A_{\mu } + \partial _{\mu }\varPsi \), where \(\varPsi \) is an arbitrary scalar function, so that the corresponding electromagnetic tensor is \(F^\mathrm{new}_{\mu ' \nu '} = \partial _{\nu '} A_{\mu '} - \partial _{\mu '} A_{\nu '} = F^\mathrm{old}_{\mu ' \nu '}\). Similarly, in a linearised theory of general relativity, the gauge transformation (18) will preserve the components of the Riemann tensor, i.e., \(R^\mathrm{new}_{\alpha \beta \mu \nu } = R^\mathrm{old}_{\alpha \beta \mu \nu } + \mathcal {O}(R^2)\).

- (a)
*orthogonality condition*: four components of the amplitude tensor can be specified since the Lorenz gauge implies that \(\varvec{A}\) and \(\varvec{\kappa }\) are orthogonal, i.e., \(A_{\mu \nu } \kappa ^{\nu } = 0\). - (b)
*choice of observer*: three components of the amplitude tensor can be eliminated after selecting the infinitesimal displacement vector \(\xi ^{\mu } = iC^{\mu } \exp (i\kappa _{\alpha }x^{\alpha })\) so that \(A^{\mu \nu }u_{\mu } =0\) for some chosen four-velocity vector \(\varvec{u}\). This means that the coordinates are chosen so that for an observer with four-velocity \(u^{\mu }\) the gravitational wave has an effect only in spatial directions.^{2} - (c)
*traceless condition*: one final component of the amplitude tensor can be eliminated after selecting the infinitesimal displacement vector \(\xi ^{\mu } = iC^{\mu } \exp (i\kappa _{\alpha }x^{\alpha })\) so that \(A^{\mu }_{\ \,\mu } = 0\).

*(a), (b)*and

*(c)*define the so-called

*transverse–traceless*(TT)

*gauge*, which represents a most convenient gauge for the analysis of gravitational waves. To appreciate the significance of these conditions, consider them implemented in a reference frame which is globally at rest, i.e., with four-velocity \(u^{\alpha } = (1,0,0,0)\), where the amplitude tensor must satisfy:

- (a)i.e., the spatial components of \(h_{\mu \nu }\) are$$\begin{aligned} A_{\mu \nu } \kappa ^{\nu } = 0 \qquad \Longleftrightarrow \qquad \partial ^j h_{ij} = 0, \end{aligned}$$(22)
*divergence-free*. - (b)i.e., only the spatial components of \(h_{\mu \nu }\) are$$\begin{aligned} A_{\mu \nu } u^{\nu } = 0 \qquad \Longleftrightarrow \qquad h_{\mu t} = 0 , \end{aligned}$$(23)
*nonzero*, hence the*transverse*character of the TT gauge. - (c)i.e., the spatial components of \(h_{\mu \nu }\) are$$\begin{aligned} A^{\mu }_{\ \, \mu } = 0 \qquad \Longleftrightarrow \qquad h=h^{j}_{\ j} = 0 , \end{aligned}$$(24)
*trace free*hence the*trace-free*character of the TT gauge. Because of this, and only in this gauge, \({\bar{h}}_{\mu \nu } = h_{\mu \nu }\).

### 2.2 Making sense of the TT gauge

*A*and

*B*on a geodesic motion and how this separation changes in the presence of an incident gravitational wave (see Fig. 1). For this purpose, let us introduce a coordinate system \(x^{\hat{\alpha }}\) in the neighbourhood of particle

*A*so that along the worldline of the particle

*A*the line element will have the form

*A*, the affine connections vanish (i.e., \(\varGamma ^{\hat{j}}_{{\hat{\alpha }} {\hat{\beta }}}=0\)) and the covariant derivative in (31) can be replaced by an ordinary total derivative. Furthermore, because in the TT gauge the coordinate system \(x^{\hat{\alpha }}\) moves together with the particle

*A*, the proper and the coordinate time coincide at first order in the metric perturbation [i.e., \(\tau =t + \mathcal {O}((h^{^\mathrm{TT}}_{\mu \nu })^2)\)]. As a result, equation (31) effectively becomes

*A*, the particle

*B*is seen oscillating with an amplitude proportional to \(h^{^\mathrm{TT}}_{{\hat{j}} {\hat{k}}}\).

Note that because these are transverse waves, they will produce a local deformation of the spacetime only in the plane orthogonal to their direction of propagation. As a result, if the two particles lay along the direction of propagation (i.e., if \({\varvec{n}} \parallel {\varvec{\kappa }}\)), then \(h^{^\mathrm{TT}}_{{\hat{j}} {\hat{k}}} x^{\hat{j}}_{_\mathrm{B}}(0) \propto h^{^\mathrm{TT}}_{{\hat{j}} {\hat{k}}} \kappa ^{\hat{j}}_{_\mathrm{B}}(0) = 0\) and no oscillation will be recorded by *A* [cf. Eq. (22)].

*z*-direction. In this case

*linearly*polarized plane waves or in two

*circularly*polarized ones. In the first case, and for a gravitational wave propagating in the

*z*-direction, the polarization

*tensors*\(+\) (“plus”) and \(\times \) (“cross”) are defined as

*tensors*describing the two states of circular polarization and indicate with \(\varvec{e}_{_\mathrm{R}}\) the circular polarization that rotates clockwise (see Fig. 3)

### 2.3 The quadrupole formula

The quadrupole formula and its domain of applicability were mentioned in Sect. 1, and some examples of its use in a numerical simulation are presented in Sect. 8. In practice, the quadrupole formula represents a low-velocity, weak-field approximation to measure the gravitational-wave emission within a purely Newtonian description of gravity.^{3} In practice, the formula is employed in those numerical simulations that either treat gravity in an approximate manner (e.g., via a post-Newtonian approximation or a conformally flat metric) or that, although in full general relativity, have computational domains that are too small for an accurate calculation of the radiative emission.

In what follows we briefly discuss the amounts of energy carried by gravitational waves and provide simple expressions to estimate the gravitational radiation luminosity of potential sources. Although the estimates made here come from analogies with electromagnetism, they provide a reasonable approximation to more accurate expressions from which they differ for factors of a few. Note also that while obtaining such a level of accuracy requires only a small effort, reaching the accuracy required of a template to be used in the realistic detection of gravitational waves is far more difficult and often imposes the use of numerical relativity calculations on modern supercomputers.

*q*the electrical charges and

*x*their separation, is easily estimated to be

*N*point-like particles of mass \(m_{_{A}}\) (\(A=1,2,\ldots ,N\)), in fact, the total mass dipole and its first time derivative are

*N*charges \((q_1, q_2, \ldots , q_N)\).

Although extremely simplified, expressions (48) and (50) contain the two most important pieces of information about the generation of gravitational waves. The first one is that the conversion of any type of energy into gravitational waves is, in general, not efficient. To see this it is necessary to bear in mind that expression (46) is in geometrized units and that the conversion to conventional units, say cgs units, requires dividing (46) by the very large factor \(c^5/G \simeq 3.63 \times 10^{59}\ \mathrm{erg\ s}^{-1}\). The second one is contained in the last expression in Eq. (50) and that highlights how the gravitational-wave luminosity can also be extremely large. There are in fact astrophysical situations, such as those right before the merger of a binary system of compact objects, in which \(\sqrt{\langle v^2\rangle } \sim 0.1\,c\) and \(R \sim 10\,R_{_\mathrm{S}}\), so that \(L_\mathrm{mass-quad} \sim 10^{51}\ \mathrm{erg\ s}^{-1} \sim 10^{18}\, L_{\odot }\), that is, \(10^{18}\) times the luminosity of the Sun; this is surely an impressive release of energy.

#### 2.3.1 Extensions of the quadrupole formula

*R*from the source in terms of the quadrupole wave amplitude \(A_{20}\) (Zanotti et al. 2003)

Expressions (52)–(58) are strictly Newtonian. Yet, these expression are often implemented in numerical codes that are either fully general relativistic or exploit some level of general-relativistic approximation. More seriously, these expressions completely ignore considerations that emerge in a relativistic context, such as the significance of the coordinate chosen for their calculation. As a way to resolve these inconsistencies, improvements to these expressions have been made to increase the accuracy of the computed gravitational-wave emission. For instance, for calculations on known spacetime metrics, the gravitational potential in expression (52) is often approximated with expressions derived from the metric, e.g., as \(\Phi = (1 - g_{rr})/2\) (Zanotti et al. 2003), which is correct to the first Post-Newtonian (PN) order. Improvements to the mass quadrupole (53) inspired by a similar spirit have been computed in Blanchet et al. (1990), and further refined and tested in Shibata and Sekiguchi (2004), Nagar et al. (2005), Cerdá-Durán et al. (2005), Pazos et al. (2007), Baiotti et al. (2007), Dimmelmeier et al. (2007), Corvino et al. (2010).

Making use of a fully general-relativistic measurement of the gravitational-wave emission from a neutron star oscillating nonradially as a result of an initial pressure perturbation, Baiotti et al. (2009) concluded that the various quadrupole formulas are comparable and give a very good approximation to the phasing of the gravitational-wave signals. At the same time, they also suffer from systematic over-estimate [expression (61)] or under-estimates of the gravitational-wave amplitude [expressions (60), and (62)–(63)]. In all cases, however, the relative difference in amplitude was of 50 % at most, which is probably acceptable given that these formulas are usually employed in complex astrophysical calculations in which the systematic errors coming from the microphysical modelling are often much larger.

## 3 Basic numerical approaches

### 3.1 The 3+1 decomposition of spacetime

At the heart of Einstein’s theory of general relativity is the equivalence among all coordinates, so that the distinction of spatial and time coordinates is more an organisational matter than a requirement of the theory. Despite this “covariant view”, however, our experience, and the laws of physics on sufficiently large scales, do suggest that a distinction of the time coordinate from the spatial ones is the most natural one in describing physical processes. Furthermore, while not strictly necessary, such a distinction of time and space is the simplest way to exploit a large literature on the numerical solution of hyperbolic partial differential equations as those of relativistic hydrodynamics. In a generic spacetime, analytic solutions to the Einstein equations are not known, and a numerical approach is often the only way to obtain an estimate of the solution.

Following this principle, a decomposition of spacetime into “space” and “time” was already proposed in the 1960s within a Hamiltonian formulation of general relativity and later as an aid to the numerical solution of the Einstein equations in vacuum. The basic idea is rather simple and consists in “foliating” spacetime in terms of a set of non-intersecting spacelike hypersurfaces \(\varSigma :=\varSigma (t)\), each of which is parameterised by a constant value of the coordinate *t*. In this way, the three spatial coordinates are split from the one temporal coordinate and the resulting construction is called the 3+1 *decomposition* of spacetime (Misner et al. 1973).

*foliation*\(\varSigma \), we can introduce a timelike four-vector \(\varvec{n}\) normal to the hypersurface at each event in the spacetime and such that its dual one-form \(\varvec{\varOmega } :=\varvec{\nabla } t\) is parallel to the gradient of the coordinate

*t*, i.e.,

*A*a constant to be determined. If we now require that the four-vector \(\varvec{n}\) defines an observer and thus that it measures the corresponding four-velocity, then from the normalisation condition on timelike four-vectors, \(n^\mu n_\mu =-1\), we find that

*lapse*function, it measures the rate of change of the coordinate time along the vector \(n^\mu \) (see Fig. 4), and will be a building block of the metric in a 3+1 decomposition [cf. Eq. (72)].

*spatial projection operator*(or

*spatial projection tensor*)

*time projection operator*(or

*time projection tensor*)

*shift vector*and will be another building block of the metric in a 3+1 decomposition [cf. Eq. (72)]. The decomposition of the vector \(\varvec{t}\) into a timelike component \(n\varvec{\alpha }\) and a spatial component \(\varvec{\beta }\) is shown in Fig. 4.

Because \(\varvec{t}\) is a coordinate basis vector, the integral curves of \(t^{\mu }\) are naturally parameterised by the time coordinate. As a result, all infinitesimal vectors \(t^{\mu }\) originating at a given point \(x_0^i\) on one hypersurface \(\varSigma _t\) would end up on the hypersurface \(\varSigma _{t+dt}\) at a point whose coordinates are also \(x_0^i\). This condition is not guaranteed for translations along \(\varOmega _{\mu }\) unless \(\beta ^\mu =0\) since \(t^{\mu }t_{\mu } = g_{tt}= -\alpha ^2 + \beta ^{\mu }\beta _{\mu }\), and as illustrated in Fig. 4.

*line element*in a 3+1 decomposition as

When defining the unit timelike normal \(\varvec{n}\) in Eq. (65), we have mentioned that it can be associated to the four-velocity of a special class of observers, which are referred to as *normal* or *Eulerian observers*. Although this denomination is somewhat confusing, since such observers are not at rest with respect to infinity but have a coordinate velocity \(dx^i/dt = n^i =-\beta ^i/\alpha \), we will adopt this traditional nomenclature also in the following and thus take an “Eulerian observer” as one with four-velocity given by (71).

*Lorentz factor*

*W*of \(\varvec{u}\) as measured by \(\varvec{n}\) (de Felice and Clarke 1990)

### 3.2 The ADM formalism: 3+1 decomposition of the Einstein equations

The 3+1 decomposition introduced in Sect. 3.1 can be used not only to decompose tensors, but also equations and, in particular, the Einstein equations, which are then cast into an initial-value form suitable to be solved numerically. A 3+1 decomposition of the Einstein equations was presented by Arnowitt et al. (2008), but it is really the reformulation suggested by York (1979) that represents what is now widely known as the *ADM formulation* (see, e.g., Alcubierre 2008; Gourgoulhon 2012 for a detailed and historical discussion). As we will see in detail later on, in this formulation the Einstein equations are written in terms of purely spatial tensors that can be integrated forward in time once some constraints are satisfied initially.

Here, we only outline the ADM formalism, and refer to the literature for the derivation and justification. Further, it is important to note that the ADM formulation is, nowadays, not used in practice because it is only weakly hyperbolic. However, the variables used in the ADM method, in particular the three-metric and the extrinsic curvature, are what will be needed later for gravitational-wave extraction, and are easily obtained from the output of other evolution methods (see discussion in Sects. 5, 7).

Instead of the ADM formalism, modern simulations mainly formulate the Einstein equations using: the BSSNOK method (Nakamura et al. 1987; Shibata and Nakamura 1995; Baumgarte and Shapiro 1999); the CCZ4 formulation (Alic et al. 2012), which was developed from the Z4 method (Bona et al. 2003, 2004; Bona and Palenzuela-Luque 2009) (see also Bernuzzi and Hilditch 2010 for the so-called Z4c formulation and Alic et al. 2013 for some comparisons); or the generalized harmonic method (Pretorius 2005) (see also Baumgarte and Shapiro 2010; Rezzolla and Zanotti 2013 for more details).

*three-dimensional covariant derivative*\(D_i\). Formally, this is done by projecting the standard covariant derivative onto the space orthogonal to \(n^\mu \), and the result is a covariant derivative defined with respect to the connection coefficients

^{4}Similarly, the three-dimensional Riemann tensor \({^{{(3)}}}\!R^{i}_{\ \,j k\ell }\) associated with \(\varvec{\gamma }\) has an explicit expression given by

*extrinsic curvature*\(K_{ij}\), which is purely spatial. Loosely speaking, the extrinsic curvature provides a measure of how the three-dimensional hypersurface \(\varSigma _t\) is curved with respect to the four-dimensional spacetime. For our purposes, it is convenient to define the extrinsic curvature as (but note that other definitions, which can be shown to be equivalent, are common)

Overall, the six equations (89), together with the six equations (85) represent the time-evolving part of the ADM equations and prescribe how the three-metric and the extrinsic curvature change from one hypersurface to the following one. In contrast, Eqs. (90) and (91) are constraints that need to be satisfied on each hypersurface. This distinction into evolution equations and constraint equations is not unique to the ADM formulation and is indeed present also in classical electromagnetism. Just as in electrodynamics the divergence of the magnetic field remains zero if the field is divergence-free at the initial time, so the constraint equations (90) and (91), by virtue of the Bianchi identities (Alcubierre 2008; Bona et al. 2009; Baumgarte and Shapiro 2010; Gourgoulhon 2012; Rezzolla and Zanotti 2013), will remain satisfied during the evolution if they are satisfied initially (Frittelli 1997). Of course, this concept is strictly true in the continuum limit, while numerically the situation is rather different. However, that issue is not pursued here.

Two remarks should be made before concluding this section. The first one is about the gauge quantities, namely, the lapse function \(\alpha \) and the shift vector \(\beta ^i\). Since they represent the four degrees of freedom of general relativity, they are not specified by the equations discussed above and indeed they can be prescribed arbitrarily, although in practice great care must be taken in deciding which prescription is the most useful. The second comment is about the mathematical properties of the time-evolution ADM equations (89) and (85). The analysis of these properties can be found, for instance, in Reula (1998) or in Frittelli and Gómez (2000), and reveals that such a system is only weakly hyperbolic with zero eigenvalues and, as such, not necessarily well-posed. The weak-hyperbolicity of the ADM equations explains why, while an historical cornerstone in the 3+1 formulation of the Einstein equations, they are rarely used in practice and have met only limited successes in multidimensional calculations (Cook et al. 1998; Abrahams et al. 1998). At the same time, the weak hyperbolicity of the ADM equations and the difficulty in obtaining stable evolutions, has motivated, and still motivates, the search for alternative formulations.

### 3.3 Gravitational waves from \(\psi _4\) on a finite worldtube(s)

*x*,

*y*,

*z*) are approximately Cartesian, then spherical polar coordinates are defined using Eqs. (399) and (400). However, in general these coordinates are not exactly spherical polar, and in particular the radial coordinate is not a surface area coordinate (for which the 2-surface \(r=t=\) constant must have area \(4\pi r^2\)). We reserve the notation

*r*for a surface area radial coordinate, so the radial coordinate just constructed will be denoted by

*s*. Then the outward-pointing radial unit normal \(\varvec{e}_{s}\) is

*s*and

*r*, and in spherical polar coordinates \((t,r,\theta ,\phi )\) (Fig. 5)

*r*as is feasible). Often \(\lim _{r\rightarrow \infty }r\psi _4\) is denoted by \(\psi _4^0(t,\theta ,\phi )\), but that notation will not be used in this section. These issues are discussed further in Sect. 6, but for now we will regard gravitational waves, and specifically \(r\psi _4\), as properly defined only in a spacetime whose metric can be written in a form that tends to the Minkowski metric, and for which the appropriate definition of the null tetrad is one that tends to the form Eq. (96), as \(r\rightarrow \infty \).

#### 3.3.1 Extracting gravitational waves using \(\psi _4\) on a finite worldtube

“3+1” numerical simulations are restricted to a finite domain, so it is not normally possible to calculate exactly a quantity given by an asymptotic formula (but see Sects. 6, 7). A simple estimate of \(r\psi _4\) can be obtained by constructing coordinates \((s,\theta ,\phi )\) and an angular null tetrad vector \(\varvec{m}\) as discussed at the beginning of Sect. 3.3. Then \(r\psi _4\) can be evaluated using Eq. (104) on a worldtube \(s=\) constant, and the estimate is \(r\psi _4=s\psi _4\) or alternatively \(r\psi _4=\psi _4 \sqrt{A/4\pi }\), where *A* is the area of the worldtube at time *t*. This approach was first used in Smarr (1977), and subsequently in, for example, Pollney et al. (2007), Pfeiffer (2007) and Scheel et al. (2009). This method does not give a unique answer, and there are many variations in the details of its implementation. However, the various estimates obtained for \(r\psi _4\) should differ by no more than \({{\mathcal {O}}}(r^{-1})\).

The quantity \(\psi _4\) has no free indices and so tensorially is a scalar, but its value does depend on the choice of tetrad. *However*, it may be shown that \(\psi _4\) is first-order tetrad-invariant if the tetrad is a small perturbation about a natural tetrad of the Kerr spacetime. This result was shown by Teukolsky (1972, 1973); see also Chandrasekhar (1978), and Campanelli et al. (2000). Briefly, the reasoning is as follows. The Kinnersley null tetrad is an exact null tetrad field in the Kerr geometry (Kinnersley 1969). It has the required asymptotic limit, and the vectors \(\ell ^\alpha \), \(n_{_{[NP]}}^\alpha \) are generators of outgoing and ingoing radial null geodesics respectively. In the Kerr geometry \(C^\mathrm{[Kerr]}_{\alpha \beta \mu \nu }\ne 0\), but using the Kinnersley tetrad all \(\psi _n\) are zero except \(\psi _2\). Thus, to first-order, \(\psi _4\) is evaluated using the perturbed Weyl tensor and the background tetrad; provided terms of the form \(C^\mathrm{[Kerr]}_{\alpha \beta \mu \nu }n_{_{[NP]}}^\alpha \bar{m}^\beta n_{_{[NP]}}^\mu \bar{m}^\nu \), where three of the tetrad vectors take background values and only one is perturbed, are ignorable. Allowing for those \(\psi _n\) that are zero, and using the symmetry properties of the Weyl tensor, all such terms vanish. This implies that the ambiguity in the choice of tetrad is of limited importance because it is a second-order effect; see also Campanelli and Lousto (1998); Campanelli et al. (1998). These ideas have been used to develop analytic methods for estimating \(\psi _4\) (Campanelli et al. 2000; Baker et al. 2000a; Baker and Campanelli 2000; Baker et al. 2001). Further, the Kinnersley tetrad is the staring point for a numerical extraction procedure.

In practice the spacetime being evolved is not Kerr, but in many cases at least far from the source it should be Kerr plus a small perturbation, and in the far future it should tend to Kerr. Thus an idea for an appropriate tetrad for use on a finite worldtube is to construct an approximation to the Kinnersley form, now known as the quasi-Kinnersley null tetrad (Beetle et al. 2005; Nerozzi et al. 2005). The quasi-Kinnersley tetrad has the property that as the spacetime tends to Kerr, then the quasi-Kinnersley tetrad tends to the Kinnersley tetrad. The method was used in a number of applications in the mid-2000s (Nerozzi et al. 2006; Campanelli et al. 2006; Fiske et al. 2005; Nerozzi 2007).

*a*is a small relative error of order \(a^2/r^2\).

#### 3.3.2 Extracting gravitational waves using \(\psi _4\) in practice: the extrapolation method

*r*should be a surface area coordinate (although often requiring this property is not important). We assume that the data available is \(\psi _4^{\ell m}(t,s)\) obtained by decomposing \(\psi _4\) into spherical harmonic components on a spherical surface of fixed coordinate radius

*s*at a given coordinate time

*t*. Then the first step in extrapolation is to obtain \(\psi _4^{\ell m}(t_{*},r)\), where \(t_*\) is a retarted time coordinate specified in Eq. (108) below, and where

*A*the area of the coordinate 2-surface \(t=~\)constant, \(s=~\)constant. Because the spacetime is dynamic, it would be a complicated process to construct \(t_*\) exactly. Instead, it is assumed that extraction is performed in a region of spacetime in which the geometry is approximately Schwarzschild with (

*t*,

*s*) approximately standard Schwarzschild coordinates. Then

*M*is an estimate of the initial mass of the system, usually the ADM mass, and \(g^{tt},g^{ss}\) are averaged over the 2-sphere \(t^\prime =~\)constant, \(s_j=~\)constant. It is straightforward to check that if (

*t*,

*s*) are exactly Schwarzschild coordinates, then \(t_*\) is null.

^{5}In practice, the real and imaginary parts of \(\psi _4^{\ell m}\) may vary rapidly, and it has been found to be smoother to fit the data to the amplitude and phase.

^{6}For each spherical harmonic component two data-fitting problems are solved

*r*, and so for certain values of \(r_k\) it may be necessary to add \(\pm 2\pi \) to \(\arg (\psi _4^{\ell m}(t_*,r_k))\). Then the estimate is

*N*, and of the extraction spheres \(r_k\) (or more precisely, since extraction is performed on spheres of specified coordinate radius, of the \(s_k\)). The key factors are the innermost and outermost extraction spheres, i.e., the values of \(r_1\) and \(r_{_K}\), and of course the requirement that \(K>N+1\). Essentially, the extrapolation process uses data over the interval \(1/r\in [1/r_{_K},1/r_1]\) to construct an estimate at \(1/r=0\). Polynomial extrapolation can be unreliable, or even divergent, as

*N*is increased; it can also be unreliable when the distance from the closest data point is larger than the size of the interval over which the data is fitted. As a result of this latter condition, it is normal to require \(r_{_K}>2 r_1\). On the other hand, increasing \(r_{_K}\) increases the computational cost of a simulation, and decreasing \(r_1\) could mean that a higher order polynomial is needed for accurate modelling of the data at that point. A compromise is needed between these conflicting factors. Values commonly used are that

*N*is between 3 and 5, \(r_1\) is normally of order 100 M, and \(r_{_K}\) is 300 M with values as large as 1000 M reported. Typically,

*K*is about 8, with the \(1/r_k\) evenly distributed over the interval \([1/r_{_K},1/r_1]\).

#### 3.3.3 Energy, momentum and angular momentum in the waves

Starting from the mass loss result of Bondi et al. (1962), the theory of energy and momentum radiated as gravitational waves was further developed in the 1960s (Penrose 1963, 1965a; Tamburino and Winicour 1966; Winicour 1968; Isaacson 1968) and subsequently (Geroch 1977; Thorne 1980a; Geroch and Winicour 1981). Formulas for the radiated angular momentum were presented in Campanelli and Lousto (1999), Lousto and Zlochower (2007) based on earlier work by Winicour (1980); formulas were also obtained in Ruiz et al. (2007), Ruiz et al. (2008) using the Isaacson effective stress-energy tensor of gravitational waves (Isaacson 1968).

## 4 Gravitational waves in the Cauchy-perturbative approach

Black-hole perturbation theory has been fundamental not only for understanding the stability and oscillations properties of black hole spacetimes (Regge and Wheeler 1957), but also as an essential tool for clarifying the dynamics that accompanies the process of black hole formation as a result of gravitational collapse (Price 1972a, b). As one example among the many possible, the use of perturbation theory has led to the discovery that Schwarzschild black holes are characterised by decaying modes of oscillation that depend on the black hole mass only, i.e., the black hole quasi-normal modes (Vishveshwara 1970b, a; Press 1971; Chandrasekhar and Detweiler 1975). Similarly, black-hole perturbation theory and the identification of a power-law decay in the late-time dynamics of generic black-hole perturbations has led to important theorems, such as the “no hair” theorem, underlining the basic black-hole property of removing all perturbations so that “all that can be radiated away is radiated away” (Price 1972a, b; Misner et al. 1973).

The foundations of non-spherical metric perturbations of Schwarzschild black holes date back to the work of Regge and Wheeler (1957), who first addressed the linear stability of the Schwarzschild solution. A number of investigations, both gauge-invariant and not, then followed in the 1970s, when many different approaches were proposed and some of the most important results about the physics of perturbed spherical and rotating black holes established (Price 1972a, b; Vishveshwara 1970b, a; Chandrasekhar and Detweiler 1975; Zerilli 1970a, b; Moncrief 1974; Cunningham et al. 1978, 1979; Teukolsky 1972, 1973). Building on these studies, which defined most of the mathematical apparatus behind generic perturbations of black holes, a number of applications have been performed to study, for instance, the evolutions of perturbations on a collapsing background spacetime (Gerlach and Sengupta 1979b, a, 1980; Karlovini 2002; Seidel et al. 1987, 1988; Seidel 1990, 1991). Furthermore, the gauge-invariant and coordinate independent formalism for perturbations of spherically symmetric spectimes developed in the 1970s by Gerlach and Sengupta (1979b, 1979a, 1980), has been recently extended to higher-dimensional spacetimes with a maximally symmetric subspace in Kodama et al. (2000), Kodama and Ishibashi (2003), Ishibashi and Kodama (2003), Kodama and Ishibashi (2004), for the study of perturbations in brane-world models.

Also nowadays, when numerical relativity calculations allow to evolve the Einstein equations in the absence of symmetries and in fully nonlinear regimes, black hole perturbative techniques represent important tools.^{7} Schwarzschild perturbation theory, for instance, has been useful in studying the late-time behaviour of the coalescence of compact binaries in a numerical simulation after the apparent horizon has formed (Price and Pullin 1994; Abrahams and Cook 1994; Abrahams et al. 1995). In addition, methods have been developed that match a fully numerical and three-dimensional Cauchy solution of Einstein’s equations on spacelike hypersurfaces with a perturbative solution in a region where the components of three-metric (or of the extrinsic curvature) can be treated as linear perturbations of a Schwarzschild black hole [this is usually referred to as the “Cauchy-Perturbative Matching”] (Abrahams et al. 1998; Rupright et al. 1998; Camarda and Seidel 1999; Allen et al. 1998; Rezzolla et al. 1999a; Lousto et al. 2010; Nakano et al. 2015). This method, in turn, allows to “extract” the gravitational waves generated by the simulation, evolve them out to the wave-zone where they assume their asymptotic form, and ultimately provide outer boundary conditions for the numerical evolution.

This section intends to review the mathematical aspects of the metric perturbations of a Schwarzschild black hole, especially in its gauge-invariant formulations. Special care is paid to “filter” those technical details that may obscure the important results and provide the reader with a set of expressions that can be readily used for the calculation of the odd and even-parity perturbations of a Schwarzschild spacetime in the presence of generic matter-sources. Also, an effort is made to “steer” the reader through the numerous conventions and notations that have accompanied the development of the formalism over the years. Finally, as mentioned in the Introduction, a lot of the material presented here has already appeared in the Topical Review by Nagar and Rezzolla (2006).

### 4.1 Gauge-invariant metric perturbations

It is useful to recall that even if the coordinate system of the background spacetime has been fixed, the coordinate freedom of general relativity introduces a problem when linear perturbations are added. In particular, it is not possible to distinguish an infinitesimal “physical” perturbation from one produced as a result of an infinitesimal coordinate transformation (or gauge-transformation). This difficulty, however, can be removed either by explicitly fixing a gauge (see, e.g., Regge and Wheeler 1957; Price 1972a, b; Vishveshwara 1970b, a; Zerilli 1970a, b), or by introducing linearly gauge–invariant perturbations (as initially suggested by Moncrief 1974 and subsequently adopted in several applications Cunningham et al. 1978, 1979; Seidel et al. 1987, 1988; Seidel 1990, 1991).

Stated differently, the possibility of building gauge–invariant metric perturbations relies on the existence of symmetries of the background metric. In the case of a general spherically symmetric background spacetime (i.e., one allowing for a time dependence) and which has been decomposed in multipoles (see Sect. 4.2), the construction of gauge-invariant quantities is possible for multipoles of order \(\ell \ge 2\) only (Gerlach and Sengupta 1979b, a; Martín-García and Gundlach 1999; Gundlach and Martín-García 2000). In practice, the advantage in the use of gauge-invariant quantities is that they are naturally related to scalar observables and, for what is relevant here, to the energy and momentum of gravitational waves. At the same time, this choice guarantees that possible gauge-dependent contributions are excluded by construction.

Of course, this procedure is possible if and only if the background metric has the proper symmetries under infinitesimal coordinates transformation; in turn, a gauge-invariant formulation of the Einstein equations for the perturbations of a general spacetime is not possible. Nevertheless, since any asymptotically flat spacetime can in general be matched to a Schwarzschild one at sufficiently large distances, a gauge-invariant formulation can be an effective tool to extract physical information about the gravitational waves generated in a numerically evolved, asymptotically flat spacetime (Abrahams et al. 1998; Rupright et al. 1998; Camarda and Seidel 1999; Allen et al. 1998; Rezzolla et al. 1999a) (see also Sect. 5.6 for additional implementational details). The following section is dedicated to a review of the mathematical techniques to obtain gauge-invariant perturbations of a the Schwarzschild metric.

### 4.2 Multipolar expansion of metric perturbations

*t*,

*r*) and \(\mathsf{S}^2\) is the 2-sphere of unit radius and coordinates \((\theta , \phi )\). As a result, the perturbations can be split “ab initio” in a part confined to \(\mathsf{M}^2\) and in a part confined on the 2-sphere \(\mathsf{S}^2\) of metric \(\varvec{\gamma }\). Exploiting this, we can expand the metric perturbations \(\varvec{h}\) in multipoles referred to as “odd” or “even-parity” according to their transformation properties under parity. In particular, are

*odd*(or

*axial*) multipoles those that transform as \((-1)^{\ell +1}\), under a parity transformation \((\theta , \phi ) \rightarrow (\pi -\theta , \pi +\phi )\), while are

*even*(or

*polar*) those multipoles that transform as \((-1)^{\ell }\). As a result, the metric perturbations can be written as

*t*,

*r*) only, and where we have omitted the indices \(\ell , m\) on \(h,h_{a}^{(\mathrm{o})}\)for clarity.

*G*(with the indices \(\ell , m\) omitted for clarity) are the coefficients of the even-parity expansion, are also functions of (

*t*,

*r*) only.

Note that while a generic perturbation will be a mixture of odd and even-parity contributions, we will exploit the linearity of the approach to handle them separately and simplify the treatment. In the following two sections we will discuss the form the Einstein equations (145) assume in response to purely odd and even-parity perturbations over a Schwarzschild background. In particular, we will show how the three odd-parity coefficients of the expansion in harmonics of the metric, i.e., \(h_{a}^{(\mathrm{o})},\,h\), and the seven even-parity ones, i.e., \(H_0,\,H_1,\,H_2,\, h_0^{(\mathrm{e})},\,h_1^{(\mathrm{e})}\, K,\, G\), can be combined to give two gauge-invariant master equations, named respectively after Regge and Wheeler (1957) and Zerilli (1970), each of which is a wave-like equation in a scattering potential.^{8}

Although our attention is here focussed on the *radiative* degrees of freedom of the perturbations (i.e., those with \(\ell \ge 2\)) because of their obvious application to the modelling of sources of gravitational waves, a comment should be made also on lower-order multipoles. In particular, it is worth remarking that the monopole component of the metric for a vacuum perturbation (i.e., with \(\ell = 0\)) is only of even-parity type and represents a variation in the mass-parameter of the Schwarzschild solution. On the other hand, the dipole component of the even-parity metric for a vacuum perturbation (i.e., with \(\ell =1\)) is of pure-gauge type and it can be removed by means of a suitable gauge transformation (Zerilli 1970b). This is not the case for a dipolar odd-parity metric perturbation, which can instead be associated to the introduction of angular momentum onto the background metric.

### 4.3 Gauge-invariant odd-parity perturbations

### 4.4 Gauge-invariant even-parity perturbations

*k*and \(\chi \) are defined as Gundlach and Martín-García (2000, 2001)

## 5 Numerical implementations of the Cauchy-perturbative approach

In the previous section we have reviewed the derivation of the equations describing the evolution of perturbations of nonrotating black holes induced, for instance, by a nonzero stress-energy tensor. These perturbations have been assumed to be generic in nature, needing to satisfy only the condition of having a mass-energy much smaller than that of the black hole. The solution of these equations with suitable initial conditions completely specifies the reaction of the black hole to the perturbations and this is essentially represented by the emission of gravitational waves.

As mentioned in Sect. 4.1, the importance of the gauge-invariant variables used so far is that they are directly related to the amplitude and energy of the gravitational-wave signal measured at large distances. The purpose of this Chapter is to review the steps necessary to obtain the relations between the master functions for the odd and even-parity perturbations and the “plus” and “cross” polarisation amplitudes \(h_+, h_\times \) of a gravitational wave in the TT gauge. In practice, and following the guidelines tracked in Cunningham et al. (1978, 1979), we will derive an expression for the perturbation metric \(\varvec{h}\) equivalent to that obtained in the standard TT gauge on a Minkowski spacetime and relate it to the odd and even-parity master functions \(\varPsi ^{(\mathrm{o})}\) and \(\varPsi ^{(\mathrm{e})}\).

*radiation gauge*. In practice, this amounts to requiring that that components \(h_{\hat{\theta }\hat{\theta }}\), \(h_{\hat{\phi }\hat{\phi }}\) and \(h_{\hat{\theta }\hat{\phi }}\) are functions of the type \(f(t-r)/r\) (i.e., they are outgoing spherical waves), while all the other components have a more rapid decay of \(\mathcal {O}(1/r^2)\). Finally, we need to impose the condition that the metric is traceless modulo higher order terms, i.e., \(h_{{\hat{\theta }}{\hat{\theta }}} + h_{\hat{\phi }\hat{\phi }}= 0 + \mathcal {O}(1/r^2)\).

In the following sections we will discuss the asymptotic expressions from odd- and even-parity perturbations, and how to implement the Cauchy-perturbative approach to extract gravitational-wave information within a standard numerical-relativity code.

### 5.1 Asymptotic expressions from odd-parity perturbations

*r*the metric asymptotes that of a flat spacetime, i.e., \(e^{b}\sim e^{-b}\sim 1\), we have

*r*are

*h*has the dimensions of a length squared, both \(h_{+}\) and \(h_{\times }\) are dimensionless. Next, we need to relate the perturbation

*h*to the odd-parity master function \(\varPsi ^{(\mathrm{o})}\). To do so, we follow the procedure outlined in Cunningham et al. (1978), and note that (cf. Eq. (III-20)

^{9})

*h*and \(h_0^{(\mathrm o)}\) imply that \(\varPsi ^{(\mathrm{o})}\sim \mathcal {O}(1)\), i.e., in the wave-zone \(\varPsi ^{(\mathrm{o})}\) has the dimensions of a length, behaves as an outgoing-wave, but it does not depend explicitly on

*r*.

*h*at large distances we can write

#### 5.1.1 The master function \(Q^{(\mathrm{o})}\)

*h*proceeds in a way similar to the one discussed above. In the radiation gauge and at large distances from the black hole, we can use relation (179) in the definition (154) with \(h_2=-2h\), so that

*r*at leading order and Eq. (186) can be integrated to give

### 5.2 Asymptotic expressions from even-parity perturbations

*G*with the even-parity function \({\varPsi }^{(\mathrm{e})}\). Firstly, it is easy to realize that the even-parity metric projected onto the tetrad, \(h_{\hat{\mu }\hat{\nu }}^{(\mathrm{e})}\), is such that

*r*and can be neglected. Furthermore, from the transverse traceless condition

*K*and

*G*

### 5.3 Asymptotic general expressions

It is also useful to underline that while expression (201) resembles the corresponding expression (10) of Kawamura and Oohara (2004), it is indeed different. Firstly, because in Kawamura and Oohara (2004) the Moncrief function is adopted for the odd-parity part of the perturbations and hence, modulo a normalisation factor, the function \(\varPsi ^{(\mathrm{o})}\) appearing there corresponds to our function \(Q^{(\mathrm{o})}\) [cf. expression (154)]. Secondly, because with this choice for the odd-parity perturbations a time derivative is needed in the asymptotic expression for the gravitational-wave amplitudes [cf. the discussion in the derivation of Eq. (189)]. As a result, expression (10) of Kawamura and Oohara (2004) (which is also missing the distinction between the real and imaginary parts) should really be replaced by expression (202). A similar use of the Moncrief function for the odd-parity part is present also in Shibata et al. (2003), Shibata and Sekiguchi (2003), Shibata and Sekiguchi (2005), where it is employed to calculate the gravitational-wave content of numerically simulated spacetimes.

### 5.4 Energy and angular momentum losses

*short-wave*approximation),

*i.e.,*whenever the wavelength of the gravitational-wave field is small when compared to the local radius of curvature of the background geometry. In practice, gravitational radiation from isolated systems is of high frequency whenever it is far enough away from its source.

*h*being now the trace of \(h_{\hat{\mu }\hat{\nu }}\). As a result, the energy per unit time and angle carried by the gravitational waves and measured by a stationary observer at large distance is given by

*f*(

*t*,

*r*). Similarly, when using the odd-parity Moncrief function one obtains

### 5.5 A commonly used convention

A rather popular choice for the gauge-invariant master functions has found successful application in the extraction of the gravitational-wave content of numerically simulated spacetimes (Abrahams and Price 1996b, a; Abrahams et al. 1998; Rupright et al. 1998; Rezzolla et al. 1999a). For instance, the convention discussed below has been implemented in the Cactus computational toolkit (Camarda and Seidel 1999; Allen et al. 1998), a diffused and freely available infrastructure for the numerical solution of the Einstein equations (Allen et al. 1999; Cactus 2016). Numerous tests and applications of this implementation have been performed over the years and we refer the reader to Camarda and Seidel (1999), Allen et al. (1998), Font et al. (2002), Baiotti et al. (2005) for examples both in vacuum and non-vacuum spacetimes.

*real*numbers, that \(h_+\) is

*only*even-parity and \(h_\times \) is

*only*odd-parity.

*i*-direction can be computed from the Isaacson’s energy-momentum tensor and can be written in terms of the two polarization amplitudes as Favata et al. (2004)

*electric*) multipoles are indicated with \(I_{\ell m}\) and the odd-parity (or

*magnetic*) ones with \(S_{\ell m}\), and are related to our notation by

*real*numbers and are obtained as the real and imaginary part of the right-hand side of Eq. (226).

### 5.6 Implementation summary

All of the material presented in the previous sections about the gauge-invariant description of the perturbations of a Schwarzschild black hole has laid the ground for the actual implementation of the Cauchy-perturbative extraction method in numerical-relativity calculations. We recall that the goal of the Cauchy-perturbative method is that of replacing, at least in parts of the three-dimensional numerical domain, the solution of the full nonlinear Einstein’s equations with the solution of a set of simpler linear equations that can be integrated to high accuracy with minimal computational cost. In turn, this provides an unexpensive evolution of the radiative degrees of freedom, the extraction of the gravitational-wave information, and, if needed, the imposition of boundary conditions via the reconstruction of the relevant quantities at the edge of the three-dimensional computational domain.

**N**in Fig. 7) will comprise an isolated region of spacetime where the gravitational fields are strong and highly dynamical. In this region, indicated as \(\mathcal {S}\) in Fig. 7, the full nonlinear Einstein equations must be solved. Outside of \(\mathcal {S}\), however, in what we will refer to as the perturbative region \(\mathcal {P}\), a perturbative approach is not only possible but highly advantageous. Anywhere in the portion of \(\mathcal {P}\) covered by

**N**we can place a two-dimensional surface, indicated as \(\varGamma \) in Fig. 7, which will serve as the surface joining numerically the highly dynamical strong-field region \(\mathcal {S}\) and the perturbative one \(\mathcal {P}\). In practice, it is easier to choose this surface to be a 2-sphere of radius \(r_{_\varGamma }\), where \(r_{_\varGamma }\) can either be the local coordinate radius, the corresponding Schwarzschild radial coordinate, or some more sophisticated radial coordinate deduced from the local values of the metric (cf. discussion in Sect. 4.2).

^{10}It is important to emphasize that the 2-sphere \(\varGamma \) need not be in a region of spacetime where the gravitational fields are weak or the curvature is small. In contrast to approaches which matched Einstein’s equations onto a Minkowski background (Abrahams and Evans 1988, 1990), the matching is here made on a Schwarzschild background, so that the only requirement is that the spacetime outside of \(\mathcal {S}\) approaches a Schwarzschild one. Of course, even in the case of a binary black-hole merger, it will be possible to find a region of spacetime, sufficiently distant from the black holes, where this requirement is met to the desired precision (Price and Pullin 1994; Abrahams and Cook 1994; Abrahams and Price 1996b, a; Abrahams et al. 1995).

In a practical implementation of the Cauchy-perturbative approach (Rupright et al. 1998; Rezzolla et al. 1999a), a numerical code provides the solution to the full nonlinear Einstein equations everywhere in the three-dimensional grid **N** except at its outer boundary surface \(\varvec{B}\). At the extraction 2-sphere \(\varGamma \), a different code (i.e., the perturbative module) “extracts” the gravitational wave information and transforms it into a set of multipole amplitudes which are chosen to depend only on the radial and time coordinates of the background Schwarzschild metric (Rupright et al. 1998; Rezzolla et al. 1999a).

**L**covering the perturbative region \(\mathcal {P}\) and ranging between \(r_{_\varGamma }\) and \(r_{_A} \gg r_{_\varGamma }\). From a computational point of view, this represents an enormous advantage: with a few straightforward transformations, the computationally expensive three-dimensional evolution of the gravitational waves via the nonlinear Einstein equations is replaced with a set of one-dimensional linear equations that can be integrated to high accuracy with minimal computational cost. Although linear, these equations account for all of the effects of wave propagation in a curved spacetime and, in particular, automatically incorporate the effects of backscatter off the curvature.

Note that as a result of this construction, (and as shown in Fig. 7), the perturbative region \(\mathcal {P}\) is entirely covered by a one-dimensional grid L and only partially by a three-dimensional grid in the complement to \(\mathcal {S}\) in **N**. The overlap between these two grids is essential. In fact, the knowledge of the solution on \(\mathcal {P}\) allows the perturbative approach to provide boundary conditions at the outer boundary surface \(\varvec{B}\) and, if useful, Dirichlet data on every gridpoint of **N** outside the strong region \(\mathcal {S}\). This is also illustrated in Fig. 8, which represents a one-dimensional cut of Fig. 7, and highlights the difference between the asymptotic values of the gravitational waves extracted at the boundary *A* of the one-dimensional grid (filled blue circles) with and the boundary values that can be instead specified (i.e., “injected”) on the outer boundary surface *B* of the three-dimensional grid.

The freedom to specify boundary data on a 2-surface of arbitrary shape as well as on a whole three-dimensional region of **N** represents an important advantage of the perturbative approach over similar approaches to the problem of gravitational-wave extraction and imposition of boundary conditions.

In what follows we briefly review the main steps necessary for the numerical implementation of the Cauchy-perturbative approach in a numerical-relativity code solving the Einstein equations in a 3+1 split of spacetime. This approach, which follows closely the discussion made in Rupright et al. (1998), Rezzolla et al. (1999a), basically consists of three steps: *(1) extraction* of the independent multipole amplitudes on \(\varGamma \); *(2) evolution* of the radial wave equations (247)–(249) on **L** out to the distant wave zone; * (3) reconstruction* of \(K_{ij}\) and \(\partial _t K_{ij}\) at specified gridpoints at the outer boundary of **N**. We next discuss in detail each of these steps.

#### 5.6.1 Perturbative expansion

The first step is to linearize the Einstein equations around a static Schwarzschild background by separating the gravitational quantities of interest into background (denoted by a tilde) and perturbed parts: the three-metric \(\gamma _{i j} = \widetilde{\gamma }_{i j} + h_{i j}\), the extrinsic curvature \(K_{i j} = \widetilde{K}_{i j} + \kappa _{i j}\), the lapse \(N = \widetilde{N} + \alpha \), and the shift vector \(\beta ^i = \widetilde{\beta }^i + v^i\). Note that the large majority of modern numerical-relativity codes implement the BSSNOK (Nakamura et al. 1987; Shibata and Nakamura 1995; Baumgarte and Shapiro 1999) or the CCZ4 (Alic et al. 2012) formulation of the Einstein equations. As mentioned in Sect. 3.2, in these formulations, the extrinsic curvature tensor is not evolved directly, but rather a traceless tensor extrinsic curvature tensor related to a conformal decomposition of the three-metric (Alcubierre 2008; Bona et al. 2009; Baumgarte and Shapiro 2010; Gourgoulhon 2012; Rezzolla and Zanotti 2013). Of course, also in these formulations it is possible to reconstruct the physically related extrinsic curvature tensor \(K_{ij}\) and we will therefore continue to make use of \(K_{ij}\) hereafter.

#### 5.6.2 Extraction

**N**as a solution to Einstein’s equations. Assuming that

**N**uses topologically Cartesian coordinates,

^{11}the Cartesian components of these tensors are then transformed into their equivalents in a spherical coordinate basis and their traces are computed using the inverse background metric, i.e., \(H = \widetilde{\gamma }^{i j} K_{i j}\), \(\partial _t H = \widetilde{\gamma }^{i j} \partial _t K_{i j}\). From the spherical components of \(K_{i j}\) and \(\partial _t K_{i j}\), the independent multipole amplitudes for each \((\ell ,m)\) mode are then derived by an integration over the 2-sphere:

#### 5.6.3 Perturbative evolution

Once the multipole amplitudes, \((a_\times )_{_{\ell m}}\), \((a_+)_{_{\ell m}}\), \((h)_{_{\ell m}}\) and their time derivatives are computed on \(\varGamma \) in the timeslice \(t=t_0\), they are imposed as inner boundary conditions on the one-dimensional grid. Using a suitably accurate integration scheme, the radial wave equations (247)–(249) can be evolved for each \((\ell , m)\) mode forward to the next timeslice at \(t=t_1\). The outer boundary of the one-dimensional grid is always placed at a distance large enough that background field and near-zone effects are unimportant, and a radial Sommerfeld condition for the wave equations (247)–(249) can be imposed there. The evolution equations for \(h_{i j}\) [Eq. (235)] and \(\alpha \) [Eq. (236)] can also be integrated using the data for \(K_{i j}\) computed in this region. Note also that because \(h_{i j}\) and \(\alpha \) evolve along the coordinate time axis, these equations need only be integrated in the region in which their values are desired, not over the whole region **L**.

Of course, the initial data on **L** must be consistent with the initial data on **N**, and this can be determined by applying the aforementioned extraction procedure to the initial data set at each gridpoint of **L** in the region of overlap with **N**. In the latter case, initial data outside the overlap region can be set by considering the asymptotic fall-off of each variable.

#### 5.6.4 Reconstruction

An important side product of the evolution step discussed above is that outer boundary values for **N** can now be computed, although, to the best of our knowledge, this procedure has not been implemented yet as a way to obtain outer boundary conditions. In particular, for codes using the BSSNOK (Nakamura et al. 1987; Shibata and Nakamura 1995; Baumgarte and Shapiro 1999) or the CCZ4 (Alic et al. 2012) formulation of the Einstein equations, it is sufficient to provide boundary data only for \(K_{i j}\), since the interior code can calculate \(\gamma _{i j}\) at the outer boundary by integrating in time the boundary values for \(K_{i j}\).

In order to compute \(K_{i j}\) at an outer boundary point of **N** (or any other point in the overlap between **N** and \(\mathcal {P}\)), it is necessary to reconstruct \(K_{i j}\) from the multipole amplitudes and tensor spherical harmonics. The Schwarzschild coordinate values \((r,\theta ,\phi )\) of the relevant gridpoint are first determined. Next, \((a_\times )_{_{\ell m}}\), \((a_+)_{_{\ell m}}\), and \((h_{_{\ell m}})\) for each \((\ell ,m)\) mode are interpolated to the radial coordinate value of that point. The dependent multipole amplitudes \((b_\times )_{_{\ell m}}\), \((b_+)_{_{\ell m}}\), \((c_+)_{_{\ell m}}\), and \((d_+)_{_{\ell m}}\) are then computed using the constraint equations (240). Finally, the Regge–Wheeler tensor spherical harmonics \((\hat{e}_1)_{i j}\)–\((\hat{f}_4)_{i j}\) are computed for the angular coordinates \((\theta ,\phi )\) for each \((\ell ,m)\) mode and the sum in Eq. (238) is performed. This leads to the reconstructed component of \(\kappa _{i j}\) (and therefore \(K_{i j}\)). A completely analogous algorithm can be used to reconstruct \(\partial _t K_{i j}\) in formulations in which this information is needed.

It is important to emphasize that this procedure allows one to compute \(K_{i j}\) at any point of **N** which is covered by the perturbative region. As a result, the numerical module can reconstruct the values of \(K_{ij}\) and \(\partial _t K_{i j}\) on a 2-surface of arbitrary shape, or any collection of points outside of \(\varGamma \).

## 6 Gravitational waves in the characteristic approach

The formalism for expressing Einstein’s equations as an evolution system based on characteristic, or null-cone, coordinates is based on work originally due to Bondi (1960) and Bondi et al. (1962) for axisymmetry, and extended to the general case by Sachs (1962). The formalism is covered in the review by Winicour (2005), to which the reader is referred for an in-depth discussion of its development and the associated literature.

Most work on characteristic evolution uses, or is an adpatation of, a finite difference code that was originally developed at the University of Pittsburgh and has become known as the PITT null code. The early work that eventually led to the PITT code was for the case of axisymmetry (Isaacson et al. 1983; Bishop et al. 1990; Gómez et al. 1994), and a general vacuum code was developed in the mid-1990s (Bishop et al. 1996b, 1997b; Lehner 1998, 1999, 2001). Subsequently, the code was extended to the non-vacuum case (Bishop et al. 1999b, 2005), and code adaptations in terms of variables, coordinates and order of accuracy have been investigated (Gómez 2001; Gómez and Frittelli 2003; Reisswig et al. 2007, 2013a). Spectral, rather than finite difference, implementations have also been developed, for both the axially symmetric case (de Oliveira and Rodrigues 2009) and in general (Handmer and Szilágyi 2015). One potential difficulty, although in practice it has not been important in characteristic extraction, is the development of caustics during the evolution, and algorithms to handle the problem have been proposed (Stewart and Friedrich 1982; Corkill and Stewart 1983). There are also approaches that use outgoing null cones but for which the coordinates are not Bondi–Sachs (Bartnik 1997; Bartnik and Norton 2000).

Shortly after the publication of the Bondi and Bondi–Sachs metrics and formalism, the idea of conformal compactification was introduced. This led to the well-known asymptotic description of spacetime, and the definitions of asymptotic flatness, past, future and spacelike infinity (\(I^+,I^-,I^0\)), and of past and future null infinity (\({\mathcal {J}}^-,\mathcal {J}^+\)) (Penrose 1963); see also Penrose (1964, 1965b) and Tamburino and Winicour (1966); and the reviews by Adamo et al. (2012) and Frauendiener (2004). The key result is that gravitational radiation can be defined unambiguously in an asymptotically flat spacetime only at null infinity. The waves may be expressed in terms of the Bondi news \({{\mathcal {N}}}\) (see Eq. (271) below), the Newman–Penrose quantity \(\psi _4\), or the wave strain \((h_+,h_\times )\).

After a characteristic code has been run using a compactified radial coordinate as in Eq. (259), the metric is known at \(\mathcal {J}^+\), and so it would seem to be straightforward to calculate the emitted gravitational radiation. Unfortunately, this is not in general the case because of gauge, or coordinate freedom, issues. The formulas do take a very simple form when expressed in terms of coordinates that satisfy the Bondi gauge condition in which the asymptotic flatness property is obviously satisfied, and for which conditions set at \(\mathcal {J}^+\) are propagated inwards along radial null geodesics. However, in a numerical simulation that is not the case: coordinate conditions are fixed on an extraction worldtube (in the case of characteristic extraction), or perhaps on a worldline (Siebel et al. 2003) or ingoing null hypersurface, and then propagated outwards to \(\mathcal {J}^+\). The result is that the geometry at and near \(\mathcal {J}^+\) may appear very different to one that is foliated by spherical 2-surfaces of constant curvature. Of course, the Bondi gauge and the general gauge are related by a coordinate transformation, and formulas for \({{\mathcal {N}}}\) and \(\psi _4\) are obtained by constructing the transformation.

An explicit formula in the general gauge for the news was obtained in Bishop et al. (1997b) (“Appendix 2”); and a calculation of \(\psi _4\) was reported in Babiuc et al. (2009), but the formula produced was so lengthy that it was not published. These formulas have been used in the production of most waveforms calculated by characteristic codes. An alternative approach, in which the coordinate transformation is explicit, rather than partially implicit, was suggested (Bishop and Deshingkar 2003) but has not been further used or developed. Recently, a formula for the wave strain \((h_+,h_\times )\), which is the quantity used in the construction of templates for gravitational-wave data analysis, was derived (Bishop and Reisswig 2014). An important special case is that of the linearized approximation, in which deviations from the Bondi gauge are small. The resulting formulas for \({{\mathcal {N}}}\), \(\psi _4\) and \((h_+,h_\times )\), are much simpler and so much easier to interpret than in the general case. Further these formulas are widely used because the linearized approximation often applies to the results of a waveform computation in a realistic scenario.

We set the context for this section by summarizing the Einstein equations in characteristic coordinates, and outlining the characteristic evolution procedure. The focus of this section is formulas for gravitational waves, and we next present the formulas in the simplest case, when the coordinates satisfy the Bondi gauge conditions. Much of the remainder of the section will be devoted to formulas for gravitational waves in the general gauge, and will include a discussion of conformal compactification. This section makes extensive use of spin-weighted spherical harmonics and the eth formalism, which topics are discussed in “Appendix 2”.

### 6.1 The Einstein equations in Bondi–Sachs coordinates

*u*label these hypersurfaces, \(\phi ^{^{_A}}\)\((A=2,3)\) be angular coordinates labelling the null rays, and

*r*be a surface area coordinate. In the resulting \(x^\alpha =(u,r,\phi ^{^{_A}})\) coordinates, the metric takes the Bondi–Sachs form

*V*by \(V=r+W_c r^2\). It should be noted here that different references use various notations for what is here denoted as \(W_c\), and in particular (Bishop et al. 1997b) uses

*W*with \(W :=r^2W_c\). As discussed in Sect. 1, we represent \(q_{_{AB}}\) by means of a complex dyad \(q_{_{A}}\), then \(h_{_{AB}}\) can be represented by its dyad component \(J:=h_{_{AB}}q^{^{_A}}q^{^{_{_{B}}}}/2\). We also introduce the fields \(K :=\sqrt{1+J \bar{J}}\) and \(U :=U^{^{_A}}q_{_{A}}\). The spin-weight

*s*of a quantity is defined and discussed in section “Spin-weighted fields” in “Appendix 2”; for the quantities used in the Bondi–Sachs metric

^{12}See also section “Computer algebra” in “Appendix 3”. Here, in order to make the discussion of the Einstein equations precise but without being overwhelmed by detail, we give the equations in vacuum in the linearized case, that is when any second-order term in the quantities \(J, \beta , U, W_c\) can be ignored. The Einstein equations are categorized into three classes, hypersurface, evolution, and constraint. The hypersurface equations are

*J*given on \(u=0\), and with boundary data for \(\beta ,U,\partial _r U,W_c,J\) satisfying the constraints given on \(\varGamma \) (Fig. 9). (In characteristic extraction, the data satisfies the Einstein equations inside \(\varGamma \), and so the issue of ensuring that the boundary data must satisfy the characteristic constraint equations does not arise). The hypersurface equations are solved to find \(\beta ,U,W_c\), and then the evolution equation gives \(\partial _u J\) and thence

*J*on the “next” null cone. See Kreiss and Winicour (2011) and Babiuc et al. (2014) for a discussion of the well-posedness of the problem.

*r*by means of a transformation \(r \rightarrow x=f(r)\) where \(\lim _{r \rightarrow \infty } f(r)\) is finite. In characteristic coordinates, the Einstein equations remain regular at \(\mathcal {J}^+\) under such a transformation. In practice, in numerical work the compactification is usually

*r*, but since we will have expressions involving the compactified radial coordinate and spherical harmonics such a notation would be confusing). Starting from the Bondi–Sachs metric Eq. (250), we make the coordinate transformation (260) to obtain

*J*and

*U*terms.

### 6.2 The Bondi gauge

*H*is

*H*are related by

### 6.3 General gauge

*u*and \(\phi ^{^{_A}}\) only. Conditions on the coefficients are found by applying the tensor transformation law

### 6.4 The gravitational-wave strain

*A*.

### 6.5 Conformal compactification

Here we give only a brief introduction to this topic, as these matters are discussed more fully in many standard texts and reviews, e.g., Wald (1984) and Frauendiener (2004). We have made a coordinate compactification, resulting in the metric and null tetrad being singular at \(\rho =0\), which is therefore not included in the manifold. Thus, quantities are not evaluated at \(\rho =0\), but in the limit as \(\rho \rightarrow 0\). Introducing a conformal transformation has the advantage that this technical issue is avoided and \(\mathcal {J}^+\) at \(\rho =0\) is included in the manifold; but also that the resulting formulas for \({{\mathcal {N}}}\) and \(\psi ^0_4\) are simpler. (Of course, it should be possible to use the asymptotic Einstein equations to simplify expressions derived in physical space, but due to the complexity of the formulas this approach has not been adopted).

*conformally invariant*with weight

*n*where

*n*is the power of \(\omega \) in the additional factor; thus the metric tensor is conformally invariant with weight \(-2\). In practice, it is not necessary to establish a relation of the form Eq. (303) to prove conformal invariance. The key step is to be able to show that a tensor quantity \(T^{a\cdots }_{b\cdots }\) satisfies \(\hat{T}^{a\cdots }_{b\cdots } =\rho ^{-n}T^{a\cdots }_{b\cdots }\), then conformal invariance with weight

*n*easily follows. In Eq. (303) the error term \({{\mathcal {O}}}(\rho )\) is shown explicitly, although it turns out to be irrelevant since the relation is evaluated at \(\rho =0\). This is generally the case, so from now on the error terms will not be taken into account; the one exception will be in the News calculation Sect. 6.5.1 which in places involves off-\(\mathcal {J}^+\) derivatives (since \(\partial _\rho {{\mathcal {O}}}(\rho ) ={{\mathcal {O}}}(1)\)).

*n*-dimensions

*J*,

*K*is derived in Gómez et al. (1997), and where \(h^{^{_{AB}}}\nabla _{_{A}}\nabla _{_{B}}\log (\omega )\) is given in terms of the \(\eth \) operator in Eq. (B1) of Bishop et al. (1997b). Eq. (312) is a nonlinear elliptic equation, and in practice is not actually solved. However, it will be used later, when considering the linearized approximation, since in that case it has a simple analytic solution.

#### 6.5.1 The news \({{\mathcal {N}}}\)

^{13}

The attentive reader may have noticed that the derivation above used \(\tilde{\rho }=\rho \omega \) rather than \(\tilde{\rho }=\rho (\omega +\rho A^\rho )\), so that \(\partial _\rho \omega \) should not be taken as 0 but as \(A^\rho \). However, the corrections that would be introduced remain \({{\mathcal {O}}}(\rho )\) since (a) \(\hat{m}_{_{[G]}}^1=0\), (b) in Eq. (319) the term \(\hat{g}^{11}A^\rho \) contained in \(\hat{g}^{1\gamma }\partial _\gamma \omega \) is \({{\mathcal {O}}}(\rho ) A^\rho \), and (c) in Eq. (321) the term \(F^A F^B \hat{\varGamma }^1_{_{AB}}A^\rho \) contained in \(F^A F^B \hat{\varGamma }^\gamma _{_{AB}}\partial _\gamma \omega \) is also \({{\mathcal {O}}}(\rho ) A^\rho \).

#### 6.5.2 The Newman–Penrose quantity \(\psi ^0_4\)

### 6.6 Linearized case

In the linearized case the Bondi–Sachs metric variables \(\beta ,J,U,W_c\) and the coordinate transformation variables \(u_0,A^u,(\omega -1),A^\rho ,x^{^{_A}}_0,A^{^{_A}}\) are regarded as small. Algebraically, the approximation is implemented by introducing a parameter \(\epsilon =\) max\((|\beta |,|J|,|U|,|W_c|)\) in a neighbourhood of \(\mathcal {J}^+\). Then, the metric variables are re-written as \(\beta \rightarrow \epsilon \beta \) etc., and quantities such as \({{\mathcal {N}}},\psi ^0_4\) are expressed as Taylor series in \(\epsilon \) with terms \({{\mathcal {O}}}(\epsilon ^2)\) ignored, leading to considerable simplifications. It is common practice to assume that the error in the approximation is about \(\epsilon ^2\). While computational results do not contradict this assumption, a word of caution is needed: no work on establishing a formal error bound for this problem has been reported.

*J*into spherical harmonic components

*A*,

*H*is subject to the gauge freedom \(H\rightarrow H^\prime = H + H_{_G}\), where \(H_{_G}=\eth ^2 u_{_G} (x^{_A})\). Decomposing

*H*into spherical harmonics, \(H=\sum \,{}_2Y^{\ell \,m}H_{\ell \,m}\), it follows that

*H*were obtained by time integration of the news \({{\mathcal {N}}}\), in this case appearing as an arbitrary “constant” of integration \(f(\tilde{x}^{_A})\).

## 7 Numerical implementations of the characteristic approach

As first steps towards CCM in relativity, it was implemented for the model problem of a nonlinear scalar wave equation (Bishop et al. 1996a, 1997a) without any symmetries, and for the Einstein equations with a scalar field under the condition of spherical symmetry (Gómez et al. 1996; Lehner 2000). There has been a series of papers on CCM under axial symmetry (Clarke and d’Inverno 1994; Clarke et al. 1995; d’Inverno and Vickers 1997; d’Inverno et al. 2000; d’Inverno and Vickers 1996; Dubal et al. 1995, 1998). A detailed algorithm for CCM in relativity in the general case was presented in Bishop et al. (1999a). The stable implementation of matching is quite a challenge, and this goal has not yet been achieved (Szilágyi et al. 2000; Szilágyi 2000); although a stable implementation without symmetry has been reported with the Einstein equations linearized and using harmonic “3+1” coordinates (Szilágyi and Winicour 2003; Szilágyi et al. 2002). The issue of progress towards CCM is much more fully discussed in the review by Winicour (2005).

*r*is a surface area coordinate, so it cannot be expressed explicitly in terms of the “3+1” coordinates. This complicates the matter in two ways

- 1.
The coordinate transformation has to be made in two steps, firstly to a null coordinate system in which the radial coordinate is an affine parameter on the outgoing null radial geodesics, and secondly to Bondi–Sachs coordinates.

- 2.
In general \(\varGamma \) is not a worldtube of constant

*r*, so setting data at the innermost radial grid point of the Bondi–Sachs system requires special care.

### 7.1 Worldtube boundary data

- 1.
It filters out high frequency noise.

- 2.
It greatly simplifies the process of interpolation onto a regular angular grid.

*R*. Of course the above is a simple coordinate-specific, rather than a geometric, definition; but in practice the definition has worked well and \(\varGamma \) has not exhibited any pathologies such as becoming non-convex. As indicated above, the extraction code will need the full four-metric and its first-derivatives, and it is a matter of choice as to whether conversion from the “3+1” variables (such as lapse, shift and three-metric) to four-metric is performed in the “3+1” code or in the extraction routine; for simplicity, this discussion will be on the basis that the conversion is performed in the “3+1” code. The conversion formulas are given in Eq. (75). The time derivatives of the four-metric could be found by finite differencing, but the results are likely to be less noisy if they can be expressed in terms of other variables in the “3+1” code—e.g., the “1+log” slicing condition and the hyperbolic \(\tilde{\varGamma }\)-driver condition (Pollney et al. 2011), if being used, would mean that time derivatives of the lapse and shift are known directly, and the time derivative of the three-metric may be obtainable from the extrinsic curvature. For the spatial derivatives of the four-metric, it is sufficient to calculate and write to file only the radial derivative, since \(\partial _x\), \(\partial _y\), \(\partial _z\) can later be reconstructed from the radial derivative and angular derivatives of the spherical harmonics which are known analytically. The radial derivative in terms of the Cartesian derivatives is

*A*, whether scalar, vector or tensor, is decomposed as

*A*, the idea is to decompose

*A*into a product of spherical harmonics and Chebyshev polynomials in

*r*. More precisely, we consider a “thick” worldtube \(R_1<R<R_2\) and the idea is to seek coefficients \(A_{k,\ell ,m}\) such that we may write

### 7.2 Reconstruction from spectral modes

### 7.3 Transformation to null affine coordinates

*q*,

*p*given by Eq. (406) for \(r=R_\varGamma \).

Although the above is given in terms of stereographic angular coordinates (*q*, *p*), rather than general angular coordinates \(\phi ^{^{_A}}\), the formulas that follow will not be specific to stereographic coordinates.

*u*,

*q*,

*p*) to be constant on each null geodesic generator. Algebraically, given \((u,\lambda ,q,p)\), the “3+1” coordinates are given by

*q*,

*p*,

*u*) grid in the time direction, \(\partial _\xi \ell ^{(0)}{}^{\mu }\) can easily be found by analytic differentiation or finite differencing.

### 7.4 Null affine metric

### 7.5 Metric in Bondi–Sachs coordinates

### 7.6 Starting up the null code at the worldtube

As already mentioned, a difficulty faced is that Eq. (372) gives metric quantities on the worldtube \(\varGamma \), which is not in general a hypersurface at a constant value of the *r*-coordinate. The original method for tackling the problem makes use of a Taylor series in \(\lambda \) (Bishop et al. 1999a), and has been implemented in Szilágyi et al. (2000), Szilágyi (2000), Babiuc et al. (2005), Reisswig et al. (2009) and Reisswig et al. (2010). Recently, a method that uses a special integration algorithm between the worldtube and the first characteristic grid-point, has been proposed and tested (Babiuc et al. 2011a, b). Both approaches are outlined below.

#### 7.6.1 Taylor series method

*A*,

*A*represents \(J,\beta ,U\) and \(W_c\).

*A*needs to be written as a function of null affine coordinates, so that \(\partial _{\lambda }A\) can be evaluated. Also, using Eqs. (365) and (367) evaluated on the worldtube, we need to find the value of \(\lambda \) at which \(r(\lambda )=r_i\), where \(r_i\) is an

*r*-grid-point near the worldtube; this needs to be done for each grid-point in both the angular and time domains. The derivation of the Taylor expansions is straightforward, with second \(\lambda \)-derivatives eliminated using the Einstein equations (Bishop et al. 1999a). The results are

#### 7.6.2 Special evolution routine between the worldtube and the first radial grid-point

*J*is known at all grid-points of the Bondi–Sachs coordinate system on the given null cone, either as initial data or from evolution from the previous null cone. A mask is set to identify those radial grid-points for which \(x_i-x_\varGamma < \varDelta x\), and these points will be called “B points”. The special algorithm is concerned with setting data at the points \(i=B+1\), called “B+1 points”. The first hypersurface equation in the hierarchy is the one for \(\beta \), and is the simplest one to hadle. The algorithm is

*r*. Consequently, the right hand sides of these equations are evaluated at the B+1 points rather than at the points mid-way between \(r_{{\scriptscriptstyle B}+1}\) and \(r_{\scriptscriptstyle \varGamma }\). Schematically, the hypersurface equations are of the form \((r^n A)_{,r}=f\), and the algorithm is

Since the value of *r* varies on the worldtube, it may happen that the angular neighbour of a B+1 point is a B point. Thus, the code must also set data for the metric variables at the B points, even though much of this data will not be needed.

### 7.7 Initial data

The above discussion has shown how data should be set at, or on a neighbourhood of, the inner worldtube \(\varGamma \), but in order to run a characteristic code data for *J* is also required on an initial null cone \(u=\) constant. Earlier work has adopted the simplistic but unphysical approach of just setting \(J=0\), assuming that the error so introduced would quickly be eliminated from the system. Babiuc et al. (2011a) and Bishop et al. (2011) investigated the matter. It was found that the error due to simplistic initial data is usually small, but it can take a surprisingly long time, up to 800 M, until saturation by other effects occurs. In terms of observations by a gravitational-wave detector, the effect of the error in search templates is not relevant. However, if a signal is detected, the effect would be relevant for accurate parameter estimation at large SNR (signal to noise ratio), but no quantitative estimates have been given.

### 7.8 Implementation summary

The issues summarized here are: (1) setting up a characteristic code that starts from the output of a “3+1” code; (2) estimating the gravitational waves from metric data in a compactified domain output by a characteristic code; (3) estimating quantities derived from the gravitational waves, i.e., the energy, momentum and angular momentum.

#### 7.8.1 Setting worldtube boundary data for the characteristic code

- 1.
Within the “3+1” code, write a routine that uses Eq. (342) to perform a spectral decomposition of the three-metric, lapse and shift, and outputs the data to file.

- 2.
In a front-end to the characteristic code, write a routine that reads the data from the file created in the previous step, and reconstructs the four-metric and its first derivatives at the angular grid-points of the extraction worldtube.

- 3.
Construct the generators \(\ell ^\alpha \) of the outgoing null cone using Eq. (351), and then the Jacobian \(\partial x_{\scriptscriptstyle [C]}^{\mu }/\partial x_{\scriptscriptstyle [N]}^{\alpha }\) as a series expansion in the affine paramenter \(\lambda \), for each angular grid-point on the worldtube.

- 4.
As described in Sect. 7.4, construct the null affine metric \(g_{{\scriptscriptstyle [N]}\alpha \beta }\) and its first \(\lambda \)-derivative at the angular grid-points of the extraction worldtube; then construct the contravariant forms \(g_{\scriptscriptstyle [N]}^{\alpha \beta }\) and \(\partial _\lambda g_{\scriptscriptstyle [N]}^{\alpha \beta }\).

- 5.
From Eq. (365), determine the surface area coordinate

*r*and its first derivatives at the angular grid-points of the extraction worldtube. - 6.
Construct the Jacobian \(\partial x_{\scriptscriptstyle [B]}^{\mu }/\partial x_{\scriptscriptstyle [N]}^{\alpha }\), and thus the Bondi–Sachs metric \(g_{\scriptscriptstyle [N]}^{\alpha \beta }\) and then the metric coefficients \(\beta ,J,U,W_c\) at the angular grid-points of the extraction worldtube.

- 7.
Implement either of the special start-up procedures described in Sect. 7.6.

- 8.
The construction of a characteristic code is not described in this review, but see section “Numerical codes” in “Appendix 3” for information about the availability of such codes.

#### 7.8.2 Estimation of gravitational waves

*r*) as the radial coordinate, but that is unlikely to apply in practice. In the case that the radial code coordinate is

*x*given by Eq. (259), the relation between \(\partial _x\) and \(\partial _\rho \) at \(\mathcal {J}^+\) is

\(\phi _0^{_A}(u,x^{_A})\). Solve the evolution problem Eq. (281) with initial data \(\phi _0^{_A}(0,x^{_A})=0\). This initial condition assumes that the initial data for

*J*has been set with \(J=0\) at \(\mathcal {J}^+\).\(\omega (u,x^{_A})\). Either solve the evolution problem Eq. (282) with initial data \(\omega (0,x^{_A})=1\), or evaluate the explicit formula Eq. (287).

\(u_0(u,x^{_A})\). Solve the evolution problem Eq. (280). In this case, there is a gauge freedom to set the initial data \(u_0(0,x^{_A})\) arbitrarily.

#### 7.8.3 Energy, momentum and angular momentum in the waves

## 8 A comparison among different methods

The quadrupole formula, including various modifications, leading to the wave strain \((h_+,h_\times )\);

\(\psi _4\) (fixed radius) and \(\psi _4\) (extrapolation), leading to the Newman–Penrose quantity \(\psi _4\);

Gauge-invariant metric perturbations, leading to the wave strain \((h_+,h_\times )\);

Characteristic extraction, leading to the wave strain \((h_+,h_\times )\), the gravitational news \({\mathcal {N}}\), or the Newman–Penrose quantity \(\psi _4\).

*Physical problem motivating the simulation*. The most appropriate method for extracting gravitational waves is affected by how the result is to be used. It may be that only moderate accuracy is required, as would be the case for waveform template construction for use in searches in detector data; on the other hand, high accuracy would be needed for parameter estimation of an event in detector data at large SNR. Further, the purpose of the simulation may be not to determine a waveform, but to find the emitted momentum of the radiation and thus the recoil velocity of the remnant.*Domain and accuracy of the simulation*. The domain of the simulation may restrict the extraction methods that can be used. All methods, except that using the quadrupole formula, require the existence of a worldtube, well removed from the domain boundary, on which the metric is Minkowskian (or Schwarzschild) plus a small correction. As discussed in Sect. 3.3.2, extrapolation methods need these worldtubes over an extended region. Further, the accuracy of the simulation in a neighbourhood of the extraction process clearly limits the accuracy that can be expected from any gravitational-wave extraction method.*Ease of implementation of the various extraction methods*. All the methods described in this review are well understood and have been applied in different contexts and by different groups. Nevertheless, the implementation of a new gravitational-wave extraction tool will always require some effort, depending on the method, for coding, testing and verification.*Accuracy of the various extraction methods*. Theoretical estimates of the expected accuracy of each method are known, but precise data on actual performance is more limited because suitable exact solutions are not available. In a simulation of a realistic astrophysical scenario, at least part of the evolution is highly nonlinear, and the emitted gravitational waves are oscillatory and of varying amplitude and frequency. On the other hand, exact solutions are known in the linearized case with constant amplitude and frequency, or in the general case under unphysical conditions (planar or cylindrical symmetry, or non-vacuum). One exception is the Robinson–Trautman solution (Robinson and Trautman 1962), but in that case the gravitational waves are not oscillatory and instead decay exponentially.Thus, in an astrophysical application, the accuracy of a computed waveform is estimated by repeating the simulation using a different method; then the difference between the two waveforms is an estimate of the error, provided that it is in line with the theoretical error estimates. In some work, the purpose of comparing results of different methods is not method testing, but rather to provide validation of the gravitational-wave signal prediction. The only method that is, in principle, free of any systematic error is characteristic extraction, but the method was not available for general purpose use until the early 2010s. It should also be noted that there remains some uncertainty about factors that could influence the reliability of a computed waveform (Boyle 2016).

### 8.1 Comparisons of the accuracy of extraction methods

Nagar et al. (2005) investigates various modifications of the standard quadrupole formula in comparison to results obtained using gauge-invariant metric perturbations for the case of oscillating accretion tori. Good results are obtained when back-scattering is negligible, otherwise noticeable differences in amplitude occur.

Balakrishna et al. (2006) computes \(\psi _4\) (fixed radius) and gauge-invariant metric perturbations for gravitational waves from boson star perturbations, but detailed comparisons between the two methods were not made.

Pollney et al. (2007) compares gauge-invariant metric perturbations to \(\psi _4\) (fixed radius) extraction for the recoil resulting from a binary black hole merger. It was found that results for the recoil velocities are consistent between the two extraction methods.

Shibata et al. (2003) and Baiotti et al. (2009) compare gauge-invariant metric perturbations, modified quadrupole formula and \(\psi _4\) (fixed radius) extraction for a perturbed neutron star. While the results are generally consistent, each method experienced some drawback. The gauge-invariant method has a spurious initial junk component that gets larger as the worldtube radius is increased. In \(\psi _4\) extraction, fixing the constants of integration that arise in obtaining the wave strain can be a delicate issue, although such problems did not arise in this case. The generalized quadrupole formula led to good predictions of the phase, but to noticeable error in the signal amplitude.

Reisswig et al. (2009, 2010) compare \(\psi ^0_4\) from characteristic extraction and from \(\psi _4\)-extrapolation for Binary Black Hole (BBH) inspiral and merger in spinning and non-spinning equal mass cases. The “3+1” evolution was performed using a finite difference BSSNOK code (Pollney et al. 2011). A comparison was also made in Babiuc et al. (2011a) for the equal mass, non-spinning case. Recently, a more detailed investigation of the same problem and covering a somewhat wider range of BBH parameter space, was undertaken (Taylor et al. 2013) using SpEC for “3+1” evolution (Szilágyi et al. 2009).

These results lead to two main conclusions. (1) The improved accuracy of characteristic extraction is not necessary in the context of constructing waveform templates to be used for event searches in detector data. (2) Characteristic extraction does provide improved accuracy over methods that extract at only one radius. The \(\psi _4\)-extrapolation method performs better, and there are results for \(\psi ^0_4\) that are equivalent to characteristic extraction in the sense that the difference between the two methods is less than an estimate of other errors. However, that does not apply to all modes, particularly the slowly varying \(m=0\) “memory” modes.

A study of gravitational-wave extraction methods in the case of stellar core collapse (Reisswig et al. 2011) compared characteristic extraction, \(\psi _4\) extraction (fixed radius), gauge-invariant metric perturbations, and the quadrupole formula. In these scenarios, the quadrupole formula performed surprisingly well, and gave results for the phase equivalent to those obtained by characteristic extraction, with a small under-estimate of the amplitude. However, quadrupole formula methods fail if a black hole forms and the region inside the horizon is excised from the spacetime. The gauge-invariant metric perturbation method gave the poorest results, with spurious high frequency components introduced to the signal. In characteristic extraction and \(\psi _4\) extraction the waveform was obtained via a double time integration, and the signal was cleaned up using Fourier methods to remove spurious low frequency components.

It is only very recently (Bishop and Reisswig 2014) that a method was developed in characteristic extraction to obtain the wave strain directly instead of via integration of \({{\mathcal {N}}}\) or \(\psi ^0_4\). That work also compared the accuracy of the waveform obtained to that found from integration of \(\psi _4^0\) using \(\psi _4\)-extrapolation, in two cases—a binary black hole merger, and a stellar core collapse simulation. When comparing the wave strain from characteristic extraction to that found by time integration of \(\psi ^0_4\), good agreement was found for the dominant (2,2) mode, but there were differences for \(\ell \ge 4\).

## Footnotes

- 1.
Note that the “bar” operator can in principle be applied also to the trace so that \({\bar{h}}=-h\).

- 2.
Note that the orthogonality condition fixes three and not four components since one further constraint needs to be satisfied, i.e., \(\kappa ^{\mu } A_{\mu \nu } u^{\nu } = 0\).

- 3.
Of course no gravitational waves are present in Newton’s theory of gravity and the formula merely estimates the time variations of the quadrupole moment of a given distribution of matter.

- 4.
- 5.
The values of \(r_k\) may vary with \(t_*\) since the extraction spheres are constructed to be of constant coordinate radius

*s*. - 6.
In the case of non-oscillatory modes, usually with \(m=0\), fitting to the real and imaginary parts of \(\psi _4^{\ell m}\) is preferred.

- 7.
All of our discussion hereafter will deal with perturbative analyses in the time domain. However, a hybrid approach is also possible in which the perturbation equations are solved in the frequency domain. In this case, the source terms are given by time-dependent perturbations created, for instance, by the motion of matter and computed by fully nonlinear three-dimensional codes (Ferrari et al. 2006).

- 8.
These results were originally obtained by Regge and Wheeler (1957) and by Zerilli (1970a, b) in a specific gauge (i.e., the Regge–Wheeler gauge). Subsequently, the work of Moncrief showed how to reformulate the problem in a gauge-invariant form by deriving the equations from a suitable variational principle (Moncrief 1974).

- 9.
We recall that in the notation of Cunningham et al. (1978) \(\widetilde{\psi } = \varLambda (\varLambda -2)\varPsi ^{(\mathrm o)}\), and the multipoles in Cunningham et al. (1978) are related to ours as \(\widetilde{h}_2 = h_2 = -2h\), \(\widetilde{h}_{0} = h^{(\mathrm o)}_{0}\) and \(\widetilde{h}_{1} = h^{(\mathrm o)}_{1}\).

- 10.
Note that in principle the gauge invariant quantities are independent of radius [cf. Eq. (219)]. In practice, however, their amplitudes may reach the correct asymptotic value only at sufficiently large distances. For this reason the extraction is in practice performed at different extraction radii and the amplitudes compared for convergence to an asymptotic value (Rupright et al. 1998; Rezzolla et al. 1999a).

- 11.
This is a standard choice in modern numerical-relativity codes but there are no restrictions on the choice of the coordinate system.

- 12.
There is a misprint in Eq. (A3) of the journal version of the reference, which has been corrected in the version on the arXiv, and also in Reisswig et al. (2013a).

- 13.
This result applies only to conformal space, not to physical space.

## Notes

### Acknowledgements

N.T.B. thanks the National Research Foundation, South Africa, for financial support, and the Max Planck Institute for Gravitational Physics (Albert Einstein Institute), the Institute for Theoretical Physics, Frankfurt, and the Inter-University Centre for Astronomy and Astrophysics, India, for hospitality while this article was being completed. The authors thank Alessandro Nagar and Christian Reisswig for comments on the article. Partial support comes from “NewCompStar”, COST Action MP1304, from the LOEWE-Program in HIC for FAIR, from the European Union’s Horizon 2020 Research and Innovation Programme under Grant Agreement No. 671698 (call FETHPC-1-2014, project ExaHyPE), and from the ERC Synergy Grant “BlackHoleCam: Imaging the Event Horizon of Black Holes” (Grant 610058).

## Supplementary material

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