Linearised conformal Einstein field equations

The linearisation of a second-order formulation of the conformal Einstein field equations (CEFEs) in Generalised Harmonic Gauge (GHG), with trace-free matter is derived. The linearised equations are obtained for a general background and then particularised for the study linear perturbations around a flat background -- the inversion (conformal) representation of the Minkowski spacetime -- and the solutions discussed. We show that the generalised Lorenz gauge (defined as the linear analogue of the GHG-gauge) propagates. Moreover, the equation for the conformal factor can be trivialised with an appropriate choice for the gauge source functions; this permits a scri-fixing strategy using gauge source functions for the linearised wave-like CEFE-GHG, which can in principle be generalised to the nonlinear case. As a particular application of the linearised equations, the far-field and compact source approximation is employed to derive quadrupole-like formulae for various conformal fields such as the perturbation of the rescaled Weyl tensor.


Introduction
Although nowadays gravitational wave effects can be studied using the full non-linear Einstein field equations, linear perturbation theory remains an active area of research, e.g. black hole perturbation theory, quasinormal modes, post-Newtonian and post-Minkowskian expansions. As it is well-known, the concept of gravitational radiation can be traced back to 1918 to the study of the linearisation of the Einstein field equations around flat spacetime and the derivation of the celebrated quadrupole formula. However, a rigorous formulation of the notion of gravitational radiation had to wait for further development of mathematical aspects of the theory put forward by Bondi, Sachs, Newman and Penrose and others in the decade of 1960 -see [1,2] for a historical review. The notion of null-infinity (and more generally the notion of a conformal boundary) developed in the aforementioned theory proved crucial for the modern understanding of gravitational radiation.
Null-infinity I corresponds to an idealisation of the asymptotic region of the physical spacetime (M,g) and it is defined through a conformally related manifold (M, g) where g = Ξ 2g so that I can be identified with the region where Ξ = 0 -but ∇ a Ξ ̸ = 0. Unfortunately, the Einstein field equations (EFEs) are not conformally invariant so evaluating quantities at I is non-trivial. To circumvent this problem, there are different approaches aimed at specific goals. For instance, the hyperboloidal approach in their different renditions [3,4,5,6,7,8,9] represents a compromise between the preservation of standard formulations of the EFEs used in Numerical Relativity and the inclusion of I at the expense of dealing with formally singular (but numerically tractable) equations. On the other hand, there exists another strand of research that stems from taking further the conformal approach while insisting on having formally regular equations. This conformally regular approach was initiated in 1981 when H. Friedrich derived in [10] a reformulation of the Einstein field equations known as the conformal Einstein field equations (CEFEs) -see [11,12] for an exhaustive discussion. One of the distincive features of the CEFEs is that the conformal factor is a variable, hence the location of the conformal boundary is not fixed a priori. Strategies for fixing the conformal boundary, or scri-fixing, have been introduced for the CEFEs in [13] for vacuum and [14] for trace-free matter, but they require the existence of a congruence of conformal curves.
Null-infinity is one of the central concepts that go into some of the most important open problems in General Relativity such as the weak cosmic censorship conjecture and the global stability analysis of spacetimes. Additionally, it is crucial for gravitational waves: due to the non-localisability of gravitational radiation, it is only rigorously defined at null-infinity. The CEFEs are a reformulation of the Einstein field equations that incorporates Penrose's conformal compactification into the initial value problem in General Relativity. Hence, the systematic study of these equations is important for the deeper understanding of gravitational radiation and related effects such as the memory effect. However, despite many advances in the mathematical analysis of spacetimes and numerical evolutions with the CEFEs [15,16,14,17,18,19,20,21], the application of the CEFEs to physical problems has been surprisingly limited. This is evident in (and possibly a result of) the fact that the application of linear perturbation theory to the CEFEs has not been studied in the existing literature. Such an analysis is needed to make contact with standard (linear) metric approaches to characterising gravitational radiation. This article aims to provide a first step in this direction by studying the linearisation of the CEFEs in a form that resembles standard formulations of the linearised Einstein field equations. This article represents step in exploiting the CEFEs for physical applications. In the current article, we focus on linearisations around flat spacetime however in the future we plan to extend our analysis into more interesting backgrounds and derive the conformal (meaning derived from the Conformal Einstein field equations) counterpart of black hole perturbation theory and post-Newtonian expansions.
The core of the CEFEs are the Bianchi identities, which provide a set of evolution and constraint equations for the Weyl curvature (coupled with other fields in the non-linear case). One approach to the linearised problem concerns the study of the spin-2 equation in a fixed background spacetime, which can be thought as the linearisation of the Weyl sector of the CEFEs in spinorial form. The spin-2 equation on a particular background known as the Minkowski i 0 -cylinder has been analysed in [22,23,15]. The spin-2 equation ∇ A ′ A ϕ ABCD = 0, although elegant, looks very different to the standard linearisation of metric formulations of the Einstein field equations. Hence, in this article, we study the linearisation not of the original set of CEFEs but rather we start from a metric and second-order formulation of the CEFEs which is closer in spirit to standard hyperbolic reductions of the EFEs in generalised harmonic gauge (GHG) employed in Numerical Relativity. This non-linear wave-like formulation of the CEFEs was originally derived in the vacuum case in [24] and has been extended for the case of trace-free matter in [25]. In this paper, we linearise these equations around a general background and then the solutions to the linearised equations around flat spacetime are studied. As a concrete application of the linearisation, we derive, in the far-field approximation, quadrupole-like formulae for the conformal fields, obtaining in particular a quadrupole-like formula for the perturbation of the rescaled Weyl tensor. As a byproduct, we show how the gauge source functions can be chosen so that the conformal factor remains unperturbed. This can be regarded as a "scri-fixing" strategy for the linearised CEFEs for trace-free matter. We then argue that the same strategy carries over to the non-linear wave-CEFEs in the context of the hyperboloidal initial value problem, providing a simpler alternative to the scri-fixing strategy in [14] based on conformal curves.

Notations and conventions
The signature convention for (Lorentzian) spacetime metrics will be (−, +, +, +). Latin indices from the first half of the alphabet a, b, c, , ... will be used as abstract tensor indices while Greek indices will denote spacetime coordinate indices, taking values from the set {0, 1, 2, 3}. If an adapted coordinate system (with coordinates x µ ) is introduced, x 0 = t represents the time coordinate, and Latin indices from the second half of the alphabet represent spatial indices, taking values from the set {1, 2, 3}. Symmetrisation and antisymmetrisation of tensors will be denoted with round and square brackets and taking the trace-free part of a tensor will be denoted by adding {} on the relevant indices. For instance T (ab) := 1 2 (T ab + T ba ), T [ab] := 1 2 (T ab − T ba ) and The curvature conventions are fixed by the relation Although we will occasionally refer to expressions in spinorial form (as much of the literature on the CEFEs employs spinor notation), all calculations in this article are performed in tensor notation.
Organisation of the paper Section 2 gives a brief summary of the CEFEs and the wave-like hyperbolic reduction of [25]. This is included to give a concise reference to the reader and to provide the non-linear context of the equations studied in this paper. Section 3 contains the linearisation of the wave-CEFEs in GHG around a general background. Section 4 particularises the equations for a flat background and discusses the perturbed solutions via Green's functions methods and, in the far-field approximation, quadrupole-like formulae for the conformal fields are derived.

The conformal Einstein field equations
The conformal Einstein field equations (CEFEs) are a reformulation of the Einstein field equations (EFEs) with the aim of dynamically implementing R. Penrose's conformal approach. The CEFEs were originally introduced in [26] and encode a set of differential conditions satisfied by the geometry of a conformal extension (M, g) of a spacetime (M,g) (with g = Ξ 2g ) satisfying the EFEs. To differentiate between these two, the pair (M, g) is called the unphysical spacetime while the pair (M,g) is called the physical spacetime. The feature that distinguishes the CEFEs from other reformulations of the EFEs is that (for trace-free matter) the CEFEs are formally regular at Ξ = 0 -see [11,26,27]. To see this and concisely introduce the CEFEs, it is convenient to define the following zero-quantities -see [11] for an extensive discussion: where Ξ is the conformal factor, s is the Friedrich scalar, L ab is the Schouten tensor, R a bcd is the Riemann tensor, d a bcd is the rescaled Weyl tensor, T ab is the rescaled energy-momentum tensor with trace T := g ab T ab and T abc is the rescaled Cotton tensor. The zero-quantities Z ab , Z a , δ abc , λ abc , Z and M a are defined simply as a bookkeeping device in the sense that the CEFEs are satisfied when a collection of fields {g ab , Ξ, s , L ab , d abcd , T ab , T abc } solve the equations Remark 1. Observe that the only singular terms in the zero-quantities (1) appear with the trace of the unphysical energy-momentum tensor, specifically in equation (1g). However for trace-free matter all the equations implied by the zero-quantities (1) are formally regular at Ξ = 0. The geometric variables are defined via where R ab and R denote the Ricci tensor and Ricci scalar of (M, g) while C a bcd is the conformally invariant Weyl tensor. Given a solution to the CEFEs, it can be verified that the associated physical metricg ab = Ξ −2 g ab will satisfy the Einstein field equations: whereT ab is the physical energy-momentum tensor, λ is the cosmological constant, andR ab and R are the respective Ricci tensor and Ricci scalar. The matter variables in (1) are related to their physical counterparts as: HereT ab is the physical energy-momentum tensor and∇ is the Levi-Civita connection ofg ab . A comprehensive discussion and derivation of the CEFEs can be found in [11].
Remark 2. Though the conceptually clean setup is the vaccum case, we include matter primarily for the purpose of deriving quadrupole-like formulae in subsection 4.5.
Assumption 1. From this point onward, it will be assumed that only trace-free matter models are in consideration. Notice that the conditions T = 0 and T {ab} = T ab simplify some of the matter terms in the zero-quantities (1).
We now consider some properties of the rescaled Cotton tensor under the trace-free matter assumption. Under this assumption, the rescaled Cotton tensor can be written in terms of the unphysical energy-momentum tensor as Observe as well that the Cotton tensor has the symmetries T abc = T [ab]c and T [abc] = 0. Furthermore, the trace-free assumption and∇ aT ab = 0 implies that ∇ a T ab = 0, ∇ c T ab c = 0.
Additionally, using again equations (9), (10) and exploiting Z ab = 0 gives Remark 3. The Ricci scalar R is not determined by the CEFEs and in fact, it encodes the conformal gauge freedom. Given two conformal extensions of the same physical spacetime g = Ξ 2g andǧ =Ξ 2g one has thatǧ = κ 2 g with κ :=Ξ/Ξ and κ ≃ O(1) at the conformal boundary. Hence, their Ricci scalars are related via Observe that ifŘ is considered as a given scalar function in (M, g) then, this equation can always be solved locally for κ. Thus, the Ricci scalar in the CEFEs is a gauge quantity that is called the conformal gauge source function -see [11] for further discussion.
In this article, we will adhere to the metric formulation of the CEFEs which consists of using the definition of the Schouten tensor to obtain an equation for the metric. Namely, from one substitutes the expression for the Ricci tensor as second derivatives of the metric respect to some coordinate basis x µ and reads L µν as a source term in the resulting equation. The source term L µν is then coupled to the rest of the variables via equations (2).

Remark 4.
In view that the Ricci scalar is the conformal gauge source function, to have a clean split between gauge quantities and non-gauge quantities one could opt for using the trace-free Ricci tensor as variable instead of L ab , The factor 1/2 in its definition is conventional and it is put so that Φ ab corresponds to the tensorial counterpart of the trace-free Ricci spinor Φ AA ′ BB ′ of the Newman-Penrose formalism. In this case, the evolution equation for the metric is read of from For convenience, we employ such a splitting, and will use the trace-free Ricci tensor rather than the Schouten tensor as a dynamical variable.
There are different hyperbolic reduction strategies to turn the tensorial expressions (1) into partial differential equations. In the next section, we revisit a hyperbolic reduction that is most appropriate for the aims of this article.

Remark 5.
A reader familiar with scalar-tensor and f (R) theories (the two being related for certain choices of scalar potential) might draw parallels between the physical and unphysical variables we employ here and the conformally related Einstein and Jordan frames (frames referring to variable choices) in those theories-see [28,29,30], for reviews on the topic. Although in the present case, we are not studying a modified theory of gravity but rather a conformal version of standard general relativity, it would be interesting to consider how one might relate the techniques employed in modified gravity with the approach explored here-in essence, we are treating the curvature as an independent dynamical variable, resembling the procedure for transforming between f (R) and scalar-tensor theory. Of course, one difficulty with such a program is the absence of a variational principle directly yielding the CEFEs (ideally one in which the CEFE variables are treated as independent); this will be explored this in future work.

Geometric wave equations
A first derivation of the metric conformal Einstein field equations as a set of wave equations was given in [24] for the vacuum case and in [25] for the case of trace-free matter. Our discussion differs in the use of the trace-free Ricci tensor instead of the Schouten tensor; we do this to make transparent the split between gauge and non-gauge quantities. Since we only make a simple variable change, the derivation will not be repeated here. The geometric wave equations read where □ = ∇ a ∇ a denotes the geometric wave operator. Observe that these are tensorial in the same way as expressions (1) and they do not include an equation for the metric. When recast in this form, the original set of equations (1) form constraints on initial data; the propagation of the constraints was shown in [25]. A further discussion of these wave equations as an explicit system of second order hyperbolic PDEs is given in Appendix B -see also [25].

Remark 6.
To obtain a set of equations for the matter fields encoded in T ab and T abc , an explicit matter model is required. Namely, given a matter model consisting on some fields τ = {τ 1 , ..., τ n } so that T ab = T ab (τ ) -and hence T abc = T abc (τ , ∇τ ) using equation (9)one has to derive wave equations for each of the fields encoded in τ and their derivatives.
To give a concise and general discussion, the wave equations for the matter fields will not be presented as this is a case-dependent analysis and can be revisited for a number of trace-free matter models in [25].

Wave equation for metric
The wave equation for the metric is derived from the expression of the trace-free Ricci tensor in some fiduciary coordinate system x µ . Here and in what follows, Γ µ αβ will denote the Christoffel symbols of the Levi-Civita connection ∇ of g in the coordinate basis x µ . First, one defines the GHG-constraint as where Γ µ := g αβ Γ µ αβ are the contracted Christoffel symbols and H µ are the coordinate gauge source functions. Recall that Γ µ = −∇ α ∇ α x µ so that setting C µ = 0 is equivalent to imposing the generalised harmonic gauge condition with the above definitions, the reduced Ricci tensor R µν is defined as A customary calculation -see for instance [31,32,33,34,8]-shows that the reduced Ricci tensor can be expressed in terms of derivatives of the metric as where ✔ := g αβ ∂ α ∂ β is the standard reduced wave operator. Similarly, recalling that the Ricci scalar of g encodes the conformal gauge freedom, one introduces where F is the conformal gauge source function. Imposing the constraint C = 0 is equivalent to choosing a representative from the conformal class [g] in the same way that imposing C µ = 0 is equivalent to choosing the coordinates to satisfy equation (17) -see Remark 3. Therefore, using equation (14) and imposing the GHG-coordinate and conformal gauge constraints we obtain the following reduced wave equation for the components of the unphysical metric in the x µ coordinates: One can perform a similar procedure on the remaining wave equations (15) to obtain them in their reduced form, but since it is not absolutely necessary for the discussion that follows, that discussion is provided in Appendix B.

The Conformal Einstein field equations in the linear approximation
In this section the linearisation of the wave equations (15) is obtained. To set up the notation regarding linearisation, first, we outline the general procedure. Consider a one-parameter family of fields ϕ(ε) which satisfy an equation where E is some (non-linear) differential operator. Then, the fieldφ := ϕ(0) will be called the background solution as it satisfies the equation Eφ = 0. Let D denote the linearisation operator: Then, using that Dφ = 0, one obtains Lδϕ = 0 where L is a linear operator acting on δϕ := dϕ dε | ε=0 . Issues of linearisation stability (the existence and correspondence between exact and linearised solutions) will not be addressed here.
For the ongoing discussion ϕ = (φ, T ), where φ encodes all the geometric variables while f and T respectively encode the gauge and matter variables. Namely, for the geometric sector φ, one is considering an approximate solution of the form For the gauge sector f one has the split For the matter sector T one has the split From this point forward (unless otherwise stated), indices will be raised and lowered with the background metricg µν .

Equation for the metric perturbation
One of the advantages of the wave formulation of the CEFEs is that the metric sector of the equations resembles conventional formulations of General Relativity. This can be seen clearly even in the non-linear equation (21) where this equation is formally identical to the standard (non-conformal) Einstein field equations in generalised harmonic gauge with a source term in this case given by −4Φ ab − 1 2 F g ab which could be thought conceptually as some artificial "geometric matter term". To exploit the latter viewpoint let us define so that equation (14) agnostically reads: Expressing the equation for the metric in this way is advantageous since one can follow the classical discussion for linearising the Einstein field equations with respect to a general background g ab -see for instance Sec. 7.5 of [35]. Since a few gauge transformations are needed and we want to reserve the symbol δg µν for the last transformation, we first decompose the metric as g µν =g µν + h µν . Then, a straightforward linearisation of equation (28) gives where□ :=∇ µ∇ µ with∇ denoting the Levi-Civita connection ofg. Then, upon commuting covariant derivatives and defining the trace-reversed metric perturbationĥ µν : where δW α α :=g αβ δW αβ , andR µναβ andR µν are the Riemann and Ricci tensors ofg, which can be written in terms of the rescaled Weyl tensor and trace-free Ricci tensors via the decomposition: To avoid unnecessarily long expressions, background curvature terms will not be expanded out. Notice that equation (30) contains only divergences ofĥ µν and derivatives of the trace are absent.
To obtain hyperbolic equations, the divergence terms∇ αĥν α are removed by employing a slight generalisation of the procedure outlined in [35]; one begins by recognising that the quantity q µν := h µν + 2∇ (µ ξ ν) is a gauge transformation of the metric perturbation h µν . Introduce its trace-reversed version δg µν =q µν := q µν − 1 2 q c cg µν . Since the vector field ξ ν is arbitrary, one can choose a gauge in which ξ ν satisfies an inhomogeneous wave equation. In particular, the divergence of δg µν :∇ may be rewritten as the following wave equation: where F µ is a given set of functions of the coordinates x α and the evolved fields (but not their derivatives), which we call the Lorenz gauge source functions. These are related to the linearisation of the coordinate gauge source functions H µ as discussed in Remark 7. If ξ ν satisfies the above wave equation, then δg µν satisfies the gauge (32). The gauge condition in (32) is a linear analogue of the generalised harmonic gauge condition, and as such we called it the generalised Lorenz gauge -a similar definition is given also in [36] in the non-conformal context assuming a flat background. Working in the generalised Lorenz gauge, we obtain the following equation for the metric perturbation: subject to the gauge condition (32).
Remark 7. The analogy between the Lorenz and Harmonic gauge is more subtle when dealing with a non-flat background since the direct linearisation of the generalised harmonic condition C µ := Γ µ + H µ = 0 renders Comparing equations (35) and (32) one concludes that : Observe that even for the harmonic caseH µ = δH µ = 0 equation (35) reduces to retrieving the Lorenz gauge condition only if the background connection vanishesΓ µ αβ = 0.
To complete the discussion now we look at the expression for δW µν . It follows from the definition of W ab in equation (27) that Recalling that the conformal gauge constraint has been imposed C := R − F = 0, observe that the linearisation of this condition leads to the following constraint: For consistency, the background conformal gauge source function and the background Ricci scalar should be identified:R =F . Then, with this identification and using equations (38) the linearisation of the conformal constraint reads The latter equation states that although Φ µν is trace-free, in general, its perturbation may not be. Nonetheless, the trace of the perturbation is fixed by equation (40). In particular, notice that if the background is flat, this condition reduces to δΦ µ µ = 0. To obtain the final expression for the metric perturbation equation is then enough to substitute equation (38) into equation (34). Although the resulting equation for δg µν looks more complicated than the usual expression for the metric perturbation this is mainly because the linearisation was performed with respect to a general background and due to the use of the generalised Lorenz gauge. Later on, when the background is fixed the expressions presented in this section will simplify considerably.

Equations for the perturbation of other conformal fields
The linearisation of the rest of the conformal fields can be obtained systematically. The main calculation involves obtaining a suitable expression for the linearisation of the geometric wave operator acting on each of the conformal fields. To do the calculation in the scalar sector, first notice that for a scalar field φ one has that whereφ denotes a background quantity and δφ the perturbation. Performing the same gauge transformations of subsection 3.1 for the metric perturbation h µν and imposing the generalised Lorenz gauge one obtains: A direct calculation using the latter expression and exploiting that the background fieldsφ satisfy the CEFEs in the form of equations (15) as well as equations (2), give the following equation for the perturbation of the conformal factor 3T µν δg µν (43) A similar calculation gives the linearised wave equation for the Friedrich scalar. The explicit expression is lengthy and has been put in Appendix A. For tensor fields the calculation is slightly more involved. To give an abridged discussion consider now the case of the linearisation of □w µ , where w a is some covector field. A direct calculation gives, whereẘ µ denotes a background quantity and δw µ the associated perturbation. Substituting the term□h µν using equation (29) and rewriting the equation in terms of the trace-reversed metric perturbationĥ µν renders Commuting covariant derivatives on the second term in the right-hand side of equation (45), performing the gauge transformation described in subsection 3.1 and imposing the generalised Lorenz gauge one obtains where L F denotes the Lie-derivative along F µ . Extending the latter calculation on tensors of higher valence is a long but straighforward calculation. The latter leads to the following linearised equations for the conformal fields: Here we recall that ϕ schematically represents the conformal fields, and the gauge sources are schematically denoted as f ; containing the Lorenz gauge source functions F µ as well asF and δF . The semicolon separates the arguments which do not contain the evolved perturbation variables explicitly. Observe that only the equation for δΦ µν contains second derivatives of δF which restricts the allowed choices for δF -see Remark 9. The expressions for H g µν , and H Ξ are given in the respective equations (34) and (43 4 Analysis of wave solutions around flat spacetime 4

.1 Fixing the background solution
The discussion given in the last section is general in the sense that neither the background solution ϕ nor the gauge source functions f have been fixed. The first observation to be made is that, although the expression for H d µναβ is complicated, in fact, if one restricts the analysis to the vacuum case and a conformally flat background (so thatd µναβ = 0 butΦ µν ̸ = 0 andR ̸ = 0), then the equation for the perturbation of the rescaled Weyl tensor reduces to and this equation decouples from the rest. This is to be expected since as one way to study linearised gravity is through the spin-2 equation for which the metric perturbation plays no role -see [37,38]. Since the aim of this article is to give a first analysis into the linearised version of the CEFEs, it is relevant to now examine the complete system even if some of the equations decouple as in the case described above.
The simplest choice of background is one in which the background geometry is flat. Naively, one would think that this forces the background conformal factor to be constant,Ξ = 1, and the equations trivialise to the standard non-conformal case; this would defeat the purpose of considering the CEFEs in the first place, as one would not be able to incorporate the conformal boundary in such an analysis. Fortunately, there exists a non-trivial conformal transformation that maps the Minkowski spacetime into itself such that spatial infinity and null infinity for the physical Minkowski spacetime are respectively mapped to the origin and the lightcone through the origin of the (conformally transformed) unphysical Minkowski spacetime. The evaluation of the field at the conformal boundary of the physical spacetime corresponds to evaluation at finite coordinate locations in the unphysical spacetime. We point out that such a conformal transformation does not necessarily yield a full compactification, as some points (such as the physical origin) are mapped to infinity -see Figure 1. This choice for a flat background, which we call the inversion representation of the Minkowski spacetime, is the simplest non-trivial case (in the sense that it can incorporate the conformal boundary) that one can analyse in the conformal setup. Therefore, the following analysis can be regarded as the conformal counterpart of the standard discussion of linearised (physical) Einstein field equations around flat spacetime. The inversion representation of the Minkowski spacetime is a compelling choice not only for its simplicity, but also because it is related to the i 0 -cylinder representation of the Minkowski spacetime (see for instance [15,39,40,41,42,43]) which hasd µναβ = 0,R = 0 with a non-trivial trace-free Ricci tensorΦ µν ̸ = 0. While an analysis on the i 0 -cylinder background would be of great interest, the sources H in the equations for the fields other than δd µναβ (see Appendix A) become cumbersome whenΦ µν ̸ = 0; this will be pursued in future work.

The inversion conformal representation of the Minkowski spacetime
The inversion representation of the Minkowski spacetime described in the preceding paragraph is a standard conformal representation detailed in [44,40,41,43], which we now summarise. In what follows (R 4 ,η) will denote the (physical) Minkowski spacetime. Letxμ = (t,xĩ), represent physical Cartesian coordinates. The associated coordinate basis vectors and covectors are denoted by ∂μ and dxμ, respectively. The components of an arbitrary abstract tensor S ab in the physical Cartesian coordinate basis are denoted by Sμν. The Minkowski metric reads whereημν = diag(−1, 1, 1, 1). Defining the physical radial coordinate viaρ 2 = δĩjxĩxj where δĩj = diag(1,1,1) and considering an arbitrary choice of coordinates on S 2 the physical Minkowski metric can be written asη witht ∈ (−∞, ∞),ρ ∈ [0, ∞) where σ denotes the standard metric on S 2 . We now introduce the unphysical Cartesian coordinates x µ = (t, x i ), and denote the associated vector and covector basis with ∂ µ and dx µ , respectively. As before, the components of an arbitrary tensor S ab in the unphysical Cartesian coordinate basis will be denoted as S µν . The relationship between the physical and unphysical Cartesian coordinates is given by: where δμ µ = diag(1, 1, 1, 1), X 2 = 1/X 2 and η µν = diag(−1, 1, 1, 1). This coordinate transformation is valid in the complement of the lightcone at the origin in the physical Minkowski spacetime whereX 2 > 0.
Remark 8. The notational reason for having a tilde over the physical coordinate indices is to distinguish the physical and unphysical coordinate bases denoted by ∂ µ and ∂μ. The latter requires the use of δμ µ in the expressions (50) to keep the tilded and untilded indices balanced. This notation is inspired by that of the dual foliation formalism of [45].
From the relation between the physical and the (unphysical) inversion coordinate bases one can construct the Jacobians To identify a suitable conformal metric, it is enough to compute η µν dx µ ⊗dx ν = η µν Jμ µ Jν ν dxμ ⊗ dxν. Introducing η = η µν dx µ ⊗ dx ν , we write this compactly as: with Ξ = X 2 . This demonstrates that the inversion (unphysical) Minkowski spacetime can be recast as a conformal transformation of the physical Minkowski spacetime. Although, in most of the upcoming discussion, all tensor components will be expressed in just one coordinate basis (the unphysical one), notice that, the relation between the components of a tensor S ab in the physical and the unphysical bases is then given by To complete the discussion, one can construct an unphysical spherical polar coordinate system. Upon introducing ρ 2 = δ ij x i x j , a calculation shows that the unphysical (conformal) metric η and conformal factor Ξ read with t ∈ (−∞, ∞) and ρ ∈ [0, ∞). Notice that spatial infinity i 0 of the physical Minkowski spacetime (R 4 ,η) is mapped to the origin (t = 0, ρ = 0) in (R 4 , η). Also, future and past null infinity I ± of the physical Minkowski spacetime are mapped to the lightcone passing through the origin. In other words, introducing the unphysical retarded and advanced times u := t − ρ and v := t + ρ, future/past null infinity I ± is located at v = 0 and u = 0 respectively. To round up the discussion between the relation the physical and unphysical coordinates, here we record that

The background solution
We now fix the background solutionφ to be the Minkowski inversion background discussed in the previous subsection, 4.2. This yields the following expressions for the background fields the latter implies that the curvatureR µναβ = 0 vanishes which, in terms of its irreducible decomposition implies thatΦ Since one has thatR =F , this fixes the background conformal gauge source function. Nonetheless, the remaining gauge source functions encoded in F µ and δF need not vanish and will (along with matter perturbation quantities) be left unspecified. This gives the following equations Remark 9. The Lorenz gauge source functions F µ are allowed to depend on the variables δϕ as they appear only with first derivatives in equations (58). However, the conformal gauge source function δF is only allowed to depend on the coordinates due to the second derivatives of δF in the equation (58d).

Remark 10.
Although for simplicity we have fixed the notation in subsection 4.2 so that x µ denotes the unphysical Cartesian system of coordinates, in fact, equations (58) are not bound to these coordinates, the only assumption used was that the background Riemann curvature R µ αβν vanishes. In other words, in deriving equation (58), the vanishing of the (background) Christoffel symbolsΓ α µν has not been assumed, consequently, formally identical equations to (58) will hold in any coordinate system. Nonetheless, the unphysical Cartesian system x µ will be assumed in the discussion of subsection 4.4 so that one may apply standard techniques such as expressing the solutions in terms of Greens functions and multipolar expansion approaches.

The matter relations
The matter equations depend on the particular matter model in question, however, since the background matter terms are assumed to vanish, the direct linearisation of the general relations, combined with the symmetries of T abc and T ab described in section 2, yield a set of formally identical expressions that the perturbations must satisfy: Also, one has the relations:

Propagation of the generalised Lorentz gauge
Tracing equation (58d) and using equations (59) shows that Recall that the linearisation of the conformal gauge constraint, translates in the present case, to the condition This, in particular, simplifies equation (58a): The condition (61) is trivially propagated by equation (60). Namely, if initial data is given such that δΦ µ µ = 0 and∇ n δΦ µ µ = 0 on Σ, where∇ n := n µ∇ µ and n µ denotes the normal to a spacelike hypersurface Σ. Then, by existence and uniqueness of solutions to the flat-wave equation (60), one has where U is an open set and D + (Σ) denotes the future domain of dependence of Σ. This result can be understood as the propagation of the conformal constraint in the linear setting. To show that generalised Lorenz gauge condition (32) propagates one needs to use the contracted Bianchi identity. In the CEFE language this is encoded in equation (1c). Hence, for completeness and future reference, we linearise the CEFEs as defined in equations (1) and (2) and record the result in Remark 11.
Remark 11. Using equation equation (12) to express (1) using the trace-free Ricci tensor and linearising around a flat background gives where where the expressions have been written using q µν instead of δg µν to avoid long expressions.
Recall that δg µν is simply the trace-reversed version of q µν .
Remark 12. Observe that if δF = 0 is set then δΦ µν is a TT-tensor (transverse and tracefree). Furthermore, if the Lorenz gauge condition is chosen, F µ = 0, equation (68a) looks formally identical to the textbook linearisation Einstein equations with an geometricartificial matter term δΦ µν .
Remark 13. The initial data for the wave equations (68) is not free as it needs to satisfy equations (63) on Σ. The propagation of the (first order) CEFEs by the wave-CEFEs has been proven at the non-linear level in [25] and the result at the linear level follows from linearisation of the subsidiary system obtained in [25]. Also, a procedure to obtain initial data for the conformal wave CEFEs at the non-linear level has been outlined in [25]: one starts by solving the conformal Einstein constraint equations on Σ which constitute the spatial parts of the zero-quantities (1). Then, the time derivatives of the evolved fields on Σ can be read from the zero-quantities using the solution to the conformal Einstein constraints. An analogous procedure can be performed for the linear case to obtain initial data for the wave equations (68). Observe that from equations (63d) and (63e) one only needs to consider their trace-free parts as the traces give the wave equations (68b) and (68a), respectively.

Scri-fixing gauge
A difficulty with the standard CEFEs concerns the fact that in general, the conformal boundary is not fixed with respect to the coordinates. Although there exist a generalisation of the CEFEs, the extended CEFEs (see for instance [11]) that allows to have an explicit expression the conformal factor, this approach is based on gauge adapted to a congruence of conformal geodesics, called the conformal Gaussian gauge. The latter requires the use of more general type of connections known as Weyl connections and it is in general difficult to write explicitly the line element of physically relevant solutions in the coordinates adapted to the conformal Gaussian gauge. Here, we give an alternative simpler approach based on exploiting the gauge source functions in the wave formulation of the standard CEFEs. In the linear case, the key is to find a gauge choice such that equation (68b) is homogeneous. This amounts to choosing the Lorenz gauge source functions such that F µ satisfies Thus, if trivial initial data is given to equation (68b) on the initial hypersurface then the trivial solution δΞ = 0 is obtained in the development. Hence, Ξ =Ξ and that the location of the conformal boundary of the perturbed linearised solution coincides with that of the background spacetime. Observe that to solve equation (69) one needs∇ µΞ ̸ = 0 and although this condition does not hold at i 0 , this is not a problem for foliations that end at a cut C of I (e.g. the hyperboloidal and null foliations) -see Figure 1.
In the non-linear case one can proceed in a similar way. The non-linear version of the wave equation for the conformal factor reads: where ✔ := g µν ∂ µ ∂ ν and x µ is a fiduciary coordinate system -see Appendix B. One can follow the same approach as in the linear case and think of Ξ as a given function of the coordinates and solve for the gauge source functions. Although this approach is of limited use for the Cauchy problem as ∇ µ Ξ vanishes at i 0 , this is not an issue for foliations which end at a cut C of I for which ∇ µ Ξ ̸ = 0. As a concrete example and to make contact with the hyperboloidal framework of [3,4,5,6,7] consider an unphysical coordinate system x µ = (τ, r, ϑ A ) with A = 1, 2, related to physical coordinatesxμ = (t,r, ϑ A ) according to the standard hyperboloidal prescriptiont = τ + H(r) and r = r Ω(r) where H and Ω are known as the height and compression functions. If one sets Ξ = Ω(r) then equation (70) fixes one of the components of the coordinate gauge source functions H µ as: where Ω ′ := dΩ dr . If the slice is hyperboloidal then Ω ′ ̸ = 0 at I . Remark 14. In [3,5,6] a similar gauge-fixing is developed, however in the previous work the evolution equations are formally singular while in the CEFE approach of this paper the equations are formally regular. Also, the general strategy put forward here is not bound to the hyperboloidal and asymptotically flat set-up as a similar scri-fixing procedure could be adapted to study spacetimes with de-Sitter (and anti-de-Sitter) asymptotics using CEFEs similar to the case of [46] and [17] where the gauge was fixed using, instead, a congruence of conformal geodesics.

The vacuum equations
In this subsection an analysis of the linearisation of the wave CEFEs in vacuum is given (later, we extend the discussion to the case where matter is included and with a more general gauge choice). To start the discussion in the simplest possible set up, consider the vacuum case with vanishing gauge source functions F µ = δF = 0 so that the equations read □ δΦ µν = 0, (74e) subject to the constraints imposed by equations (63) with vanishing matter terms. Note that we have split the metric perturbation into its trace δg µ µ and its trace-free part δg {µν} . In this manner, one can identify the variables that satisfy homogeneous equations.

Solutions to the homogeneous equations
The solutions of homogeneous wave equations in flat spacetime are widely known and understood, but since the initial data are subject to the initial constraints (63), it is perhaps appropriate to briefly discuss how one might write the solutions explicitly in terms of the initial data. For a field δQ H satisfying a homogeneous wave equation□δQ H = 0, the solution may be written in the form [47]: where dΣ ′ λ = (1/3!)ϵ λαβγ dx ′α ∧ dx ′β ∧ dx ′γ is the surface element on Σ, the underline denotes evaluation of a quantity on Σ, and the quantity D(x, is the difference between the retarded (−) and advanced (+) Green's functions, defined as and satisfying the inhomogeneous equation□G ± (x, (75) is a solution of (74d-74f) can be seen by noting that D(x, x ′ ), being the difference of two Green's functions, is a solution of the homogeneous wave equation. It is straightforward to show that D(x, x ′ ) = 0 if x 0 = x 0′ , and one can also show [47] that D(x, x ′ ) satisfies the properties: where ∆t := x 0 − x ′0 . Equation (77b) follows from the fact that for a given |⃗ x − ⃗ x ′ |, D(x, x ′ ) is by construction an odd function in ∆t. Given (77), one can verify that the coefficients in (75) do in fact coincide with the initial data (though note that (75) only depends on the time derivatives of the fields). Of course, as indicated earlier, the initial data for the fields must satisfy the vacuum version of (63). Given the simplicity of the vacuum wave equations and their resemblance to wave equations in ordinary Minkowski space it is tempting to write the solutions in terms of plane waves. While one can certainly obtain solutions to the wave equations in this manner, a difficulty arises from the fact that even in the vacuum case, the constraints (63) are nontrivial, since the background expression forΞ in the inversion coordinates is explicitly coordinate dependent. As a result, solutions satisfying the constraints do not have a clean mode separation in a Cartesian Fourier expansion (though perhaps an alternative mode decomposition may be appropriate).

Solutions to the inhomogeneous equations and including matter
For completeness, we discuss the solutions of equations (74a) and (74b). One may begin by assuming that one has in hand initial data satisfying the constraints (63). We note that the structure of the vacuum equations has a hierarchy: one can first solve for δs, δΦ µν , δd µναβ , and δg µ µ , then use the result to solve for δg {µν} and δΞ. Schematically, one may then write the inhomogeneous wave equations for the latter two in the following form (where S is a source that depends on the background field expressions and δs, δΦ µν , and δg µ µ ): which yields a particular solution (with x 0 = t, x ′0 = t ′ ): where δQ H0 (x) is a solution to the homogeneous equation with initial conditions chosen so that the particular solution δQ I0 (x) has trivial initial conditions δQ I0 (x) = 0 and ∂ t δQ I0 (x) = 0 on Σ; this can in principle be done by evaluating the integral term and its derivative on Σ to obtain −δQ H0 (x ′ ) and −∂ t δQ H0 (x ′ ). Finally, to obtain the solution corresponding to the prescribed initial data, one can then add to δQ I0 (x) a homogeneous solution of the form in equation (75).
Since we are interested in the initial value problem, we have chosen the retarded Green's function (79). This is (as the reader might be aware) because G − (x, x ′ ) is only nontrivial for x 0 > x ′0 , so that for a given point x, the field contributions from the integral in (79) depend on the values of the source in the past of x (specifically over the past light cone of x). We note that the procedure described for the inhomogeneous equation can be straightforwardly extended to the full set of non-vacuum equations (68), subject to the constraints (63) on the initial data. As in the vacuum case, one has a hierarchy in which one can first specify the matter sources, then solve for the variables δs, δΦ µν , δd µναβ , and δg µν , and finally solve for the conformal perturbation δΞ using the previously obtained results. Of course, the preceding discussion glosses over the problem of solving the matter equations, however such analysis is matter-model dependent. Hence in the present analysis they are simply regarded as given sources in the equations.

The far-field approximation and quadrupole-like formulae
As emphasised before, an appealing property of the wave formulation of the CEFEs is that the metric sector of the equations looks as the standard non-conformal Einstein field equations with some artificial-geometric matter term. This suggests that some of the standard techniques and approximation methods can be applied to the linearised wave CEFEs such as solving them in the far-field regime and obtaining quadrupole-like formulae.

The far-field approximation
To derive the far-field approximation, we focus our attention not to the homogeneous part of the solution to equation (78) but to the part generated by the sources. For simplicity, we omit nontrivial homogeneous solutions and δQ H0 (x) in the particular solution (which corresponds to an appropriate choice of initial data), so that the solution consists only of the integral in equation (79). Upon evaluating the time integral, one obtains: This is of course just the standard expression one can find in standard textbooks-see [48]. To obtain a far-field expression in the unphysical set-up we start by recalling that the origin in the inversion (unphysical) Minkowski spacetime corresponds to spatial infinity i 0 for the physical Minkowski spacetime so the far-field region in physical Minkowski spacetime corresponds to a neighbourhood of the origin in the inversion Minkowski spacetime. Let p ′ and p be a points in the physical Minkowski spacetime with physical Cartesian coordinates (0,x ′i ) and (0,x i ) respectively. Expressed in unphysical Cartesian coordinates, these points correspond to (0, x ′i ) and (0, x i ). Hence, the distance r between p ′ and p can be expressed through the vector In terms of the physical coordinates, imposing the far-field approximation means | ⃗ x| >> | ⃗ x ′ |. Consequently, Therefore, in the far-field approximation one can write In particular, for the case where δQ is either δΦ µν , δg µν or δd αβµν one can derive quadrupole-like formulae. To do so, it is necessary to compute the divergence of the source terms appearing in equations (68a), (68d) and (68e). Let's define Recall that the fields encoded inφ satisfy the CEFEs, hence the background conformal factor satisfiesZ ab = 0 as defined in equation (1a), which for the flat background case reduces to ∇ µ∇νΞ =sg µν . A direct calculation using the latter expression and the matter relations (59) shows that∇ The first two equations simply encode equations (66) and (65) in a divergence-like format. The third equation in (87) shows that equation (63c) propagates.
Remark 16. For the scalar variables δs and δΞ, the leading order source terms can be evaluated directly. In the case of rank-2 tensors, a reason for reducing the sources to quadrupole form is to make the field dependent on a single (the time-time) component of the source tensor.

A quadrupole-like formula for δΦ µν and δg µν
Here, we derive quadrupole-like formulae for δΦ µν and δg µν , which we discuss in tandem. Since the source term S µν contains δΦ µν [cf. equation (84)] first one solves for δΦ µν and then for δg µν . Let δQ µν = δΦ µν or δg µν and F µ = 1 8∇ µ δF or F µ , and σ µν = S µν or S µν , respectively. Exploiting that equations (64) and (32) can be collectively written as∇ ν δQ µν = F µ one has that Thus, once δQ ij is determined, δQ 0i can be obtained by formally integrating the second equation in (88). In turn, δQ 00 is obtained by integrating the first equation in (88). For the particular case of δΦ µν one could instead use that δΦ µ µ = 0 to solve algebraically for δΦ 00 . Thus, one can reduce the discussion to that of obtaining a quadrupole-like formula for δQ ij . Recall that the far-field approximation reads If one were to restrict the calculations to gauge source functions satisfying□F µ =□∇ µ δF = 0 (in particular, F µ = δF = 0) then the integrand would satisfy∇ ν σ µν = 0 leading to an expression which is formally identical to the quadrupole formula in the non-conformal set-up for the physical metric perturbation -see for instance [48,35]. However, it is instructive to keep the gauge source function terms since it will serve as model for the derivation of the quadrupole-like formula for the rescaled Weyl tensor where the divergence of the source term S µναβ cannot be set to zero by gauge considerations -see the last equation in (87). Here, we only assume that the gauge source functions are specified. The first two equations (87) in the current notation read∇ ν σ µν =□F µ , which in components read Integrating∇ i (σ ik x ′j ) over a region Ω ⊂ Σ and using Gauss' law gives the following identity where we have used∇ i x ′j = δ i j and ds i is the surface element of ∂Ω.
Assumption 2. It will be assumed that the matter fields encoded in S µν and S µναβ , as defined in equation (84), have compact support in Ω ⊂ R 3 .

Remark 17.
Recall that□Φ µν = S µν , then under assumption 2 it follows that δΦ µν has compact support in Ω. Furthermore, choosing the gauge source function F µ appropriately, one can then assume that that S µν [as given in equation (84)] has compact support in Ω.
Using assumption 2 and Remark 17, equations (91) and (90) give Similarly, integrating∇ k (σ 0k x ′i x ′j ) over a region Ω and proceeding as before, making use of Gauss' law one obtains Then using Remark 17 and equations (93) and (90) renders Combining equations (92) and (94) one gets therefore, using the far-field approximation (89) gives where As discussed earlier, the gauge source functions could be set to zero, but in certain applications, it can be useful to consider general gauge source functions. Of course, one may wish to implement a scri-fixing gauge for this situation, but such a gauge may be incompatible with the assumption that S µν has compact support in Ω. However, the purpose of the present analysis is to provide a template for deriving the quadrupole formula for the Weyl tensor, in which such a scri-fixing gauge can be implemented.
Remark 18. The reader might wonder whether quadrupole-like formulae are appropriate for trace-free energy-momentum tensors, which often corresponds to that of radiation. The concern is that in order to implement the far-field approximation, one implicitly assumes a slow-motion approximation, which is inconsistent with radiation propagating at the speed of light. However, geons [49,50,51] and photon stars [52,53,54,55] provide examples of stationary compact sources formed from self-gravitating radiation, which may at the very least provide useful toy models (the extension of our results to the case of nonvanishing trace will be discussed elsewhere). Nevertheless, the primary reason for the preceding analysis is to provide a template for deriving the quadrupole formula for the rescaled Weyl tensor, which can be straightforwardly generalised to the case with trace. This is discussed further in Remark 19.

A quadrupole-like formula for the rescaled Weyl tensor perturbation
We now turn to the case of the rescaled Weyl tensor δQ αβµν = δd αβµν . Here, the source term S αβµν is given entirely in terms of matter fields -see equation (84). Expanding in components the divergence equation (87) one has Considering the case σ = j in equation (87) one has Integrating∇ k (S kµjν x ′i ) over Ω and using Gauss' law gives Using assumption 2 the boundary term in the left-hand side of (100) vanishes. Solving for the integral with S iµjν and substituting∇ k S kµjν using equation (99), gives the following expression (after symmetrising the spatial indices): Now, considering the case σ = 0 in equation (87) one has Integrating∇ k (S 0µkν x ′i x ′j ) over Ω and using Gauss' law gives Using assumption 2 to remove the boundary terms and exploiting the pair-interchange-symmetry of S µναβ to rewrite∇ k S 0µkν in using equation (102) renders Together, equations (101) and (104) give To employ equation (105) one needs to perform an electric-magnetic decomposition of the tensors involved in the current discussion. Consider any tensor W abcd with the symmetries of the Weyl tensor and let n a be a normalised timelike vector such that n a n a = −1. Define the associated projector as h a b := g a b + n a n b . Then, the electric part E ab and magnetic part B ab of W abcd are given by where B ab = − 1 2 ϵ cd ef W abef with ϵ abcd is the (4-dimensional) volume-form of g ab . In practice (particularly when working in an adapted frame), instead of working directly with B ab it is simpler to use which is related to B ab via B ab = − 1 2 B acd ϵ b cd where ϵ abc := ϵ f abc n f is the (3-dimensional) volume-form of h ab -see [11,32] for further discussion on the electric-magnetic decomposition. In the present case, since the background is flat, one can simply take the normal vector to be n ν = δ 0 ν so that the electric-magnetic split of d µναβ simply correspond to The wave equation for δd µναβ , which compactly written reads□δd µναβ = S µναβ , can then be split into the wave equations for d ij and d ijk : where S ij := S i0j0 and S ijk := S i0jk . Using the far-field approximation expression (83) and equation (105) then gives where In particular, setting µ = 0 and ν = 0 one gets the quadrupole-like expression for δd ij and similarly taking µ = 0 and ν = k for δd ijk . Proceeding as described before and exploiting the symmetries of S µναβ (those of the Weyl tensor) and those of δT αµν [cf. equation (59)] renders Recall that the rescaled Cotton tensor T αµν can be expressed in terms of the derivatives of the unphysical energy momentum tensor. Therefore equation (112) allows one to express the perturbation to the rescaled Weyl tensor in the far-field approximation in terms derivatives of the perturbation to the unphysical energy momentum tensor δT µν and background quantities via equation (59b).

Remark 19.
Although the focus of this article has been in the conformal spacetime setup -and hence restricted to trace-free matter models-an analogous calculation can be performed directly in the physical spacetime, lifting then the trace-free matter restriction, obtaining a quadrupole-like formula for perturbations of the Weyl tensor. Observe that, for the derivation of the wave equation satisfied by the rescaled Weyl tensor, the specific form of the rescaled Cotton tensor (in terms of the unphysical energy-momentum tensor) is never used. It follows that for the physical set-up, one gets a formally identical equation. The wave equation for the Weyl tensor has been derived for instance in [56] and [57]. Therefore, the analysis of section 4.5.3 would still hold replacing the rescaled Weyl tensor d abcd with the Weyl tensor C abcd and the rescaled Cotton tensor T abc by its physical counterpartT abc which written in terms of the physical energy-momentum reads: Remark 20. We note that the expression for S µναβ is independent of the choices for the gauge source function. For this reason, one can choose the scri-fixing gauge independently of the assumption that S µναβ has compact support in Ω; the quadrupole-like formula we obtain for the rescaled Weyl tensor is therefore compatible with any gauge choice, in particular the scri-fixing gauge introduced earlier.

Summary and discussion
In this article, we have presented the linearisation of the wave-like hyperbolic reduction of the CEFEs in a GHG gauge for a general background. As a first step toward studying the linear perturbations of spacetimes using the conformal CEFE approach, the equations have been particularised to the case where the background is the inversion (unphysical) Minkowski spacetime. Conceptually, when studying perturbations of the physical Minkowski spacetime, one can in principle use any conformally flat metric so thatd µναβ = 0. However having a non-vanishingΦ µν makes the equations quite cumbersome. Hence, as a first analysis, we have opted to use as a background the inversion Minkowski spacetime for whichΦ µν =R = 0. For this background, we discuss the propagation of the gauge (the generalised Lorenz gauge and the linear version of the conformal constraints). The fact that the chosen background is flat permits the use of techniques similar to those of the classical methods in physical spacetime to study gravitational waves. In this manner, we have obtained the conformal CEFE counterpart of the classical linearisation of the (physical) Einstein field equations. Nevertheless, one may wish to consider linearising around non-flat conformal representations of the Minkowski spacetime. A particularly interesting choice is the i 0 -cylinder representation of Minkowski spacetime, as this is particularly useful for studying the gravitational field close to the critical sets I ± -the region where null-infinity I and spatial infinity i 0 meet. Although this analysis has been done via the spin-2 equation ∇ A ′ A ϕ ABCD = 0 in [22] (see also [41,43] for a discussion of the solutions of wave equations in this background) where it has been shown that the (linearisation of rescaled Weyl spinor) spin-2 field develops logarithmic terms, it would be of interest to see how these logarithmic terms manifest in the metric, which is the variable employed in most perturbation schemes in physics (such as the post-Minkowskian expansions). Another interesting class of backgrounds are Petrov type D spacetimes such as Kerr, so that a conformal Teukolsky-like equation may be obtained; however, a different linearisation procedure (such as that of Chandrasekhar [58]), may be more appropriate in that case. These avenues will be pursued in future work. More generally, the present work represents a first step into studying linear perturbations of spacetimes of physical interest (such as black holes) within the CEFE framework.
A particularly useful result of this paper is the development of a scri-fixing strategy using gauge source functions for CEFEs. Our strategy provides an alternative to more abstract gauges based on conformal curves and conformal geodesics [14], one that is particularly well-suited for numerical implementation. Though the strategy is initially developed for the linearised equations (which are expected to be good approximation in the weak field region of asymptotically flat spacetimes of physical interest), the generalisation to the nonlinear case, as discussed in the main text, is straightforward. A minor limitation of this scri-fixing strategy in the asymptotically flat setting is that the initial data must be prescribed on slices that intersect future null infinity (as opposed to spatial infinity), but our scri-fixing strategy fits well within the hyperboloidal approach to Numerical Relativity.
Finally, we have constructed quadrupole-like formulae for the CEFEs linearised around a flat background. While the quadrupole-like formulae for δg µν and δΦ µν require gauge source functions with compact support and a trace-free energy-momentum tensor, the quadrupole-like formula we obtain for the Weyl tensor is independent of gauge choice and is formally identical to the case where the energy-momentum tensor has nonvanishing trace. Since the Weyl tensor yields an unambigious notion of gravitational radiation at future null infinity, our result provides a familiar prescription for computing gravitational waveforms at null infinity generated by weakly gravitating sources in the linear approximation.

B Reduced wave equations
In this appendix, we extend the analysis leading up to (21) to the other wave equations listed in (15) for the sake of completeness. In particular, we wish to recast (15) explicitly as a system of second order hyperbolic equations. This non-linear system of wave equations was originally derived in [25] using slightly different set of variables and a different reduced wave operator.
First observe that in general, the geometric wave operator acting on a tensor will give rise to derivatives of the Christoffel symbols. Since the metric is a variable of the system, this is problematic for hyperbolicity. In particular, for a covector w a one has In this appendix, indices are raised and lowered with the metric g µν . Observe that the second term in the right-hand side of equation (120) represents second derivatives of the metric. However, one can replace these derivatives of the Christoffel symbols by noticing the identity Exploiting the identity (121) a direct calculation gives where L Γ is the Lie-derivative along Γ a . Using the definition of the conformal and the GHGconstraints one can rewrite the last expression as where L C and L H are the Lie derivative along C a and H a respectively and Q(w) µ := 2Φ µν w ν − 2Γ νµα ∇ α w ν − 2Γ α µ β Γ ναβ w ν (124) One can generalise the latter calculation for tensor of type (0, n) as follows Then, by imposing the conformal and GHG-constraints are satisfied so that C = 0 and C a = 0 one effectively has effectively removed the problematic terms for hyperbolicity: □w µ1...µn = ✔ w µ1...µn + n 4 F w µ1...µn + L H w µ1...µn + Q(w) µ1...µn Applying the latter rule on each of the fields appearing on the geometric wave equations one obtains a set of hyperbolic non-linear wave equations which read as follows.
For the components of the trace-free Ricci tensor: For the components of the rescaled Weyl tensor: For the Friedrich scalar: For the conformal factor: Remark 21. In vacuum, the hyperbolicity of the wave equations (129) is clear. However, for the case with matter T ab = T ab (τ , ∇τ ), one could at first instance question their hyperbolicity due to the presence of first derivatives of the rescaled Cotton tensor T abc which encode second derivatives of the energy-momentum tensor T ab -hence the equations contain two (or more) derivatives of the matter fields τ . Nonetheless, one can still ensure hyperbolicity constructing wave equations for τ and ∇τ so that T abc is expressed in terms of evolved fields only. For instance, in the case of the conformally invariant scalar field T ab = T ab (ϕ, ∇ϕ) one needs to construct wave equations for ϕ and a reduction variable φ = ∇ϕ -see [25].
Since this is a case-dependent analysis and this has been done in [25] for a number of matter models, further discussion is omitted.
In [25] the reduced wave operator is defined so that the lower order contributions (along with terms involving the gauge source functions) are absorbed into the definition of the operator ■ in such a way that the reduced equations look formally identical to the geometric ones. Here the standard reduced wave operator ✔ := g αβ ∂ α ∂ β used in most discussions of generalised harmonic gauge in Numerical Relativity was employed instead. The initial data for the reduced wave equations (129) have to satisfy Z µν | Σ = 0, Z µ | Σ = 0, δ µνσ | Σ = 0, λ µνσ | Σ = 0, Z| Σ = 0.
where Σ ⊂ M is a spacelike hypersurface on which the initial data is prescribed. It should also be noticed that the GHG-constraint C µ and the conformal constraint C are not independent, in fact a calculation shows C = ∇ µ C µ . Propagation of the gauge and a full discussion of the propagation of the constraints [i.e. propagation of the zero-quantities (130)] has carefully been done in [25] and will not be reproduced here. The aim of revisiting the derivation of the reduced wave equations (129) is simply to provide context and the non-linear analogue of the linearised equations derived in Section 3.