Calculation of asymptotic charges at the critical sets of null infinity

The asymptotic structure of null and spatial infinities of asymptotically flat spacetimes plays an essential role in discussing gravitational radiation, gravitational memory effect, and conserved quantities in General Relativity (GR). Bondi, Metzner and Sachs (BMS) established that the asymptotic symmetry group for asymptotically simple spacetimes is the infinite-dimensional BMS group. Given that null infinity is divided into two sets: past null infinity I− and future null infinity I+, one can identify two independent symmetry groups: BMS− at I− and BMS+ at I+. Associated with these symmetries are the so-called BMS charges. A recent conjecture by Strominger suggests that the generators of BMS− and BMS+ and their associated charges are related via an antipodal reflection map near spatial infinity. To verify this matching, an analysis of the gravitational field near spatial infinity is required. This task is complicated due to the singular nature of spatial infinity for spacetimes with non-vanishing ADM mass. Different frameworks have been introduced in the literature to address this singularity, e.g. Friedrich’s cylinder, Ashtekar-Hansen’s hyperboloid and Ashtekar-Romano’s asymptote at spatial infinity. This paper reviews the role of Friedrich’s formulation of spatial infinity in the investigation of the matching of the spin-2 charges on Minkowski spacetime and in the full GR setting. This article is part of a discussion meeting issue ‘At the interface of asymptotics, conformal methods and analysis in general relativity’.


Introduction
In classical General Relativity (GR), isolated systems are commonly described by asymptotically flat spacetimes, with a metric approaching the Minkowski metric far from the source.In this setting, the influential work of R. Penrose [1,2] offers a geometrical approach to the studies of isolated systems.In particular, Penrose's notion of "asymptotic simplicity" identifies spacetimes with a conformal extension similar to that of Minkowski spacetime, implying the existence of null infinity I , comprised of two disjoint sets: future null infinity I + and past null infinity I − .The universal fields shared by asymptotically simple spacetimes allow us to identify the infinite-dimensional asymptotic symmetry group known as the BMS group [3], named after Bondi, Metzner and Sachs.However, Penrose's notion of asymptotic simplicity is not concerned with the behaviour of the gravitational field at spatial infinity, which is a crucial ingredient in the discussion of conserved quantities in GR [4].
One of the challenging aspects of studies of the asymptotic structure at spatial infinity is the singular conformal structure at spatial infinity i 0 for spacetimes with non-vanishing Arnowitt-Deser-Misner (ADM) mass.Different formulations of spatial infinity [5,6,7] can be used to resolve the structure of the gravitational field in this region.Of particular importance to this article is Friedrich's formulation of spatial infinity, initially introduced in [7] with the goal of obtaining a regular initial value problem at spatial infinity for the so-called conformal Einstien field equations.This representation of spatial infinity is linked to the conformal properties of spacetimes, and it introduces a blow-up of the spatial infinity point i 0 to a cylinder (−1, 1) × S 2 commonly known as the cylinder at spatial infinity I.The cylinder I touches the endpoints of past and future null infinities I ± at the critical sets I ± = {±1} × S 2 .This representation of spatial infinity is useful for relating quantities at the critical sets I ± to initial data on a Cauchy hypersurface --see, e.g, [8,9].Other equally significant contributions to the studies of the asymptotic structure at spatial infinity are Ashtekar-Hansen's and Ashtekar-Romano's formulations of spatial infinity -see [5,6].While the relation between Friedrich's and Ashtekar-Romano's formulations was established in [10], the link between Friedrich's formulation and Ashtekar-Hansen's remains unexplored.Ashtekar-Hansen's and Ashtekar-Romano's formulations introduce the Spi group, denoting the infinite-dimensional asymptotic symmetry group at spatial infinity, with a structure similar to the BMS group at null infinity.
One physical motivation for studying symmetries is the prospect of defining conserved quantities (also known as Noether charges or simply charges) in an isolated system, e.g., energy, momentum and angular momentum.For fields on a fixed background, conserved quantities are defined by considering the integral of the local energy-momentum tensor contracted with Killing fields of the background spacetime over a Cauchy hypersurface.Given the dual role of the spacetime metric in GR, describing both geometrical and physical aspects of the theory, one can only define such conserved quantities in the asymptotic limit, where the background and physical fields can be studied separately -see [11].As noted in [12], there had been a clear distinction in earlier work in defining conserved quantities associated with asymptotic symmetries at null and spatial infinity.In particular, the Hamiltonian formulation was prominently used in the derivation of conserved quantities at spatial infinity [13,14,15] compared to null infinity.The challenge in defining "conserved quantities" at null infinity using a standard Hamiltonian formulation is that symplectic current can be radiated away at null infinity, and thus, generically, there exists no Hamiltonian generating BMS transformations.However, the general prescription in [12] allows one to define charges associated with asymptotic symmetries even in situations where the Hamiltonian does not exist -see also [16,17].This prescription was used in [18] to derive explicit expressions for the charges associated with BMS symmetries at null infinity.Finally, note that "conserved quantities" at null infinity are not exactly conserved for general dynamical spacetimes.Instead, a charge associated with a BMS symmetry will have a non-vanishing flux through null infinity I ± [12].
The discussion in this article is motivated by a recent conjecture by Strominger [19] suggesting that BMS symmetries and their associated charges can be linked to soft theorems [20,19,21] and the gravitational memory effect [22,23,24].This link is based on the so-called matching problem, i.e., the idea that the BMS groups at past and future null infinities (BMS − at I − and BMS + at I + ) can be matched by an antipodal reflection map near spatial infinity.The matching of these symmetries leads to a global diagonal asymptotic symmetry group in GR, implying that the incoming flux of an asymptotic charge at I − would be equal to the outgoing flux of the corresponding charge at I + .Generically, the matching of BMS + and BMS − and their associated charges requires an analysis of the gravitational field and the charges near spatial infinity.One significant challenge in this analysis is the singular nature of the conformal structure at spatial infinity i 0 , further highlighting the importance of the different representations of spatial infinity [5,25,6,7] in the discussion of the matching problem.In recent years, numerous articles discussed the asymptotic symmetry group at spatial infinity [26,27,28,29,30,31] and their matching with the asymptotic charges at null infinities [26,32,33,34,35,36].On Minkowski spacetime, the matching of supertranslation asymptotic charges has been investigated for the spin-1 field in [26] and the spin-2 field in [32].For spacetimes satisfying Ashtekar-Hansen's definition of asymptotic flatness at null and spatial infinity (see [5] or [34] for precise definition), the matching of supertranslation asymptotic charges for the spin-1 and the gravitational field was shown in [33,34].Moreover, the matching for Lorentz charges was also investigated in [35] using Ashtekar-Hansen's formulation of spatial infinity.
The purpose of this paper is to provide a streamlined presentation of the calculation of asymptotic charges at the critical sets and their matching using Friedrich's formulation of spatial infinity.The expressions of the asymptotic charges used in this article are adapted from [34], which agrees with the general prescription of conserved quantities given in [12].The use of Friedrich's formulation allows us to express the supertranslation asymptotic charges at I ± in terms of initial data given on a Cauchy hypersurface, and to show that the matching of the asymptotic charges at I ± follows from certain regularity conditions on the free initial data.The full analysis of the asymptotic charges in a full GR setting using Friedrich's formulation will be presented elsewhere.However, the main results can be summarised as follows: For the generic initial data set given in [37], the asymptotic charges (as defined in [34]) associated with BMS supertranslation symmetries are well-defined at I ± if and only if the initial data satisfy extra regularity conditions.The regularity conditions can be imposed on the free conformal initial data.Finally, given initial data that satisfy the regularity conditions, the asymptotic charge Q l,m associated with a given harmonic Y l,m at I + is related to the corresponding asymptotic charge at I − by: The structure of this paper is as follows: In Section 2, the calculation of the spin-2 supertranslation asymptotic charges on Minkowski spacetime using Friedrich's formulation is reviewed.We start by introducing Friedrich's representation and Friedrich-gauge (F-gauge) of spatial infinity on Minkowski spacetime in Section 2.1.Since the asymptotic charges are expressed in terms of the so-called Newman-Penrose gauge (NP-gauge), Section 2.2 provides a brief discussion of the transformation from the NP-gauge to the F-gauge.Finally, Section 3 presents a brief description of the tools and techniques used to analyse the supertranslation asymptotic charges in full GR.

Notations and conventions
This article will use tensors and spinors separately in various calculations.The following indices will be used: • a, b, c, . ..: spacetime abstract tensorial indices.
The components of a tensor T ab with respect to a tensorial frame {e a } are defined as Similarly, if {o, ι} is a spin basis defined by then the components of a spinor ξ A with respect to the spin frame {ϵ A } are given by The spin basis {o, ι} satisfies o, ι = 1, where ., . is the antisymmetric product defined by Here, ϵ AB is the antisymmetric ϵ-spinor that can be regarded as a raising/lowering object for spinor indices.

The spin-asymptotic charges on Minkowski spacetime
In this section, we summarise the calculation of supertranslation asymptotic charges for the spin-2 field as presented in [38].Given the role of Friedrich's representation of spatial infinity in this calculation, let us begin by introducing the F-gauge on Minkowski spacetime.

The Minkowski spacetime in the F-gauge
To introduce the F-gauge on Minkowski spacetime (R 4 , η), start with the Minkowski metric η in the standard Cartesian coordinates . This metric can be written in terms of the standard spherical coordinates where θ A is a choice of coordinates on S 2 and σ is the standard round metric on S 2 .Now, define X2 ≡ ηµν xµ xν = t2 − ρ2 , where x0 ≡ t and ρ2 ≡ (x 1 ) 2 + (x 2 ) 2 + (x 3 ) 2 .Then, it is clear to see that spatial infinity is contained in the domain D (See Figure 1) defined as The conformal metric η = Ξ 2 η, with Ξ = X−2 produces a point compactification of the physical spacetime (R 4 , η), where all the points at infinite spatial distances in the physical spacetime (R 4 , η) are mapped to the spatial infinity point i 0 in (R 4 , η).The metric η can be written explicitly as where To introduce Friedrich's blow-up of spatial infinity, define a new time coordinate τ = t/ρ and the rescaling From this, the relation between η and η can be written as In this representation, one sees that the spacetime metric η is singular at ρ = 0 while the intrinsic metric on the ρ = const.hypersurfaces have a well-defined limit as ρ → 0 and is given by Given the above, define the conformal extension (M, η) with then introduce the following subsets of the conformal boundary (Θ = 0) -see Figure 2.
past and future null infinity , the cylinder at spatial infinity the critical sets at of null infinity and where I 0 is the intersection of I with the initial hypersurface S * ≡ {τ = 0}.In subsequent discussions, we will refer to (I, q) as the cylinder at spatial infinity.Moreover, it will be convenient to introduce a frame basis {e a } adapted to Friedrich's cylinder at spatial infinity on Minkowski spacetime, the so-called F-gauge frame.Start with the Minkowski metric η given by (1).It is straightforward to see that the metric on the hypersurfaces Q τ,ϱ of constant ρ and τ is the standard metric on S 2 .Then, introduce the complex null frame Now, the F-gauge frame {e AA ′ } and their dual {ω AA ′ } can be defined as follows where e AA ′ is obtained from e a by contraction with the Infeld-van der Waerden symbols σ a AA ′ .So, e AA ′ ≡ σ a AA ′ e a .The dual frame ω ± satisfy In terms of the above frame fields, the metric η can be written as

Remark 1. A special class of conformal curves, known as conformal geodesics, play a major role in
Friedrich's representation of spatial infinity -see [10] for details.For example, on Minkowski spacetime, the curves of constant ρ and θ A describe a non-intersecting congruence of conformal geodesics on M, suggesting that the congruence of conformal geodesics starting at ρ = 0 coincide with the cylinder at spatial infinity.One of the remarkable properties of conformal geodesics is that they specify a canonical conformal factor Θ [39,40], and an F-gauge metric g given by where g is the metric on a vacuum Einstein spacetime ( M, g).On Minkowski spacetime, the canonical conformal factor associated with the above-mentioned congruence of conformal geodesics on Minkowski is equivalent to Θ given in (2).
Further discussion of Friedrich's blow-up of spatial infinity and the F-gauge on more general spacetimes will be postponed for later sections.

NP-gauge to F-gauge
As mentioned in the introduction, the supertranslation asymptotic charges at I ± are generally expressed in terms of the NP-gauge, comprised of certain conformal gauge conditions, certain coordinates and an orthonormal frame field {e • a } satisfying certain frame gauge conditions.A general description of these gauge conditions was given in [8], along with a prescription of the transformation between the NP-gauge and the F-gauge.The main observation in [8] is that the NP-gauge frame is adapted to null infinity I ± while the F-gauge frame is adapted to Cauchy hypersurfaces.Additionally, the NP conformal gauge conditions, as described in [8], imply that a metric g • satisfying those conditions will be related to the F-gauge metric g by with θ satisfying a linear ordinary differential equation, which can be solved on the generators of I ±see [8].Given the above, the NP-gauge orthonormal frame {e The general framework in [8] was used in [9] to obtain an explicit transformation between the NP-gauge and the F-gauge on Minkowski spacetime.In particular, the results in [9] show that θ = 1 and where ω is an arbitrary real number that encodes the spin rotation of the frames on S 2 .Given the above transformation between the NP-gauge and the F-gauge on Minkowski spacetime, the asymptotic charges can be evaluated on the critical sets I ± given solutions for the spin-2 field equations.

The spin-2 charges in the F-gauge
The goal of this section is to obtain an expression of the supertranslation asymptotic charges that can be evaluated at the critical sets I ± .First, introduce the NP null tetrad (l a , n a , m a , ma ) as Then, let W • abcd denote a Weyl-like tensor, i.e., a tensor with symmetries of the Weyl tensor, and define abcd is the left Hodge dual of W • abcd .Then, the spinorial counterpart of W • abcd can be decomposed in terms of the symmetric spin-2 spinor ψ • ABCD as Following the discussion in [34], the asymptotic charges associated with smooth functions λ on S 2 can be written as where C denotes a cross-section of I ± .From ( 7) and ( 8), it can be shown that the charges Q can be written as To evaluate the charges at the critical sets, one must obtain an expression for Q in terms of the F-gauge.The transformation from the NP-gauge to the F-gauge on Minkowski spacetime, discussed in the previous section, implies that where ψ 2 ≡ ψ 0011 .Thus, the final expression of the charges in the F-gauge is given by To evaluate this expression at I ± , the next step is to obtain a solution for ψ2 using the field equations.

The spin-2 field equations
The spinorial spin-2 field equation can be written as ∇ A A ′ ψ ABCD = 0. Applying −2∇ E A ′ , the wave equation satisfied by ψ ABCD can be written as where □ ≡ ∇ AA ′ ∇ AA ′ is the D'Alembertian operator.To analyse the solutions for this equation in a neighbourhood of spatial infinity, assume that the components ψ n of the spin-2 spinor can be expanded near ρ = 0 in terms of spin-weighted spherical harmonics n Y l,m as where a n;l,m : R → C.
Remark 3. The expansion (11) is consistent with the estimates developed in [41] that demonstrates that for a certain non-trivial class of initial data, the components ψ n can be expanded as In the above, the coefficients ψ Using (11) and substituting in (10), one obtains second order ordinary differential equations for the coefficients a n;l,m (τ ) (1 − τ 2 )ä 2;l,m − 2τ ȧ2;l,m + l(l + 1)a 2;l,m = 0, Given that the expression of the charges ( 9) is written in terms of ψ 2 , one only requires the solution for (12c) in order to evaluate Q at τ = ±1.Equation (12c) is a Jacobi differential equation, with a solution that can be expressed in terms of hypergeometric functions -see [42].However, a simpler expression can be obtained by using the differential equation solver of the Wolfram Language.In particular, it can be shown that for l ≥ 0 and −l ≤ m ≤ l, the solution a 2;l,m is given by where P l (τ ) is the Legendre polynomial of order l and Q l (τ ) is the Legendre function of the second kind of order l.The constants A l,m and B l,m can be expressed in terms of the initial data for a 2;l,m .Since Q l (τ ) is proportional to ln (1 + τ ) and ln (1 − τ ), the solution (13) will diverge logarithmically near τ = ±1, unless B l,m = 0.
To obtain a well-defined solution for a 2;l,m at the critical sets, the constant B l,m is required to vanish.Note that B l,m can be written in terms of a 2;l,m (0) and ȧ2;l,m (0) as where Γ denotes the Gamma function.Then, the observation that the coefficient of a 2;l,m (0) vanishes for even l while the coefficient of ȧ2;l,m (0) vanishes for odd l suggests the following regularity conditions: Lemma 1.The solution (13) is well-defined at I ± if and only if: 1. a 2;l,m (0) = 0 for odd l, and 2. ȧ2;l,m (0) = 0 for even l.
These regularity conditions can be expressed in terms of freely specifiable data as shown in [38].Making use of ( 9), ( 11) and ( 13) and by choosing initial data satisfying Lemma 1 and λ = Y l,m , the charge Q l,m associated with Y l,m at I ± can be written as where (a n ) * ≡ a n;l,m (0).The main conclusions from the above discussion are 1.For generic boosted initial data, the charges Q are not well-defined in the limits of spatial infinity, i.e., at the critical sets I ± .
2. Boosted initial data satisfying Lemma 1 allows us to obtain well-defined expressions for Q at the critical sets.
3. The antipodal matching of the charges is obtained naturally in this formalism.In particular, we have

The asymptotic charges in full GR
The process of the calculation of the asymptotic charges for the spin-2 field at the critical sets presented in the previous section can be extended to the full GR setting.For this, assume that ( M, g) is a spacetime satisfying the vacuum Einstein field equations i.e.
where Rab is the Ricci tensor associated with the Levi-Civita connection ∇ of g.The conformal rescaling implies transformation laws for the physical fields e.g. the curvature tensor Ra bcd , the Schouten tensor Lab etc.It follows that the vacuum Einstein field equations are not conformally invariant and that the field equations implied by (17) cannot be analysed at the conformal boundary Ξ = 0 since the conformal Ricci tensor R ab is singular at the points where Ξ = 0.If Ca bcd denotes the Weyl tensor, then the Bianchi identity can be written in terms of the Levi-Civita connection associated with g as If one defines the rescaled Weyl tensor d a bcd ≡ Ξ −1 Ca bcd , then equation ( 18) can be written as Exploiting the symmetries of the rescaled Weyl tensor implies Our calculations of the asymptotic charges rely on Friedrich's extended conformal field equations [43,44,45,39] written in terms of a Weyl connection ∇ satisfying where f a is an arbitrary 1-form.The explicit form of these equations will not be necessary for this article, interested readers can refer to Chapter 8 in [39].The extended conformal field equations yield differential equations to be solved for the g-orthonormal frame fields {e a }, the components of the Weyl connection coefficients Γa b c , the Schouten tensor Lab and the rescaled Weyl tensor d a bcd .One significant feature of the extended conformal field equations is that they exhibit gauge freedom indicated by the fact that there are no equations to fix the conformal factor Ξ and the Weyl connection ∇.To fix this gauge freedom, one can make use of the so-called conformal Gaussian gauge, based on conformal geodesics, that allows us to write the field equations as a symmetric hyperbolic system in which the evolution equations reduce to a transport system along the conformal geodesics.Given the field equations in this gauge, it is possible to obtain a spinorial version of these equations to be analysed near spatial infinity.
The above discussion highlights one of the key tools of conformal methods in GR.The following section will introduce Friedrich's regular initial value problem for the conformal field equations.

Friedrich's regular initial value problem
The purpose of this section is to briefly introduce Friedrich's formulation in full GR.As mentioned in the introduction, the aim of Friedrich's formulation is to introduce a regular initial value problem for the conformal field equations near spatial infinity.An extensive discussion of this framework is provided in [7,8].In this framework, the spacetime ( M, g) is assumed to be the development of some asymptotically Euclidean and regular [11,39] initial data ( S, h, K).In particular, the initial data ( S, h, K) is said to be an asymptotically Euclidean and regular manifold if there exists a 3-dimensional smooth compact manifold (S ′ , h ′ ) with a point i ∈ S ′ , a diffeomorphism Φ from S ′ \ {i} onto S and a conformal factor Ω ′ which is analytic on S ′ and satisfying i) To apply this, start with the initial data satisfying the Hamiltonian and momentum constraints as introduced in [37]: ), there exists a vacuum initial data set ( h, K) such that the components of h and K with respect to the standard Euclidean coordinate chart (x α ) have the following asymptotics: where A, {B α } 3 α=1 are some constants and r = (x 1 ) 2 + (x 2 ) 2 + (x 3 ) 2 .Then, define the inverse coordinates (y α ) and the conformal factor Ω ′ as so that the components of the conformal initial data h ′ = Ω ′2 h and K ′ = Ω ′ K can be expanded around ϱ = (y 1 ) 2 + (y 2 ) 2 + (y 3 ) 2 = 0 as The O(ϱ) term in (22a) can be made to vanish by performing a coordinate transformation from (y α ) to normal coordinates (z α ) [46].Then, the term O(ϱ 2 ) can be removed by performing a further conformal transformation Here, l ′ αβ (i) denotes the components of the Schouten tensor associated with h ′ in normal coordinates (z α ) evaluated at i(ϱ = 0).If h ′(0) αβ is the metric at i and |z| 2 ≡ h ′(0) αβ z α z β , then the components of the conformal initial data h = ϖ 2 h ′ and K = ϖK ′ can be written as where ϑ α = z α /|z|.The initial data ( h, K) will be referred to as the conformal normal initial data.It can be shown, using the conformal constraint equations, that the initial data for the components of the conformal Schouten tensor Lαβ and the electric and magnetic parts of the Weyl tensor, dαβ and dαβγ , respectively, are singular at |z| = 0. To introduce regular initial data, one must introduce a further conformal rescaling as suggested in [7] with κ = O(|z|).Let ρ = |z|, then the conformal factor Ω can be expanded around ρ = 0 as where Π 3 [Ω] is written in terms of the angular coordinates ϑ α , the constant A, the function α and its derivatives with respect to ϑ α .The conformal rescaling (24) introduces the conformal metric h = κ −2 h.Then, if {e i } is an h-orthonormal frame, one can show and Hence, the initial data (h, K) for the conformal field equations are regular at |z| = 0.One of the advantages of using the conformal Gaussian gauge mentioned in the last section is that it implies a conformal factor Θ that can be written in terms of initial data.Following [40,7], we have where τ refers to the parameter along the conformal geodesics used to construct the conformal Gaussian system and ω = 2Ω |h(dΩ, dΩ)| .
Remark 4. In the following, SU(2, C) refers to the special unitary group of degree 2 over complex numbers.We also use SU(S) to refer to the bundle of normalised spin frames over a manifold S with structure group SU(2, C).
The basic idea of the blow-up of the point i involves replacing i with the space of directions pointing out of i.In other words, the blow-up of i is a certain subspace of the tangent space at i, which is diffeomorphic to S 2 .In Friedrich's formulation, rather than working with tensor frames, the blow-up of i is achieved by considering a certain subset of the bundle of the normalised spin frames SU(S ′ ) with structure group SU(2, C) -see [8] for details.In this picture, the blow-up of i is diffeomorphic to S 3 while its quotient by U(1) is diffeomorphic to S 2 .The extra dimension in this blow-up corresponds to the choice of a phase parameter given that the choice of the spin frame is not unique.More precisely, consider a fixed spin frame {ϵ A } at i and t ∈ SU(2, C), the transformed spin frame ϵ A (t) ≡ t A B ϵ B can be extended to an open ball B a (i) in S ′ of radius a centred at i by parallel propagation along an h-geodesic starting at i.If ρ is the affine parameter along the geodesic, then for a fixed t, the propagated spin frame can be written as with the following subsets past and future null infinity (27a) , the cylinder at spatial infinity (27b) the critical sets of null infinity (27c) and To relate the structures on the fibre bundle to the spacetime manifold ( M, g) satisfying ( 16), let (M, g) denote a smooth conformal extension such that i) Θ > 0 and g = Θ 2 g on M, ii) Θ = 0 and dΘ ̸ = 0 on I ± a .Now let N ⊂ M denote the domain of influence of B a (i) \ {i}, then the projection map π′ from M a,κ to N can be factored as where M ′ a,κ ≡ M a,κ /U(1) is implied by the action of U(1) on SU(2, C).From ( 26), the map π′ 1 is given by the identity on the R × R component, and, under the identification SU(2, C) = S 3 , by the Hopf fibration on the SU(2, C) component.In other words, π′ 1 maps M a,κ onto R × R × S 2 .Finally, note that the spin frames ϵ A (ρ, t) can be extended to the spacetime M a,κ by a certain propagation law along the conformal geodesics orthogonal to the S a , where S a can be thought of as the initial hypersurface on M a,κ , i.e.
The propagated spin frames ϵ A (τ, ρ, t) are determined at any p ∈ M a,κ \ (I ∪ I + ∪ I − ) up to a multiplication factor that corresponds to the action of U(1) on SU(M).
Remark 5. Friedrich's formulation involves encoding the F-gauge conditions in the initial data and the properties of the fields appearing in the conformal field equations.For further discussions of Friedrich's formulation and the F-gauge, readers are referred to [7,8].Remark 6. Fields on M a,κ can be decomposed in terms of complex-valued functions T m j k : SU(2, C) → C, closely related to the standard spin-weighted harmonics on S 2 -see e.g.[8].The analysis of the conformal field equations can be carried out on M a,κ and their solutions can be projected onto R × R × S 2 and used to evaluate BMS asymptotic charges at the critical sets.

The supertranslation asymptotic charges in full GR
To introduce BMS asymptotic charges at I ± , let d • abcd denote the rescaled Weyl tensor in the NP-gauge and introduce the spinorial counterpart d • AA ′ BB ′ CC ′ DD ′ which can be decomposed as follows where ϕ • ABCD is a symmetric valence 4 spinor.Given the above, the asymptotic charges associated with smooth functions f on S 2 can be written as where C is some cross-section of I ± and ε 2 is its area element, σ •ab is the shear tensor, N • ab is the news tensor and Using ( 29) and ( 7), one gets Moreover, the term involving σ •ab N • ab can be written in terms of the NP-connection coefficients [47,48,49], whose explicit form depends on whether we are considering the asymptotic charges at I + or I − .In particular, σ Here, ∆ ≡ n a ∇ • a and σ • , µ • , γ • , λ • , ρ • , ϵ • are the NP-connection coefficients defined as In the above, μ• , γ• , ρ• and ε• refer to the complex conjugates of µ • , γ • , ρ • and ϵ • , respectively.To evaluate the expression of the charges (30) at the critical sets I ± , one must find a transformation between the NP-gauge frame and the F-gauge frame in full GR.Following [8], a general transformation between a NP-gauge spin frame {ϵ • A } and an F-gauge spin frame {ϵ A } is parameterised by a conformal factor θ and an SL(2, C) transformation matrix implying transformations for φ• 2 and the NP-connection coefficients (34).The expressions for these will not be presented here.
As we are interested in evaluating the expressions of the charges at I ± , an asymptotic solution for the conformal field equations is analysed, given the initial data prescribed in the previous section.Given the zero-order solution, asymptotic expansions for the conformal factor θ and the transformation matrices Λ B A are obtained, following [8].If ϕ 0 , ϕ 1 , ϕ 2 , ϕ 3 , ϕ 4 denote the components of the rescaled Weyl tensor in the F-gauge, then the explicit transformation from the NP-gauge to the F-gauge implies 1. Contributions to Q| I ± from ϕ 0 , ϕ 1 , ϕ 3 , ϕ 4 are at most O(ρ).
2. The background term σ •ab N • ab does not contribute to Q| I ± at zero order in ρ.
Hence, the asymptotic charges at I ± are determined by f and the zero-order solution of ϕ 2 , i.e., 2 ).
Given that the equation for ϕ (0) 2 is equivalent to the equation for ψ 2 on Minkowski spacetime, the solution for ϕ (0) 2 will develop a logarithmic singularity at I ± unless our initial data satisfy certain regularity conditions.The explicit form of these regularity conditions will be presented in a later article as well as the final expression of Q| I ± .The main result is that given initial data that satisfy our regularity conditions, one can show that Q| I ± are fully determined by Π 3 [Ω].Moreover, if the initial data are chosen to satisfy the regularity conditions, the asymptotic charges Q l,m associated with a given harmonic Y l,m at I + and I − are related by:

Conclusions
This article addresses the matching of the asymptotic charges associated with supertranslation symmetries in the context of an initial value problem using Friedrich's formulation of spatial infinity.The results in this paper demonstrate that the zero-order solution of ϕ 2 develops logarithmic singularities at I ± given the prescribed initial data in Section 3.1.Therefore, Q| I ± are only well-defined if extra regularity conditions are imposed on our initial data.An upcoming article will present the explicit form of these regularity conditions.A significant consequence of this result is that the matching of the BMS asymptotic charges, as defined in this article, is not feasible for generic asymptotically flat spacetimes unless these spacetimes are the development of initial data satisfying certain regularity conditions.

Figure 1 :
Figure 1: (a) The domain D containing spatial infinity, (b) The domain D on the conformal diagram of Minkowski spacetime.

Figure 2 :
Figure 2: A diagram of the neighbourhood of spatial infinity in Friedrich's representation.In this representation, the spatial infinity point i 0 is blown up to a cylinder I connecting past null infinity I + and future null infinity I − .The critical sets I ± represents the sets where I touches I ± .The set I 0 represents the intersection of the cylinder I with the initial hypersurface {τ = 0}.

) Remark 2 .d
The • notation indicates that W • abcd (or W • abcd ), ϵ • AB are in the NP-gauge.Given that θ = 1 on Minkowski spacetime, one has W • abcd = W abcd .However, this is not true for general spacetimes, i.e., the Weyl tensor C • abcd associated with g • will be related to C abcd associated with g by C • abcd = θ 2 C abcd .Moreover, the ϵ-spinor in the NP-gauge ϵ • AB will be related to ϵ AB by ϵ • AB = θϵ AB .If one wishes to obtain a relation between C • abcd and C abcd , where C • abcd ≡ C • abcd e and C abcd ≡ C abcde a a e b b e c c e d d , then one makes use of the transformation between C • abcd and C abcd as well as the transformation between the NP-gauge frame {e • a } and the F-gauge frame {e a } given by e • a = θ −1 Λ b a e b , where Λ b a ∈ O(1, 3).

n
are explicitly known functions of τ and the angular variables which are smooth for τ ∈ (−1, 1) and whose regularity at τ = ±1 can be controlled in terms of the initial data.The reminder satisfies R p [ψ n ] ∈ C m for p ≥ m + 6 for ρ near 0 and τ ∈ [−1, 1].