The extended Conformal Einstein field equations with matter: the Einstein-Maxwell field

A discussion is given of the conformal Einstein field equations coupled with matter whose energy-momentum tensor is trace-free. These resulting equations are expressed in terms of a generic Weyl connection. The article shows how in the presence of matter it is possible to construct a conformal gauge which allows to know \emph{a priori} the location of the conformal boundary. In vacuum this gauge reduces to the so-called conformal Gaussian gauge. These ideas are applied to obtain: (i) a new proof of the stability of Einstein-Maxwell de Sitter-like spacetimes; (ii) a proof of the semi-global stability of purely radiative Einstein-Maxwell spacetimes.


Introduction
The Einstein conformal field equations are a powerful tool to prove statements concerning the stability of vacuum spacetimes -see e.g. [5]. These methods have been extended to deal with the case of the gravitational field coupled to the Maxwell and Yang-Mills fields [7]. In [8] a more general version of the vacuum conformal field equations has been developed. These extended conformal field equations are written in terms of a Weyl connection. The extra gauge freedom incorporated in this representation of the equations allows the construction of gauge systems based on conformal structures of the spacetime. As it so often happens in this type of considerations, a judicious gauge choice based on geometrical considerations can greatly simplify the analysis in question. An example of the gauge choices that can be employed are the conformal Gaussian gauge systems introduced in [8] -see as well [10,11]. The extended conformal field equations in conjunction with conformal Gaussian systems have been used, among other things: to provide an existence proof of anti-de Sitter spacetimes [8]; to construct a representation of spatial infinity allowing for a regular finite initial value problem at spatial infinity [9]; to provide a new proof of the global stability of the de Sitter spacetime and the semi-global stability of Minkowski spacetime [16]; and to provide a semi-global stability result of purely radiative vacuum spacetimes [17].
The common feature in the applications described in the previous paragraph is that one is, ultimately, concerned with solutions to the vacuum Einstein field equations (with or without a cosmological constant). A key feature of conformal Gaussian systems in vacuum spacetimes is that the conformal geodesics upon which they are constructed render a canonical conformal factor which provides a priori knowledge about the location of the conformal boundary of the spacetime. This property is, however, lost if one considers conformal geodesics on non-vacuum spacetimes. In this article we show it is possible to get around this difficulty if one considers a more general class of conformal curves to construct gauge systems. As in the case of conformal geodesics in vacuum spacetimes, the conformal curves provide again a canonical conformal factor which is known prior to evolution.
As an application of the ideas described in the previous paragraph, in this article we will consider initial value problems for spacetimes (M,g µν ) with cosmological constant λ satisfying the Einstein-Maxwell field equations where∇ denotes the Levi-Civita connection of the metricg µν ,R µν ,R are the associated Ricci tensor and Ricci scalar, andF µν denotes the Maxwell tensor -the conventions for the geometric quantities used above, will be set out in detail in Section 2. The discussion of the solutions to equations (1a)-(1b) will be carried out in terms of a conformally rescaled, unphysical metric g µν related to the physical metricg µν according to The gauge systems based on the new class of conformal curves are used to provide a new (simpler) proof of the existence and stability of Einstein-Maxwell de Sitter-like spacetimes. We also provide a stability proof of purely radiative Einstein-Maxwell spacetimes. These particular applications lead us to consider the extended Einstein conformal field equations with matter. To the best of our knowledge, this is the first time these equations are considered. As a simplifying technical assumption, our general considerations will be restricted to matter models with a trace-free stress-energy tensor -a property satisfied by the electromagnetic field. Although the particular examples to be considered are only concerned with the Einstein-Maxwell equations, most of our discussion can be adapted to trace-free perfect fluids (sometimes also called conformal fluids) -this will be discussed in future work.

Outline of the article
We start by summarising our conventions and the basic ideas behind the notion of conformal rescaling in Section 2. The conventions follow closely those used in references [16,17]. Section 3 presents a brief review of the notion of Weyl connection and the transformation formulae for the connection and the Schouten tensor. Section 4 gives the formulation of the extended conformal field equations with matter in both a frame and a spinorial formalism. Section 5 introduces the concept of conformal curves and the associated generalised conformal Gaussian systems. These gauge systems are instrumental in our subsequent analysis as combined with the extended conformal field equations, they render hyperbolic reductions for which the location of the conformal boundary is know a priori. Section 6 provides a discussion of the procedure of hyperbolic reduction for the geometric part of the extended conformal field equations in generalised Gaussian systems. Section 7 is concerned with the matter part of the field equations, which in the case under consideration is given by the Maxwell field. Section 8 summarises the key structural properties of the evolution equations implied by the conformal field equations with a view to applications involving existence and stability results. Section 9 discusses the so-called propagation of the constraints. Section 10 is concerned with the first application of the methods developed in the article: a new proof of the stability of Einstein-Maxwell spacetimes which have a global structure similar to that of the de Sitter spacetime. Finally, Section 11 provides a second application: a stability result for Einstein-Maxwell radiative spacetimes. This result generalises the analysis for the purely vacuum case carried out in [17].

Basics and conventions 2.1 The curvature of the physical spacetime manifold
Throughout this article we work with a spacetime (M,g µν ), whereg µν , (µ, ν = 0, 1, 2, 3) is a Lorentzian metric with signature (+, −, −, −). We will denote by∇ the Levi-Civita connection ofg µν -that is, the unique torsion-free connection that preserves the metricg µν . As in the introduction, letR µνλρ ,R µν andR denote, respectively, the Riemann curvature tensor, the Ricci tensor and the Ricci scalar of the Levi-Civita connection∇. The conventions for the curvature used in this article are such that For the Riemann tensor one has the decompositioñ whereC µ νλρ denotes the conformal Weyl tensor ofg µν , while the trace parts are given in terms of the Schouten tensor defined byP In terms of an arbitrary stress-energy tensorT µν , it is given bỹ

Conformal rescalings
Letg µν and g µν be two Lorentzian metrics which are conformally related according to equation (2). Let [g] denote the conformal class ofg µν . Two invariants of the conformal class are the tensor and the conformal Weyl tensorC µ νλρ = C µ νλρ . The Levi-Civita covariant derivative of the metric g µν will be denoted by ∇. In the sequel, it will be convenient to consider a frame e k , k = 0, 1, 2, 3 which is orthonormal with respect to the metric g µν . That is, g(e i , e j ) = η ij .
In what follows frame components are always taken with respect to the frame e k . In particular, ∇ i ,∇ i will denote the covariant derivatives in the direction of e i . Let Furthermore, let Γ i j k ,Γ i j k denote the connection coefficients of ∇,∇ with respect to the frame e i . One has that An analogous decomposition to that of equation (4) holds for the Riemann tensor R µνλρ .

Weyl connections
In this article, we will also consider connections∇ (not necessarily Levi-Civita) which respect the conformal structure of the conformal class [g], in the sense that for some 1-forms b µ and f µ . One has that We shall write the above equations aŝ The fact thatg µν and g µν are assumed to be conformally related implies The Riemann and Ricci tensors of the Weyl connection∇ are defined in an analogous way to (3) and will be denoted byR µνλρ andR µν respectively. The analogue of the decomposition (4) is given byR where the Schouten tensor of∇, denoted byP µν , is given bŷ Alternatively, the latter decompositions could have been written using the physical metricg µν .
Transformation rules between the curvature tensors of the Weyl connection∇ and the Levi-Civita connections∇, ∇ can be found in [10]. Important for the subsequent discussion is the transformation rule for the Schouten tensor. This is given bỹ A similar expression holds between the tensors P µν andP µν by replacing b µ with f µ , namely: Finally, we introduce the Cotton-York tensor associated to the connection∇ The physical Cotton-York tensor can be expressed in terms ofT µν andT µνg Remark. It should be noted that above the definitions and decompositions are invariant under conformal rescaling. Nevertheless, when raising indices or applying contractions we have explicitly written out the metric to avoid ambiguity. In the sequel, a frame formalism will be used throughout. This choice will remove the ambiguity as the frame metric will always be η ij . Consistent with equation (7) the metric g µν and its inverse will be used throughout for raising and lowering tensorial indices.

The extended conformal field equations with matter
The idea of vacuum conformal Einstein field equations expressed in terms of the Levi-Civita connection ∇ of a conformally rescaled metric g µν and associated objects was originally introduced in [2,3,4]. The generalisation of these conformal equations to physical spacetimes containing matter was discussed in [7]. More recently, a more general type of vacuum conformal equations -the extended conformal Einstein field equations-expressed in terms of a Weyl connection∇ has been introduced -see [8]. In this section we discuss how these extended conformal field equations can be modified to discuss spacetimes with matter.

Frame formulation
As in the previous section, let e k denote a frame which orthogonal with respect to the metric g µν so that equation (7) holds. In order to discuss the extended conformal Einstein field equations, it will be convenient to depart slightly from the point of view taken in the previous section and regard, for the moment, the connection ∇ only as a metric connection with respect to g µν -i.e. ∇ λ g µν = 0. Under this assumption, the connection ∇ could have torsion, and thus it would not be a Levi-Civita connection. The connection coefficients Γ i k j of ∇ with respect to the frame e k are defined by the relation ∇ i e j = Γ i k j e k . As a consequence of having a metric connection the connection coefficients satisfy If Σ i k j = 0 so that the connection ∇ is the unique Levi-Civita connection of g µν , the connection coefficients acquire the additional symmetry Now, given the connection coefficients Γ i k j of a metric connection as above and a 1-form f µ , one can define a further connectionΓ i k j using the relation -cfr. (9b). LetΣ i j k denote the torsion of the connection∇. It follows directly that so that∇ will not be a Weyl connection unlessΣ i k j = 0.
In our subsequent discussion it will be convenient to distinguish between the geometric curvaturer k lij -i.e. the expression of the curvature related to the connection coefficientsΓ i j k -and the algebraic curvatureR k lij -i.e. the decomposition of the curvature in terms of irreducible components. One has that For ease of the subsequent discussion we introduce the following zero quantities: The interpretation of the zero quantities (15a)-(15e) is as follows: the zero quantity given by (15a) measures the torsion of the connection∇; that of (15b) relates the expression of the curvature of∇ with its decomposition in terms of irreducible components; equation (15c) is contraction of (15b) over the first two indices. It is included here for later convenience. Equations (15d) and (15e) measure the deviation from the fulfilment of the Bianchi identity.
The extended conformal Einstein field equations with matter are then given bŷ These equations yield differential conditions for the frame coefficients e i , the spin coefficientsΓ i j k , the components of the 1-form f i , the components of the Schouten tensorP ij , and the Weyl tensor C k lij , respectively. The latter need to be complemented with the energy-momentum conservation equation∇ iT ij = 0, whose particular details will depend on the matter model under consideration.
Note that in equations (16), the 1-form b k relating∇ and∇ remains unspecified. In the sequel it will be convenient to introduce the variables In terms of the latter, the last two conformal field equations then read: As in the case of b k , the newly introduced function Θ and the 1-form d k remain unspecified at this stage. They will latter be fixed by the choice of a suitable conformal gauge.
Remark 1. If the extended conformal field equations (16) are satisfied then the frame e k can be used to construct a metric g µν via the relation (7). The connection coefficientsΓ i j k give rise to a torsion-free connection, so that the connection Γ i j k given by (12) is the Levi-Civita connection of g µν . Consequently,Γ i j k defines a Weyl connection with conformal Weyl tensor given by C k lij and Schouten tensorP ij . Showing that the solution so obtained implies a solution to the Einstein field equations requires bringing into consideration gauge conditions. This will be discussed together with the propagation of the constraints in section 9.
Remark 2. As a consequence of the transformation rules for the Schouten tensor (10a)-(10b) and (11a)-(11b), the zero quantities (15a)-(15e) involved in the extended conformal field equations (16) transform covariantly (i.e. homogeneously) under a change in the conformal gauge. Thus, if they are satisfied in one gauge, then they are satisfied in all gauges.

Spinorial formulation
In the sequel we will make use of a spinorial version of the extended conformal field equations (16). The use of this type of representation leads to simplifications, in particular, when obtaining a reduced system of propagation equations. However, we will switch to a frame representation whenever it is more convenient for the discussion.
The connection between the components of a tensor with respect to an orthonormal basis and its spinorial counterpart is realised by the constant Infeld-van der Waerden symbols. In particular, let e AA ′ ,∇ AA ′ , d AA ′ denote, respectively, the spinorial counterparts of e i ,∇ i , d i . Furthermore, let Γ AA ′ BB ′ CC ′ ,Γ AA ′ BB ′ CC ′ denote, respectively, the spinorial counterpart of the connection coefficients Γ i j k ,Γ i j k . As the connection defined by Γ i j k is assumed to be metric, it follows that one can write The symmetry condition on the last pair of indices of the spin connection coefficient encodes the assumption of having a metric connection. For the spin Weyl connection coefficients one haŝ Letr AA ′ BB ′ CC ′ DD ′ denote the spinorial counterpart of the geometric curvaturer k lij . For future use we note the Ricci identity: valid for any spinor µ AA ′ and whereΣ CC ′ EE ′ DD ′ is the spinorial counterpart of the torsion. In our conventions, the geometric and algebraic curvature tensor of a general Weyl connection satisfŷ Their spinorial counterpart can be decomposed aŝ In terms of the reduced spin coefficientsΓ AA ′ BC one writes the geometric curvature (assuming that the torsion vanishes) as

The uncontracted spinorial conformal field equations
The spinorial version of the zero quantities (15a)-(15e) is given bŷ In terms of these spinorial zero quantities, the extended conformal field equations are given bŷ

The contracted spinorial conformal field equations
Equations (20a)-(20c) are antisymmetric upon interchange of a pair of indices. This structural property will be used to obtain a contracted version of the equations which will be systematically used in the sequel. The associated zero quantities are given by and their complex conjugate versions. Above the symmetries of the spinorial counterparts of d klij andỸ ijk have been exploited by writing Using the latter formulae equation (20d) reduces to its more usual form: The extended conformal field equations can be expressed in terms of these contracted zero quantities as: It is important to remark that the contracted equations (23) are fully equivalent to (21).

Conformal curves and generalised conformal Gaussian gauge systems
The advantage of considering extended conformal equations in terms of Weyl connections is that they allow to consider gauge systems based on conformally invariant objects. An example of these gauge systems are the conformal Gaussian systems introduced in [8,11]. These gauge systems are based on conformal geodesics. Conformal Gaussian systems are of great utility in the discussion of evolution problems for the conformal field equations as they provide a canonical conformal factor as well as structural simplifications in the form of the evolution equations.
It was shown in [8,11] that for vacuum spacetimes the conformal factor is quadratic in the conformal time and can be read of from the initial data of the evolution system. Accordingly, the location of the conformal boundary is know a priori. This predetermined character of the conformal factor hinges crucially on the fact that the physical spacetime is vacuum. In the sequel we show that the conformal geodesic equations in the presence of matter can be modified in such a way that one has again a conformal factor known a priori.

A class of conformal curves
Let I ∈ R be an open interval. We will consider a class of conformal curves, x µ (τ ), whose tangent vector v µ ≡ẋ µ is coupled to a 1-form b ν via the equations Equations (24a)-(24b) will be supplemented with a frame propagation equation viã In a slight abuse of terminology, we will call a triple (v µ , b ν , e µ k ) solving equations (24a)-(24b) and (26) a conformal curve, since these curves exhibit the conformally invariant behaviour described in the following lemma.
, e µ k (τ )) satisfies (24a), (24b) and (26) expressed in the connection∇ =∇ + S(h). In particular, in terms of the Weyl connection given by∇ =∇ + S(b) the conformal curve equations take the form Moreover, conformal curves are preserved as point sets under reparametrisations of τ by fractional linear transformations.
Remark. Comparing (10a)-(11b) and (25) we can see that A vector frame satisfying (26) is called Weyl propagated. It is noted that the velocity can be chosen as one of the frame vectors due to (24a). Suppose along a conformal curve we define Letting g µν be given as in (2), one finds that g(v, v) = 1. In fact, for a frame e µ k the frame metric is constant along the curve. Hence a g-orthonormal frame evolves into a g-orthonormal frame along the curve. Following an analogous discussion for conformal geodesics given in [11] one differentiates (28) twice along the curves and substitutes (24a) and (24b) to obtain Note that forH µν = λg µν the right hand side vanishes exactly. Thus, the following result holds: is a solution to the conformal curve equations (24a), (24b) and (26) with respect to the metricg µν such that x(τ ) is a timelike curve inM defined on some open interval I. Ifg µν satisfies the Einstein equations with matter then: ii) the conformal factor Θ is given for τ ∈ I by where a quantity with a subscript * is constant along x(τ ).
For vacuum spacetimes one hasJ µν = 0 and one recovers known results for conformal geodesics -see e.g. [8,11]. In the presence of matter the conformal curves curves given by equations (24a)-(24b) are no longer conformal geodesics. However, our choice ofH µν gives the same behaviour for the canonical conformal factor as in the vacuum case. We will also require the following result: ) is a conformal curve as in Lemma 2. Letg µν satisfy the Einstein field equations with matter andH µν = λg µν .
The proof of this result is a calculation analogous to the one described in the proof of Lemma 3.2 in [8]

Jacobi fields for conformal curves
Suppose we are given a congruence of conformal curves with velocity v. The separation vector η satisfies [v, η] = 0 along each conformal curve and will be referred to as a Jacobi field. Recall that Weyl connections are torsion-free, so that When the Jacobi field becomes tangent to the curve at a point p we say that p is a conjugate point. These points are of interest to us for the following reason. If we create Gaussian coordinate system by dragging spatial coordinates along the congruence beyond a conjugate point then these are ill-defined. For this reason we measure |η − g(v, η)v| 2 = (η ab η a η b ) with a, b = 1, 2, 3. As long as this quantity does not vanish our coordinate system will be well-defined.

Generalised conformal Gaussian systems
In analogy to the way conformal geodesics have been used in [8,9,16,17], to construct conformal Gaussian gauge systems, here we will use the conformal curves solving equations (24a)-(24b) and (26) to construct what we will call generalised conformal Gaussian systems.
LetS be a space-like hypersurface in the spacetime (M,g µν ). OnS we choose an initial conformal factor Θ * > 0, a frame field e k * , and a 1-form b * such that and e 0 * is orthogonal toS. For fixedH µν and given x * ∈S there exists a unique conformal curve (x µ (τ ), b ν (τ )), which for τ = 0 passes through x * and which satisfies the initial conditionṡ If all data are smooth, then in some neighbourhood U ∈S these curves define a smooth caustic free congruence covering U. Furthermore, b ν defines a smooth 1-form on U which allows to construct a Weyl connection∇. A smooth frame field e µ k and the related conformal factor Θ are obtained in U by solving the propagation equations (26) and (28) for given initial data onS. Then e µ 0 (τ ) =ẋ µ (τ ) on U and we definê The frame one obtains from solving the propagation equation is orthonormal for the metric g µν = Θ 2g µν , whileΓ 0 j k = 0. Dragging along local coordinates x a onS with the congruence and setting x 0 = τ , one obtains a coordinate system. A coordinate system, a frame field and a conformal factor constructed with the above procedure will be known as a generalised conformal Gaussian system.
We will follow the setup used in [16,17], where we use a global frame field c s (s = 0, 1, 2, 3) constructed from the coordinate vectors in such a way that c 0 = ∂ τ and c r (r = 1, 2, 3) is a constant linear combination of the spatial coordinate vectors. The frame e k then is written in terms of its expansion e k = e s k c s . The same is done for the spinorial version.
Remark. Above we have set up a generalised conformal Gaussian system in the physical spacetime (M,g µν ). However, due to the conformal invariance of conformal curves proven in Lemma1 the same gauge can also be constructed starting from an spacelike hypersurfaceŠ in a conformally related spacetime (M,ǧ µν ) such thatS ⊂Š,M ⊂Š. The initial data will be related in the obvious way, with Θ * and b * changing accordingly while the frame remain the same. We will make us of this fact later on when constructing a generalised conformal Gaussian system from hyperboloidal data given on S in the unphysical spacetime (M, g µν ).
Due to Lemma1 a generalised conformal Gaussian system is characterised on U by the explicit ρσg ρσg µν one has, by virtue of Lemma 2, that the solution of the evolution of the conformal factor is known a priori. In the sequel we will only consider trace-free matter (the Maxwell field). Consequently, The latter implies ∇ µ J µν = 0 if one defines J µν ≡ Θ −2J µν .

Hyperbolic reductions of the conformal field equations
In this section we discuss how to extract a symmetric hyperbolic system of propagation equations. For this, we resort to a space-spinor formalism -see e.g. [21]-based on the spinorial counterpart, τ AA ′ , of a timelike vector τ µ in the conformally rescaled spacetime (M, g µν ). More precisely, the vector τ µ will be taken to be parallel to the tangent vector v µ to the conformal curves described in Section 5. The normalisation condition τ µ τ µ = 2 will be used.
The reduced symmetric hyperbolic system of evolution equations is to be deduced from the following contractions of the conformal field equations together with

The space spinor formalism in brief
In what follows, we will consider spin dyads {δ A } for which the spinor τ AA ′ admits the decomposition In particular, one has that Using the spinor τ AA ′ , the gauge conditions (34) can be rewritten as whereJ BB ′ denotes the spinorial counterpart ofJ ij τ i .
The spinor τ AA ′ can also be used to obtain an unprimed version of the spinorial Weyl connection covariant derivative∇ AA ′ . More precisely, one has that: The latter, in turn, can be decomposed in its irreducible parts: whereP The differential operatorD AB is the so-called Sen connection of∇ relative to the vector field τ AA ′ .

Hyperbolic reduction of a first model equation
The Procedure and subtleties of deriving hyperbolic equations from the extended conformal equations (21) will be illustrated with a model equation.
Let M iK and N ijK = N [ij]K be two tensorial quantities, where K stands for any set of tensor or bundle indices, satisfying the equation Two derive an evolution equation we contract with τ i : Note that in our setup the connection coefficients in the expression vanish due to our gauge choice. However, the following analysis is valid without this condition and it should be observed that these terms have a polynomial form. We note that M 0K appears inside the second term. However, we can not obtain an evolution equation for M 0K from equation (6.2) since setting j = 0 in order to get the evolution equation for M 0K makes both sides reduce trivially to zero due to the skew symmetry in ij. Instead, M 0K must be determined from the symmetries of M iK (if any) or if not, then it must be regarded as free data.
The subsequent discussion of the properties of the model equation (6.2) will be carried out in the spinor formalism. From the spinorial version of (6.2) one obtains the contracted versionŝ In order to change to space spinor components it is observed that one has to contract with τ A ′ A inside the derivative. One uses that Thus, from (41a)-(41b) one obtainŝ Using the decomposition (39) for∇ AB and writing M ABK as one obtains: Making linear combinations of the latter equations one finally arrives at: The terms E ABK introduced on the right hand side are formed from the original term N ABK and connection coefficients. It will be seen that in their explicit form they are polynomial in the variables of our system. We will write the original variable, here M , in square brackets and the variables E and C are used to indicate whether the term is for the evolution or the constraint equation. (ii) m K cannot be reexpressed in terms of the m ABK in which case the former is regarded as free data which has to be specified by means of a gauge choice. This may lead to a transport equation for m ABK .
On the other hand, (42b) is a constraint equation for the components m ABK which one expects to hold at latter times if satisfied initially -the so-called propagation of the constraints. Note that, a priori, there are no constraints for m K .

Hyperbolic reduction of a second model equation
In the discussion of the propagation of the constraints a different type of model equation will be considered. In what follows we will briefly discuss its hyperbolic reduction.
Let M ijK = M [ij]K and N kijK = N [kij]K denote two tensorial quantities, where again K stands for any set of spinor indices. The model equation to be considered is given bŷ This type of equation is motivated by the observation that if ω is a 2-form, then its Lie derivative with respect to a vector field τ µ is given by where i τ ω denotes the contraction of the 2-form ω with τ µ . Now, if i τ ω = 0 (as it is the case with the the zero quantities associated with the extended conformal field equations), one finds that In what follows, let ǫ ijkl denote the components with respect to the frame e k of the volume form of the metric g µν . Now,∇ Because of the connection with the Lie derivative, it follows then that implies an hyperbolic equation for the tensorial field M ijK . The relevance of this equations to prove the propagation of the constraints depends on whether its right hand side can be casted as an homogeneous expression of other zero quantities -see Section 9.

The reduced geometric equations
Following the discussion of the model equation (6.2) in the previous section one introduces the unprimed spinorial fields e s AB ,Γ ABCD andP ABCD defined by where H (AB)CD contains quadratic terms inΓ (AB)CD andP (AB)CD and in d AB and φ ABCD . Their explicit form will not be important for our subsequent discussion.
As it will be discussed in Section 7.5, the spinorỸ ABCD for the case of the Einstein-Maxwell system contains derivatives of the Maxwell field. These terms enter in the principal of equation (44d). This feature requires us to introduce new field equations -essentially, the covariant derivative of the Maxwell spinor. The termD ABJCD in equation (44d) will lead to similar problems, for it will be seen thatJ CD is quadratic in the Maxwell spinor.
For the evolution of the Jacobi field we split its space spinor into irreducible components Then we get the evolution equations

The reduced Bianchi equation
In order to construct an evolution equation for the Weyl spinor φ ABCD we consider the zero quantityΛ Again, using the decomposition (39) one obtainŝ renders the desired reduced equation. In equation (47) we notice again the presence of the term Y (ABCD) so that the same problem arises as for the reduced equation (44d). We will thus treat this term in the same way as outline for equation (44d), in order to ensure the symmetric hyperbolicity of the system. We observe the presence of the potentially singular term Θ −1 . However, as will be seen in the sequel, this term is cancelled out by a a similar term appearing in the explicit form ofỸ ABCD .
Finally, it is noticed that the remaining content of the zero quantity Λ ABCD is contained in corresponding to the constraints associated to the Bianchi identity (20d).

The spinorial Maxwell equations
Up to this point our discussion has been completely general and irrespective of the trace-free matter models under consideration. In order to proceed further, explicit information about the matter model has to be provided -in our case the Maxwell field.

The Maxwell equations in the physical spacetime
The physical spacetime Maxwell equations (1c) are equivalent to the spinorial equatioñ where the antisymmetric Maxwell tensorF µν and the totally symmetric spinorφ AB are related to each other by the correspondencẽ The energy-momentum tensor (1b) is given in spinorial terms bỹ

The Maxwell equations in the unphysical spacetime
If upon the conformal rescaling (2) one imposes the transformation rule then one obtains that In terms of the Weyl connection∇ one has that For later use we define the space spinor For more details on the Hermitian conjugation map for space spinors, see e.g. [8].
With regards to the stress-energy tensor one has that with This last property follows from the trace-freeness property of T AA ′ BB ′ in four dimensions [7]. As a consequence of this discussion, the following zero quantity is introduced: In the sequel it will be seen that in order to obtain a symmetric hyperbolic reduction of the conformal Einstein-Maxwell equations, it is necessary to introduce the derivatives of the Maxwell field as a variable. For this, one considers for a given gauge choice a spinorial field ψ AA ′ BC =ψ AA ′ (BC) and an associated zero quantityω AA ′ BC . These two quantities are related byω and under a connection change (9b) the spinorial fieldψ AA ′ BC is adapted aŝ The zero quantityω AA ′ BC will be handled in the sequel as a constraint. In order to obtain an equation forψ AA ′ BC we adopt the strategy used in [7] and make use of the Ricci identity -cfr. equation (19)-for the Weyl connection∇ applied to the spinor φ AB : Replacing the derivatives of φ AB byψ AA ′ BC and assuming that the conformal equations (21) are satisfied one obtains the required equation: To the latter we associate the following zero quantity: Remark. It can be verified from (49) and (53) that the zero quantitiesω A ′ B ,ω A ′ ABC ,ω AA ′ BB ′ CC ′ transform homogeneously upon changes of conformal gauge. This observation is of relevance for the propagation of the constraints.

The reduced Maxwell equations
The reduced equations implied by the Maxwell equations (50) is handled in a similar way to the Bianchi identity (20d) -one considers the unprimed zero quantitŷ from where one obtains the propagation equation and the constraint ω Q Q =D P Q φ P Q − f P Q φ P Q = 0.

The reduced equations for the derivatives of the Maxwell spinor
The treatment of the equation associated with the zero quantity ω AA ′ BB ′ EF follows the model discussed in Section 6.2. In particular, one has to consider the contracted zero quantity and its complex conjugate. The procedure described in Section 6.2 then leads tô whereψ ABCD is the space spinor version ofψ AA ′ BC given bŷ The source term H

[ψ]
ABCD contains quadratic terms involving φ ABCD and ψ ABCD , φ AB andP ABCD and φ AB and φ ABCD . Some of parts of the quadratic expression involving φ AB andP ABCD lead to terms cubic in φ AB . As in the case of the reduced equations (44a)-(44d), the explicit form of the source H

[ψ]
ABCD will not be required. The reduction procedure described in the previous lines does not provide an evolution equation for the componentsψ P P CD . To get around this, we writê where ν ABCD ≡ψ (AB)CD , ν CD ≡ψ P P CD . Let also ν ≡ψ P Q P Q .
It follows then that Now, assuming thatω AB = 0,ω AA ′ BC = 0 so thatψ ABCD =∇ AB φ CD and the Maxwell equations hold, one finds that In particular, the relation (59b) allows us to express the full content of the fieldψ ABCD in terms of ν (ABCD) , ν AB and ν P Q P Q -the term ν P (AB) P being redundant. Substituting the decomposition (58) into equation (57) one obtains: Using equation (59b) one finds that Using equation (44b) to replacePf QB by H

[ψ]
AB and equation (56) to expressPφ CQ in terms of ψ ABCD and a quadratic expression in f AB and φ AB it follows from (60b) that where H

′[ψ]
BC contains the terms appearing in H BC as well as a linear combination of the terms appearing in H AB with terms quadratic in f AB andψ ABCD and terms cubic in f AB and φ AB . Equations (60a), (60c) together with (61) constitute a symmetric hyperbolic system of equations for the components ν (ABCD) , ν AB and ν of the spinorial fieldψ ABCD .

The decomposition of the physical Cotton-York tensor
In vacuum spacetimes the physical Cotton-York tensorỸ ijk vanishes. Thus, it does not appear in the Bianchi equations (15e)-(18b). In the case of trace-free matterỸ ijk is, in general, nonvanishing and carries information about the physical fields in both equations (15e) and (18b). The fieldỸ ijk can be written in terms of unphysical variables as: Substituting equations (50) and (52), recalling that η AA ′ BB ′ ≡ ǫ AB ǫ A ′ B ′ is the spinorial counterpart of η ij and assuming thatψ AA ′ BC =∇ AA ′ φ BC i.e.ω AA ′ BC = 0 one obtains From the latter one readily finds that The corresponding unprimed versionỸ ABCD can be written entirely in terms of φ AB , φ † AB , Θ,Θ and d AB , by recalling thatφ These explicit expressions will not be required in the subsequent discussion. Important to note is that due to the presence of an overall factor of Θ 2 inỸ ABCC ′ , the term Θ −1Ỹ ABCD in equation (47) is formally regular at the points where Θ = 0.

Behaviour of the field variables and the zero quantities under gauge changes
Before discussing the structural properties of the reduced conformal Einstein-Maxwell equations in Section 8 and the propagation of the constraints in Section 9 we would like to briefly highlight the topic of gauge choice and gauge invariance. The field variables and the zero quantities that have been introduced in previous chapters have all been defined for a specific choice of Weyl connection∇, metric g µν , frame {e k } and spinor dyad δ A related by (7), (8), (9b). Implied in their definitions are the transformation rules under gauge change. These rules have either been explicitly given -e.g. (11b), (49), (53) -or can be derived directly from these rules and the extended conformal field equations (16). Therefore we refrain from listing them again.
However we would like to highlight that as a particular consequence of these transformation rules, it follows that the various zero quantities defined in earlier chapters are conformally covariant. Thus if they vanish in one gauge they will also vanish in another. This will be used in the discussion of the propagation of the constraint in Section 9.

Structural properties of the reduced conformal Einstein-Maxwell equations
We summarise the analysis of Sections 6 and 7 in a form suitable for the applications that will be given in the sequel.
We introduce the notation where it is understood that υ, φ, ϕ and ψ contain only the independent irreducible components of the respective spinors. Let also u ≡ (υ, φ, ϕ, ψ) In terms of these quantities the propagation equations (44a)-(44d) can be written as: where K denotes a matrix with constant coefficients, Q(υ, υ), R(ϕ, ψ) bilinear vector value functions with constant coefficients and T(φ, ψ, υ) a trilinear vector valued function with constant coefficients. On the other hand, L is a linear matrix-valued function with coefficients depending on the coordinates. Equations (47), (56) and (60a)-(60c) can be written in the form imply real symmetric matrices if one decomposes the entries of φ, ϕ and ψ into real and imaginary parts. Hence, (62) and (63a) -(63c) give rise to a symmetric hyperbolic system for u.

Propagation of the constraints
In this section we show that the conformal constraint equations propagate by virtue of the conformal evolution equations, thus implying a solution to the whole conformal field equations. More precisely, Assume that the unknowns (υ, φ, ϕ, ψ) given on U represent a smooth solution of the reduced equations (62), (63a), (63b) and (63c) for data on V satisfying the Einstein-Maxwell conformal constraint equations. Let g µν be the metric for which the frame obtained from the unknowns υ is orthonormal and let D + (V) ⊂ U be the future domain of dependence of V with respect to g µν . Then the conformal Einstein-Maxwell field equationŝ are satisfied on D + (V) by the fields (υ, φ, ϕ, ψ). Furthermore, the metric g µν = Θ −2 g µν is a solution to the (physical) Einstein-Maxwell field equations on The proof of this result follows a combination of the techniques discussed in [7] and in [8]. We divide the proof in several steps. In order to ease the presentation, in the subsequent discussion we will use tensorial notation whenever possible. The discussion of the propagation of the constraints follows the lines of the arguments given in [7,8]. This argument requires long computations to obtain a complicated system of subsidiary equations for the various zero quantities involved in the extended conformal field equations. Since the argument is not particularly illuminating and for the sake of the presentation, we follow the spirit of previous sections of the article and present a schematic description of the procedure. In addition, we provide an alternative argument for the propagation of the constraints based on the local existence results of [7].
(a) Propagation of the constraints as a consequence of the subsidiary equations.
In the subsequent discussion it will be assumed that the reduced conformal field equations (35)-(36) are satisfied. Furthermore, it will be assumed that the gauge conditions (34) hold.
We define the following zero quantities associated to the conformal gauge: whereT ij is given by the matter model under consideration. Under the assumption that equations (35)-(36) and (34) are satisfied, a computation along the lines discussed in [8] shows that k is an homogeneous expression in the zero-quantities δ k , γ ij , ς ij andΣ i k ; H ij is an homogeneous expression in γ ij ; finally H

[ς]
ij is homogeneous inΞ k lij . The explicit form of the source terms in the above equations and the evolution equations for the other zero quantities to be discussed in the sequel will not be required in the following discussion.
The discussion of the propagation equations for the zero quantitiesΣ i j k ,Ξ k lij and∆ kij follow the model of equation (43) discussed in Section 6.3. In this case a lengthy computation shows that where H [Σ] i j k is a homogeneous expression in the zero-quantitiesΣ i k andΞ k lij ; H [Ξ]k lij is an homogeneous expression on the geometrical equations zero-quantitiesΞ k lij ,∆ kij ,Σ i j k , Λ kij and the gauge zero-quantity δ k ; finally H [∆] kij is homogeneous in the geometrical equations zero-quantities∆ kij ,Σ i k ,Λ kij , the gauge zero-quantities γ ij and δ k , and the tensorial counterpart of the spinorial matter zero-quantitiesω A ′ A ,ω AA ′ BC ,ω AA ′ BB ′ CD .
The construction of a propagation equation equation for the Bianchi equationΛ kij is slightly different. Following the discussion in [8] one considers the quantity∇ kΛ kij . A lengthy manipulation using the definition ofΛ kij and symmetries of the Weyl tensor shows that ij depends homogeneously on the geometrical zero-quantitiesΞ k lij ,Σ i j k , the gauge zero-quantity ς ij , and the tensorial counterpart of the matter zero-quantitiesω A ′ A ,ω AA ′ BC , ω AA ′ BB ′ CD . Now, the spinorial counterpart of∇ kΛ kij is given by∇ P P ′Λ P ′ P BC . A spacespinor decomposition shows that the components ofΛ A ′ ABC satisfy a symmetric hyperbolic equation. In particular, for the Bianchi constraint one has that where H

[Λ]
AB has the same dependence on zero-quantities as H ij .
Finally, for the constraints associated to the matter equations (the Maxwell field) one has that an analogous procedure to the one described in [7] renders also symmetric hyperbolic equations for the components ofω A ′ A ,ω AA ′ BC ,ω AA ′ BB ′ CD which are homogeneous in the matter zero-quantities themselves and in the geometric zero-quantities.
Summarising: in the gauge (38) and as a consequence of the reduced equations the geometrical zero-quantitiesΣ together with the Maxwell zero-quantitieŝ and the gauge zero-quantities form a symmetric hyperbolic system which is homogeneous in the zero quantities themselves. Accordingly, if the zero-quantities vanish on V ⊂ S, then the zero-quantities vanish on D + (V). Hence one has a solution to the conformal Einstein-Maxwell field equations of D + (V).
(a') An alternative argument for the propagation of the constraints.
An argument involving less computations to prove the propagation of the constraints can be obtained by directly exploiting the local existence results of [7]. In what follows we consider the extended conformal field equations (20a)-(20d) in an arbitrary gauge. We notice that if one sets f AA ′ = 0 in these equations, then from equation (17b) it follows that d AA ′ = ∇ AA ′ Θ, and hence, the arbitrary Weyl connection∇ reduces to the associated Levi-Civita connection ∇. In order to obtain the full correspondence with the conformal equations of [7] one has to prescribe equations for the conformal factor Θ and the 1-form d AA ′ . The relevant equations are given by with s ≡ 1 4 ∇ P P ′ ∇ P P ′ Θ + 1 24 R Θ. Equations (64a)-(64c) arise from the transformation rule for the Schouten tensor under conformal rescalings and from the definition of s. It should be noted that these equations are not required in the particular type of hyperbolic reduction considered in this article as both Θ and d AA ′ are fixed by the gauge of Section 5.3 -the generalised conformal Gaussian systems.
From the theory in [7] one has that given initial data satisfying the conformal constraint equations on V ⊂ S, there exists W ⊂ D + (V) in which the "standard" conformal field equations are satisfied. In particular, this implies the existence of a physical spacetime with metricg µν . The solution constructed by the procedure of [7] is unique up to conformal rescalings, coordinate transformations and a choice of frame. For the subsequent discussion we denote the conformal factor, metric and associated Levi-Civita connection thus obtained byΘ,ǧ µν and∇. The metricsǧ µν andg µν are related viǎ µν .
Now, consider on V ⊂ S initial data for the∇-version of the equations (24a)-(24b) to construct a timelike congruence of conformal curves. It follows from standard theorems on the existence of ordinary differential equations that given a set of initial data for these curves, one can always find a smaller subset W ′ ⊂ W such that the congruence is free of conjugate points. Thus, through every point p ∈ U ′ we have a unique conformal curve starting on V. In particular, one can choose initial data for the congruence so that one obtains a generalised conformal Gaussian system like the one described in Section 5. The solution to the∇-version of the conformal curve equations gives a 1-formb µ on W ′ . Furthermore, recall that given initial data on V one can construct a preferred conformal factor0 by solving the appropriate version of equation (28):0 In particular, one may choose as initial data0 * = 1 on V. This choice connects the metrič g µν with the unique metric g µν for which the tangent vectors v µ to the congruence of conformal curves are taken to be orthogonal to S and satisfy g(v, v) = 1.
By construction, the standard conformal field equations, and hence the extended conformal field equations, are satisfied in the gauge of the connection∇. As observed in Sections 3,4 and 7 the zero quantities used in the formulation of the extended conformal field equations are conformally covariant. Thus, if the extended conformal field equations are satisfied in the gauge∇, then they are also satisfied in that given by∇. As a consequence, the reduced conformal field equations associated to the constructed congruence hold. Since the solution to the reduced conformal field equations is unique, it follows that using the above initial data one must obtain the same solution for (62), (63a), (63b) and (63c) as the one constructed above. In particular, one can finally conclude that the constraint equations must be satisfied throughout W ′ .
(b) A solution to the conformal Einstein-Maxwell field equations implies a solution to the physical Einstein-Maxwell field equations.
Assume now that on U one has a solution to the extended Einstein-Maxwell conformal field equations -that is, Assume also that the additional zero quantities satisfŷ The solution to the reduced conformal field equations provides, in particular, fields where in this discussion the 1-form f AA ′ is defined via asΣ AA ′ BB ′ CC ′ = 0. The connection coefficientsΓ AA ′ BB ′ CC ′ give rise to a torsion-free connection∇. Motivated by the relation (7) one can use the frame e AA ′ and the frame metric η AA ′ BB ′ ≡ ǫ AB ǫ A ′ B ′ to construct a metric g µν . By construction e AA ′ (ǫ CD ) = 0 so that∇ Thus,∇ is a Weyl connection for the metric g µν . Motivated by (12), one defines the connection ∇ with connection coefficients then ∇ AA ′ ǫ CD = 0, so that ∇ µ g νλ = 0 -that is, ∇ is a metric connection. Using the invariance of the torsion under change of connection -cfr. equation (13)-it follows that ∇ is torsion free. Thus, because of uniqueness, ∇ must be the Levi-Civita connection of g µν .

Now, fromΞ
ABCC ′ DD ′ ≡r ABCC ′ DD ′ −R ABCC ′ DD ′ = 0, the fieldsP AA ′ BB ′ and φ ABCD on U obtained as a solution of the reduced conformal field equations can be identified, respectively, with the Schouten and Weyl spinors of the Weyl connection∇ -recall that the decomposition in terms of irreducible components is unique. Due to conformal invariance, the Weyl tensor of the Weyl connection∇ is also the Weyl tensor of the Levi-Civita connection ∇.
Motivated by the rescaling (2) we use the transformation rule (9a) to define a physical connection∇. From δ AA ′ = 0 one has that and accordingly∇ is the Levi-Civita connection of the metricg µ ≡ Θ −2 g µν . Using the transformation rule (10b) one finds that the physical Schouten spinor is given bỹ Note that since ς AA ′ BB ′ = 0, one has that Furthermore, from γ AA ′ BB ′ = 0 one finds that From the field equationsω A ′ A = 0,ω AA ′ BB ′ CC ′ = 0 and the constraintω AA ′ BC = 0, one has thatφ AB satisfies the physical Maxwell equations. Thus,T AA ′ BB ′ defined bỹ is the energy momentum tensor of the Maxwell field and the equations given in (65) are equivalent to the Einstein-Maxwell field equations.

A first application: stability of Einstein-Maxwell de Sitter-like spacetimes
The use of a gauge based on the conformal curves described in section 5 allows to directly transcribe the analysis of the conformal boundary for vacuum de Sitter-like spacetimes to the case of Einstein-Maxwell de Sitter-like spacetimes.
For the Sitter-like spacetimes one can formulate two slightly different Cauchy initial value problems: one where initial data is prescribed on a standard Cauchy hypersurface, and a second one where the data is prescribed on one portion of the conformal boundary -say, past null infinity. The de Sitter-like spacetimes that will be considered have Cauchy slices with the topology of S 3 . The construction of suitable coordinate systems and a frame vectors this type of configurations has been discussed in detail in [16,17].

Structure of the conformal boundary
Following the general ideas of [16], here we present a brief discussion of the structure of the conformal boundary of de Sitter-like Einstein-Maxwell spacetimes.

Standard Cauchy problem
If the initial hypersurface S is a standard Cauchy hypersurface one has that for some Θ * = 0. The conformal factor vanishes at One has then that Furthermore, ∇ k Θ∇ k Θ = −2λ, so that both components of null infinity are space-like.

Cauchy problem on past null infinity
In the case of an initial value problem prescribed on null infinity, one has that Θ * = 0 so that Combining (64c) and Lemma 3 one finds that g ♯ (d, d) * = −2λ and, if one sets d * = (∇Θ) * , that The conformal factor vanishes at Note that the location of I + is determined by the free dataΘ * .

Stability of Einstein-Maxwell de Sitter-like spacetimes
Combining the a priori knowledge on the structure of the conformal boundary discussed in the previous sections with the structural properties of the reduced equations (62), (63a), (63b), (63c) discussed in Section 8, Lemma 4 on the propagation of the constraints, and Kato's existence and stability theorems for symmetric hyperbolic systems [13,14,15] one obtains the following existence and stability result for de Sitter-like Einstein-Maxwell spacetimes. The proof is identical to that in [16,17] and it is omitted. Let in what followsů denote the solution to the reduced equations (62), (63a), (63b), (63c) corresponding to the (vacuum) de Sitter spacetime.
Theorem 1. Let u 0 =ů 0 +ȗ 0 be Einstein-Maxwell Cauchy (standard or at past null infinity) data for a de Sitter-like spacetime. There exists ε > 0 such that ifȗ 0 is sufficiently small, then there exists on [τ − , τ + ] × S a unique smooth solution u =ů +ȗ to the conformal propagation equations (62), (63a), (63b), (63c) such that the associated congruence of conformal curves contains no conjugate points in [τ − , τ + ]. The field u implies a smooth solution to the Einstein-Maxwell field equations with positive cosmological constant for which the sets I ± defined by (66) -in the standard Cauchy problem-or by (67) -in the Cauchy problem with data at null infinityrepresent past and future null infinity.
Remark. Note that this stability result for Einstein-Maxwell spacetimes is given with respect to a vacuum reference spacetime.

A second application: stability of Einstein-Maxwell radiative spacetimes
As a second example of our approach, we obtain a generalisation of the stability results for purely radiative spacetimes discussed in [17]. In contrast to the stability proof for de Sitter-like Einstein-Maxwell spacetimes, in this case the reference solution has a non-vanishing electromagnetic field. For the sake of conciseness most of the technical details are omitted and we only remark on those aspects of the analysis that differ from the treatment for vacuum spacetimes given in [17].

Einstein-Maxwell initial data sets with vanishing mass
In what follows, a static solution to the Einstein Maxwell solutions (an electrostatic solution) will be understood to be a triple (h αβ ,Φ,Ψ), solving the electrostatic field equations. The (negative definite) Riemannian 3-metrich αβ is the metric of the quotient manifold, and Φ, Ψ denote, respectively, the gravitational and electric potentials. Any static, asymptotically flat solution to the Einstein-Maxwell equations admits an analytic compactification of a neighbourhood of spatial infinity i -see [20]. The triple (h αβ ,Φ,Ψ) can be suitably rescaled to render another triple (h αβ , Φ, Ψ) which is analytic in a neighbourhood B a (i) and solves the conformal electrostatic field equations. Any such triple gives rise to a solution (h αβ ,Ω,Ē α ) of the (conformally rescaled) time symmetric Einstein Maxwell constraints with vanishing mass and charge -hereD andr denote, respectively, the Levi-Civita connection and Ricci scalar of the metrich αβ ; the tensorĒ α is the electric field. From (h αβ ,Ω,Ē α ) one can construct initial data for the extended conformal field equations. In particular, data for the Schouten and Weyl tensors are given, respectively by the expressions which can be shown to be analytic in B a (i). In these last expressions, C denotes the trace-free part of the tensor in parenthesis.

Construction of a reference radiative Einstein-Maxwell spacetime
Let (h αβ ,Ω,Ē α ) onS be one of the solutions to the time symmetric conformal constraint discussed in the previous subsection. For the present purposes it will be convenient to consider a conformal factorΩ which is negative -this obtained by making the obvious sign changes in the relevant equations. By constructionΩ satisfies the following asymptotic flatness conditions: We work in a suitably small neighbourhood, B a (i) ⊂S such that all the statements made in the sequel make sense. We use the coordinates x A , A = 1, 2, 3 centred at i and consider the following initial data for a congruence of conformal curves: τ * = 0,ẋ µ =n µ ,Θ * =Ω,Θ * = 0,d * ≡Θ * b * = (dΘ) * .
The coordinates x A are extended offS by dragging along the congruence of conformal curves to obtain generalised conformal Gaussian coordinates. It can be readily verified thatΘ > 0 onS. It follows that along each conformal curve the conformal factorΘ is given bȳ Define now the conformal boundary,Ī , in a natural way as the locus of points in the development of the data onS for whichΘ = 0 and dΘ = 0. It is easy to see that a conformal curve with data given by (69) passes throughĪ wheneverτ = ±ω.
Having located the conformal boundary for the evolution of data on B a (i) for the Einstein-Maxwell system, one can discuss now the existence of solutions to the propagation system given by (62), (63a), (63b), (63c). For this we extend the data on B a (i) to data on the whole ofS ≃ S 3 in the way discussed in [5,16]. Using the same methods as in [17] and Lemma 4 one obtains the following local result: The spacetime (M \Ī ,g µν ) implied by the solution to the conformal Einstein-Maxwell field equations is conformally related to an Einstein-Maxwell spacetime spacetime, (M \Ī ,Θ −2g µν ), with vanishing cosmological constant. The spacetime (M \Ī , Θ −2g µν ) is a radiative spacetime for which the setĪ + corresponds to its future null infinity, while the point i + = (0, i) ∈ {0} ×S is its future timelike infinity.
The following is an obvious corollary of theorem 2 -for details of the proof see the analogous construction in [16].

Structure of the conformal boundary
We consider now hyperboloidal data which is "close" in some suitable sense to the hyperboloidal data given by corollary 1. Using analogous arguments to the ones used in [17] one can prove the following result.
As it is customary, let I (null infinity) denote the set of points for which Θ = 0 where the conformal factor is given by (71).

A stability result for purely radiative spacetimes
The information about the conformal boundary of a hypothetical radiative Einstein-Maxwell spacetime arising from hyperboloidal data which is contained in Proposition 1 allows to readily obtain a stability result for a spacetime belonging to the class arising from Theorem 2. The proof of the following result is similar to that in [17] -see also [6,16].
Remark. The purely radiative spacetimes used as reference solutions in our analysis are not perturbations of the Minkowski spacetime. A way of seeing this is to consider the Newman-Penrose constants of the spacetime. The Newman-Penrose constants are a set of absolutely conserved quantities defined as integrals of certain components of the Weyl tensor and the Maxwell fields over cuts of null infinity -see [18,19] and [1] for the Einstein-Maxwell case. In [12] it has been shown that the value of the Newman-Penrose constants for a vacuum radiative spacetime coincides with the value of the rescaled Weyl spinor at i + -this result can be extended to the electrovacuum case using the methods of this article. For the radiative spacetimes arising from the construction of [20] it can be seen that the value of the Weyl spinor at i + is essentially the mass quadrupole of the seed static spacetime. It follows, that the Newman-Penrose constants of the radiative spacetime can take arbitrary values. On the other hand, for the Minkowski spacetime, the Newman-Penrose constants are exactly zero, and those of perturbations thereof will be small. Thus, in this precise sense, our radiative spacetimes are, generically, not perturbations of the Minkowski spacetime, unless all the Newman-Penrose constants vanish.