Conformal wave equations for the Einstein-tracefree matter system

Inspired by a similar analysis for the vacuum conformal Einstein field equations by Paetz [Ann. H. Poincar\'e 16, 2059 (2015)], in this article we show how to construct a system of quasilinear wave equations for the geometric fields associated to the conformal Einstein field equations coupled to matter models whose energy-momentum tensor has vanishing trace. In this case, the equation of conservation for the energy-momentum tensor is conformally invariant. Our analysis includes the construction of a subsidiary evolution system which allows to prove the propagation of the constraints. We discuss how the underlying structure behind these systems of equations is the integrability conditions satisfied by the conformal field equations. The main result of our analysis is that both the evolution and subsidiary equations for the geometric part of the conformal Einstein-tracefree matter field equations close without the need of any further assumption on the matter models other than the vanishing of the trace of the energy-momentum tensor. Our work is supplemented by an analysis of the evolution and subsidiary equations associated to three basic tracefree matter models: the conformally invariant scalar field, the Maxwell field and the Yang-Mills field. As an application we provide a global existence and stability result for de Sitter-like spacetimes. In particular, the result for the conformally coupled scalar field is new in the literature.


Introduction
The conformal Einstein field equations are a conformal representation of the Einstein field equations which allow to study the global properties of the solutions to equations of General Relativity by means of Penrose's procedure of conformal compactification -see e.g. [11,18] for an entry point to the literature on the subject. Crucially, a solution to the conformal Einstein field equations implies a solution to the Einstein field equations away from the conformal boundary.
A key step in the analysis involving the conformal Einstein field equations is the so-called procedure of hyperbolic reduction in which a subset of the field equations is cast in the form of a hyperbolic evolution system (the evolution system) for which known techniques of the theory of partial differential equations allow to establish well-posedness. An important ingredient in the hyperbolic reduction is the choice of a gauge -which in the case of the conformal Einstein field equations involves not only fixing coordinates (the coordinate gauge) but also the representative of the conformal class of the spacetime metric (the so-called unphysical metric) to be considered (the conformal gauge). Naturally, gauge choices should bring to the fore the physical and geometric features of the setting under consideration. In order to make contact with the Einstein field equations, the procedure of hyperbolic reduction has to be supplemented by an argument concerning the propagation of the constraints by means of which one identifies the conditions under which one can guarantee that a solution to the evolution system implies a solution to the full system of conformal equations, independently of the gauge choice. The propagation of the constraints involves the construction of a subsidiary evolution system describing the evolution of the conformal field equations disregarded in the evolution system and of the conditions representing the gauge. The construction of the subsidiary system requires lengthy manipulations of the equations which are underpinned by integrability conditions inherent to the field equations.
Most of the results concerning the conformal Einstein field equations available in the literature make use of hyperbolic reductions leading to first order symmetric hyperbolic evolution systems. This approach works best for the frame and spinorial versions of the conformal equations. Arguably, the simplest variant of the conformal Einstein field equations is given by the so-called metric conformal Einstein field equations in which the field equations are presented in tensorial form and the unphysical metric is determined by means of an unphysical Einstein field equation relating the Ricci tensor of the unphysical metric to the various geometric fields entering in the conformal equations -these can be thought of as corresponding to some fictitious unphysical matter. Remarkably, until recently, there was no suitable hyperbolic reduction procedure available for this version of the conformal field equations. In [16] Paetz has obtained a satisfactory hyperbolic procedure for the metric vacuum Einstein field equations which is based on the construction of (second order) wave equations. To round up his analysis, Paetz then proceeds to construct a system of subsidiary wave equations for tensorial fields encoding the conformal Einstein field equations (the so-called geometric zero-quantities) -and thus, he shows the propagation of the constraints. The motivation behind Paetz's approach is that the use of second order hyperbolic equations gives access to a different part of the theory of partial differential equations which complements the results available for first order symmetric hyperbolic systems -see e.g. [6,3]. Paetz's construction of an evolution system consisting of wave equations has been adapted to the case of the spinorial conformal Einstein field equations in [12]. In addition to its interest in analytic considerations, the construction of wave equations for the metric conformal Einstein field equations is also of relevance in numerical studies, as the gauge fixing procedure and the particular form of the equations is more amenable to implementation in current mainstream numerical codes than other formulations of the conformal equations.
The purpose of the present article is twofold: first, it generalises Paetz's construction of a system of wave equations for the conformal Einstein field equations to the case of matter models whose energy-momentum tensor has a vanishing trace -i.e. so-called tracefree matter. This type of matter models is of particular interest as in this case the equation of conservation satisfied by the energy-momentum is conformally invariant and the associated equations of motion for the matter fields can, usually, be shown to possess good conformal properties -see [18], Chapter 9. Second, it clarifies the inner structure of Paetz's original construction by identifying the integrability conditions underlying the mechanism of the propagation of the constraints. The motivation behind this analysis is to extend the recent work on the construction of vacuum antide Sitter-like spacetimes in [3] to the case of tracefree matter. However, we believe that the analysis we present has an interest on its own right as it brings to the fore the subtle structure of the conformal Einstein field equations.
The main results of this article can be summarised as follows: Theorem. The geometric fields in the metric conformal Einstein field equations coupled to a tracefree matter field satisfy a system of wave equations which is regular up to and beyond the conformal boundary of a spacetime admitting a conformal extension. Moreover, the associated geometric zero-quantities satisfy a (subsidiary) system of homogeneous wave equations independently of the matter model. The subsidiary system is also regular on the conformal boundary.
The precise statements concerning the above main result are contents of Lemmas 1 and 3.

Remark 1.
A remarkable property of our analysis is that it renders suitable evolution equations for the conformal fields and the zero-quantities without having to make any assumptions on the matter model except that it satisfies good evolution equations in the conformally rescaled spacetime. Thus, our discussion can be regarded as a once-for-all analysis of the evolution equations associated to the geometric part of the metric conformal field equations valid for a wide class of coordinate gauges prescribed in terms of the coordinate gauge source function appearing in the generalised wave coordinate condition.
Remark 2. The homogeneity of the subsidiary system on the geometric zero-quantities is the key structural property required to ensure the propagation of the constraints by exploiting the uniqueness of solutions to a system of wave equations. This feature is also exploited when the propagation of the gauge is studied.
The approach followed to obtain our main result is based on the identification of a family of integrability conditions associated to the conformal Einstein field equations. To the best of our knowledge, these integrability conditions have not appeared elsewhere in the literature. In our opinion this approach brings better to the fore the structural properties of the conformal Einstein field equations and, in particular, it makes the construction of the subsidiary evolution system more transparent than the brute force approach adopted in [16]. A similar strategy is also adopted to study the propagation of the gauge. In particular, by setting the matter fields to zero, our analysis subsumes the main results of [16]. Despite offering a more sleek approach to the construction of an evolution system for the conformal Einstein field equations, our analysis stills requires heavy computations which are best carried out in a computer algebra system. In the present case we have made systematic use of the suite xAct for the manipulation of tensorial expressions in Mathematica -see [15].
We supplement our general analysis of the metric conformal Einstein field equations with an analysis of the evolution and subsidiary evolution equations of some of the tracefree matter models more commonly used in the literature: the Maxwell field, the Yang-Mills field and the conformally coupled scalar field. For each of these fields we construct suitably second order wave equations for the matter fields and the associated matter zero-quantities. For the case of the Yang-Mills field, our analysis makes no assumptions on the gauge group.
As an application of our analysis, in the final section of this article we present stability results for the de Sitter spacetime for perturbations which include the Maxwell, Yang-Mills or conformally invariant scalar field. Proofs of this result for the Maxwell and Yang-Mills fields have been obtained in [9] using the spinorial version of the conformal equations and a first order hyperbolic reduction. The stability result for the conformally coupled scalar field is, to the best of our knowledge, new.

Overview of the article
In Section 2 we briefly summarise the key properties of the metric conformal Einstein field equations coupled to tracefree matter and their relation to the Einstein field equations. Section 3 provides the derivation of the geometric wave equations for the geometric fields appearing in the conformal Einstein field equations. Section 4 introduces the key notion of geometric zeroquantity and discusses the identities and integrability conditions associated to objects of this type. Section 5 is devoted to the construction of the subsidiary evolution system for the geometric zero-quantities used in the argument of the propagation of the constraints. This is, in principle, the most calculationally intensive part of our analysis. However, using the integrability conditions of Section 4 we provide a streamlined presentation thereof. In Section 6 we discuss the gauge freedom inherent to the geometric evolution systems obtained in Sections 3 and 5 and how this freedom can be used to complete the hyperbolic reduction of the equations. Section 7 establishes the consistency of the gauge introduced in the previous section, independently of the particular tracefree matter model. Section 8 provides a case-by-case analysis of three prototypical tracefree matter models -the conformally coupled scalar field (Subsection 8.1), the Maxwell field (Subsection 8.2) and the Yang-Mills field (Subsection 8.3). The discussion for each of these matter models includes the construction of suitable wave evolution equations and subsidiary evolution equations. Finally, Section 9 provides an application of the analysis developed in this article to the global existence and stability of de Sitter-like spacetimes.

Conventions
In what follows, (M,g) will denote a trace-free matter spacetime satisfying the Einstein equations. The signature of the spacetime metric is (−, +, +, +). The lowercase Latin letters a, b, c, . . . are used as abstract spacetime indices. The Greek letters µ, ν, λ, . . . will be used as spacetime coordinate indices. Our conventions for the curvature are 2 The metric conformal Einstein field equations with tracefree matter The purpose of this section is to provide a brief overview of the basic properties of the conformal Einstein field equations with tracefree matter. A more extended discussion of the properties of these equations as well as their derivation can be found in of [18], Chapter 8.

Basic relations
In what follows let (M,g ab ) denote a spacetime satisfying the Einstein field equations with matter whereR ab andR denote, respectively, the Ricci tensor and Ricci scalar of the metricg ab , λ is the Cosmological constant andT ab is the energy-momentum tensor. As a consequence of the contracted Bianchi identity one obtains the conservation law Here∇ a denotes the Levi-Civita covariant derivative of the metricg ab . Now, let (M, g ab ) denote a spacetime related to (M,g ab ) via a conformal embedding In a slight abuse of notation we write with Ξ a smooth scalar field -the so-called conformal factor.
Remark 3. Following the standard usage, we refer to (M,g ab ) as the physical spacetime while (M, g ab ) will be called the unphysical spacetime.

The unphysical energy-momentum tensor
Since equation (3) does not determine the wayT ab transforms, it will be convenient to define the unphysical energy-momentum tensor as Using the transformation rule between the Levi-Civita covariant derivatives of conformally related metrics, it readily follows that equation (2) takes the form where ∇ a is the Levi-Civita covariant derivative of g ab and T ≡ g ab T ab . It then follows that ∇ a T ab = 0 if and only if T = 0.
Assumption 1. In the remaining of this article we restrict our attention to matter models for which T = 0 so that the corresponding unphysical energy-momentum tensor T ab is divergence-free -that is,

Basic properties of the conformal Einstein field equations
The metric conformal Einstein field equations with tracefree matter have been first discussed in [9]. In terms of the notation and conventions used in this article they are given by: A detailed derivation of these equations can be found in [18]. In the above expressions L ab , s, d a bcd and T abc denote, respectively, the Schouten tensor, the Friedrich scalar, the rescaled Weyl tensor and the rescaled Cotton tensor. These objects are defined as where C a bcd is the conformally invariant Weyl tensor. Observe that T abc has the following symmetries: Relevant for the subsequent discussion is the well-known fact that the rescaled Weyl tensor has two associated Hodge dual tensors, namely: where ǫ abcd is the 4-volume form of the metric g ab . One can check that * d abcd = d * abcd . We also define the Hodge dual of T abc as * T abc ≡ 1 2 ǫ ab de T dec .
Moreover, if Assumption 1 and equation (5a) are taken into account, one obtains two additional relations: Remark 4. Equations (5a)-(5d) will be regarded as a set of differential conditions for the fields Ξ, s, L ab and d a bcd . Equation (5e) can be shown to play the role of a constraint which only needs to be verified at a single point -see e.g. [18], Lemma 8.1. A differential condition for the unphysical metric g ab will be discussed in Section 6.2 Remark 5. By a solution to the conformal Einstein field equations with matter it will be understood a collection of fields (g ab , Ξ, s, L ab , d a bcd , T ab ) satisfying equations (4) and (5a)-(5e).
The relation between the conformal Einstein field equations (5a)-(5e) and the Einstein field equations (1) is given in the following proposition -see [18], Proposition 8.1, for a proof: Proposition 1. Let (g ab , Ξ, s, L ab , d a bcd , T ab ) denote a solution to the conformal Einstein field equations with matter such that Ξ = 0 on an open set U ⊂ M. Then the metricg ab = Ξ −2 g ab is a solution to the Einstein field equations (1) with energy momentum tensor given byT ab = Ξ 2 T ab on U.

An alternative equation for d a bcd
For our purposes, it will be convenient to consider an alternative version of the conformal field equation for the rescaled Weyl tensor. This can be obtained as follows: multiplying (5d) by ǫ abf g and exploiting the identity Remark 6. This last equation is equivalent to (5d) and will be essential in sections 3 and 4 where a system of wave equations for the geometric fields and the zero-quantities associated to the equations (5a)-(5e) is discussed.

An equation for the components of the metric g ab
The system (5a)-(5e) needs to be supplemented with an equation for the metric g ab . This equation is given by the definition of the Schouten tensor, equation (6a), written in the form Here, R ab and L ab are considered as independent variables. In particular, the Ricci tensor R ab is assumed to be expressed in terms of first and second derivatives of the components of the metric whilst L ab is a field satisfying equations (5a)-(5e). This will be further discussed in Section 6 where a suitable wave equation for the components of the metric is constructed.

Remark 7.
As pointed out in [10], equation (11) can be regarded as an Einstein field equation for the unphysical metric g ab . From this point of view the geometric fields Ξ, s, L ab and d abcd can be regarded as unphysical matter fields. Accordingly, in the following we refer to equation (11) as the unphysical Einstein equation. This point of view should allow to adapt well-tested numerical methods for the Einstein field equations to the case of the conformal field equations.

The evolution system for the geometric fields
In this section we show how to construct an evolution system for the geometric fields appearing in the conformal Einstein field equations, equations (5a)-(5e). These evolution equations take the form of geometric wave equations -that is, their principal part involves the D'Alambertian ≡ ∇ a ∇ a of the conformal metric g ab .
In [16], Paetz has obtained a system of geometric wave equations for the set of conformal fields (Ξ, s, L ab , d a bcd ) in the vacuum case. Next, we present a generalisation of this result to the tracefree case: Lemma 1. The tracefree conformal Einstein field equations (5a)-(5e) imply the following system of geometric wave equations for the conformal fields: Proof. Equation (12a) is a direct consequence of (5a). Equations (12b) and (12c) result, respectively, from applying a covariant derivative to (5b) and (5c), and using the second Bianchi identity. The wave equation for d a bcd , on the other hand, requires to consider the alternative conformal field equation (10). Applying ∇ e to the latter and using equation (5d) along with the first Bianchi identity, a long but straightforward calculation yields equation (12d).

Remark 8.
It is possible to eliminate terms containing L ab from the wave equation (12d) through the generalisation of an identity obtained in [16] to the case of tracefree matter. Multiplying equation (10) by Ξ, using the definitions of d a bcd and * T abc , equation (5c) and the second Bianchi identity to simplify it, one has: Applying a further covariant derivative ∇ g to the last expression and making use of equations (5a), (5d) and (10) as well as the properties of the rescaled Cotton tensor, the following identity is obtained: Substituting this into expression (12d) we get an alternative wave equation for d a bcd : Remark 9. In concrete applications it may prove useful to express the Schouten tensor in terms of the tracefree Ricci tensor and the Ricci scalar through the formula As it will be discussed in Section 6.1, the Ricci scalar R is associated to the particular choice of conformal gauge. Thus, the decomposition (16) allows to split the field L ab into a gauge part and a part which is determined through the field equations. Keeping the simplicity of presentation in mind, except for Section 7, we do not pursue this approach further as it leads to lengthier expressions.

Zero-quantities and integrability conditions
In this section we consider a convenient setting for the discussion and book-keeping of the evolution equations implied by the conformal Einstein field equations with tracefree matter. Our approach is based on the observation that the conformal Einstein field equations constitute an overdetermined system of differential conditions for the various conformal fields. Thus, the equations are related to each other through integrability conditions -i.e. necessary conditions for the existence of solutions to the equations.

Definitions and basic properties
In this section we define the set of geometric zero-quantities (also called subsidiary variables) associated to the system of tracefree conformal Einstein field equations (5a)-(5e): In terms of the above, the conformal Einstein field equations (5a)-(5e) can be expressed as the conditions Υ ab = 0, Θ a = 0, ∆ abc = 0, Λ abc = 0, from where these fields take their name.

Properties of the zero-quantities
By definition, the zero-quantities possess the following symmetries: In particular, one can check that ∆ abc and Λ abc satisfy the identities which are useful for simplifying certain combinations of zero-quantities. Moreover, direct calculations and the second Bianchi identity show that they satisfy further identities: In the following, it will result useful to introduce a further auxiliary zero-quantity associated to equation (10) -see Remark 6: Here the second equality has been obtained through a calculation similar to the one yielding (10). From the above definition it follows that Λ ab d cd = Λ abc and that it has the symmetries

Some consequences of the wave equations
Key for our subsequent analysis is the observation that if the wave equations for the conformal fields (12a)-(12d) are satisfied, then the geometric zero-quantities satisfy further identities. These are summarised in the following lemma: Lemma 2. Assume that the wave equations (12a)-(12d) and Assumption 1 hold, then the geometric zero-quantities satisfy the identities Proof. The result follows directly from the definitions of the zero-quantities with the aid of the wave equations for the conformal fields (12a)-(12d), the second Bianchi identity and the properties of the rescaled Cotton tensor.

Integrability conditions
The zero-quantities are not independent of each other but they are related via a set of identities, the so-called integrability conditions. These relations are key for the computation of a suitable (subsidiary) system of wave equations for the zero-quantities. The procedure to obtain these relations is to compute suitable antisymmetrised covariant derivatives of the zero-quantities which, in turn, are expressed in terms of lower order objects. Following this general strategy we obtain the following: Proof. The result follows from direct calculations employing the definitions of the zero-quantities, the rescaled Cotton tensor and the first Bianchi identity.
Remark 10. Observe that these relations have right-hand sides consisting of lower order expressions which are homogeneous in the zero-quantities. This property will be relevant when suitable wave equations for these fields are derived in the next section. Equations (24a)-(24d) together with (23e) constitute the set of integrability conditions for the zero-quantities associated to the tracefree conformal Einstein field equations.

The subsidiary evolution system for the zero-quantities
An important aspect of any hyperbolic reduction procedure for the (conformal) Einstein field equations is the identification of the conditions upon which a solution to the (reduced) evolution equations implies a solution to the full set of field equations -this type of analysis is generically known as the propagation of the constraints. In practice, the propagation of the constraints requires the construction of a suitable system of evolution equations for the zero-quantities associated to the field equations.

Construction of the subsidiary system
In this section it is shown how the set of integrability conditions provides a systematic and direct way to obtain wave equations for the zero-quantities -a so-called subsidiary evolution system. The propagation of the constraints then follows from the structural properties of the subsidiary system as a consequence of the uniqueness of solutions to systems of wave equations.
Regarding Θ a , an analogous calculation using expression (24b) in conjunction with the same equations as in the previous case leads directly to a wave equation for this field. Exploiting (5c), (6d) and (24a) to simplify it one obtains: A wave equation for Λ abc can be obtained by applying ∇ d to integrability condition (24c), commuting derivatives and using (23d) to eliminate the second order derivatives. A direct but long calculation exploiting the same relations used in the previous two cases as well as (5d) and (21) yields: A wave equation for Z is readily found by direct application of ∇ a to equation (24d):

Wave equation for Λ abc
Notice that the integrability condition for Λ abc , (23e), contains derivatives of zero-quantities on both sides of the equation. This feature seems to hinder our standard approach for the construction of a subsidiary equation. Thus, in order to construct a suitable wave equation it will be necessary to exploit the symmetries of Λ abc . Applying ∇ d to the integrability condition (23e), identities (19) and (20c) lead to Observe that in the previous equation the last term on the right-hand side puts at risk the hyperbolicity of the system. However, we can exploit the antisymmetric properties of Λ abc -see equation (22). Writing it as its antisymmetric part with respect to the second and third indices a commutator naturally appears. Proceeding in this fashion a short further calculation results in

Summary
The results of this section can be summarised in the following lemma: Lemma 3. Assume that the conformal fields satisfy the wave equations (12a)-(12d). Then, the zero-quantities (17a)-(17e) associated to the conformal Einstein field equations with tracefree matter satisfy the system of geometric wave equations (25)-(29), which is homogeneous in the zero-quantities and their first derivatives.

Propagation of the constraints
As it will be discussed in detail in Section 6, the system of geometric wave equations (25)-(29) implies, in turn, a system of proper (hyperbolic) wave equations for which a theory of the existence and uniqueness of solutions is readily available -see e.g. [13]. From the latter one directly obtains the following result: Proposition 3. Assume that the zero-quantities Υ ab , Θ a , Λ abc , ∆ abc , Z and their first derivatives vanish on a fiduciary spacelike hypersurface S ⋆ of an unphysical spacetime (M, g ab ). Then the zero-quantities vanish on the domain of dependence D(S ⋆ ) of S ⋆ .
Remark 11. Working, for example, with coordinates adapted to the hypersurface S ⋆ , it can be readily checked that the completely spatial part of of the zero-quantities Υ ab , Θ a , Λ abc , ∆ abc and Z encodes the same information as the conformal Einstein constraint equations -see e.g. [18], Chapter 11. Similarly, projections with a transversal (i.e. timelike) component can be read as a first order evolution system for the geometric conformal fields. Thus, in order to ensure the vanishing of the zero-quantities on the initial hypersurface S ⋆ , one needs, firstly, to produce a solution to the conformal constraint equations -this ensures the vanishing of the spatial part of the zero-quantities. Secondly, one reads the transversal components of the zero-quantities as definitions for the normal derivatives of the conformal fields which can be readily be computed from the solution to the conformal constraints. In this way, the transversal components of the zero-quantities vanish a fortiori.

Gauge considerations
The purpose of this section is to discuss the gauge freedom inherent to the conformal Einstein field equations and the associated evolution equations. This gauge freedom is of two types: conformal and coordinate. We discusse these in turn.

Conformal gauge source functions
An important feature of the conformal Einstein field equations is that the Ricci scalar R of the metric g ab can be regarded as a conformal gauge source specifying the representative in the conformal class [g] of the (conformal) unphysical metric. Given two conformally related metrics g ab and g ′ ab such that g ′ ab = ϑ 2 g ab , the respective Ricci scalars are related to each other via If the values of R and R ′ are prescribed, the above transformation law can be recast as a wave equation for the conformal factor relating the two metrics. More precisely, one can write This wave equation can always be solved locally given suitable initial data. Thus, one can always find (locally) a conformal rescaling such that the metric g ′ ab has a prescribed Ricci scalar R ′ . Remark 12. Following the previous discussion, in what follows the Ricci scalar of the metric g ab is regarded as a prescribed function R(x) of the coordinates and we write R = R(x).

Generalised harmonic coordinates and the reduced Ricci operator
The components of the Ricci tensor R ab can be explicitly written in terms of the components of the metric tensor g ab in general coordinates x = (x µ ) as and where one defines the contracted Christoffel symbols as A direct computation then gives Following the well-known procedure for the hyperbolic reduction of the Einstein field equations, we introduce coordinate gauge source functions F µ (x) to prescribe the value of the contracted Christoffel symbols via the condition so that the coordinates x = (x µ ) satisfy the generalised wave coordinate condition -see e.g. [5,17,18]. Associated to the latter, it is convenient to define the reduced Ricci operator More explicitly, one has that Thus, by choosing coordinates satisfying the generalised wave coordinates condition (30), the unphysical Einstein equation (11) takes the form Assuming that the components L µν are known, the latter is a quasilinear wave equation for the components of the metric tensor.

The reduced wave operator
The geometric wave operator acting on tensorial fields contains derivatives of the Christoffel symbols which, in turn, contain further first order derivatives of the components of the metric tensor. The presence of these second order derivative terms is problematic as they destroy, in principle, the hyperbolicity of evolution systems involving equation (32) since they enter in the principal part of the system. However, as discussed in e.g. [16,3], the generalised wave coordinate condition (30) can be used to reduce the geometric wave operator to a proper second order hyperbolic operator.
Direct manipulations show that for a covector ω a with components ω µ with respect to a coordinate system x = (x µ ) satisfying condition (30) one has that Making the formal replacements in equation (33), one defines the reduced wave operator , acting on the components ω µ as where f λ (g, ∂g, ω, ∂ω) denotes lower order terms whose explicit form will not be required. In fact, from the previous discussion it follows that one can write An analogous discussion for covariant tensors of arbitrary rank leads to the following: Definition 1. The reduced wave operator acting on a covariant tensor field T λ···ρ is defined as where ≡ g µν ∇ µ ∇ ν . The action of on a scalar φ is simply given by Remark 13. The operator provides a proper second order hyperbolic operator for systems which involve the metric as unknown -in contrast to . Accordingly, when working in generalised harmonic coordinates, all the second order derivatives of the metric tensor can be removed from the principal part of geometric wave equations.

Summary: gauge reduced evolution equations
The discussion of the previous sections leads us to consider the following gauge reduced system of evolution equations for the components of the conformal fields Ξ, s, L ab , d abcd and g ab with respect to coordinates x = (x µ ) satisfying the generalised wave coordinate condition (30): Remark 14. The reduced system of evolution equations (35a)-(35d) is a system of quasilinear wave equations for the fields Ξ, s, L µν , d µνλρ and g µν . More explicitly, one has that where X, S, F µν , D µνλρ and G µν are polynomial expressions of their arguments. Strictly speaking, the system is a system of wave equations only if g µν is known to be Lorentzian. The basic existence, uniqueness and stability results of systems of the above type have been given in [13] -these results are the second order analogues of the theory developed in [14] for symmetric hyperbolic systems. The basic theory for initial-boundary value problems can be found in [4,7].
Remark 15. A system of evolutions equations expressed in terms of the reduced wave operator (rather than in terms of the geometric wave operator ) will be said to be proper.

Propagation of the gauge
This section is devoted to study the consistency of the conformal and coordinate gauge introduced in Section 6 by constructing a system of homogeneous wave equations for a set of subsidiary fields. This aims to extend the analysis in Section 5 of [3] for the vacuum case.

Basic relations
Consider a set of coordinates x = (x µ ). Let g µν denote the components of a metric g ab in these coordinates. Similarly, R µν denotes to the components of the associated Ricci tensor R ab , while R is the corresponding Ricci scalar. Then, we now investigate the requirements for R and R µν to coincide, respectively, with R(x) and R µν . In addition, we also need to investigate the conditions under which Φ µν corresponds to the symmetric tracefree part of R µν -see equation (16). This can be expressed as the vanishing of the following gauge subsidiary fields: In the following we make the following assumption: Assumption 2. Let T µν and T µνλ be, respectively, the components of a tracefree energy momentum tensor with vanishing divergence and its associated rescaled Cotton tensor. Let g µν and Φ µν , with Φ µ µ = 0, be solutions to the following equations: As a direct consequence of equation (37a) one can find that the gauge zero-quantities (36a)-(36c) are not independent of each other. Simple calculations yield Furthermore, equation (31) and Definition 1 lead to Remark 16. Equations (38a)-(38b) show that if Q µ = 0 then Q and Q µν automatically vanish.
In this sense, we will consider Q µ as the basic gauge zero-quantity of the system.

The gauge subsidiary evolution system
In this subsection we obtain a system of homogeneous wave equations for the gauge subsidiary variables. This will be achieved via exploiting the properties of the so-called Bach tensor which will play the role of an integrability condition for the system.

The Bach tensor
Let g ab be a 4-dimensional metric. The Bach tensor is defined as: From this definition it is easy to verify that B ab is symmetric and tracefree. Additionally, it satisfies the following identity, independently of the Einstein field equations: Remark 17. A straightforward calculation shows that the Bach tensor can be expressed in terms of the geometric zero-quantities as It follows from this that if g ab is a solution to the conformal field Einstein equations then the Bach tensor does not vanish unless T ab = 0.

Wave equations for the gauge subsidiary variables
The Bach tensor can be conveniently expressed in terms of the gauge subsidiary quantities. Terms containing R µν and R can be rewritten according to definitions (36a) and (36c) along with (38a) and (39a). This results in: An expression for Φ µν can be obtained combining (37b) and (39b). Notice also that this term is the only one containing contributions from the matter component. Direct substitution yields where we have grouped the terms depending on T µν as Commuting covariant derivatives and making further suitable substitutions, a lengthy calculation gives In the following we introduce the auxiliary field Taking the divergence of the last equation and after some direct manipulations, equations (38a)-(38b) and (41) imply that where Q and Q stand, respectively, for Q a and Q, and for simplicity H ν represents a homogeneous function of its arguments. On the other hand, we can rewrite the term ∇ µ N µν -which encodes the matter contributions-in a suitable way. Using the symmetries of T abc in conjunction with equations (36c), (39a) and the geometric zero-quantities we obtain so the wave equation for M µ takes the schematic form Lastly, a wave equation for Q is required to close the system. This can be obtained by direct application of the operator along with the aid of equations (36a), (38b) and (39a), resulting in Remark 18. The gauge subsidiary evolution system, equations (42)-(44), is homogeneous in M a , Q a , Q, Υ ab , Λ abc and their first derivatives.
The previous discussion leads to the following result: Lemma 4. Assume that the quantities M µ , Q µ , Q, Υ µν and Λ µνλ along with their first covariant derivatives vanish on a fiduciary hypersurface S ⋆ . Then the unique solution to the system (42)-(44) on an enough small slab of S ⋆ corresponds to Q = 0, Q µ = 0 and M µ = 0, which in turn implies that Q µν = 0.

Evolution equations for the matter fields
Having settled the analysis of the geometric part of the conformal Einstein equations with tracefree matter, we now proceed to investigate the evolution and subsidiary equations associated to a number of suitable matter models: the conformally coupled scalar, Maxwell and Yang-Mills fields.

The conformally coupled scalar field
It is well-known that the equation∇ a∇ aφ = 0, whereφ is a scalar field, is not conformally invariant. This deficiency can be healed by the addition of a term involving the coupling with the Ricci scalar -namely, a direct computation shows that equation (45) implies In what follows, for convenience, equation (45) will be know as the conformally coupled wave equation -or conformally invariant wave equation. The energy-momentum associated to the wave equation (46) is given by so that ∇ a T ab = 0 holds if equation (46) is satisfied. It can be readily verified that T ab , as given by the expression above, is tracefree.

Remark 19.
Observe the presence of the Schouten tensor L ab in (47). Moreover, notice that T ab involves second derivatives of the scalar field. This, in turn, leads to third order derivatives of the matter field in the expression of the rescaled Cotton tensor -cf. equation (6d). This difficulty will be addressed in the sequel.
Remark 20. The conformally coupled scalar field is related to the standard scalar field satisfying the wave equation through a transformation originally due to Bekenstein [1]. Thus, in principle, the theory for the conformally coupled scalar developed in this section can be rephrased in terms of the standard scalar field.

Auxiliary fields and the evolution equations
We start by observing that the third order derivative terms in the expression of the rescaled Cotton tensor for the conformally invariant scalar field are of the form ∇ [a ∇ b] ∇ c φ. Using the commutator of covariant derivatives, these terms can be transformed into first order derivative terms according to the formula Thus, one is left with an expression for the Cotton tensor containing, at most, second order derivatives.
In order to eliminate second order derivative terms in the rescaled Cotton tensor which, potentially, could destroy the hyperbolic nature of the wave equations (35a)-(35d), one needs to promote the first and second derivatives of φ as further (independent) unknowns. Accordingly, we define Following the previous discussion, and exploiting equation (5c), one can write the rescaled Cotton tensor for the conformally coupled scalar field as Next, suitable evolution equations for φ a and φ ab will be obtained by constructing a set of integrability conditions for these fields. Firstly, the identity ∇ a φ b = ∇ b φ a represents an integrability condition for φ a . A wave equation then readily follows after applying ∇ b and using equation (46): On the other hand, an integrability condition for φ ab can be obtained directly from its definition: Then, applying ∇ c to this relation and using equations (5c), (5d), (46) and (50), a straightforward calculation leads to: Remark 21. In equation (51) it is understood that the rescaled Cotton tensor T bca is expressed in terms of the auxiliary fields φ a and φ ab according to (49) and thus it does contains not second or higher derivatives of the fields.
Remark 22. When coupling the wave equations (46), (50) and (51) to the system (35a)-(35d) satisfied by the geometric conformal fields it is understood that the geometric wave operator is replaced by its reduced counterpart as discussed in Section 6.2.1.

Subsidiary equations
To verify the consistency of our approach in dealing with the higher order derivative terms in the rescaled Cotton tensor for the conformally invariant scalar field we introduce the following zero-quantities: (52b) Now we proceed to construct suitable wave equations for these fields.
A wave equation for Q a can be obtained in a straightforward way. Applying to the definition (52a) and with the help of relations (46) and (50) a short calculation yields: Similarly, applying to equation (52b), commuting covariant derivatives and using the definitions of the geometric zero-quantities one obtains Remark 23. Observe that the system of wave equations (53) and (54) is homogeneous in Q a , Q ab , ∆ abc and Λ abc . Thus, it follows from general uniqueness results for solutions to wave equations that if these quantities and their derivatives vanish on an initial hypersurface S ⋆ , then necessarily Q a = 0 and Q ab = 0 at least on a small enough slab around S ⋆ .

Summary
The analysis of the conformally coupled scalar field can be summarised in the following manner: Proposition 4. The system of equations (35a)-(35d) with rescaled Cotton tensor given by (49), together with the conformally coupled wave equation (46) and the auxiliary system (50)-(51) written in terms of the reduced wave operator consitute a proper system of quasilinear wave equations -see Remark 15.

The Maxwell field
The Maxwell equations expressed in terms of the antisymmetric Faraday tensorF ab are given bỹ The associated energy-momentum tensor of the Maxwell field takes the form which can be verified to be trace and divergencefree.
It is well-known that the Maxwell equations are conformally invariant. Defining an unphysical Faraday tensor F ab as F ab ≡F ab , it follows that the physical Maxwell equations (55a)-(55b) imply that with the associated unphysical Maxwell energy-momentum tensor given by Alternatively, defining the Hodge dual F * ab of the Faraday tensor as the second unphysical Maxwell equation (56b) can be written as:

Auxiliary field and the evolution equations
A geometric wave equation for the Faraday tensor can be obtained from differentiation of the Maxwell equation (56b), which represents a natural integrability condition for this field. Commuting covariant derivatives and applying equation (56a) a short calculation results in From equation (6d) it follows that the rescaled Cotton tensor contains first derivatives of F ab . This puts at risk the hyperbolicity of the system (35a)-(35d). In order to deal with this problem we introduce the auxiliary variable satisfying F abc = F a [bc] . By virtue of equation (56b), it also follows that F [abc] = 0. In terms of this quantity, it can be readily checked that the rescaled Cotton tensor for the Maxwell field takes the form From definition (61) it follows that F abc possesses two independent divergences: ∇ a F abc is simply the right-hand side of wave equation (60) whilst the other is given by as a direct calculation confirms. In order to obtain an integrability condition for F abc consider the expression 3∇ [d F |a|bc] . Commuting covariant derivatives and using the first Bianchi identity for the Weyl tensor, a straightforward calculation results in: A geometric wave equation can be obtained applying ∇ d to the last expression and commuting derivatives. Using equations (8), (5c), (5d), (10), (63) as well as the symmetries of d abcd and T abc to simplify it, a long but direct calculation yields: This equation can be further simplified via a pair of observations. Firstly, multiplying equation (13) by F dg the following auxiliary identity is found: Secondly, from equation (10) we have the following relations: Combining them we readily obtain the identity Making use of the identities (66) and (67), the wave equation for F abc takes a simpler form: As remarked in the case of the conformally invariant scalar field, the geometric operator is replaced by when equations (60) and (68) are coupled to the system (35a)-(35d).

Subsidiary equations
In order to complete the discussion of the Maxwell field it is necessary to construct suitable evolution equations for the zero-quantities Here M abc possesses the symmetries Also, one can verify the following identities: Remark 24. Following the spirit of the discussion of the previous section, the zero-quantities M a and M abc encode Maxwell equations (56a) and (56b), respectively, while Q abc does so for the auxiliary equation (61).
Equation for M a . Observe that equation (71b) works as an integrability condition for M a . Applying ∇ b and using (71a) one obtains where in the second equality the definition of F * ab and equation (69b) have been used. From this definition it can be easily checked that M * a is divergencefree which, in turn, implies an integrability condition. More explicitly, we have Applying ∇ d to (74), commuting derivatives and noticing that ∇ [a ∇ |d| M bc] d vanishes by virtue of equation (71b) and the first Bianchi identity, a straightforward calculation leads to Equation for Q abc . A wave equation for the field Q abc will be obtained by direct application of the operator. Employing definitions (69a), (69c), along with equations (17c), (17d), (60) and (68) one obtains the expression In order to show that the terms not containing zero-quantities vanish, first observe that the first Bianchi identity implies that On the other hand, multiplying definition (21) by F de , a short calculation yields the auxiliary identity From the last two expressions it follows then that Remark 25. Geometric wave equations (72), (75) and (77) are crucially homogeneous in M a , M abc , Q abc and Λ abc . Thus, if these quantities and their first covariant derivatives vanish on an initial hypersurface S ⋆ it can be guaranteed that there exists a unique solution on an enough small slab of S ⋆ and it corresponds to M a = 0 and M abc = 0 and Q abc = 0.

Summary
The previous discussion about the coupling of the Maxwell field with the conformal Einstein field equations can be summarised as follows: Proposition 5. The system of wave equations (35a)-(35d) with rescaled Cotton tensor given by (62) together with the wave equations (60) and (68) written in terms of the wave operator is a proper quasilinear system of wave equations for the Einstein-Maxwell system.

Yang-Mills field
The Yang-Mills field is the last example of a tracefree matter model we study in this paper. Due to its similarities with the Faraday field analysed in the previous subsection, some of the calculations will not be showed in their totality as they are analogous to the aforementioned mater model. However, one of the features of the Yang-Mills field is that in order to obtain a hyperbolic reduction of the equations one needs to introduce a set of gauge source functions fixing the divergence of the gauge potential. The consistency of this gauge choice will be analysed towards the end of the section.

Basic equations
The Yang-Mills field consists of a set of gauge potentialsÃ a µ and gauge fieldsF a µν where the indices a, b, . . . take values in a Lie algebra g of a Lie group G.
The equations satisfied by the fieldsÃ a µ andF a µν are: denote the structure constants of the Lie algebra g. The structure constants satisfy the Jacobi identity In addition, the gauge fields satisfy the Bianchi identitỹ Conformal invariance. The Yang-Mills equations are well-known to be conformally invariant. More precisely, defining the unphysical fields F a ab ≡F a ab , A a a ≡Ã a a , a direct computation under the conformal rescaling (3) renders the unphysical Yang-Mills equations with energy-momentum tensor given by Finally, it will result useful to introduce the dual of F a ab defined as: Remark 26. Due to the form of the energy-momentum tensor given in (80), first derivatives of F a ab will appear in the rescaled Cotton tensor, putting at risk the hyperbolicity of the system (35a)-(35d). As in the case of the Maxwell field, this will make necessary the introduction of an auxiliary quantity.

Evolution equations for the Yang-Mills fields
Suitable wave equations for the Yang-Mills fields can be obtained by a procedure analogous to the one used for the Maxwell field. Accordingly, in order to cast such relations as wave equations, we introduce the auxiliary field F a abc ≡ ∇ a F a bc .
In addition, the construction of a geometric wave equation for the Yang-Mills gauge potentials requires the introduction of gauge source functions f a (x) depending in a smooth way on the coordinates and fixing the value of the divergence of the potential. More precisely, in the following we set ∇ a A a a = f a (x).
Wave equation for the field strength. The Yang-Mills Bianchi identity, equation (79c), represents an integrability condition for the field strength tensors F a ab . Differentiating it and making use of the identity (78) along with the Yang-Mills equations (79a)-(79c), a straightforward calculation results in Equation for the gauge potential. Equation (79a) provides a natural integrability condition for the gauge potential field. After applying ∇ b , commuting derivatives and using equation (79b) one arrives to: Equation for the auxiliary field. A suitable wave equation for the field F a abc can be obtained from its definition. Using this and equation (79c), some manipulations yield: Proceeding as in the case of the wave equation for F a abc , as well as using the Jacobi identity and definitions (87a)-(87d) a lengthy calculation results in: Similar to the two previous matter models, when the equations (84)-(86) are coupled to the system of wave equations for the geometric fields, the operator is to be replaced by its counterpart .

Subsidiary equations
The next step in the analysis of the Yang-Mills field is the introduction of the corresponding subsidiary quantities and the consequent construction of suitable geometric wave equations for them. Define the following set of zero-quantities: Notice that, unlike the Maxwell field analysis, an additional field M a ab must be considered due to the introduction of the gauge potential A a a . Combining (87c) and (87d) an auxiliary relation is directly obtained-namely From these definitions, it follows that M a abc and M a ab possess the symmetries Furthermore, direct calculations show that the Yang-Mills zero-quantities satisfy the relations . Commuting covariant derivatives, substituting expressions (87c), (87d) and exploiting the Jacobi identity for the structure constants one obtains the integrability condition Applying ∇ c to the last equation a short calculation using equations (90a) and (90c) yields: Equation for M a a . Equation (90c) constitutes an integrability condition for the field M a a . A suitable wave equation can be obtained by first applying ∇ c , commuting derivatives and observing that ∇ c ∇ a M a b ac = ∇ [c ∇ a] M a b ac . Then, using definitions (87a)-(87d) along with (90a), (90b), (91), the Jacobi identity, and an appropriate substitution of (88), a long but straightforward computation results in: Equation for M a abc . In a similar fashion to the approach adopted for the Maxwell field zeroquantity M abc , and in order to simplify the calculations, we introduce the Hodge dual of M a abc : Here, the second equality has been obtained with help of definition (81) along with (87c). With this expression we compute the divergence of M * a a . Making use of (87b) and the Jacobi identity, a calculation yields: In terms of non-dual objects this takes the form of an integrability condition: Then, a suitable wave equation can be obtained applying ∇ d and commuting derivatives. After a long calculation in which definitions (87a)-(87d), equations (88)-(91) and the Jacobi identity are employed, one finds that: Equation for Q a abc . Similar to the case for the Maxwell field, a wave equation for Q a abc can be obtained applying the operator directly to its definition. Since the identity used in the deduction of equation (77) has the same form for the Yang-Mills strength field, a completely analogous procedure can be followed. A long computation gives:

Propagation of the gauge
In this subsection we show the consistency of the introduction of the gauge source functions f a (x) into the analysis of the propagation of the constraints for the Yang-Mills potential. For this purpose we introduce the zero-quantity P a defined as: The computation of a wave equation for this field is straightforward: first, a short calculation employing equations (85), (87a), (87b) and (90b) gives From here, application of a further covariant derivative results directly in Remark 27. Geometric wave equations (92), (93), (96), (97) and (99) are homogeneous in M a a , M a ab , M a abc , Q a abc , P a , Λ abc and their first covariant derivatives. Then, if these fields vanish on an initial hypersurface S ⋆ , it can be guaranteed that there exists a unique solution on an enough small slab of S ⋆ and this solution is the trivial one.

Summary
The previous discussion about the Yang-Mills field coupled to the conformal Einstein field equations leads to the following statement: Proposition 6. The system of wave equations (35a)-(35d) with energy-momentum tensor given by (80) coupled to wave equations (84)-(86) written in terms of the operator is a proper quasilinear system of wave equations for the Einstein-Yang-Mills system.

Applications
The purpose of this section is to provide a direct application of the analysis of the evolution systems and subsidiary equations associated to the conformal Einstein field equations coupled to tracefree matter. Arguably, the simplest applications of our analysis to a problem of global nature is that of the existence and stability of de-Sitter like spacetimes. The original stability result of this type, for vacuum perturbations, was carried in [8]. For the sake of conciseness of the presentation and given that the key technical details have been discussed in the literature -see e.g. [18], Chapter 15-here we pursue a high-level presentation in the spirit of [11].
In order to present the result, it is recalled that one of the key features of the conformal Einstein field equations is that they are regular up to the conformal boundary. This property is also satisfied by the conformally coupled scalar field equation, the Maxwell equations and the Yang-Mills equations. Thus, they admit initial data prescribed on spacelike hypersurfaces describing the conformal boundary of spacetime. In an analogous way to the Einstein field equations, the metric conformal Einstein field equations admit a 3+1 decomposition with respect to a foliation of spacelike hypersurfaces. The equations in this decomposition which are intrinsic to the spacelike hypersurfaces are known as the conformal Einstein constraint equations -see e.g. [18], Chapter 11. When evaluated at a spacelike hypersurface representing the conformal boundary of a de Sitter-like spacetime, these equations simplify considerably and a procedure to construct the solutions to these equations is available -see [18], Proposition 11.1 for the vacuum case; this result can be generalised to include tracefree matter models. From the geometric side, the freely specifiable data in this construction is given by the intrinsic metric of the conformal boundary and a TT-tensor prescribing the electric part of the rescaled Weyl tensor. The initial data obtained by this type of construction will be known as asymptotic de Sitter-like initial data. The component of the conformal boundary where the asymptotic de Sitter-like data is prescribed can be either the future or the past one. In the following, for convenience, we restrict the discussion to the case of the past component of the conformal boundary.
For asymptotic initial data sets of the type described in the previous paragraph one has the following result: Theorem 1. Consider (past) asymptotic de-Sitter initial data for the Einstein field equations with positive Cosmological constant coupled to any of the following matter models: (i) the conformally coupled scalar field, (ii) the Maxwell field, (iii) the Yang-Mills field.
Then one has that: (a) The initial data determines a unique, maximal, globally hyperbolic solution to the Einstein field equations which admits a smooth de Sitter-like conformal future extension.
(b) The set of initial data sets leading to developments which admit smooth conformal extensions to both the future and past is an open set (in the appropriate Sobolev norm) of the set of asymptotic initial data.
Proof. We only provide a sketch of the proof as the strategy is similar to the one followed in the proof of the stability of the Milne spacetime in [12]. A version of the proof which uses first order symmetric hyperbolic systems can be found in [18], Chapter 15, for symmetric hyperbolic systems.
The first main observation is that the conformal representation of the (vacuum) de Sitter spacetime in terms of the embedding into the Einstein cylinder gives rise to a solution to the conformal Einstein field equations. Coordinates (x) = (t, x) can be chosen so that the two components of the conformal boundary are located at t = ± 1 2 π. For this representation the Ricci scalar takes the value −6 and the conformal factor is given byΞ = cos t. In the following we denote byů this solution to the conformal equations and byů ⋆ its restriction to the hypersurface t = − 1 2 π which corresponds to the past conformal boundary I − . We will look for solutions to the conformal evolution equations of the form u =ů +ȗ with initial data given by u ⋆ =ů ⋆ +ȗ ⋆ . The fieldsȗ andȗ ⋆ describe the (nonlinear) perturbations. Substituting this into the evolution equations one obtains a system of quasilinear equations for the components ofȗ which can be written schematically as g µν (x) +g µν (x;ȗ) ∂ µ ∂ νȗ = F (x;ȗ, ∂ȗ). (100) In the above expressiong µν denote the components of the contravariant metric on the Einstein cylinder. The above equation is in the form for which the local existence and Cauchy stability theory of quasilinear wave equations as given in, say, [13] applies. Initial data for the system (100) is of the form (ȗ ⋆ , ∂ tȗ⋆ ). The size of the initial data is encoded in the expression (ȗ ⋆ , ∂ tȗ⋆ ) S 3 ,m ≡ ȗ ⋆ S 3 ,m + ∂ tȗ⋆ S 3 ,m where S 3 ,m denotes the standard Sobolev norm of order m ≥ 4 on a manifold which is topologically S 3 . If the initial data (ȗ ⋆ , ∂ tȗ⋆ ) is sufficiently small then the contravariant metric on I − given byg µν (x ⋆ ) +g µν (x ⋆ ;ȗ ⋆ ) is Lorentzian -this property is preserved in the evolution. Now, as the background solutionů is well defined and smooth on the whole of the Einstein cylinder -in particular, up to t = π for which one has thatΞ| t=π = −1. It follows from the Cauchy stability statements in [13] that if (ȗ ⋆ , ∂ tȗ⋆ ) S 3 ,m is sufficiently small then the solution will exists up to t = π. By restricting, if necessary, the size of the data one has that Ξ| t=π = −1 +Ξ t=π < 0.
From the above observation it can be argued that the function Ξ =Ξ +Ξ over the Einstein cylinder becomes zero on a spacelike hypersurface which lies between the times t = 0 and t = π. This hypersurface corresponds to the future conformal boundary (I − ) arising from the data (ȗ ⋆ , ∂ tȗ⋆ ) on I − .
Once the existence of a global solution to the evolution system has been established, one makes use of the uniqueness of solutions to systems of quasilinear wave equations to prove the propagation of the constraints. To this end one observes that if the initial data satisfies the conformal constraints at the past conformal boundary I − , then a calculation shows that the zero-quantities and their normal derivatives also vanish on I − . As the subsidiary evolution system is homogeneous on the zero-quantities, it follows that its unique solution must be the trivial (i.e. vanishing) one. Thus, one has obtained a global solution to the conformal Einstein field equations. From the general theory of the conformal Einstein field equations -see e.g. Proposition 8.1 in Chapter 8 of [18]-this solution implies, in turn, a solution to the Einstein field equations with positive Cosmological constant having de Sitter-like asymptotics.
Remark 28. The above theorem is a global stability result for the de Sitter spacetime under perturbations involving a conformally coupled scalar field, a Maxwell field or a Yang-Mills field as (trivially) the de Sitter spacetime can be constructed from asymptotic initial data. Thus, for a suitably small neighbourhood of asymptotic de Sitter data all data in the neighbourhood give rise to global solutions.
Remark 29. The cases (ii) and (iii) -the Maxwell and Yang-Mills fields, have been studied using first order symmetric hyperbolic systems in [9]. However, the case (i) -the conformally coupled scalar field-has, hitherto, not been considered in the literature.
Remark 30. The theory in [13] is the analogue for systems wave equations of the theory for symmetric hyperbolic systems developed in [14]. A version of the key existence and Cauchy stability result in [13] given in the form used in Theorem 1 can be found in the Appendix of [12].

Concluding remarks
The global existence and stability result presented in Theorem 1 is the simplest application of the analysis of the second order conformal evolution equations developed in this article. A further application is to the construction of anti-de Sitter-like spacetimes with tracefree matter models following the strategy implemented in [3] -this construction will be presented elsewhere [2]. The theory developed in this article should also allow to obtain matter generalisation of the existence results for characteristic initial value problems considered in [6].
More crucially, the analysis in this article should also pave the road for numerical simulations of spacetimes with tracefree matter in the conformal setting. The use of the metric conformal Einstein equation in conjunction with a coordinate gauge prescribed in terms of generalised wave condition provides a formulation of the evolution equations for the conformal fields which can be regarded as a (unphysical) reduced Einstein equation with (unphysical) matter described by the conformal factor, Friedrich scalar, Ricci tensor and the rescaled Weyl tensor. Viewed in this way, one can readily adapt the pletora of numerical know-how that has been developed in the numerical simulations of the Einstein field equations. A further discussion of this idea can be found in [10].