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Asymptotic simplicity and static data

The present article considers time symmetric initial data sets for the vacuum Einstein field equations which in a neighbourhood of infinity have the same massless part as that of some static initial data set. It is shown that the solutions to the regular finite initial value problem at spatial infinity for this class of initial data sets extend smoothly through the critical sets where null infinity touches spatial infinity if and only if the initial data sets coincide with static data in a neighbourhood of infinity. This result highlights the special role played by static data among the class of initial data sets for the Einstein field equations whose development gives rise to a spacetime with a smooth conformal compactification at null infinity.


Introduction
The analysis of the structure of spatial infinity using the Einstein conformal field equations and the construction known as the cylinder at spatial infinity -see [8,11,19,18,20,21,22]-suggests that static initial data sets play a special role among the class of initial data sets for the vacuum Einstein field equations whose development has a smooth conformal compactification at null infinity. This issue is of fundamental importance in the construction of so-called asymptotically simple spacetimes. In connection with this expectation, the following theorem has been proved in [22]: Theorem 1. Consider a time symmetric initial data set for the Einstein vacuum field equations which is conformally flat near infinity. The solutions to the regular finite initial value problem at spatial infinity is smooth through the critical sets where null infinity touches spatial infinity if and only if the data is exactly Schwarzschildean in a neighbourhood of infinity.
The context of this theorem is better appreciated if one recalls that the Schwarzschild spacetime is the only static spacetime with conformally flat slices -see [11]. The use of conformally flat data sets in the analysis of [22] is a convenient technical assumption. Otherwise, the full complexity of time symmetric initial data sets would make it impossible to identify useful structures. The analysis in [22] builds upon the original analysis in [8] and the computer algebra calculations of [19], to make generic assertions about a certain type of asymptotic expansions for the conformal Einstein field equations made possible by the framework of the cylinder at spatial infinity.
The purpose of the present article is to consider an extension of Theorem 1 to the case of more general -non-conformally flat-time symmetric initial data sets. Again, explicit computer algebra calculations carried out in [20] suggest the type of result that one can expect to prove. A fundamental difficulty in the way of constructing a generalisation to Theorem 1 is to obtain a parametrisation of time symmetric initial data sets for which it is simple to decide whether the data under consideration is static or not.
The properties of time symmetric initial data sets for the vacuum Einstein field equations, (S,h ij ), to be analysed in the present article are best discussed using a point compactification of the initial hypersurface S and an associated conformally rescaled metric h ij -the conformal metric. The 3-manifold S contains singled out points {i 1 , i 2 , . . . } representing the points at infinity of the 3-manifoldS. Our analysis will be local to one of these points, so it will be assumed, without loss of generality, that there is only one of them. This point will be deonted by i. In [6] it has been shown that static initial data sets satisfy a certain regularity condition involving the Cotton tensor and its higher order derivatives -see equation (6) in the main text. This property is, however, not enough to fully assert whether an initial data set is static -initial data sets like those of Misner [14] and Brill-Lindquist [3] satisfy the regularity condition as they are conformally flat, but clearly they do not, in general, give rise to static spacetimes. Further conditions need to be imposed on the initial data to obtain a static development. This gap between initial data sets satisfying the regularity condition and initial data sets which are exactly static at spatial infinity is still to be understood 1 .
The present article gets around the difficulty exposed in the previous paragraph by considering a restricted class of initial data sets for which it is simple to decide whether they are actually static or not. This class of initial data sets is constructed by looking at solutions to the equation giving rise to the conformal factor ϑ relating the 3-metrich ij and h ij , the Yamabe equationsee equation (4) in the main text. In a suitably small neighbourhood of infinity, the solutions to the Yamabe equation can be split into its massless and massive parts. The massless part contains the information of the local geometry in a neighbourhood of i, whereas the massive part contains information of global nature-in particular the mass. The class of initial data sets to be used in the present article takes the solution of the Yamabe equation for static data and adds to it a further massive term that does not contribute to the mass so to obtain a new solution to the Yamabe equation -this can be done because of the linearity of the setting. This new solution to the Yamabe equation implies, in turn, a new solution to the constraint equations in a neighbourhood of infinity with the same conformal metrich ij as a static initial data set. We say that these solutions to the constraint equations have a static massless part. It can be verified that this class of initial data sets satisfies the regularity condition of [6] -cfr. also equation (6) of the main text. This observation is of relevance in the present article as it has been shown in [8] that this condition is necessary for solutions to the regular finite initial value problem of the conformal Einstein field equations to extend smoothly through the critical sets where null infinity touches spatial infinity.
For the class of time symmetric initial data sets for the Einstein vacuum field equations discussed in the previous paragraph one can prove the following generalisation of Theorem 1: Theorem. Given an initial data set with static massless part, the solution to the regular finite initial value problem at spatial infinity is smooth through the critical sets if and only if the data is exactly static in a neighbourhood of infinity.
In other words, the smoothness of the development through the critical sets forces the extra massive part added to the (background) static initial data to vanish. A more precise version of this result will be given in the main text.
As in the case of the assumption of conformal flatness made in [22], the use of initial data sets with a static massless part is a useful technical assumption which allows to identify relevant structures in the conformal field equations. It is clear that not all time symmetric initial data sets admitting an analytic conformal compactification at infinity have a static massless part. A general version of the main theorem of this article can only be obtained once one knows what extra conditions have to be imposed on a generic time symmetric initial data set to have a static massless part -cfr. similar remarks in the previous paragraphs. This task requires learning how to exploit to the maximum extent the conformal gauge freedom implicit in the conformal Ansatz. As the conformal metric, h ij , encodes all the freely specifiable information of a time symmetric initial data set, the extra requirements will have to be in the form of conformally invariant conditions on the conformal class.
The proof of the main theorem builds upon the analysis of the conformally flat case discussed in [22]. This analysis relied heavily on the use of computer algebra methods to transform the transport equations implied by the conformal field equations at the cylinder at spatial infinity into a system of reduced ordinary differential equations for which explicit solutions can be computed for any order of the expansion. The approach in the present article consists in conveniently grouping the various terms appearing in the transport equations in Schwarzschildean terms and deviations-from-Schwarzschild terms. The former are formally identical to terms appearing in [22] and thus, assertions about their smoothness can be readily given. It turns out that most of the terms that one needs to consider are Schwarzschildean terms. It is only in the last step of the argument that deviation terms arise. As it will be seen, their presence indicates that the extra massive part that has been added to the static data has to vanish up to a certain order -thus, putting into action an inductive argument from which the main theorem is obtained. Remarkably, essentially all the computer algebra required for this argument has already been performed in [22].
Outline of the article Section 2 discusses some general properties of time symmetric solutions to the Einstein constraint equations in the conformal setting. It also introduces the class of time symmetric initial data sets that will be used in our subsequent analysis -initial data sets with a static massless part. Section 3 gives a concise summary of the framework of the cylinder at spatial infinity and of the so-called F-gauge. Its main purpose is to introduce the notation to be used in the rest of the article. It also provides an overview of the key properties of the transport equations implied by the conformal Einstein field equations at spatial infinity. Section 4 briefly discusses the key result of the construction of the cylinder at spatial infinity for static spacetimes -namely, that the structures are as smooth as they can be. Section 5 discusses key properties of initial data sets which are static up to a certain order -the results will be used extensively in the sequel. Section 6 contains the core of our analysis: a discussion of the properties of solutions to the transport equations at the cylinder at spatial infinity for data which is static up to a certain order. The results presented in this section take the form of an induction argument which leads, ultimately, to our main theorem in Section 7.

Notation and conventions
The present article is concerned with the asymptotic properties of spacetimes (M,g µν ) solving the Einstein vacuum field equationsR The metricg µν will be assumed to have signature (+, −, −, −) and µ, ν, . . . are spacetime indices taking the values 0, . . . , 3. The spacetime (M,g µν ) will be thought of as the development of a time symmetric initial data set (S,h ij ) whereS is an asymptotically Euclidean hypersurface.
The metrich ij will be taken to have signature (−, −, −). The indices i, j, . . . will be spatial ones taking the values 1, 2, 3. The spinorial conventions of [15,16] will be adopted. The present article draws heavily from the analysis in [8,11,22] so we have followed the notation and conventions of these references as closely as possible.

A class of time symmetric data
For time symmetric initial data sets (S,h ij ) the Einstein vacuum field equations (1) imply the constraint equationr = 0, onS, wherer denotes the Ricci scalar of the metrich ij .
Our analysis will be local to a neighbourhood of infinity. Hence, without loss of generality, only one asymptotically flat end will be assumed. The asymptotic flatness of the time symmetric initial data (S,h ij ) will be expressed in terms of conditions on a conformally rescaled manifold. For this, it will be assumed that there is a 3-dimensional, orientable, smooth compact manifold S, a metric h ij , a point i ∈ S, a diffeomorphism Φ : S \ {i} →S and a function Ω ∈ C 2 (S) ∩ C ∞ (S \ {i}) with the properties where D j denotes the Levi-Civita covariant derivative of the 3-metric h ij . For the sake of simplicity the last condition will be written as h ij = Ω 2h ij so that S \ {i} is identified withS. If assumptions (3a)-(3c) are satisfied, the pair (S,h ij ) will be said to be asymptotically Euclidean and regular. Suitable punctured neighbourhoods of the point i are mapped to the asymptotic end ofS.

Asymptotically Euclidean and regular data
The Hamiltonian constraint, equation (2), together with the boundary conditions (3a)-(3c) imply on B a (i) the Yamabe Equation where δ(i) denotes the Dirac delta distribution with support on i while ∆ and r correspond, respectively, to the Laplacian and the Ricci scalar of the conformal metric h ij . For later use we define ω = 2Ω |D k ΩD k Ω| .
It is well known -see e.g. [8,11]-that if a is suitably small, then the solutions to the Yamabe equation (4) on B a (i) are of the form for some asymptotically Cartesian coordinates x i . The terms U/|x| and W will be known, respectively, as the massless and massive parts of ϑ. The function U , the Green function, satisfies the equation and describes the local geometry in B a (i). The function W satisfies the equation annd contains information of global nature. In particular, W (i) = m/2, where m is the ADM mass of the initial data set.
A rescaling , Ω → Ω ′ = θ 2 Ω, with a smooth positive factor θ satisfying θ(i) = 1 leaves the physical metrich ij = Ω −2 h ij unchanged. However, it implies the transitions where |x ′ | is given in terms of the h ′ -normal coordinates. There are several possibilities to fix this conformal gauge freedom. For the purpose of the present analysis it turns out that a good choice is that of the so-called conformal normal (cn)-gauge introduced in [8]. In the following definition let l ij denote the Schouten tensor of the metric h ij .
Definition 1. The metric h ij is said to be in the cn-gauge if given a solution (x(λ), b(λ)) to the 3-dimensional conformal geodesic equationṡ with initial conditions Remark. Let h ij be analytic in a neighbourhood of i. Assuming that a is sufficiently small, there exists on B a (i) a unique analytic rescaling h ij → h ′ ij = θ 4 h ij for which h ′ ij is in the cn-gauge. The metric and connection remain unchanged at i. As it will be discussed in the sequel, the practical advantage of the cn-gauge is that it renders simpler multipolar expansions for various quantities of interest.

Asymptotically static data
Let (S,h ij ) denote a static initial data set given in the cn-gauge, and let ϑ ≡Ů |x| +W , denote the corresponding solution to the Yamabe equation (4) in a suitably small neighbourhood B a (i). The static initial data set can be specified entirely in terms of its multipole moments -see e.g. [12]. This important fact will, however, not be exploited here. As a consequence of the analysis in [11,12] one has the following: Proposition 1. Let (S,h ij ) be a static initial data set and let B a (i) be a suitably small neighbourhood of i. If the conformal metrich ij satisfies the cn-gauge B a (i), then there exist normal coordinates x i such thath ij ,Ů andW are analytic in the neighbourhood.
Remark 1. Given a sequence of multipoles for a static solution, one can always assume without loss of generality, that the dipolar terms vanish -this amounts to the gauge choice of working in the centre of mass. It can be readily verified that this assumption has the consequence that static data in the cn-gauge satisfiesW The latter form of the functionW will be assumed in the sequel.
Remark 2. IfŮ = 1 andW = m/2 in B a (i), then the static initial data set corresponds to initial data for the Schwarzschild spacetime.
The following observation will be crucial in our subsequent analysis. LetW satisfy Clearly, due to linearity one has that ϑ =Ů |x| +W +W (7) is also a solution to the Yamabe equation (4) with the same boundary conditions in B a (i). Due to the analyticity ofh ij , the functionW will also be analytic. Time symmetric initial data sets, (S,h ij ), constructed out of the conformal metrich ij and the conformal factor ϑ given by equation (7) will be said to have a static massless part. Moreover, following the ideas in [22], we make the following ad hoc definition: Definition 2. A time symmetric initial data set in the cn-gauge will be said to be static up to order p • if and only if there exists coordinates x i in a suitable neighbourhood B a (i) of infinity such that the solution ϑ to the Yamabe equation (4) is of the form whereŮ /|x| andW are, respectively, the massless and massive parts of a static initial data set.

Remark.
It can be shown that in the cn-gauge the polynomials are harmonic with respect to the flat Laplacian. That is, the constant tensors are trace-free with respect to the flat metric δ ij .

Properties of data which is static up to a certain order
Consistent with Definition 2, we write Ω =Ω +Ω, ω =ω +ω, whereΩ andω are, respectively, the static parts of Ω and ω (obtained by settingW = 0). Their non-static partsΩ andω are then obtained viȃ The following result is obtained by direct computation observing expression (7): be an initial data set which is static up to order p • ≥ 2. Then Let b ij denote the Cotton-Bach tensor of the conformal metric h ij of an initial data set which is static up to a certain order in the sense of Definition 2. Crucially, one has that: Lemma 2. Let (S,h ij ) be an initial data set which is static up to order p • ≥ 2. Then where C denotes the operation of taking the symmetric trace-free part.
Proof. As already discussed, Proposition 2 shows that the property holds for exactly static initial data sets. In [8] it has been shown that this property concerns only the massless part of time symmetric data sets. The result follows then by observing that an initial data set which is static up to a certain order has the same massless part as the one of a static initial data set so that b ij =b ij .

The cylinder at spatial infinity
In [8] a representation of the region of spacetime close to null infinity and spatial infinity has been introduced -see also the comprehensive discussion in [11]. The standard representation of this region of spacetime depicts i 0 as a point. In contrast, the representation introduced in [8] depicts spatial infinity as a cylinder -the cylinder at spatial infinity. This construction is briefly reviewed for the case of time symmetric initial data sets with an analytic conformal metric in a neighbourhood B a (i) of infinity. The reader is referred to [8,11] for a thorough discussion of the details -see also [22].

The Manifold C a
Starting from the initial hypersurface S, the construction introduced in [8] makes use of a blowup of the point i ∈ S to the 2-sphere S 2 . This blow-up requires the introduction of a particular bundle of spin-frames over B a . Consider the (conformally rescaled) spacetime (M, g µν ) obtained as the development of the time symmetric initial data set (S, h ij ). Let SL(S) be the set of spin dyads δ = {δ A } A=0,1 on S which are normalised with respect to the alternating spinor ǫ AB in such a way that ǫ 01 = 1. Let τ = √ 2e 0 , where e 0 is the future g-unit normal of S and τ AA ′ its spinorial counterpart. The spinor τ AA ′ enables the introduction of space-spinors -sometimes also called SU (2) spinors, see [1,4,17]. It defines a sub-bundle SU (S) of SL(S) with structure group SU (2, C) and projection π onto S. Given a spinorial dyad δ ∈ SU (S) one can define an associated vector frame e a , a = 1, 2, 3. We shall restrict our attention to dyads related to frames {e j } j=0,··· ,3 on B a such that e 3 is tangent to the h-geodesics starting at i. LetȞ denote the horizontal vector field on SU (S) projecting to the radial vector e 3 . The fibre π −1 (i) ⊂ SU (S) (the fibre "over" i) can be parametrised by choosing a fixed dyad δ * and then letting the group SU (2, C) act on it. Let (−a, a) ∋ ρ → δ(ρ, t A B ) ∈ SU (S) be the integral curve to the vectorȞ satisfying δ(0, t A B ) = δ(t A B ) ∈ π −1 (i). With this notation one defines the set which is a smooth submanifold of SU (S) diffeomorphic to (−a, a) × SU (2, C). It follows that the projection map π of the bundle SU (S) maps C a into B a . The manifold C a inherits a number of structures from B a . In particular, the solder and connection forms can be pulled back to smooth 1-forms on C a satisfying the structure equations which relate them to the curvature form. The explicit form of the structure equations will not be required here. As S is 3-dimensional, the full content of its curvature can be expressed in terms of the spinorial counterpart of the trace-free Ricci tensor s ABCD and the Ricci scalar r.
In the sequel t A B ∈ SU (2, C) and ρ ∈ R will be used as coordinates on C a . Consequently, one has thatȞ = ∂ ρ . Vector fields X ± , X relative to the SU (2, C)-dependent part of the coordinates can be introduced by requiring the commutation relations and by requiring that they commute withȞ = ∂ ρ . More importantly, it can be seen that for p ∈ B a \ {i} the projections of the fields ∂ ρ , X ± span the tangent space at p. Given these vector fields, define the frame c AB = c (AB) by with constant spinors x AB , y AB and z AB given by and analytic spinor fields satisfyingč For the frame c AB , the connection coefficients, γ ABCD , are given by

An orthonormal basis for functions on SU(2, C)
with j, k = 0, . . . , m and m = 1, 2, 3, . . .. The subindex expression (A1···Am) k means that the indices are symmetrised and then k of them are set equal to 1, while the remaining ones are set to 0. Details about the properties of these functions can be found in [5,8]. The functions √ m + 1T j m k form a complete orthonormal set in the Hilbert space L 2 (µ, SU (2, C)), where µ denotes the normalised Haar measure on SU (2, C). The action of the differential operators X ± on the functions T k m j is given by In the sequel, we will need to linearise products of the form T i m k . To this end, we use the formula: with µ = min{i 1 , i 2 , j 1 + j 2 , k 1 + k 2 } and and C(i, j; k, l; m, n) the Clebsch-Gordan coefficients of SU (2, C).

Normal expansions at I 0
In the sequel, we will consider the lift of analytic fields defined on B a to C a . In particular, the lift of |x| is ρ. More generally, let ξ A1B1···A l B l denote a spinorial field on B a . Denote, again, by ξ A1B1···A l B l its lift to C a . Denote by ξ j = ξ (A1B1···A l B l )j , 0 ≤ j ≤ l its essential components. The function ξ j has spin weight s = l − j and a unique expansion of the form with complex coefficients ξ j,p;2q,k . More generally, we shall consider symmetric spinorial fields ξ A1···Ar on C a with independent components ξ j = ξ (A1···A2r )j , 0 ≤ j ≤ 2r, and spin-weight s = r−j which do not descend to analytic spinor fields on B a . In this case one has that where one has a priori that 0 ≤ |r − j| ≤ q(p). An expansion of the latter form will be said to be of type q(p).

Particular results concerning expansion types
We recall the following result in [8]: Lemma 3. Assuming the cn-gauge and if for some non-negative integer k it holds then we have the following expansion types: In addition, we have that The particular structure of the expansions dictated by this lemma greatly simplifies the subsequent analysis. Important for our subsequent analysis is the particular form of the lift to C a of the function W appearing in the conformal factor ϑ for data which is static up to order p • . Using the methods of [8] one finds that where the coefficientsw p;2q,k are related to the constant tensors w i1···ip via contractions with Infeld-van der Waerden symbols. In particular, it can be seen that The key observation in equation (10) is that the terms of order O(ρ p•+1 ), O(ρ p•+2 ), O(ρ p•+3 ) are formally identical to the ones appearing in the corresponding expansions for the functionW of data which is Schwarzschildean up to order p • (W = m/2). In other words, w p•+1;q,k = 0, q = 0, . . . p • + 1, k = 0, . . . q, w p•+2;q,k = 0, q = 0, . . . p • + 2, k = 0, . . . q, w p•+3;q,k = 0, q = 0, . . . p • + 3, k = 0, . . . q.

The spacetime Friedrich gauge
The formulation of the initial value problem near spatial infinity presented in [8] employs gauge conditions based on timelike conformal geodesics. The conformal geodesics are curves which are autoparallel with respect to a Weyl connection -i.e. a torsion-free connection which is not necessarily the Levi-Civita connection of a metric. An analysis of Weyl connections in the context of the conformal field equations has been given in [7]. In terms of this gauge based on conformal geodesics -which shall be called the Friedrich gauge or F-gauge for short-the conformal factor of the spacetime can be determined explicitly in terms of the initial data for the Einstein vacuum equations. Hence, provided that the congruence of conformal geodesics and the fields describing the gravitational field extend in a regular manner to null infinity, one has complete control over the location of null infinity. This can be ensured by making B a suitably small. In addition, the F-gauge renders a particularly simple representation of the propagation equations. Using this framework, the singular initial value problem at spatial infinity can be reformulated into another problem where null infinity is represented by an explicitly known hypersurface and where the data are regular at spacelike infinity. The construction of the bundle manifold C a and the blowing up of the point i ∈ B a to the set I 0 ⊂ C a , briefly described in section 3.1, are the first steps in the construction of this regular setting. The next step is to introduce a rescaling of the frame bundle so that fields that are singular at I 0 become regular.
Following the discussion of [8] assume that given the development of data prescribed on B a , the timelike spinor τ AA ′ introduced in section 3.1 is tangent to a congruence of timelike conformal geodesics which are orthogonal to B a . The canonical conformal factor rendered by this congruence of conformal geodesics is given in terms of an affine parameter τ of the conformal geodesics by with ω as given by equation (5) and where Ω = ϑ −2 and ϑ solves the Yamabe equation (4) -see [7,8,10]. The function κ > 0 expresses the remaining conformal freedom in the construction. It will be taken to be of the form κ = κ ′ ρ, with κ ′ analytic, κ ′ (i) = 1. Associated to the conformal factor Θ there is a 1-form d µ from which the Weyl connection can be obtained. In spinorial terms, one has that for conformally flat data The function κ in the conformal factor Θ, induces a scaling δ A → κ 1/2 δ A of the spin frame. Accordingly, one considers the bundle manifold C a,κ = κ 1/2 C a of scaled spinor frames. Using C a,κ one defines the set which, assuming that the congruence of null geodesics and the relevant fields extend adequately, can be identified with the development of B a up to null infinity -that is, the region of spacetime near null and spatial infinity. In addition, one defines the sets: which will be referred to as, respectively, the cylinder at spatial infinity, the critical sets and future and past null infinity. In order to coordinatise the hypersurfaces of constant parameter τ , one extends the coordinates (ρ, t A B ) off C a,κ by requiring them to be constant along the conformal geodesics -i.e. one has a system of conformal Gaussian coordinates.
Remark. For the purpose of the analysis carried out in this article it turns out that the most convenient choice of the function κ in the conformal factor Θ of equation (11) is This leads to considerable simplifications in all the relevant expressions. From this point onwards, this choice will always be assumed.

The evolution equations
On the manifold M a,κ it is possible to introduce a calculus based on the derivatives ∂ τ and ∂ ρ and on the operators X + , X − and X. The operators ∂ ρ , X + , X − and X originally defined on C a can be suitably extended to the rest of the manifold by requiring them to commute with the vector field ∂ τ . In order to derive the propagation equations, a frame c AA ′ and the associated spin connection coefficients Γ AA ′ BC of the Weyl connection ∇ will be used. The gravitational field is, in addition, described by the spinorial counterparts of the Schouten tensor of the Weyl connection, Θ AA ′ BB ′ , and of the rescaled Weyl tensor, φ ABCD -see [7,8,11]. Let φ i ≡ φ (ABCD)i . In the present gauge, the information of the spacetime spinors c µ AA ′ , Θ AA ′ BB ′ and Γ AA ′ BC is encoded, respectively, in space spinors c µ AB , Θ ABCD and Γ ABCD -see [11] for the detailed relation between the two sets of spinors.
In what follows, we will arrange the independent components of the spinorial fields c µ AB , Θ ABCD and Γ ABCD in the vector υ, and those of φ ABCD in the vector φ. Suitable field equations for the fields contained in υ and φ can be obtained from the first and second Cartan structure equations, the Bianchi identity of the conformally rescaled spacetime (M, g µν ), and the Bianchi identity of the physical spacetime (M,g µν ) -see e.g. [9,11] for details. A solution to the equations thus constructed implies a solution to the vacuum Einstein field equations. The vector υ has 45 independent complex components, while φ has 5 independent complex components. Using the F-gauge it can be shown that the extended conformal field equations given in [8] imply the following evolution equations for the unknowns υ where K and Q denote, respectively, linear and quadratic constant matrix-valued functions with constant entries, and L is a linear matrix-valued function with coefficients depending on the coordinates and such that L| ρ=0 = 0. For the unknowns φ, the Bianchi identity ∇ AA ′ φ ABCD = 0 implies, respectively, a set of propagation and constraint equations of the form: where E denotes the 5 × 5 unit matrix, A µ (c) and B µ (c), µ = 0, . . . , 3, are, respectively, 5 × 5 and 3×5 constant matrix-valued linear functions depending on the frame field coefficients c µ AB . Finally, F(Γ) and G(Γ) denote constant matrix-valued linear functions of the connection coefficients Γ ABCD .

Initial data for the evolution equations
For quick reference, we include here the expressions for the initial data for the curvature components appearing in equations (14)-(15b). These expressions can be deduced from the conformal constraint equations -see [8]. On C a,κ one has that: where, Ω denotes the lift to C a,κ of the conformal factor obtained from equation (7). The expressions for the frame and connection coefficients do not involve the function Ω, and thus their detailed expressions will not be required here.

The transport equations
Consider now the system (14)-(15a) with data given on C a,κ , and let be given a neighbourhood W of C a,κ in M a,κ on which a unique smooth solution of the Cauchy problem exists. From the point of view of the propagation equations, the subset W ∩ I is a regular hypersurface. Introduce the notation Due to the property L| ρ=0 = 0, equations (14) decouple from equations (15a) and can be integrated on W ∩ I using the observation that the restriction of the initial data to I 0 coincides with Minkowski data. The solutions thus obtained extend analytically to the whole of I and in particular to the critical sets I ± . The set I turns out to be a total characteristic of the system (14)-(15a) in the sense that the whole system reduces to an interior system on I. Moreover, the constraint equations (15b) also reduce to an interior system on I. As mentioned before, this feature is a consequence of the fact that the unphysical metric g µν determined by a solution to the conformal field equations degenerates as ρ → 0.
A crucial structural property is that so that the matrix E + A 0 (c) which is positive definite for |τ | < 1 degenerates at I ± . Understanding the effects of this degeneracy is the main motivation behind the analysis in the present article and that in [22].
In order to analyse the effects of the degeneracy of the matrix E + A 0 (c) we will consider the hierarchy of transport equations which is obtained by repeated application of the operator ∂ ρ to equations (14)-(15b) and then evaluation on I. By this procedure one obtains interior systems for the quantities υ (p) ≡ ∂ (p) ρ υ| I , φ (p) ≡ ∂ (p) ρ φ| I . The transport equations take the following form for p ≥ 1: Note that the non-homogeneous terms in the equations (18a)-(18c) depend on υ (p ′ ) , φ (p ′ ) for 0 ≤ p ′ < p. Thus, if their values are known, then equations (18a)-(18b) constitutes an interior system of linear equations for υ (p) and φ (p) . The principal part of these equations is universal, in the sense that it is independent of the value of p. If the initial data on C a,κ for the system (14)-(15a) is analytic -as it is the case in the present analysis-then suitable initial data for the interior system (18a)-(18b) can be obtained by repeated ρ-differentiation and evaluation on I 0 .
The language of jets is natural in the present context. For p = 0, 1, 2, . . . and any sufficiently smooth (possibly vector valued) function f defined on M a,κ , the sets of functions {f (0) , f (1) , . . . , f (p) } on I will be denoted by J We note the following result of [22] which will be used in the sequel.

Decomposition in spherical harmonics
Our analysis requires decomposing the entries of the vector unknowns υ (p) and φ (p) in terms of the functions T j i k . Following the discussion in [22], given a vector u (p) = (u (p) 1 , . . . , u (p) N ) with entries in R × S 2 and non-negative integers q and k = 0, . . . , 2q one defines the sector S q,k [u (p) ] as the collection of coefficients u i,p;2q,k ≡ (2q + 1) where s is the spin-weight of u i . Furthermore, one defines With this notation, a sector will be said to vanish if S q [u (p) ]. For convenience in the discussion, we introduce the following terminology: • A coefficient u i,p;2q,k will be said static if it depends only on the mass m and on the terms in the expansions ofŮ andW . We will write u i,p;2q,k =ů i,p;2q,k .
• The coefficient u i,p;2q,k will be said to be Schwarzschildean if it only depends on m. In our gauge, Schwarzschildean terms can only arise in the sectors S 0 .
• A coefficient u i,p;2q,k will be said to be non-static if it depends on the terms in the expansion ofW . We will write u i,p;2q,k =ȗ i,p;2q,k .
The structure of the transport equations for the elements of the various sectors S q,k [υ (p) ] and S q,k [φ (p) ] has been discussed in detail elsewhere -see [8,11,22]. In particular, the degeneracy at the critical sets I ± implied by expression (17) is only relevant for sectors with p ≥ 2. Thus, in the sequel it will always be assumed that p ≥ 2. If we denote by ′ differentiation with respect to τ , the relevant transport equations are of the form: The terms f p;2q,k , g p;2q,k , h p;2q,k are calculated from the left hand sides of equations (18a)-(18c) using formula (9). Detailed expressions for certain values of the multiindex (p; q, k) will be given in the sequel.

The cylinder at spatial infinity for static spacetimes
As mentioned in the introduction, static initial data sets are expected to play a privileged role among the class of time symmetric initial data sets with a development which is asymptotically simple. This point of view is supported by the following result proved in [11] showing that the construction of the cylinder at spatial infinity described in the previous section is for static spacetimes as smooth as it is to be expected.
Theorem 2. For static vacuum solutions which are asymptotically flat, the construction of the cylinder at spatial infinity is analytic in the sense that there exists an a > 0 and a frame for which all the fields, including the rescaled conformal Weyl tensor extend to analytic fields on some neighbourhood N of I in M κ,a . This statement depends neither on the particular choice of the conformal gauge used to prescribe the (analytic) free datum on S nor on the choice of κ.
Essential for our analysis is the following corollary: Corollary 1. The solutions to the transport equations (18a)-(18c) for time symmetric static data given in the cn-gauge extend analytically through I ± for all orders p. Moreover, the solutions to the transport equations are polynomial in τ .
Proof. Once analyticity has been asserted, the polynomial dependence of the solutions with respect to τ follows from an analysis of the possible solutions to the reduced equations -cfr. the discussion in section 6.2 and 6.3 of [22].

Properties of data sets which are static up to a given order
In what follows we will discuss some properties of initial data sets which are static up to a certain order -in the sense of definition 2. These properties have mainly to do with the multipole structure of quantities associated to these initial data sets.
Given a quantity u, we will systematically write it as where, in the terminology of the previous section,ů andȗ denote, respectively, its static and non-static parts. For quantities on the initial hypersurface C a,κ we similarly write u 0 =ů 0 +ȗ 0 .
Let as before, c and Γ denote, respectively, the parts of the vectorial unknown υ containing the independent components of the frame and connection coefficients. As mentioned in section 3.6, the expressions for the value of the frame and connection coefficients on the initial hypersurface C a,κ are independent of Ω. Thus, one has that: Lemma 5. For data which is static up to order p • ≥ 0 one has that Crucial for our analysis is the behaviour of the data for the curvature spinors Θ ABCD and φ ABCD . It is observed that from where it follows that the leading terms in these expressions are Schwarzschildean. The first non-Schwarzschildean contributions arise at orders O(ρ 5 ) and O(ρ 3 ), respectively -see e.g. [13].
Using the expansions (20a)-(20b) together with formulae (16a)-(16b) one obtains after a long but straightforward computation the following: Lemma 6. For data which is static up to order p • ≥ 2 one has that the curvature fields on C a,κ satisfyΘ An inspection of the explicit computations in [13] one obtains the following result.

Moreover, the termsΘ
Lemma 7. For a data which is static up to order p • ≥ 3 one has that (1) have only contributions to the sector S 0 . On the other hand, the coefficients of (2) , contain contributions to the sectors S 0 and S 2 .
Remark. In the previous lemma L denotes the linear matrix-valued function appearing in equation (14). It depends on the conformal factor Θ as given by (11) and on the 1-form d AB as given by (12).
A lenghty but straightforward computation using formulae (11) and (12) renders the following lemma.

Solutions to the transport equations for data which is static up to a certain order
In this section we consider a time symmetric initial data set which is static up to order p = p • + 1.
Lemma 9. For initial data which is static up to order p = p • , one has that In other words, the first deviations from static evolution appear in φ (p•+1) .
As this last equation is homogeneous in the unknownυ (p) , the initial conditionυ (p) 0 = 0 implies υ (p) = 0. A similar argument leads to the following equation forφ (p) : Again, the homogeneity of this last equation, together with the initial conditionφ (p) 0 = 0 renders φ (p) = 0. Thus, one has thatυ (p) = 0,φ (p) = 0, 0 ≤ p ≤ p • . The argument can be repeated foȓ The ideas of the proof of Lemma 9 will be used to study with some detail the solutions to the sector S p•+1 transport equations for the orders p • + 1, p • + 2, p • + 3 and p • + 4.

6.1
The transport equations at order p = p • + 1 From Lemma 9 one knows that υ (p•+1) =υ (p•+1) . Thus, at this order one only needs to study the solutions to the Bianchi transport equations.
Combining this analysis with Corollary 1 one finds the following result: Proposition 3. The solution to the transport equations at spatial infinity at order p • + 1 for data which is static up to order p • are polynomial in τ . Hence, they extend analytically through τ = ±1.

6.2
The transport equations at order p = p • + 2 Using Lemma 4 one finds that the solutions υ (p•+2) to the order p • + 2 υ-transport equations are polynomial in τ given that υ (p) and φ (p) for 0 ≤ p ≤ p • + 1 are polynomial in τ . However, we require more precise information.

The transport equations at order p = p • + 3
We now adapt the procedure discussed in the previous section to the analysis of the order p • + 3 transport equations. As in the previous order the polynomial dependence in τ follows directly from Lemma 4 once one knows that the entries in φ (p•+2) are polynomial. Further detailed information will follow from the analysis of the sector S p•+1 .
As a consequence of the analysis of the transport equations for the orders p • + 1 and p • + 2 one has that the entries in depend only on m and the coefficientsw p•+1;2(p•+1);k . As in the analysis of the orders p • + 1 and p • + 2 one has the crucial observation: Observation 3. The equations (24a)-(24c) and their initial conditions are formally identical to the order p • + 3 transport equations for initial data sets which are Schwarzschildean up to order p • .

6.4
The transport equations at order p = p • + 4 Finally, using similar methods, we discuss the order p • +4 transport equations. As in the previous order the polynomial dependence in τ follows directly from Lemma 4 once one knows that the entries in φ (p•+3) are polynomial. The rest of the analysis is more subtle than at lower orders. The reason for this is twofold: (i) As a consequence of Lemma 6 one finds thatφ ABCD has a multipolar structure which is more complicated than that of data which just deviates from Schwarzschild data.
(ii) The discussion of the transport equations for the sectors in S p•+1 involves non-trivial multiplications of the functions T j i k . Evidence from explicit calculations suggests that the solutions to the Bianchi transport equations at this order are, generically, not smooth at the critical sets.
The crucial observation is the following: Observation 4. Equations (25a)-(25c) and the data (27) satisfying (29) are formally identical to the order p • + 4 transport equations analysed in [22] for data which are Schwarzschildean up to order p • . These equations have no solution which is smooth at the critical sets.

The main result
The discussion in the previous sections is summarised in the following result.
Proposition 7. Given a time symmetric initial data set which in a neighbourhood B a of infinity is static up to order p = p • (in the sense of Definition 2), the solutions to the transport equations for the orders p = p • + 1, p • + 2, p • + 3 are polynomial in τ , and hence, extend smoothly through the critical sets I ± . On the other hand, the solutions at order p = p • + 4 contain logarithmic singularities which can be avoided if an only if the initial data is, in fact, static up to order p = p • + 1.
As in the case of the analysis given in [22], one can use the previous result to implement an induction argument in which the explicit computer algebra calculations of [19] play the role of the base step to obtain the main result on this article. Theorem 3. The solution to the regular finite initial value problem at spatial infinity for time symmetric initial data of the form given by Definition 2 is smooth through I ± if and only if the restriction of the data to I 0 coincides with the restriction of static data at every order -that is, if their jets of order p coincide for all p. Furthermore, the analyticity of the setting implies that the data is exactly static in a neighbourhood of infinity.
As already mentioned in the introduction, in order to complete the analysis of the solutions of the transport equations at the critical sets for general analytic time symmetric initial data sets, one needs (conformally invariant) conditions which reduce the initial data set to data with a static massless part in the sense discussed in Section 2. This problem will be analysed elsewhere.