A comparison of Ashtekar's and Friedrich's formalisms of spatial infinity

Penrose's idea of asymptotic flatness provides a framework for understanding the asymptotic structure of gravitational fields of isolated systems at null infinity. However, the studies of the asymptotic behaviour of fields near spatial infinity are more challenging due to the singular nature of spatial infinity in a regular point compactification for spacetimes with non-vanishing ADM mass. Two different frameworks that address this challenge are Friedrich's cylinder at spatial infinity and Ashtekar's definition of asymptotically Minkowskian spacetimes at spatial infinity that give rise to the 3-dimensional asymptote at spatial infinity $\mathcal{H}$. Both frameworks address the singularity at spatial infinity although the link between the two approaches had not been investigated in the literature. This article aims to show the relation between Friedrich's cylinder and the asymptote as spatial infinity. To do so, we initially consider this relation for Minkowski spacetime. It can be shown that the solution to the conformal geodesic equations provides a conformal factor that links the cylinder and the asymptote. For general spacetimes satisfying Ashtekar's definition, the conformal factor cannot be determined explicitly. However, proof of the existence of this conformal factor is provided in this article. Additionally, the conditions satisfied by physical fields on the asymptote $\mathcal{H}$ are derived systematically using the conformal constraint equations. Finally, it is shown that a solution to the conformal geodesic equations on the asymptote can be extended to a small neighbourhood of spatial infinity by making use of the stability theorem for ordinary differential equations. This solution can be used to construct a conformal Gaussian system in a neighbourhood of $\mathcal{H}$.


Introduction
The theory of isolated systems plays a central role in astrophysical applications of Einstein's theory of General Relativity. A particularly influential approach to this theory is through Penrose's definition of asymptotic simplicity -see e.g. [22,29]: Definition 1. A vacuum spacetime (M,g) is said to be asymptotically simple if there exists a smooth, oriented, time-oriented and causal spacetime (M, g) and a C ∞ function Ξ on M such that: (i) M is a manifold with boundary I ≡ ∂M; While in Minkowski spacetime i 0 is a regular point of the conformal structure, it is well known that for spacetimes with non-vanishing (ADM) mass, one has a singularity of the conformal structure -see e.g. [29], Chapter 20. This fact makes the analysis of the gravitational field particularly challenging. To address this difficulty, it is necessary to introduce a different representation of spatial infinity which, in turn, allows to suitably resolve the structure of the gravitational field in this region. In particular, to avoid having to deal with directional dependent limits at spatial infinity, it is natural to blow up the point i 0 to a 2-sphere.
The hyperboloid at spatial infinity. In an attempt to overcome the difficulties posed by the fact that the classic definition of asymptotic simplicity (see Definition 1 above) makes no reference to the behaviour of the gravitational field at spatial infinity, in [3] Ashtekar & Hansen introduced a new definition of asymptotic simplicity in both null and spacelike directions -see also [2]. This definition (asymptotically empty and flat at null infinity and spatial infinity, AEFANSI ) combines the conditions on the conformal extension given in the definition of asymptotical flatness with further assumptions on the conformal factor at spatial infinity. Essentially, it is further required that spatial infinity admits a so-called point compactification -see, e.g. [29], Section 11.6 for a discussion of this notion. The strategy behind the approach put forward in [3] is to make use of spacetime notions rather than, say, making use of the initial value problem. Their definition of asymptotic flatness allows them to blow up i 0 to the timelike unit 3-hyperboloid -the hyperboloid at spatial infinity. The sections of the hyperboloid give the blowing up of i 0 to 2-spheres while time direction along the (timelike) generators of the hyperboloid can be, roughly speaking, associated with all the possible ways in which the asymptotic region of a Cauchy hypersurface can be boosted. The definition of AEFANSI spacetimes is geared towards the discussion of asymptotic symmetries and conserved quantities at spatial infinity.
In a slightly different context, in [6,5] the hyperboloid at spatial infinity is used, in conjunction with the vacuum Einstein field equations to obtain asymptotic expansions of the gravitational field near spatial infinity in negative powers of a radial coordinate. The main observation is that the field equations give rise to a hierarchy of linear equations for the coefficients of the expansion. It turns out that this hierarchy can be solved, provided the initial data satisfies certain conditions. This work can be regarded as the first serious attempt to investigate the consistency between the Einstein field equations and geometric notions of asymptotic flatness.
The notion of the hyperboloid at spatial infinity has been revisited in [4] where the definition of asymptotically Minkowskian spacetimes at spatial infinity (AMSI) has been introduced -see Definition 2 in Section 4 of the main body of the present article. As in the analysis of [3], the approach in [4] makes use of spacetime structures and, thus, it is not geared towards the discussion of initial value problems. However, in contrast to the definition of AEFANSI spacetimes in [3], the definition of AMSI spacetimes focuses entirely on spatial infinity and, thus, it makes no assumptions about the properties of null infinity. A consequence of the concept of AMSI spacetimes is the existence of a 3-dimensional manifold H, the asymptote at spatial infinity, which generalises the notion of the hyperboloid at spatial infinity.
Friedrich's cylinder at spatial infinity. In [10], Friedrich puts forward an alternative conformal representation of the region of spacetime in the neighbourhood of spatial infinity -the F-gauge. The aim of this representation is the formulation of a regular initial value problem at spatial infinity for the conformal Einstein field equations. This initial value problem is key in the programme to analyse the genericity of asymptotically simple spacetimes and relies heavily on the properties of certain conformal invariants (conformal geodesics), and it is such that both the equations and the initial data are regular at the conformal boundary. A central structure in this representation of spatial infinity is the cylinder at spatial infinity which, in broad terms, corresponds to the blow-up of the traditional point at spatial infinity to a 2-sphere plus a time dimension -hence, the cylinder at spatial infinity. In contrast to the hyperboloid at spatial infinity, the cylinder at spatial infinity has a finite extension in the time direction. The endpoints of the cylinder, the critical sets, correspond to the points where spatial infinity meets null infinity. Thus, the F-gauge is ideally suited to the analysis of the connection between the behaviour of the gravitational field at spatial infinity and radiative properties. This idea has been elaborated in [14] to express the Newman-Penrose constants in terms of the initial data for the Einstein field equation and to study the behaviour of the Bondi mass as one approaches spatial infinity -see [23]; also [28]. The key property of the cylinder at spatial infinity, which allows connecting properties of the Cauchy initial data in a neighbourhood of spatial infinity with the behaviour near null infinity, is that the cylinder at spatial infinity is a total characteristic of the conformal Einstein field equations -that is, the associated evolution equations reduce in its entirety to a system of transport equations on this hypersurface. This property precludes the possibility of prescribing boundary data on the cylinder but allows computing a particular type of asymptotic expansions which makes it possible to understand the role of certain pieces of the initial data have on the regularity properties of the gravitational at null infinity -the so-called peeling behaviour ; see [10,14,25,24,26,27,17].
The F-gauge used in Friedrich's representation of spatial infinity is a gauge prescription based on the properties of conformal geodesics -i.e. a pair consisting of a curve and a covector along the curve satisfying conformally invariant properties. A remarkable property of conformal geodesics is that in Einstein spaces, they give rise to a canonical factor that has a quadratic dependence on the parameter of the curve -see [15,29]. The resulting expression depends on certain initial data, which can be chosen in such a way that the curves reach the conformal boundary and that no caustics are formed. In this way, one can obtain a conformal Gaussian gauge in which coordinates and a frame defined on an initial hypersurface are propagated along the conformal geodesics. In effect, this procedure provides a canonical way of obtaining conformal extensions of Einstein spaces. In the context of the initial value problem for the conformal Einstein equations, one has a gauge in which the location of the conformal boundary is known a priori.
Melrose-type compactifications. In the context of this introduction, it is worth mentioning the proof of the stability of Minkowski spacetime by Hintz & Vasy in which the existence of spacetimes with polyhomogeneous asymptotic expansions is established -see [20]-makes use of of a compactification of spacetime into a manifold with boundary and corners. This procedure is inspired by the methods of Melrose's geometric scattering theory programme -see e.g. [21]. This construction has a very strong resemblance to that introduced by Friedrich in [10]. The precise relation between these two seemingly connected representations of spatial infinity will be elaborated elsewhere.

Main results of this article
A cursory glance at Ashtekar's and Friedrich's constructions of spatial infinity suggests that they should be closely related. That this is the case has been part of the longstanding folklore of mathematical relativity. It is the purpose of this article to establish, in a rigorous way, the connection between these two representations of spatial infinity.
In order to carry out the objective outlined in the previous paragraph, we first analyse this relation for Minkowski spacetime, where everything can be explicitly computed. The outcome of this analysis is that Ashtekar's and Friedrich's representations are related to each other through a conformal transformation which preserves the asymptotic behaviour of the metric along spatial infinity but compactifies the time direction. In particular, the hyperboloid at spatial infinity is compactified into the 1 + 2 Einstein Universe in the same way as the de Sitter spacetime is compactified into the 1+3 Einstein Universe. The transformation between the two representations can be expressed in terms of properties of conformal geodesics -although in the case of Minkowski spacetime, because of its simplicity, this does not play an essential role.
The intuition gained in the analysis of Minkowski spacetime is, in turn, used to establish the relation between Ashtekar's notion of an asymptote at spatial infinity given by the definition of an AMSI spacetime -see Definition 2 in the main text. As the definition of an AMSI spacetime does not imply the existence of null infinity, to carry out our analysis, an extra assumption is required -see Assumption 1 in the main text. In essence, we assume that a sufficiently regular null infinity can be attached to the neighbourhood of spacetime around Ashtekar's asymptote. Our main result, whose proof is based on a stability argument for the solutions to the conformal geodesic equations in a neighbourhood of the asymptote is the following: Theorem. Given an AMSI spacetime that can be conformally extended to null infinity, there exists a sufficiently small neighbourhood of the asymptote at spatial infinity in which is possible to construct conformal Gaussian coordinates based on curves which extend beyond null infinity.
Associated with the conformal Gaussian system whose existence is ensured by the above statement, there exists a conformal factor that provides the precise relation between Ashtekar's representation of spatial infinity and F-gauge representation.

Outline of the article
This article is structured as follows. In Section 2 we provide a brief summary of the conformal methods that are required in the analysis of this article. These include conformal geodesics, conformal Gaussian systems, the conformal Einstein field equations and their constraints. For a full account of the associated literature, the reader is referred to the monograph [29]. Section 3 provides a discussion of the various representations of spatial infinity on Minkowski spacetime. In particular, it contains the explicit connection between the F-gauge and Ashtekar's representation of spatial infinity. Furthermore, it also contains a discussion on how this connection can be expressed in terms of conformal geodesics. Section 4 reviews Ashtekar's definition of asymptotically Minkowskian spacetimes at spatial infinity (AMSI) and the associated notion of an asymptote. It further contains a discussion of the consequences of this definition in light of the conformal Einstein field equations and the associated constraint equations on timelike hypersurfaces. Subsection 4.4 provides a detailed discussion of the relation between Ashtekar's asymptote and Friedrich's cylinder viewed as intrinsic 3-manifolds. Section 5 contains the main analysis in the article -namely, the construction of a conformal Gaussian gauge system in a neighbourhood of an asymptote. It also provides a more detailed statement of the main theorem of this article -Theorem 1. In addition, the article contains several appendices. Appendix A provides the stability theorem for solutions to a system of ordinary differential equations and their existence times which is used in the analysis of the solutions to the conformal geodesic equations in a neighbourhood of the asymptote. Appendix B provides details of the proof of a technical lemma (Lemma 4) on the regularity of geometric fields at the asymptote. Finally, Appendix C provides details on a certain class of conformal geodesics on Minkowski spacetime.

Notations and conventions
In what follows a, b, c . . . will denote spacetime abstract tensorial indices, while i, j, k, . . . are spatial tensorial indices ranging from 1 to 3. By contrast, a, b, c, . . . and i, j, k, . . . will correspond, respectively, to spacetime and spatial coordinate indices.
The signature convention for spacetime metrics is (+, −, −, −). Thus, the induced metrics on spacelike hypersurfaces are negative definite. The analysis of this article involves several conformally related metrics. To differentiate between them we adhere to the following conventions: the tilde (˜) is used to denote metrics satisfying the physical Einstein field equations; the overline (¯) is used to denote metrics in the F-gauge; metrics without an adornment (e.g. g) are in the conformal gauge given by the Definition 2 of a spacetime asymptotically Minkowskian at spatial infinity (AMSI); metrics in a gauge with compact asymptote are denoted by a grave accent (`). Finally, metrics with a hat (ˆ) correspond to conformal representations in which spatial infinity corresponds to a point. An index-free notation will be often used. Given a 1-form ω and a vector v, we denote the action of ω on v by ⟨ω, v⟩. Furthermore, ω ♯ and v ♭ denote, respectively, the contravariant version of ω and the covariant version of v (raising and lowering of indices) with respect to a given Lorentzian metric. This notation can be extended to tensors of higher rank (raising and lowering of all the tensorial indices). The conventions for the curvature tensors will be fixed by the relation Also, one can write the decomposition of the curvature tensor as where C c dab is the Weyl tensor and L bd is the Schouten tensor of the metric g ab .

Tools of conformal methods
The purpose of this section is to introduce the tools of conformal methods that will be required in this article.

General conventions and notation
In the following assume that (M,g) denotes a spacetime satisfying the vacuum Einstein field equationR Following standard usage, we call the pair (M,g) the physical spacetime, while any conformally related spacetime (M, g) with g ≡ Ξ 2g , will be referred to as the unphysical spacetime.
Then the conformal factor Θ can be written as a quadratic polynomial in terms of τ , i.e.
where λ is the cosmological constant.
Remark 1. In subsequent discussions, the cosmological constant λ is taken to be zero.

Conformal Gaussian systems
Conformal geodesics can be used to construct the so-called Conformal Gaussian Systems in which coordinates and adapted frames are propagated off an initial hypersurface S. One constructs a conformal Gaussian system by initially introducing a g-orthonormal Weyl-propagated frame {e a } along a congruence of conformal geodesics (x(τ ), β(τ )) on (M,g = Ξ −2 g) and choosing the time coordinate such that e 0 = ∂ τ . Then if (x i ) denotes the coordinates of a point p on S, one can propagate the spatial coordinates off S by requiring them to be constant along the conformal geodesic intersecting S at p. Then the conformal Gaussian system is given by (τ, x i ).

Conformal field equations
The vacuum conformal Einstein field equations on the unphysical spacetime are given by where d a bcd = Ξ −1 C a bcd is the rescaled Weyl tensor and s is the Friedrich's scalar given by The conformal Einstein field equations constitute a set of differential equations for the fields Ξ, s, L ab and d a bcd . A solution to the vacuum conformal field equations (7a)-(7e) implies, whenever Ξ ̸ = 0, a solution to the vacuum Einstein field equations (3) on the physical spacetime.

Conformal constraint equations
For later use, we give here the constraint equations implied by the conformal Einstein field equations (7a)-(7e) on a timelike hypersurface T . These equations can be obtained through a projection formalism and details of the derivation can be found in [29].
Let n a denote the unit-normal to the timelike hypersurface T . Then, the projection tensor is given by Then introduce the shorthand notation Given the above, the conformal Einstein constraint equations in vacuum can be written in terms of q ab , the extrinsic curvature K ab , the Schouten tensor L ab and ω, denoting the restriction of the conformal factor Ξ to T In the above, the conformal factor ω is assumed to be constant on T , i.e., D a ω = q a b ∇ b ω = 0. Moreover, the symbol ⊥ indicates a contraction with the unit normal n a , so the terms d a⊥b⊥ and d a⊥bc correspond, essentially, to the electric and magnetic parts of the rescaled Weyl tensor d abcd with respect to the normal n a -respectively.
1. The constraint equations (9) can be obtained from the conformal field equations (7) by various contractions with the projection tensor q a b and the unit-normal n a . For example, equations (9a) and (9b) are obtained by contractions of equation (7a) with q c a q d b and q c a n b , respectively.
2. All tensors intrinsic to T , e.g., q and K, will have vanishing components in the normal direction.

Representations of spatial infinity in Minkowski spacetime
In this section, we review several representations of spatial infinity for Minkowski spacetime. This analysis will provide insight and motivate the analysis in curved spacetimes.
In the following let (R 4 ,η) denote Minkowski spacetime. Let (x) = (x µ ) denote the standard Cartesian coordinates and write x 0 =t, etc. We will also make use of spherical coordinates (t,ρ, θ A ) where (θ A ) denotes some choice of spherical coordinates over S 2 . In terms of the above, one has thatη where η µν ≡ diag(1, −1, −1, −1), and σ is the standard round metric over S 2 .

Spatial infinity as point
In first instance, we consider the standard representation of spatial infinity as a point. Intuitively, the region in Minkowski spacetime associated with spatial infinity is contained in the domaiñ -the complement of the light cone through the origin, see Figure 1. Now, introducing the inversion coordinates y = (y µ ) defined by The latter suggests introducing the conformal factor Ξ ≡ 1/X 2 so that The conformal boundary defined by Ξ = 0 decomposes into the sets The sets I + (I − ), can be shown to be the endpoints of future (past) null geodesics while spatial geodesics end up in the point i 0 -spatial infinity, located in this representation at the origin. Observe that I + and I − do not contain the whole of null infinity, only the part of the conformal boundary close to spatial infinity -this is a peculiarity of this conformal representation.
Remark 3. In literature, I + and I − are usually used to denote the whole of future and past null infinity, respectively. In this setting, in a slight abuse of notation, we use I + and I − to denote the parts of the conformal boundary close to spatial infinity.

The cylinder at spatial infinity
A different representation of spatial infinity can be obtained from the rescalinḡ In this representation it is convenient to introduce a new time coordinate τ via the relation with the contravariant metric given bȳ Introducing the coordinate ϱ ≡ − ln ρ, the metriceη can be rewritten as From this metric and observing equation (12) and (11), one can readily see that spatial infinity i 0 , which corresponds to the condition ρ = 0, lies at an infinite distance as measured by the metric η.
It follows from the previous discussion that one can writē Consistent with the above one can define the set which gives rise to a conformal extension (M,η) of Minkowski spacetime. In this representation, the following sets play an important role in our discussion: and Moreover, future and past null infinity are given by: Remark 4. In the following, we call the above conformal representation of the neighbourhood of spatial infinity the F-gauge horizontal representation.
Remark 5. Although the metric (14) is singular on I, it induces the Lorentzian 3-metric which is regular. This metric can be regarded as the 1 + 2-dimensional version of Einstein's universe metric. In particular, its Ricci tensor is proportional to the metric -i.e. one has an Einstein space.

Conformal geodesics
A central aspect of the conformal representation of Minkowski spacetime given by (M,η) is its relation to conformal geodesics. More precisely, one has the following: for fixed (ρ ⋆ , θ A ⋆ ) ∈ S ⋆ constitutes a non-intersecting congruence of conformal geodesics inM. The details of the calculations providing the proof of the above lemma can be found in Appendix C.
Remark 6. In particular, the pair as given by the relations in (17) is a solution to theḡ-conformal geodesic equations.
Remark 7. In the following, in a slight abuse of notation, we identify the affine parameter s of the conformal geodesics with the time coordinate τ . The conformal geodesics given by Lemma 1 have tangent vector given by ∂ τ .
The conformal factor Θ given in equation (15) can be deduced from the solution to the conformal geodesic equation given by Lemma 1 and Proposition 1. More precisely, writing the canonical conformal factor in terms of the parameter s as withΘ ⋆ andΘ ⋆ given by the relations in (6). From Lemma 1, it readily follows thaṫ Thus, to recover the conformal factor in (15) one identifies the parameters s and τ .
Remark 8. While on the one hand one has Θ ⋆ | ρ=0 = 0, on the other hand dΘ| ρ=0 ̸ = 0. Thus, the choice of Θ ⋆ in this conformal representation is not that one of a point compactification. More generally, if Ω is a conformal factor giving rise to a point compactification of an asymptotic end of an asymptotically Euclidean manifold (cf. the conditions in (10)) the prescription of Θ ⋆ is of the form κ −1 Ω with κ a smooth function of the form κ = κρ and κ(i 0 ) = 1.

Gauge freedom
The conformal representation of the Minkowski spacetime described in the previous subsections can be generalised to obtain a description in which null infinity does not coincide with hypersurfaces of constant τ . For this, instead of relation (13) one rather considers with κ a smooth function of the spatial coordinates. This leads to the conformal factor In this case, the neighbourhood of spatial infinity is given bȳ while null infinity is described by the sets . Remark 9. The general F-gauge representation of Minkowski spacetime is related to the horizontal representation associated with the line element (14) via a conformal transformation which is the identity on I. Moreover, also notice that the parameter of the conformal geodesics in both representations do not coincide and are related to each other via a reparametrisation (Möbius transformation).

Remark 10.
A key difference between the horizontal representation line element (14) and the more general F-gauge line element (18) is that the former metric loses rank (i.e. degenerates) all along null infinity (I + ) while the latter does it only at the critical sets (I ± ). Thus, horizontal representation provides a slightly more singular representation of the neighbourhood of spatial infinity. This singular behaviour does not play a role in the subsequent discussion of this article. Accordingly, given its relative analytic simplicity, all further discussions of the F-gauge are done in the horizontal representation.

The hyperboloid at spatial infinity
Following the discussion in [4] a different (albeit related) representation of spatial infinity as an extended set can be obtained by considering hyperbolic coordinates (ψ, χ, θ, φ), related to the standard Cartesian coordinates (x µ ) via the relations It readily follows thatρ Accordingly, the hypersurfaces of constant ψ are timelike hyperboloids -see Figure 2. As in Section 3.1 one has that so that, as before, spatial infinity is contained in the regioñ It follows that the hyperbolic coordinates only cover the domainD -see Figure 1. In hyperbolic coordinates, one can writeη asη is the (Lorentzian) metric of the unit hyperboloid. This metric can be obtained from the pullback ofη to the hypersurface defined by the conditionρ 2 −t 2 = 1.

Standard representation of spatial infinity
In order to discuss the behaviour near spatial infinity it is convenient to introduce a new coordinate ζ ≡ 1 ψ .

It follows then thatη
Thus, it seems natural to introduce a conformal factor of the form Ω ≡ ζ 2 , so as to obtainη This leads to the standard representation of spatial infinity as a point. This is shown by noting that the metric of the 2-surfaces of constant ζ is given by ζ 2 ℓ. Hence, the set given by ζ = 0 has zero volume and is forced to be a single point by the choice of conformal factor Ω = ζ 2 . Moreover, one can confirm that at ζ = 0, we have Ω = 0, dΩ = 0, Hess Ω = −2η.

The hyperboloid at spatial infinity
In the present case, it is better to define the conformal factor This particular rescaling readily connects with Friedrich's framework already discussed in Section 3.4.
Consider in the following the timelike hyperboloids defined by the condition ζ = ζ • where ζ • is a constant. The key observation in [4] is that although the conformal metric (22) is singular at ζ = 0, the conformal 3-metric q = H 2q = ℓ, is well defined. Observe also, that the contravariant 4-dimensional metric is well defined -although, it losses rank at the set where ζ = 0. In addition to the above, observe that The later suggests introducing a rescaled unit normal vector through the relation Remark 11. Starting directly from the (singular) metric η one readily finds that the unit normal covector is given by ζ −1 dζ. Moreover, one has that which, despite being well defined at ζ = 0 vanishes. As it will be seen in the following, vectors with this behaviour at infinity play a key role in Friedrich's framework of spatial infinity.
Remark 12. In the following, the timelike hyperboloid H described by the condition ζ = 0 together with the induced metric q = ℓ will be known as the hyperboloid at spatial infinity of Minkowski spacetime. This hyperboloid is an example of the general notion of asymptote at spatial infinity introduced in the definition of an asymptotically Minkowskian spacetime at spatial infinity (AMSI) -see Definition 2 in Section 4.

Relating the Ashtekar and F-gauge construction in the Minkowski spacetime
In this subsection, we obtain the explicit relation between the description of spatial infinity in terms of the hyperboloidal coordinates and that based on the F-gauge. The procedure for relating these two representations will serve as a template for an analogous computation in more general classes of spacetimes.

From the hyperboloid at infinity to the cylinder at infinity
In order to relate the pair (H, ℓ) corresponding to Ashtekar's hyperboloid at spatial infinity to the pair (I,q) withq as given by equation (16), it is observed while the former can be thought of as a 1 + 2 version of the de Sitter spacetime, the latter is a 1 + 2 version of the Einstein static Universe -see Remark 5. As both metrics ℓ andq are Einstein, their Cotton tensor vanishes and thus, they are conformally flat. Accordingly, ℓ andq are conformally related.
In order to find the conformal factor relating ℓ andq, we follow the same procedure used to show that the (4-dimensional) de Sitter spacetime can be conformally embedded in the (4dimensional) Einstein static Universe -see e.g. Section 6.3 in [29]. More precisely, starting from the metric ℓ of the unit hyperboloid, introduce the coordinate transformation given by the relation dχ = cosh χdτ.
It follows readily that from where one can indeed see that ℓ andq are conformally related. In particular, it can be shown that cosh χ = sec τ .
Remark 13. It follows from the previous discussion that Friedrich's cylinder at spatial infinity is indeed a time compactified version of Ashtekar's hyperboloid. In particular, the critical sets I ± correspond to the limits χ → ±∞.

Relating the neighbourhoods of spatial infinity
Now, to relate the two constructions away from spatial infinity we recall that Now, as ζ = 1/ψ, a direct computation then gives that so that ζ = ρ sech χ.
Moreover, from the rescalings in (23) it follows that In terms of coordinates, one has Observing that χ = arctanh τ it follows that Notice that ϖ| S⋆ = 1. Moreover, ϖ → ∞ as τ → ±1. Thus, ϖ gives rise to a conformal representation of Minkowski spacetime that does not include null infinity.
Remark 14. Using equation (24) and setting ψ = ψ • where ψ • is a constant. One has that (tanh χ, ψ −1 • cosh χ) describes a curve in the (τ, ρ)-plane -see the Figure 3. Observe, in particular, Accordingly, the hyperboloids of constant ψ never reach the conformal boundary -i.e. I ± ; see Figure 3. Hence, this representation of spatial infinity cannot be used to study the effects of gravitational radiation and the relation between asymptotic charges at null infinity and conserved quantities at spatial infinity.

Conformal geodesics and construction of a conformal Gaussian system
The unphysical metricη is conformally related to the physical metricη via the conformal factor Θ = ρ(1 − τ 2 ). Thus, a solution to the conformal geodesic equations onη implies a solution to the conformal geodesic equation on the physical metricη. Recalling that the pair given by x(τ ) = (τ, ρ ⋆ , θ A ⋆ ) andβ = dρ/ρ satisfies the conformal geodesic equations for the metricη, it follows, using the coordinate transformation to hyperbolic coordinates given by (19a)-(19d), that the tangent vector to the curve x(τ ) is given in the (ζ, χ, θ A ) coordinates bẏ Figure 3: Example of one of the timelike hyperboloids used in Ashtekar's representation of spatial infinity for the Minkowski as seen from the point of view of the F-gauge. Observe that the hyperboloid asymptotes the sets described by the conditions τ = ±1. Notice, however, that in this description the hyperboloid at spatial infinity is compact and corresponds to the portion of the vertical axis between the values τ = −1 and τ = 1.
Then, (ẋ, β) provides a solution to the conformal geodesic equation, where β is given by where it has been used that Setting up a conformal Gaussian system. To conclude this discussion, we show how to compute a conformal Gaussian system on top of Ashtekar's conformal representation of spatial infinity in Minkowski spacetime. As we have already shown that we have a non-intersecting congruence of conformal geodesics, one can invoke Proposition 1 in Section 2.1 so that one can associate a conformal factor along each of the curves of the congruence and where s is the natural parameter of the curves. The coefficients in the above expression satisfy the relationṡ Consistent with the conformal metric (22), for Λ ⋆ one prescribes Λ ⋆ = ζ. For the covectorβ one would like to prescribe a behaviour of the formβ ⋆ = 2dρ/ρ. Note that the above form of β is consistent with the conformal geodesic solution obtained on the F-gauge construction with β ∝ 1/ρ. Also, this type of behaviour near spatial infinity is obtained from the conformal factor Ω = ζ 2 = 1/ψ 2 ∼ 1/r 2 , which renders the standard point compactification of spatial infinity. Consistent with this discussion setβ = d ln Ω = 2 ζ dζ.
it follows from the above discussion thatΛ ⋆ = 0 as ⟨dζ, ∂ τ ⟩ = 0. Also, notice that whereh is the metric on hypersurfaces of constant χ. A quick computation gives that From the latter it follows then thatΛ ⋆ = −2ζ. Accordingly, the conformal factor associated to the congruence of conformal geodesics has the form Finally, definingη ≡ Λ 2η so thatβ =β − d ln Λ, one concludes thatβ ⋆ = dζ/ζ. This shows the consistency of the prescription of initial data for the covectorβ given above.

Asymptotically Minkowskian spacetimes at spatial infinity
In [4] the notion of spacetimes which are asymptotically Minkowskian at spatial infinity has been introduced. We want to analyse this definition in light of Friedrich's framework. Hence, we briefly review the relevant definitions and properties.

Definitions
Following the discussion in [4], in the following we will consider the following definition: (ii) the fields q ≡ Ω 2 g + Ω −4 dΩ ⊗ dΩ and N ≡ Ω −4g♯ (dΩ, ·) admit smooth limits to H. In particular, the pullback of q (to be denoted again by q) to H is also well defined and has signature (+ − −); In addition, if H has the topology of R × S 2 then (M,g) is said to be asymptotically flat at spatial infinity. Moroever, if H is geodesically complete with respect to q we say that the spacetime is asymptotically Minkowskian at spatial infinity (AMSI).
Notation. In the following, for convenience, we will make use of the symbol ≃ to denote equality on H. With this notation the conditions in point (i) of the Definition 2 are written as

Properties and consequences of the definition of an AMSI spacetime
In the following, we explore some direct consequences of Definition 2, which will be used repeatedly in the rest of this article. Accordingly, throughout, we assume that one has a vacuum AMSI spacetime (M,g).
For conceptual clarity, let ζ denote a coordinate such that in a neighbourhood of H one has Ω = ζ.
It follows then that the metric q ≡ ζ 2 g + ζ −4 dζ ⊗ dζ , has a smooth limit as ζ → 0. From this assumption, it follows that q can be written as withq independent of ζ and,q(∂ ζ , ·) = 0. The tensorq has smooth components such that q = o(ζ).
The statement (25) can be regarded as a zeroth-order Taylor expansion of the metric q with respect to ζ. From the above, it follows that the physical metricg has, close to H, the form Following the analogy of Minkowski spacetime, define the conformal metric g ≡ Ω 2g , so that one has g = − 1 ζ 2 dζ ⊗ dζ + q +q), -cf. the line element (22) for the conformal Minkowski spacetime in hyperboloidal coordinates.
Remark 17. As in the case of Minkowski spacetime, the conformal metric (26) is singular at ζ = 0. Dealing with this singular behaviour will be the main challenge in the subsequent analysis of this section.
Remark 18. As it will be seen, the metric (26) can be further specialised by choosing coordinates on H so thatq = ℓ = dχ ⊗ dχ − cosh 2 χσ, the metric of the unit timelike hyperboloid.

The conformal constraint equations on H
In this section, we discuss the implications of Definition 2 on the conformal Einstein constraint equations (9a) (9h) when evaluated on the asymptote H. In this way, we systematically recover the conditions satisfied by the gravitational field on the hyperboloid as discussed in [4].
In order to evaluate the conformal Einstein constraints on H, we first consider the equations (9a)-(9h) on timelike hypersurfaces for which the conformal factor Ω = ζ is constant. On these hypersurfaces, one has, in adapted coordinates, that D a ζ = 0 and D a D b ζ = 0. Following the discussion in [4], define ≡ N a ∇ a Ω.
Note that the unit normal is related to Ashtekar's normal by n a = νN a with ν ≡ Ω − 1 2 = O(Ω) i.e. as ζ → 0, we have |ν| ≤ M |Ω|, where M is a positive constant. From (8), we have Σ = ν . Taking the limit of equations (9a)-(9h) as ζ → 0, one obtains the following equations on H: As in the previous section, the symbol ≃ is used to indicate an equality which holds at H, and these equations are to be understood as the limit as ζ → 0. For example, equation (27a) is to be understood as lim ζ→0 D a = 0.
The conformal field equations imply an additional constraint D a D b Ω ≃ 0, which is satisfied identically on H. In addition to the above relations, one can also consider the Gauss-Codazzi and Codazzi-Mainardi equations on H. Assuming vacuum, these give the relations where l ab denotes the Schouten tensor of the metric q ab and one uses q ab to raise and lower indices on H.
A direct consequence of the above relations is that is constant on H. Moreover, from the above relations, one obtains the following: Lemma 2. For a spacetime (M,g) satisfying Definition 2, it follows that on the asymptote H, one has the relation r ab ≃ 2 q ab .
In other words, (H, q) is an Einstein space.
Proof. Starting from the transformation rule for the Schouten tensor contracting with q a c q b d and using K ab = −q a c q b d ∇ c n d and ∇ c Ξ∇ c Ξ = Ξ 2 , the relation between L ab andL ab can be written as The first term on the right-hand side vanishes for a spacetime satisfying Definition 2. Substituting into the Gauss-Codazzi equation (28a) and using K ab ≃ − 1 2 q ab , we get l ab = q ab .
Then, defining the 3-dimensional Schouten tensor as l ab ≡ r ab − 1 4 rq ab , we can confirm that r ab = 4 q ab . Making use of the conformal gauge freedom, can be redefined so that r ab = 2 q ab .
In addition, one has the following peeling-type behaviour for the components of the Weyl tensor: Cylinder at Spatial infinity (R × S 2 ,q) ς ω Proof. Starting with the definition of the electric and magnetic part of the Weyl tensor where * C acbd is the left Hodge dual of C acbd . Using the decomposition of the curvature tensor (2) and after a lengthy calculation, one can write where L n denotes the Lie derivative in the direction of the unit normal n a . Then making use of the transformation law of the Schouten tensor and the fact thatL ab = 0 and taking the limit as ζ → 0 and again using K ab ≃ − 1 2 q ab , it can be shown that lim ζ→0 E ab = 0 and lim ζ→0 B ab = 0.

Relating a general asymptote to Friedrich's cylinder
The purpose of this section is to show that, as in the case of Minkowski spacetime, the 3dimensional Lorentzian manifold (H, q) is conformally related to the 3-dimensional Einstein Universe (Friedrich's cylinder) (R × S 2 ,q ≡ dτ ⊗ dτ − σ).
In the following, we assume one is given an asymptote H as given by Definition 2. From Lemma 2, it follows that the intrinsic Ricci tensor r ij can be written as is constant on H. In order to find the conformal transformation between q andq, we proceed in two steps: (i) First, we show that the intrinsic metric q ij is conformally related to the 1 + 2-dimensional Minkowski spacetime metricq ij ; and (ii) use the fact that the 1 + 2dimensional Minkowski spacetime is conformally related to the 3-dimensional Einstein cylinder. The composition of these two conformal rescalings gives the relation between q andq -see Figure  4.

From the Einstein metric to 1 + 2-Minkowski spacetime
We begin by observing that if q is conformally related to the 1 + 2-Minkowski metricq via a rescaling of the form q = ς 2q , then the Schouten tensors of q andq are related via Now, defining then we can rewrite equation (29) as Given thatl ij = 0, one finds that Multiplying by q il and expanding the covariant derivative in local coordinates x = (x α ), one ends up with the expression This is an overdetermined partial differential equation for the components of the covector α.
To ensure the existence of a solution to this equation, we make use of Frobenius theorem -see e.g. [8], Appendix C. Specifically, for a system of partial differential equations of N dependent variables given in terms of n independent variables . . , n, A = 1, ..., N, the necessary and sufficient condition to find a unique solution α A = α A (x) is given by In the case of equation (31), one has that n = 3 and N = 3. Thus, making the identification ψ A α → ∂ α α A and after a lengthy calculation, we find that equation (32) can be rewritten as with r l kji denoting the components of the Riemann tensor of q. Now, in 3 dimensions, the Riemann tensor is completely determined by the Schouten tensor -more precisely Using this expression for the Riemann tensor, together with the condition l ij = q ij , valid for an Einstein space, in equation (33), we find that the integrability condition is automatically satisfied. Accordingly, there exists a solution α α to equation (31). From equation (30), one can show that D [i α j] = 0, so α j is a closed covector. Hence, it is locally exact. Thus, one can guarantee the existence of a conformal factor locally relating the 3-dimensional Lorentzian metric q and the 1 + 2-Minkowski metricq.

From the 1 + 2-Minkowski spacetime to Friedrich's cylinder
The next step is to find the conformal factor relating the 1 + 2-Minkowski to the cylinder at spatial infinity. Starting with the 1 + 2-Minkowski metric in the form we can write this metric in terms of double null coordinates u ≡ t − r and v ≡ t + r as If we introduce the two coordinate transformations whereq is the metric of Friedrich's cylinder and ω is given by ω ≡ 2 cos U cos V ≡ cos T + cos R.

Summary
The discussion of the previous subsections can be summarised as follows: The metric q of an asymptote H satisfying Definition 2 is conformally related to the standard metric of Friedrich's cylinder R × S 2 .
In the following, we make this relation more precise by recasting the neighbourhood of the asymptote H in terms of the F-gauge in which the cylinder at spatial infinity is described.

Conformal Gaussian gauge systems in a neighbourhood of an asymptote
In this section, we provide the main analysis of the article: the construction of a conformal Gaussian system in a (spacetime) neighbourhood of the asymptote H satisfying Definition 2. More precisely, we show that Definition 2 provides enough regularity in the conformal geometric fields to run a stability argument to show that the neighbourhood of the asymptote H can be covered by a non-intersecting congruence of conformal geodesics extending up to null infinity (and beyond). This congruence is used, in turn, to build on top a representation of spatial infinityà la Friedrich.
The construction proceeds in various steps and mimics, to some extent, the analysis of the conformal extension of static and stationary spacetimes given in [12,1].

A compactified version of the asymptote H
The construction of a conformal Gaussian system is based on the conformal representation of the asymptotic region of the spacetime (M,g) given by the metric (26). Moreover, using arguments similar to those used in Section 4.4 it can be assumed, without loss of generality, that the metric q is, in fact, the metric of the unit hyperboloid ℓ. Accordingly, in the following, we consider a metric of the form In order to construct a conformal Gaussian system in a neighbourhood of the asymptote H, it is convenient to consider a representation in which the time dimension has a compact extension.
There are several ways of doing this; however, for the present purposes, probably the most convenient approach is to mimic the discussion of the relation between the F-gauge and hyperboloid representations of spatial infinity for Minkowski spacetime given in Section 3.4.2.
In the following introduce new coordinates (τ ,ρ,θ A ) in the line element (34) via the relations This coordinate transformation is formally identical to the one used in Section 3.4.2, but the geometric interpretation does not follow through as the vector field ∂ τ is no longer tangent to a congruence of conformal geodesics. Observe thatτ = 0 if and only if χ = 0. A computation then shows that g = ϖ 2η +q, This suggests introducing a new conformal metricg vià whereη is the expression for the Minkowski metric in the F-gauge, equation (14), with the replacements τ →τ and ρ →ρ. Observe that In view of the above expression, we make further assumptions independently of Definition 2: has a suitably regular limit asτ → ±1 (i.e. χ → ±∞).
(ii) Moreover, it is required that Remark 19. The above assumption is, in fact, a statement about the regularity of the conformal metric in the sets where spatial infinity meets null infinity. A programme to analyse this issue has been started in [10] -see also [29], Chapter 20 for further discussions on the subject.
Remark 20. Condition (ii) above ensures thatg(dτ , dτ ) = 0 for τ = ±1, so that the conformal boundary is null as it is to be expected. Suitably regular in the present context means that the limit is assumed to be sufficiently regular for the subsequent argument to hold. The analysis of how stringent these conditions are or how this can be encoded goes beyond the scope of this article.

Regularisation of the conformal fields
The metricg shares with g the property of being singular at the asymptote H. In order to deal with this difficulty, we follow the spirit of the analysis of [12], Section 6, and consider a frame {c a }, a = 0, 1, +, −, with where (∂ + , ∂ − ) is a complex null frame on S 2 . The associated coframe {α a } is given by In terms of this frame, the standard round metric of S 2 is given by Clearly, ⟨α a , c b ⟩ = δ b a . However, notice that {c a } is notg-orthogonal. In the following, we ignore the complications arising from the fact that there is no globally defined basis over S 2 as there are standard methods to deal with them -see e.g. [11,1,16]. The componentsg ab ≡g(c a , c b ) ofg with respect to the frame {c a } are given bỳ Similarly, definingg ab ≡g ♯ (α a , α b ) one finds that In the following, letΓ a b c ≡ ⟨ω b ,∇ a c c ⟩ denote the connection coefficients of the Levi-Civita connection of the metricg with respect to the frame {c a }, andL ab the components of the associated Schouten tensor. A detailed computation leads to the following: The details of the proof of the above result can be found in Appendix B.

Analysis of theg-conformal geodesic equations
In the following, we consider the conformal geodesic equations for the metricg in terms of the basis {c a } and study their solution in a neighbourhood of the asymptote H with the aim of establishing the existence of a conformal Gaussian system.

The equations
The conformal geodesic for the metricg is given by a spacetime curve x µ (τ ) = τ (τ ),ρ(τ ),θ A (τ ) with tangent vector v(τ ) and a 1-formβ(τ ) along the curve such that: where equation (36a) gives the definition of the tangent vector. In order to analyse these equations, Using the above expansions, equation (36a) gives the components From equations (36b) and (36c) one gets

The initial data
Following the discussion of the conformal geodesics on Minkowski spacetime, in the sequel, the initial data of the curve x µ (τ ) is chosen so thatẋ isg-normalised and orthogonal to the hypersurface S ⋆ ≡ {τ = 0}.
The normal covector to this hypersurface is given by dτ . Then, it follows that the unit normal vector is given byǹ Thus, on S ⋆ , one has thatǹ .)) .

Remark 21.
By varying the starting point p ∈ S ⋆ subject to the conditionρ ⋆ (p) ∈ [0,ρ • ) ensures that a neighbourhood of H can be covered by conformal geodesics. In the next subsection, it will be shown that, possibly by reducingρ • this congruence is non-intersecting.
Remark 22. The curves of the congruence extend beyond I ± as their existence interval extend beyond [−1, 1].

The construction of the conformal Gaussian system
In this section, we discuss how the congruence of conformal geodesics obtained in Lemma 5 implies the existence of a conformal Gaussian system in a neighbourhood of the asymptote H.

General considerations
Given a point p ∈ S ⋆ with coordinates x(p) = (ρ(p),θ A (p)) such thatρ(p) <ρ • , these coordinates are propagated off S ⋆ by requiring them to be constant along the unique conformal geodesic with data as given in Lemma 6 passing through p. In order to differentiate between the propagated coordinates and those on S ⋆ , we denote the former by (ρ, θ A ). Points along the conformal geodesic passing through p are labelled using the parameter τ of the curve. In this way, by varying the point p and as long as the congruence of conformal geodesics is non-intersecting, one obtains conformal Gaussian coordinatesx = (τ, ρ, θ A ).

Analysis of the Jacobian
In order to ensure that the collection (τ, ρ, θ A ) gives rise to a well-defined coordinate system in a neighbourhood of H, we need to consider the Jacobian determinant of the change of coordinates (τ, ρ, θ A ) → (τ ,ρ,θ A ).
In order to ease the presentation in the following, we restrict the discussion to the transformation between non-angular coordinates (τ ,ρ) → (τ, ρ). The full analysis follows in a similar manner at the expense of lengthier computations. Writing τ =τ (τ, ρ),ρ =ρ(τ, ρ), the associated Jacobian determinant is given by In other words, the partial derivative ∂ρ/∂ρ is constant on H. To evaluate the constant, it is observed that by constructionρ = ρ on S ⋆ . Accordingly, one concludes that It then follows from the above argument that By continuity, it follows that the Jacobian determinant is non-zero in a neighbourhood of H.
In order to ensure that this condition holds, it may be necessary to reduce the constantρ • in Lemma 6.
Remark 24. It follows from the fact that the Jacobian of the transformation between the coordinates (τ ,ρ,θ A ) and (τ, ρ, θ A ) is non-zero that the congruence of conformal geodesics in a neighbourhood of H is non-intersecting as the intersection of curves would indicate a singularity in the coordinate system given by (τ, ρ, θ A ).

The conformal factor associated with the congruence
The existence of a congruence of conformal geodesics in a neighbourhood of the asymptote H ensures that the family of conformal factors given by Proposition 1 is well defined. This, in turn, completes the construction of a conformal Gaussian system in a neighbourhood of H. Consistent with the discussion of the initial data for the congruence of conformal geodesics in Subsection 5.3.2, one has that Θ ⋆ = Ω ⋆ = ρ.
Moreover, from (39) and (40), it follows thaṫ ForΘ ⋆ , leth ♯ denote the intrinsic contravariant metric on S ⋆ , written explicitly as Then, one hasΘ The initial data forβ impliesh Hence,Θ Observe that Then, the canonical conformal factor Θ, defined in the neighbourhood of H where the congruence is non-intersecting, is given by The conformal boundary at Θ = 0 can be identified by the conditions Solving for τ gives .
It can be shown that so that τ → ±1 as ρ → 0. Now, let .
Thus, one defines, in analogy to the case of Minkowski spacetime in the F-gauge (see Section 3.2) the set Finally, future and past null infinities I ± can be defined as Remark 25. The set I (the cylinder at spatial infinity) coincides with the asymptote H.

Remark 26.
Observe that in contrast to the representation of Minkowski in Section 3.2, the representation obtained in this section is not horizontal in the sense that the location of null infinity is not given by the condition τ = ±1. A horizontal representation can be obtained by considering a more general initial conformal factor of the form Θ ⋆ = κρ, κ| ρ=0 = 1, with κ suitably chosen.

Main statement
We summarise the discussion of the previous subsections in the following theorem, which is a more detailed version of the statement presented in the introductory section: A The basic stability theorem for ordinary differential equations depending on a parameter In the following let X ′ = F(t, X, λ), denote an initial value problem for a N -dimensional vector-value ordinary differential equation depending on a parameter λ. One has the following result -see [19], Theorem 2.1 in page 94 and Corollary 4.1 in page 101.
Moreover, if F(t, X, λ) is of class C m , m ≥ 1 on E, then X(t, t ⋆ , X ⋆ , λ) is of class C m on its domain of existence.
So, it follows from the above theorem that if F satisfies the conditions of the theorem above, and the problem X ′ = F(t, X, 0), has a solution with existence interval t ∈ (ω − , ω + ), then any other solution to (42) with λ sufficiently close to 0 will have, at least the same existence interval. In the above statement the functions ω + and ω − can take the values ∞ and −∞.
In the following, it will be shown that the connection coefficientsΓ a b c are regular atρ = 0. The analysis of the connection coefficients requires further specification of the coordinates. In the following, we will make use of Gaussian coordinates of the asymptote H -that is, the coordinates (τ ,θ A ) on H are propagated on the bulk of the spacetime using the relation: where ∂ i ≡ {∂τ , ∂θ A }. Using this condition, one can readily verify that The other components of the connection coefficients are not determined by the gauge condition (44) but can be directly computed using Kulkarni's formula (43). One finds that Thus, the connection coefficientsΓ a b c are regular with respect to the coordinates (τ ,ρ) in a neighbourhood of H.

B.0.2 The components of the Schouten tensor
The Schouten tensorsL ab andL ab are related by the formulà L ab −L ab = − 1 Θ∇ a∇b Θ + 1 2Θ 2∇ c Θ∇ c Θg ab , where Θ ≡ρ(1 −τ 2 ) is the conformal factor relatingg andg. In view of the Einstein field equations, we have thatL ab = 0 so that Remark 27. The previous expression is singular atρ = 0. We need to compute the components L ab with respect to {c a }.
A direct computation shows that Thus, the first term on the right-hand side of (45) is regular. Moreover, so that the second term in (45) is regular. Hence, the componentsL ab are regular with respect to the coordinates (τ ,ρ) in a neighbourhood of H. In conclusion, the components ofL ab coincide, on H, with those of the Schouten tensor of Minkowski spacetime in the F-gauge.

C Conformal geodesics on Minkowski spacetime
In this section, we show that the vector ∂ τ in the F-gauge representation of the Minkowski spacetime of Section 3.2 is a conformal geodesic -see Lemma 1. Thus, it should satisfy the equations∇ In particular, one has thatη (∂ τ , ∂ τ ) = 1.
Equation (46b) can be shown to be satisfied identically.