On “hyperboloidal” Cauchy data for vacuum Einstein equations and obstructions to smoothness of Scri

Lars Andersson Department of Mathematics Royal Institute of Technology S–10044 Stockholm e–mail: larsa@math.kth.se

Piotr T. Chruściel^{∗} Centre for Mathematics and its Applications Australian National University G.P.O. Box 4, Canberra A.C.T. 2601 e–mail: pxc851@huxley.anu.edu.au

September 6, 1999

Abstract

The relationship between the geometric properties of “hyperboloidal” Cauchy data for vacuum Einstein equations at the conformal boundary of the initial data surface and between the space–time geometry is analyzed in detail. We prove that a necessary condition for existence of a smooth or a polyhomogeneous Scri (i.e., a Scri around which the metric is expandanble in terms of r^{−j} log^{i} r rather than in terms of r^{−j}) is the vanishing of the shear of the conformal boundary of the initial data surface. We derive the “boundary constraints” which have to be satisfied by an initial data set for compatibility with Friedrich’s conformal framework. We show that a sufficient condition for existence of a smooth Scri (not necessarily complete) is the vanishing of the shear of the conformal boundary of the initial data surface and smoothness up to boundary of the conformally rescaled initial data. We also show that the occurrence of some log terms in an asymptotic expansion at the conformal boundary of solutions of the constraint equations is related to the non–vanishing of the Weyl tensor at the conformal boundary.

1 Introduction

In the papers of Bondi et al. [3], Sachs [18] and Penrose [17] which laid the foundations of our present understanding of the gravitational radiation, some rather strong hypotheses about the asymptotic behaviour of the gravitational field in lightlike directions were postulated. In a recent study by ourselves [1] (cf. also [2]) of the asymptotic properties of solutions of constraint equations on spacelike hypersurfaces intersecting “Scri” transversally it has, however, been observed that generic Cauchy data constructed in such a setting by the “conformal method” failed to be smoothly extendable, after appropriate rescalings, to the conformal boundary^{1} due to the occurrence of some logarithmic terms in an asymptotic expansion of the fields near the conformal boundary. This behaviour was observed even though the “seed fields” used in the conformal method were smoothly extendable to the conformal boundary. The aim of this paper is to analyze in detail the obstructions to smoothness of the solutions of the constraint equation constructed as in [1]. In order to present our results it is useful to define two tensor fields σ^{±} defined on the connected component under consideration ∂M_{i} of the conformal boundary ∂M of the initial data surface:

∗On leave of absence from the Institute of Mathematics, Polish Academy of Sciences, Warsaw. Supported in part by KBN grant # 2 1047 91 01.

σ_{AB}^{±} ≡ (λ_{AB} − \frac{h^{CD}λ_{CD}}{2} h_{AB}) ± (K_{AB} − \frac{h^{CD}K_{CD}}{2} h_{AB})|_{∂M_{i}} ,  (1.1)

which we shall call the shear tensors of ∂M_{i}, where h_{AB} is the induced metric on ∂M_{i}, K_{AB} is (the restriction to ∂M of) the extrinsic curvature (K_{ij}) of M in the unphysical space–time (V, γ), and λ_{AB} is the extrinsic curvature of ∂M_{i} in (M, g). In the case of Cauchy data constructed as described in Section 2, our results linking the geometry of the boundary of the initial data surface with the geometry of the resulting space–time can be summarized as follows:

1. Suppose that neither σ^{+} nor σ^{−} vanishes. Then there exists no development of the initial data with a smooth or polyhomogeneous Scri.

2. Suppose that σ^{+} ≡ 0 or σ^{−} ≡ 0; changing the time orientation if necessary we may without loss of generality assume that σ^{+} ≡ 0:

(a) Suppose moreover that K_{1A}^{log}|_{∂M_{i}} ≢ 0, where K_{1A}^{log} denotes “the logarithmic part” of K_{1A}, cf. Section 3 for details. If there exists a development with a polyhomogeneous Scri, then it is essentially polyhomogeneous, i.e., no development with a smooth Scri exists.

(b) Suppose instead that K_{1A}^{log}|_{∂M_{i}} ≡ 0.

i. Suppose moreover that K_{11}^{log}|_{∂M_{i}} ≢ 0. If there exists a polyhomogeneous Scri, we cannot exclude the possibility that it is inessentially so, i.e., that it can be made smooth by an appropriate change of gauges.

ii. Suppose finally that we moreover have K_{11}^{log}|_{∂M_{i}} ≡ 0. Then there exists a development which admits a smooth conformal boudary.

It should be stressed that the results linking the log terms with the non–vanishing of the Weyl tensor proved here show that the occurrence of shear and of at least some of the log terms in asymptotic expansions of physical fields at Scri is not an artefact of a bad gauge choice, including the choice of a conformal factor, or of a pathological choice of the initial data hypersurface (within the class of uniformly bounded from above and uniformly bounded away from zero, locally C^{2}, conformal factors and C^{1} deformations of the initial data hypersurface which fix ∂M): if Scri is not shear–free (by which we mean that eq. (3.11) does not hold), then no conformal transformation will make it shear free. Similarly if the Weyl tensor does not vanish at ∂M, then no gauge change in the above sense will make it vanish (cf. Section 3 for a more detailed discussion).

1It has similarly been observed by Christodoulou and Klainerman [6] that generic space–times constructed by evolving “small” Cauchy data asymptotically flat in spacelike directions seem to display asymptotic behaviour incompatible with the Bondi–Sachs–Penrose (BSP) conditions (and this result has apparently been recently rigorously established: D. Christodoulou, private communication). Since the Christodoulou–Klainerman theorem is proved under quite different conditions its conclusions are, however, somewhat different from the ones obtained here.

Let us mention that results somewhat similar to those obtained here have been recently derived in a Bondi–Sachs type setting in [7].

The overall picture that emerges from the results of [6, 1, 7] and from the results here is that the usual hypotheses of smoothness of Scri are overly restrictive. These results seem to indicate that the correct setup for an analysis of the gravitational radiation is that of polyhomogeneous rather than smooth functions on the conformally completed manifold, i.e., functions that have asymptotic expansions in terms of powers of x and log x rather than of x only. It should, however, be stressed that even though the fact that the physical fields (g̃, K̃) satisfy the constraint equations guarantees the existence of some vacuum development (V, γ̃), it is by no means obvious that in the case, say, σ^{+} ≡ 0, the existence of some kind of compactification of (M, g̃, K̃) implies the existence of some useful conformal completion of (V, γ̃). The question of existence of conformal completions of (V, γ̃) with three dimensional boundaries is the most important unsolved problem of the present theory.

This paper is organized as follows: In Section 2 we review the results of [1] about existence and structure of solutions of constraint equations satisfying “rough” hyperboloidal conditions^{2}. In Section 3 we study the relationship between the Weyl tensor at the conformal boundary and some of the log terms in the Cauchy data, cf. Theorem 3.1. We should stress that in that Section we do not assume that the data have been constructed by the methods of [1], rather we study a quite general class of initial data satisfying boundary conditions, which are inspired by those that would be satisfied by data constructed in [1]. Nevertheless the class of data considered is significantly larger than that described in Section 2, and we have actually tried to tailor the boundary conditions to be the weakest possible for our analysis to go through without some significant supplementary difficulties. The argument in Section 3 is essentially a careful reexamination of the argument presented by Geroch in [12] assuming a smooth Scri. In that Section we also derive some results which are necessary for further applications, cf. Theorem 3.2. It seemed of interest to rederive the results of Section 3 in a more direct manner, using directly the constraint equations: this is done in Section 4. In that Section we show that the rather indirect four–dimensional methods of Section 3 can be replaced by an analysis which is conceptually more straightforward but much more tedious from a calculational point of view, using 3+1 and 2+1+1 decompositions of the fields. In Section 5 we adress the question of existence of a time evolution of the initial data which possesses a polyhomogeneous or smooth Scri. Theorem 5.1 is our main existence result, while Proposition 5.1 is our main non–existence result. Various conventions and definitions are given in Appendices A and E. In Appendix B we show that initial data with a “shear–free” conformal boundary can be embedded in a space–time which is asymptotically Minkowskian in light–like directions. In Appendix C the notion of the extrinsic curvature of a null hypersurface is shortly discussed. Some calculational arguments needed in section 3 can be found in Appendix D.

2 We use the qualification “rough” for the boundary conditions of [1] to differentiate them from those that are needed in Friedrich’s evolution theorems. It is precisely the object of this paper to demonstrate with what conditions should the “rough” boundary conditions of [1] be supplemented to obtain data suitable for Friedrich’s theorems.

Acknowledgements: P.T.C. wishes to thank David Singleton and Robert Bartnik for several useful discussions and comments. He also wishes to thank the Department of Mathematics of Université de Tours for friendly hospitality during the final stage of work on this paper.

2 The conformal method

In this section we shall outline the Lichnerowicz—Choquet–Bruhat—York method for constructing solutions of the constraint equations (2.1)–(2.2) of general relativity, with asymptotic conditions appropriate to the “hyperboloidal” initial value problem [8, 9], cf. also [2, 1]. Let M̄ be a smooth, connected, Hausdorff, compact three dimensional manifold with boundary, set M = intM̄, ∂M̄ = ∂M ≡ M̄ \ M. Given “seed fields” (g_{ij}, A^{ij}), where g_{ij} is any smooth Riemannian metric on M extending smoothly to ∂M, and A^{ij} is any symmetric, traceless tensor field on M extending smoothly to ∂M, a solution (g̃, K̃) of the general relativistic constraint equations

R(g̃) = g̃_{ij}g̃_{kl}K̃^{ik}K̃^{jl} − (g̃_{ij}K̃^{ij})^{2} ,  (2.1)
D̃_{i}(K̃^{ij} − g̃_{kl}K̃^{kl}g̃^{ij}) = 0 ,  (2.2)

where D̃_{i} is the Levi–Civita connection of g̃_{ij}, and R(f) is the Ricci scalar of a metric f, satisfying the condition

K̃ ≡ g̃_{ij}K̃^{ij} = 3ϵ,  ϵ = ±1 ,  (2.3)

can be constructed by the following procedure: Let x be any defining function for ∂M, i.e. a function satisfying x ∈ C^{∞}(M̄), x ≥ 0, x(p) = 0 ⟺ p ∈ ∂M, with dx non–vanishing at ∂M. Let A^{ij} ∈ C^{∞}(M̄) be symmetric traceless and let X be any solution of the equation

D_{i} [x^{−3} (D^{i}X^{j} + D^{j}X^{i} − \frac{2}{3}D_{k}X^{k}g^{ij})] = −D_{i} (x^{−2}A^{ij}) ,  (2.4)

define

L^{ij} ≡ \frac{ω^{2}}{x^{3}} (D^{i}X^{j} + D^{j}X^{i} − \frac{2}{3}D_{k}X^{k}g^{ij}) + \frac{ω^{2}}{x^{2}} A^{ij} ,  (2.5)

where ω is a solution of the equation

ωΔ_{g}ω − \frac{3}{2}|Dω|_{g}^{2} + \frac{1}{4}ω^{2}(R(g) − |L|_{g}^{2}) + \frac{3}{2} = 0 ,  (2.6)

satisfying ω ≥ 0, ωx^{−1} — uniformly bounded on M from above and uniformly bounded away from zero, where |L|_{g}^{2} = g_{ij}g_{kl}L^{ik}L^{jl}. Setting

g̃_{ij} = ω^{−2}g_{ij} ,
K̃^{ij} = ω^{3}L^{ij} + ϵ g̃^{ij} ,

one obtains a solution of (2.2)–(2.1) satisfying (2.3). Moreover when X is chosen appropriately (cf. point 1 below) the fields (g̃_{ij}, K̃^{ij}) satisfy (in a rough way^{3}) the asymptotic conditions appropriate to the “asymptotically hyperboloidal” setting (cf. [9, 1]). In [1] the following has been shown:

3For a precise description of various boundary conditions imposed, cf. [1]. In the smooth seed fields context discussed here, the boundary condition on X is that x^{−2}X be bounded; this ensures existence and uniqueness; cf. also footnote 2.

1. For any (g, A) as above one can find a unique solution X ∈ x^{2}C^{∞}(M̄) + x^{4} log x C^{∞}(M̄) of (2.4) such that

(\frac{x}{ω} )^{2} L^{ij} = U^{ij} + x^{2} log x U_{log}^{ij}, U^{ij}, U_{log}^{ij} ∈ C^{∞}(M̄) .

Given any g there exists an open dense set (in the C^{∞}(M̄) topology) of A’s for which U_{log}^{ij}|_{∂M} ≢ 0 (however, there exists an infinite dimensional closed subspace of A’s for which U_{log}^{ij}|_{∂M} ≡ 0). If U_{log}^{ij}|_{∂M} ≡ 0, then U_{log}^{ij} ≡ 0 and thus x^{2}ω^{−2}L^{ij} ∈ C^{∞}(M̄). Let us note for further reference that in an orthonormal frame e_{i}, as described in Appendix A, if we write, in a neighbourhood of ∂M,

L^{ij} = L_{0}^{ij}(v) + xL_{1}^{ij}(v) + ⋯,

where \{v\} are coordinates on ∂M, then we have L_{0}^{1i} ≡ 0, while both L_{0}^{AB}(v)−\frac{1}{2}L_{0}^{CD}h_{CD}h^{AB} and L_{1}^{AB}(v) − \frac{1}{2}L_{1}^{CD}h_{CD}h^{AB} are freely specifiable tensor fields on ∂M.

2. For any (g, A) as above one can find a solution ω ∈ A_{phg}, where A_{phg} denotes the space of polyhomogeneous functions on M̄ (cf. Appendix E), of equation (2.6). More precisely, there exists a sequence {N_{j}}_{j=0}^{∞} with N_{0} = N_{1} = N_{2} = N_{3} = 0, N_{4} = 1 and functions ω_{i,j} ∈ C^{∞}(M̄) such that

ω ∼ ∑_{i≥0} ∑_{j=0}^{N_{i}} ω_{i,j}x^{i} log^{j} x  (2.7)

(cf. Appendix E for the meaning of “∼”). For an open dense set of (g, A)’s we have ω_{4,1}|_{∂M} ≢ 0. If ω_{4,1}|_{∂M} ≡ 0, then ω ∈ C^{∞}(M̄).

When L^{ij} ≡ 0, the obstruction to smoothness of ω has been analyzed in detail in [2]. In that reference it has been shown in particular that ω_{4,1}|_{∂M} vanishes if the Weyl tensor of the unphysical space–time metric is bounded near ∂M (cf. [2] for details). It is the object of this paper to extend that analysis to the case L^{ij} ≢ 0.

3 Weyl tensor vs log terms

In this Section we shall show that under some conditions the vanishing of the Weyl tensor at ∂M is equivalent to the vanishing of some of the log terms in the solutions of the constraint equations. The methods of this Section do not require any understanding of the origin of the log terms, which will be discussed in more detail in the next Section. It should be stressed that we do not assume the condition trK̃ = const (which has been made in [1], as described in Section 2). To obtain sharp results we will consider functions which satisfy some detailed differentiability requirements near the conformal boundary, cf. Definition 3.1 below. The reader who is not interested in sharp results might assume that all the functions appearing in the formulas below are polyhomogeneous, thus all the coefficients of expansions in terms of powers of x and log x are smooth functions of the boundary variables.

Let (M, g̃, K̃) be an initial data set, thus M is an (open) manifold, g̃ is a Riemannian metric on M and K̃ is a symmetric tensor field on M, with (g̃, K̃) satisfying the constraint equations (2.1)–(2.2). We shall also assume that M = intM̄, where M̄ is a compact manifold with boundary, and that there exists a function ω ∈ C^{0}(M̄) ∩ C_{loc}^{3}(M) such that

g̃ = ω^{−2}g on M ,
ω|_{∂M} = 0 ,

and g is a Riemannian metric on M̄ such that g ∈ C^{0}(M̄) ∩ C_{loc}^{3}(M). If we assume moreover that g̃_{ij} ∈ H_{loc}^{k+1}(M), K̃^{ij} ∈ H_{loc}^{k}(M), with some k ≥ 2, then it follows from well known theorems [14, 13, 4, 5] that there exists a vacuum space–time (V, γ̃), with γ̃ ∈ C_{loc}^{k−1}(V), together with an embedding i : M ⟶ V such that i^{∗}γ̃ = g̃, and K̃ represents the extrinsic curvature tensor of i(M) in (V, γ̃). Given an orthonormal frame ẽ_{i}, i = 1, 2, 3, for g̃ the four vectors (ẽ_{a}), a = 0, …, 3, where ẽ_{0} = η̃, the unit normal to i(M) in (V, γ̃), form an orthonormal tetrad for γ̃. Let Ω be any function on V such that

Ω ∘ i = ω ,  (3.1)

define

γ = Ω^{2}γ̃ .

In what follows we shall identify M with i(M) without explicitly mentioning this fact, and we shall always assume that (3.1) holds.

Following Geroch [12] we define

S_{ab} = R_{ab}(γ) − \frac{1}{6}R(γ)γ_{ab}  (3.2)
G^{a}_{b} = R^{a}_{b}(γ̃) − \frac{1}{6}R(γ̃)δ^{a}_{b}  (3.3)

where R_{ab}(f), R(f) denote the Ricci tensor and the Ricci scalar of the metric f; we have [12]

R_{abcd} = C_{abcd} + γ_{a[c}S_{d]b} − γ_{b[c}S_{d]a} ,  (3.4)

where R_{abcd} and C_{abcd} are the Riemann tensor and the Weyl tensor of the metric γ_{ab}, and

S_{ab} = −\frac{2}{Ω}∇_{a}∇_{b}Ω + \frac{∇^{c}Ω∇_{c}Ω}{Ω^{2}}γ_{ab} + \frac{1}{Ω^{2}} G_{ab} ,  (3.5)
(G_{ab} ≡ γ_{ac}G^{c}_{b}) ,
C_{abcd}∇^{d}Ω + Ω∇_{[a}S_{b]c} = ∇_{[a}(Ω^{−1}G_{b]c}) − Ω^{−2}γ_{c[a}G_{b]d}∇^{d}Ω ,  (3.6)
∇^{m}(Ω^{−1}C_{abcm}) = −Ω^{−2}∇_{[a}(Ω^{−1}G_{b]c}) + Ω^{−4}γ_{c[a}G_{b]d}∇^{d}Ω  (3.7)

(in this section^{4} and in Appendices A and D all indices are tetrad indices, and are raised and lowered with γ_{ab} = diag(−1, 1, 1, 1)). If γ̃ is vacuum, then clearly G^{c}_{b} = 0.

We wish to stress that above and throughout the remainder of this Section we have not assumed that any kind of well behaved conformal completion of (V, γ) exists, and that equations (3.5)– (3.7) hold whatever the extension Ω of ω to V.

4On the other hand it should be noticed that in Sections 4, 5 and in Appendix B coordinate indices are used.

The trace of equation (3.5) gives

S ≡ γ^{ab}S_{ab} = \frac{4|∇Ω|_{γ}^{2}}{Ω^{2}} − \frac{2}{Ω}□Ω + \frac{1}{Ω^{2}} γ^{ab}G_{ab} .  (3.8)

Let us assume that near a connected component ∂M_{i} of ∂M, we have

ω ∈ C^{1}(M̄), γ^{ab}G_{ab} = o(1)

and that we have chosen Ω so that along M we have

□Ω = o(ω^{−1}) .

The requirement that S = o(Ω^{−2}) gives

lim_{M∋p→∂M_{i}} |∇Ω|_{γ}^{2} = 0 ⟹ ω_{0}|_{∂M_{i}} = ±|Dω|_{g}|_{∂M_{i}} ,  (3.9)

where

ω_{0} ≡ ∇_{0}Ω|_{M} .

It follows that if we wish S_{ab} to be o(Ω^{−2}) we need to choose the boundary value of ∇_{0}Ω|_{∂M_{i}} so that (3.9) holds. The trace free part of (3.5) reads

S_{ab} − \frac{1}{4}Sγ_{ab} = −\frac{2}{Ω}[∇_{a}∇_{b}Ω − \frac{1}{4}□Ω γ_{ab}]
+ \frac{1}{Ω^{2}} (G_{ab} − \frac{1}{4}γ^{cd}G_{cd} γ_{ab}) .  (3.10)

Equation (3.10) shows that if G_{ab} = o(Ω), then a necessary condition for (V, γ̃) to admit a Scri with a C^{2}(V̄) metric is, that there exists on M functions ω_{0}, ω_{00} such that if we set

∇_{0}Ω|_{M} = ω_{0} , ∇_{0}∇_{0}Ω|_{M} = ω_{00} ,

then we have

lim_{M∋p→∂M_{i}} (∇_{a}∇_{b}Ω − \frac{1}{4}□Ω γ_{ab}) = 0 .  (3.11)

By an abuse of terminology we shall refer to (3.11) as the statement, that a smooth Scri is shear free in the “gauge” □Ω = 0.

Let us recall, that the couples (γ, Ω) and (ϕ^{2}γ, ϕΩ) determine the same physical metric γ̃ = Ω^{−2}γ. If ϕ is a twice differentiable function uniformly bounded from above on M and uniformly bounded away from zero on M, then a straightforward calculation shows that the condition (3.11) is invariant under such rescalings. As will be seen in Proposition 3.3, if the space–time under consideration admits a Scri which is a null hypersurface, then the geometric meaning of (3.11) is the vanishing of the conformal extrinsic curvature of Scri, as defined in appendix C.

We wish to show that from the Cauchy problem point of view the requirement that equation (3.11) holds will lead to a constraint on the Cauchy data: a simple 3 + 1 decomposition of (3.5) gives

S_{ij} − \frac{1}{3}g^{kl}S_{kl} g_{ij} = −\frac{2}{Ω}[(K_{ij} − \frac{1}{3}Kg_{ij})∇_{0}Ω + (D_{i}D_{j}Ω − \frac{1}{3}Δ_{g}Ω g_{ij})] +
+ \frac{1}{Ω^{2}} (G_{ij} − \frac{1}{3}g^{kl}G_{kl} g_{ij}) ,  (3.12)

where K_{ij} is the extrinsic curvature of M in the (unphysical) space–time (V, γ), K = g^{kl}K_{kl} and where Δ_{g} is the Laplacian of the metric g_{ij}. Equation (3.12) implies:

Proposition 3.1 Suppose that g ∈ C^{1}(M̄), ω ∈ C^{2}(M̄), K ∈ C^{0}(M̄), G_{ij} = o(Ω). A necessary condition for the existence of a conformal completion V̄ of (V, γ) such that γ ∈ C^{2}(V̄) is that

(K_{ij} − \frac{1}{3}Kg_{ij})|_{∂M} = −ϵ[\frac{1}{|Dω|_{g}} (D_{i}D_{j}ω − \frac{1}{3}Δ_{g}ω g_{ij})]|_{∂M} ,  (3.13)

ϵ ≡ (\frac{ω_{0}}{|Dω|_{g}} )|_{∂M} = ±1 ,

where D_{i} is the Levi–Civita connection of g_{ij}.

It is worthwile to analyze the covariance properties of eq. (3.13): recall that the quadruplets (ω, g_{ij}, K^{ij}, ∇_{0}Ω|_{M}) and (ω̄, ḡ_{ij}, K̄^{ij}, ¯∇_{0}Ω̄|_{M}), where

ω̄ = ϕ ω ,  (3.14)
ḡ_{ij} = ϕ^{2}g_{ij} ,  (3.15)
K̄^{ij} = ϕ^{−3}(K^{ij} − \frac{K}{3} g^{ij}) + (ϕ^{−1} \frac{K}{3} − ϕ^{−2}ϕ_{0})g^{ij} ,  (3.16)
¯∇_{0}Ω̄|_{M} = ∇_{0}Ω|_{M} + ϕ^{−1}ϕ_{0}ω ,  (3.17)

with ϕ ∈ C_{loc}^{2}(M), ϕ_{0} ∈ C_{loc}^{1}(M), determine the same physical Cauchy data set (g̃, K̃). If ϕ is assumed to be uniformly bounded from above on M and uniformly bounded away from zero on M, then a straightforward calculation shows that the condition (3.13) is invariant under (3.14)–(3.17). Thus, (3.13) is indeed a condition on the physical Cauchy data.

Equation (3.13) is a non–trivial condition, because it is not satisfied by generic initial data as constructed in [1]:

Proposition 3.2 For generic (g, A) ∈ C^{∞}(M̄) ⊕ C^{∞}(M̄), the solution of the constraint equations constructed by the conformal method will not satisfy (3.13).

Remarks:

1. Equation (3.12) shows that S_{ij} will blow up at ℐ = V̄ \ V as Ω^{−1} when (3.13) does not hold. This implies immediately that initial data for which (3.13) does not hold will lead to a space–time (V, γ̃) which does not admit compactifications (V, γ) with a metric γ ∈ C^{2}(V̄). It will be seen in Proposition 5.1 below that a stronger result holds: whenever (3.13) fails no conformal completion of the space–time with a polyhomogeneous Scri can exist, cf. Section 5 for details.

2. If V admits a boundary such that ℐ = V̄ \V is a null hypersurface with ∂M_{i} ⊂ ℐ, then the equation (3.18) below is equivalent to the vanishing of the conformal extrinsic curvature of ℐ on ∂M_{i} (cf. Appendix C). This together with Proposition 3.3 shows that in vacuum the equation (3.11) is equivalent to the vanishing of the conformal extrinsic curvature of ℐ on ∂M_{i}.

Proof: 2 + 1 decomposing (3.13) one finds

(K_{AB} − \frac{1}{2}h^{CD}K_{CD}h_{AB})|_{∂M_{i}} = ϵ(λ_{AB} − \frac{1}{2}λ h_{AB}) ,  (3.18)

where

λ ≡ h^{AB}λ_{AB} ,

the indices A, B, etc. run from 2 to 3, h_{AB} is the metric induced on ∂M_{i} by g, and λ_{AB} is the extrinsic curvature of ∂M_{i} in (M, g). As can be seen from what is said in Section 2 (and is analyzed in some more detail in [1]), both sides of (3.18) assume (independently) arbitrary values as (g, A) run over C^{∞}(M̄) ⊕ C^{∞}(M̄), and the result follows. □

In the Appendix B we show that Cauchy data for which (3.18) holds are asymptotically flat in a quite satisfactory sense. The question whether or not and in which sense a space–time in which (3.13) fails to hold deserves the name of asymptotically flat and/or asymptotically Minkowskian remains to be investigated (cf. [7] for a discussion of this question in a Bondi– Sachs framework).

In the remainder of this paper we shall be concerned with the case in which (3.13) holds. In order to do this we need to impose on g and K differentiability conditions consistent with the occurrence of log terms in the solutions of the constraint equations. In order to cover situations in which finite differentiability of space–time fields is appropriate we introduce the following:

Definition 3.1 For 0 < k ∈ IN the function f ∈ C_{loc}^{k}(M) ∩ C^{0}(M̄) is said to satisfy the k–condition at ∂M_{i}, where ∂M_{i} is a connected component of the boundary ∂M of M, if in local coordinates y = (x, v^{A}) as described in Appendix E we have

0 ≤ |α| ≤ k − 1 lim_{x→0} ∂_{y}^{α}(x^{k} log x f) = 0 ,

|α| = k lim_{x→0} ∂_{v}^{α}(x^{k} log x f) = 0 ,

lim_{x→0} \frac{∂_{x}^{k}(x^{k} log x f)}{log x} = k!f|_{∂M_{i}} .

Remarks:

1. A function f satisfying

∀ 0 ≤ |α| ≤ k |∂_{y}^{α}f| = o(x^{−|α|} log^{−1} x) ,

will also satisfy the k–condition. In particular if f ∈ C^{k}(M̄), then f satisfies the k– condition.

2. Functions displaying the above behaviour at ∂M_{i} appear naturally when studying the regularity of solutions of constraint equations near ∂M_{i}, under finite differentiability hypotheses on the “seed fields” used in the conformal method (cf. [1]).

3. Note that if f satisfies the k–condition, then x^{k} log x f ∈ C^{k−1}(M̄), and x^{k} f ∈ C^{k}(M̄); moreover the function x^{k} log xf behaves under differentiation and passing to the limits at ∂M_{i} in a way somewhat similar as it would if f ∈ C^{k}(M̄). An example of a function which is not in C^{1}(M̄) but which satisfies the k–condition for all k ≥ 1 is given e.g. by the functions log(x)^{−α}, with any α > 0. Functions of the form log(x)^{−α}ϕ(v/x), α > 1, where ϕ is any smooth function compactly supported in a coordinate patch on ∂M (with v – coordinates on a coordinate patch of ∂M extended to a neighbourhood of ∂M in any smooth way) also satisfy the k–condition for any k.

4. This is a condition concerning space–derivatives of f only, no time derivatives are involved.

Before stating the main result of this Section, Theorem 3.1, we need to describe a property of the initial data sets considered:

Proposition 3.3 Consider a Cauchy data set (M, g̃, K̃) for a vacuum space–time (V, γ̃) such that

g̃_{ij} = ω^{−2}g_{ij} , g_{ij} = ĝ_{ij} + x^{3} log x ĝ_{ij}^{log} ,
L̃^{ij} = ω^{3}L^{ij} , L^{ij} = L̂^{ij} + x^{2} log x L̂_{log}^{ij} ,
K̃ = K̂ + x^{2} log x K̂_{log} ,
ω = ω̂ + x^{4} log x ω̂^{log} ,

where

K̃ ≡ g̃_{ij}K̃^{ij} , L̃^{ij} ≡ K̃^{ij} − \frac{K̃}{3} g̃^{ij} ,

with ĝ_{ij} ∈ C^{3}(M̄), ω̂ ∈ C^{4}(M̄), L̂^{ij}, K̂ ∈ C^{2}(M̄), and ω̂^{log} satisfies the 4–condition, ĝ_{ij}^{log} satisfies the 3–condition, and L̂_{log}^{ij}, K̂_{log} satisfy the 2–condition at ∂M_{i}. Suppose moreover that there exists ϵ = ±1 such that

(λ_{AB} − \frac{λ}{2}h_{AB})|_{∂M_{i}} = ϵ(K_{AB} − \frac{κ}{2}h_{AB})|_{∂M_{i}} ,  (3.19)
λ ≡ h^{AB}λ_{AB} ,  κ ≡ h^{CD}K_{CD} .  (3.20)

Then:

1. there exists a function Ω ∈ C_{loc}^{4}(V) satisfying Ω|_{M} = ω such that

lim_{M∋p→∂M_{i}} |∇Ω|_{γ}^{2} = 0 ,

lim_{M∋p→∂M_{i}} (∇_{a}∇_{b}Ω − \frac{1}{4}□Ω γ_{ab}) = 0 ;

2. on M there exist functions R̂_{abcd} ∈ C^{1}(M̄) and R_{abcd}^{log} satisfying the 1–condition such that

R_{abcd}|_{M} = R̂_{abcd} + Ω log Ω R_{abcd}^{log}  (3.21)

(in particular

C_{abcd}|_{M} = Ĉ_{abcd} + Ω log Ω C_{abcd}^{log} ,  (3.22)
S_{ab}|_{M} = Ŝ_{ab} + Ω log Ω S_{ab}^{log} ,  (3.23)

with Ĉ_{abcd}, Ŝ_{ab} ∈ C^{1}(M̄), and C_{abcd}^{log}, S_{ab}^{log} satisfy the 1–condition). R̂_{abcd} and R_{abcd}^{log}, respectively Ĉ_{abcd} and C_{abcd}^{log}, can be chosen to have all the symmetries of a Riemann tensor, respectively of a Weyl tensor.

Remarks:

1. It should be stressed that we do not claim that the decompositions (3.21)–(3.23) are preserved under evolution (although we find it likely that this will be the case, at least under the conditions of Section 2). Thus, in all further calculations we assume that (3.21)–(3.23) hold on M only.

2. Since we are working in a frame formalism in this section, when ĝ_{ij}^{log} = 0 the positioning of indices on e.g. L̂_{log}^{ij} upstairs or downstairs does not affect the decomposition of L^{ij} into the log part and the regular part. If ĝ_{ij}^{log} ≠ 0 (or if one uses a coordinate formalism as in some other sections) one should use e.g. an orthonormal frame for ĝ in all the calculations (or use ĝ to raise and lower indices). This will not affect the equality of leading order part of the covariant and contravariant log part of a tensor, but will only lead to differences between some lower order terms in the covariant as compared to contravariant log part. Throughout this paper we shall freely move the index “log” in the subscript or superscript position, writing e.g. L̂_{ij}^{log} for ĝ_{ik}ĝ_{jℓ}L̂_{log}^{kℓ}, etc.

In the remainder of this paper we shall assume that the choice of the extension Ω of ω from M to V has been made in a way consistent with Proposition 3.3.

The proof of Proposition 3.3 is a straightforward 2 + 2 decomposition, and can be found in the Appendix D. Our main result is the following:

Theorem 3.1 Suppose that the hypotheses of Proposition 3.3 hold and let (V, γ̃) be a Cauchy development of (M, g̃, K̃) such that γ̃ ∈ C_{loc}^{3}(V). Let ∂M_{i} be a connected component of ∂M which is diffeomorphic to S^{2}. Then in any orthonormal frame such that e_{0} ⊥ M we have

lim_{M∋p→∂M_{i}} C_{abcd}(p) = 0 .  (3.24)

if and only if

lim_{M∋p→∂M_{i}} C_{log}^{010A}(p) = −\frac{1}{2} lim_{M∋p→∂M_{i}} S_{log}^{1A}(p) = 0 .  (3.25)

Remarks:

1. We do not assume K̃ = const, in particular there might be log terms in an asymptotic expansion of K̃ at ∂M_{i}.

2. Cauchy data (g̃, K̃) such that (3.18) holds, constructed by the conformal method from generic smooth seed fields, as described in Section 2, will behave near ∂M in a way consistent with the above assumptions. In such a case one moreover has

L̂_{log}^{AB} = −\frac{L̂_{log}^{11}}{2} h^{AB} ,  (3.26)

with ĝ_{ij}^{log} = 0, ĝ_{ij} ∈ C^{∞}(M̄), L̂^{ij} ∈ C^{∞}(M̄), while ω̂^{log} and L̂_{log}^{ij} are “polyhomogeneous” near ∂M_{i}.

3. For (g̃, K̃) ∈ H_{5}^{loc}(M)⊕H_{4}^{loc}(M) there exist vacuum developments satisfying γ ∈ C_{loc}^{3}(V).

4. The proof below is a careful reexamination of Geroch’s version of the Weyl tensor vanishing theorem of Penrose [12].

5. The condition that (V, γ̃) is vacuum can be replaced by the conditions

G_{ab}|_{M} = o(x^{2}) ,  (3.27)
[∇_{[a}(Ω^{−1}G_{b]c})]|_{M} = o(x^{2}) ,  (3.28)
[γ_{c[a} G_{b]d} ∇^{d}Ω] |_{M} = o(x^{4}) .  (3.29)

6. It should be stressed that in Theorem 3.1 we do not assume any uniform bounds on derivatives normal to M of R_{abcd}, R̂_{abcd} and R_{abcd}^{log}.

7. Our arguments below also show that for other topologies (3.25) and (3.24) are not equivalent: from what is said here and from the results of [1] it follows that there exist solutions of constraint equations with ∂M_{i} ≠ S^{2} which satisfy the hypotheses of Proposition 3.3 (in fact (g, K, ω, ω_{0}) can even be chosen to be smooth up to ∂M_{i}) and for which the space–time Weyl tensor does not vanish on ∂M_{i}.

Theorem 3.1 follows immediately from Proposition 3.3 and from the following:

Lemma 3.1 Under the hypotheses of Proposition 3.3, the only non–vanishing components of C_{abcd}|_{∂M_{i}} are (up to permutations) C_{0A0B}|_{∂M_{i}}, C_{0A1B}|_{∂M_{i}} and C_{1A1B}|_{∂M_{i}}; with

lim_{M∋p→∂M_{i}} C_{0A0B} = ϵ lim_{M∋p→∂M_{i}} C_{0A1B} = lim_{M∋p→∂M_{i}} C_{1A1B} ,

ϵ ≡ lim_{M∋p→∂M_{i}} [|Dω|_{g}^{−1}∇_{0}Ω] = ±1 .

C_{0A0B}|_{∂M_{i}} is a symmetric traceless tensor field on ∂M_{i} and satisfies the equation

D_{A} (C^{0A0B}|_{∂M_{i}}) = − lim_{M∋p→∂M_{i}} [|Dω|_{g}C_{log}^{10B0}] = \frac{1}{2} lim_{M∋p→∂M_{i}} [|Dω|_{g}S_{log}^{1B}] ,  (3.30)

where D_{A} is the Levi–Civita connexion of the metric h induced by g on ∂M_{i}.

Proof: Let Ω, R̂_{abcd}, etc., be as given by Proposition 3.3. From equation (3.6) with a, b = i, j and from (3.22)–(3.23), (3.28)–(3.29) one obtains

lim_{M∋p→∂M_{i}} C_{ijcd}∇^{d}Ω = 0 .  (3.31)

Consider the Bianchi identity

−∇_{0}R_{a0} + ∇_{i}R_{ai} = \frac{1}{2}∇_{a}R ⟺ −∇_{0}S_{a0} + ∇_{i}S_{ai} = ∇_{a}S .  (3.32)

(3.32) with a = j and (3.23) immediately give

∇_{0}S_{0j}|_{M} = O(log x) .

(3.32) inserted in (3.6) with a = c = 0 yields

[C_{0i0j}∇^{j}Ω]|_{M} = − [\frac{Ω}{2} (∇_{0}S_{i0} − ∇_{i}S_{0i})]|_{M} + o(1)
= o(1) .

From this we can conclude that

lim_{M∋p→∂M_{i}} C_{0i0j}∇^{j}Ω = 0 ,  (3.33)

which implies

lim_{M∋p→∂M_{i}} C_{0i01} = 0 .  (3.34)

Define ϵ = ±1 by (∇^{a}Ωe_{a})|_{∂M_{i}} = [|Dω|_{g}(ϵe^{0} + e^{1})]|_{∂M_{i}} (⇒ lim_{M∋p→∂M_{i}} ∇_{0}Ω = ϵ|Dω|_{g}|_{∂M_{i}}). From (3.31) we obtain

lim_{M∋p→∂M_{i}} C_{ij01} = 0 ,  (3.35)
lim_{M∋p→∂M_{i}} C_{ij0A} = ϵ lim_{M∋p→∂M_{i}} C_{ij1A} .  (3.36)

(3.34)–(3.35) imply

lim_{M∋p→∂M_{i}} C_{ab01} = 0 .  (3.37)

Consider e.g.

lim_{M∋p→∂M_{i}} C_{0323} = lim_{M∋p→∂M_{i}} [−C_{0020} + C_{0121} + C_{0222} + C_{0323}]
= lim_{M∋p→∂M_{i}} C_{0}^{a}_{2a} = 0 ,

which shows that

lim_{M∋p→∂M_{i}} C_{0ABC} = 0 .  (3.38)

(3.36) also gives now

lim_{M∋p→∂M_{i}} C_{1ABC} = 0 .  (3.39)

The identity (A.13) of Appendix A implies

lim_{M∋p→∂M_{i}} C_{1A1B} = lim_{M∋p→∂M_{i}} C_{0A0B} ,  (3.40)

so that one also obtains

lim_{M∋p→∂M_{i}} C_{2323} = lim_{M∋p→∂M_{i}} [−C_{0303} + C_{1313} + C_{2323} + C_{3333}]
= lim_{M∋p→∂M_{i}} [C^{a}_{3a3}] = 0  ⟹
lim_{M∋p→∂M_{i}} C_{ABCD} = 0 .  (3.41)

Define

Ĉ_{AB} ≡ lim_{M∋p→∂M_{i}} C_{0A0B} = lim_{M∋p→∂M_{i}} Ĉ_{0A0B} .

From (3.34)–(3.41) it follows that all nontrivial information about lim_{M∋p→∂M_{i}} C_{abcd} is contained in Ĉ_{AB}, since the only non–vanishing components of lim_{M∋p→∂M_{i}} C_{abcd} are C_{0A0B}, C_{1A0B}, C_{1A1B} (and appropriate permutations of the indices thereof), with

lim_{M∋p→∂M_{i}} C_{0A0B} = ϵ lim_{M∋p→∂M_{i}} C_{0A1B} = lim_{M∋p→∂M_{i}} C_{1A1B} = Ĉ_{AB} .

Let us also note that we have

h^{AB}Ĉ_{AB} = lim_{M∋p→∂M_{i}} [−C_{0000} + C_{1010} + C_{2020} + C_{3030}]
= lim_{M∋p→∂M_{i}} C^{a}_{0a0} = 0 .

so that Ĉ_{AB} is a symmetric traceless tensor on ∂M_{i}. From (3.7) and (3.22) it follows

lim_{M∋p→∂M_{i}} [Ω∇_{a}(Ω^{−1}Ĉ^{a0B0})] = − lim_{M∋p→∂M_{i}} [C_{log}^{a0B0}∇_{a}Ω] ,  (3.42)

since this equation does not involve derivatives of C^{abcd} in directions normal to M. For p ∈ ∂M_{i} let γ_{p} = γ_{p}(s) be an integral curve of e^{1} with γ_{p}(0) = p, thus for any function f we have \frac{d}{ds}f(γ(s)) = e_{1}(f)(γ(s)). From l’Hospital rule we have

lim_{s→0} (\frac{∇_{a}Ω Ĉ^{a0B0}}{Ω} )(γ(s)) = lim_{s→0} [\frac{e_{1}(e_{a}(Ω)) Ĉ^{a0B0} + ∇_{a}Ω e_{1}(C^{a0B0})}{e_{1}(Ω)}](γ(s)) =
= lim_{s→0} [e_{1}(Ĉ^{10B0}) + \frac{e_{1}(e_{a}(Ω)) Ĉ^{a0B0}}{e_{1}(Ω)} ](γ(s)) .  (3.43)

(3.43) shows that (3.42) does not involve derivatives of Ĉ^{a0B0} in directions transverse to ∂M_{i}. A somewhat lengthy but otherwise straightforward 2 + 2 decomposition of (3.43) gives

lim_{s→0} [Ω∇_{a}(Ω^{−1}Ĉ^{a0B0})] = D_{A}Ĉ^{AB}

− lim_{s→0} \{\frac{1}{e_{1}(Ω)}[e_{1}(e_{A}(Ω)) + ∇_{0}Ω ω_{A}^{0}_{1} + ∇_{1}Ω ω_{A}^{1}_{1}]\}Ĉ^{AB} ,

where D_{A} is the Levi–Civita connexion of the metric h induced by g on ∂M_{i}. Since lim_{M∋p→∂M_{i}} ∇_{1}∇_{A}Ω = 0, (3.42) gives

D_{A}Ĉ^{AB} = − lim_{M∋p→∂M_{i}} [|Dω|_{g}C_{log}^{10B0}] .  (3.44)

To finish the proof, let us note that (3.22) inserted in (3.6) gives

(C_{0A0d}^{log}∇^{d}Ω)|_{∂M_{i}} = −\frac{1}{2} lim_{M∋p→∂M_{i}} (\frac{1}{log Ω}∇_{0}S_{A0}) .  (3.45)

From the Bianchi identity (3.32) we have

∇_{0}S_{A0} = ∇_{i}S_{Ai} − ∇_{A}S ,

so that from (3.23) one obtains

lim_{M∋p→∂M_{i}} (\frac{1}{log Ω}∇_{0}S_{A0}) = lim_{M∋p→∂M_{i}} (\frac{1}{log Ω}∇_{1}S_{A1})
= (|Dω|_{g}S_{1A}^{log})|_{∂M_{i}} .  (3.46)

Equations (3.45) and (3.46) give

C_{0A01}^{log}|_{∂M_{i}} = −\frac{1}{2}S_{1A}^{log}|_{∂M_{i}} .

which achieves the proof. □

Proof of Theorem 3.1: Clearly, eq. (3.44) shows that Ĉ^{AB} can vanish only if C_{log}^{10B0} does. If ∂M_{i} ≈ S^{2}, the converse is also true because in that case the operator L^{AB} → D_{A}L^{AB} which maps symmetric trace free tensors to vector fields is elliptic with trivial kernel. □

Theorem 3.1 holds under more general conditions than those discussed in Section 2, since it allows for non-constant K̃, and for x^{3} log x terms in g_{ij}; moreover the structure of the log terms in K_{ij} is much more general than that which is obtained in the setting of Section 2. When g_{ij} ∈ C^{3}(M̄), the C_{log}^{01A0}|_{∂M_{i}} terms responsible for the non–vanishing of the Weyl tensor at ∂M_{i} can be related to the log terms in K_{ij} as follows:

Corollary 3.1 Under the conditions of Theorem 3.1, suppose moreover that ĝ_{ij}^{log} ≡ 0 (thus g_{ij} = ĝ_{ij} ∈ C^{3}(M̄)), and

ω_{0} = ω̂_{0} + \frac{1}{3!}ω_{0} ω^{3} log ω ω_{0}^{log} ,  (3.47)

ω̂_{0} ∈ C^{3}(M̄), ω_{0}^{log} satisfying the 3–condition. If we write

ω = ω̂ + \frac{1}{4!}ω^{4} log ω ω^{log}  (3.48)
K_{ij} = K̂_{ij} + \frac{1}{2}ω_{0}ω^{2} log ωK_{ij}^{log} ,  (3.49)

(ω̂ ∈ C^{4}(M̄), K̂_{ij} ∈ C^{1}(M̄), ω_{log} satisfying the 4–condition, K_{ij}^{log} — satisfies the 1–condition), then

C_{log}^{010A}|_{∂M_{i}} = \frac{1}{2}(|Dω|_{g}ω_{0}K_{log}^{1A})|_{∂M_{i}} ;  (3.50)

thus (3.24) holds iff K_{log}^{1A}|_{∂M_{i}} = 0.

Proof: Inserting the equation (3.49) into (3.5) one finds

S_{0A}^{log}|_{∂M_{i}} = − [|Dω|_{g}ω_{0}L_{1A}^{log}] |_{∂M_{i}} .  (3.51)
□

Theorem 3.1 and Corollary 3.1 show how the K_{1A}^{log} terms affect the space–time geometry at ∂M. In the remainder of this Section we shall consider the role of the other log terms. Let us start with an identity between the log terms in ω and K_{ij} which holds under the conditions (3.27)–(3.29).

Lemma 3.2 Under the hypotheses of Corollary 3.1 we have the identity

ω^{log}|_{∂M_{i}} = \frac{1}{2} (K^{log} − 3K_{11}^{log} − \frac{4}{3}K̃^{log}) |_{∂M_{i}} ,  (3.52)

with ω^{log} as in (3.48), K_{11}^{log} as in (3.49), and

K̃^{log} = 3 (K^{log} + ω_{0}^{log}) .  (3.53)

Remark: Equation (3.53) is consistent with (D.11) which gives

K̃ = K̂ + \frac{1}{3!}ω_{0} ω^{3} log ωK̃^{log} ,  K̂ ≡ 3ω̂_{0} + ωg^{ij}K̂_{ij} .

Thus K̃^{log} consists of the log terms in the trace of the extrinsic curvature on M of the physical space–time metric. This implies in particular that K̃^{log} is invariant under the transformations (3.14)–(3.17) (this can also be checked by a direct calculation under appropriate hypotheses on ϕ and ϕ_{0}).

Proof: Equation (3.7) gives

lim_{M∋p→∂M_{i}} (Ω∇_{a}(Ω^{−1}Ĉ^{a0cd})) = − lim_{M∋p→∂M_{i}} (∇_{i}ω C_{log}^{i0cd}) ,  (3.54)

By Lemma 3.1 and a calculation similar to that of the proof of Lemma 3.1 one finds that the right hand side of (3.54) with c = 0, d = 1 vanishes identically (this holds irrespectively of whether or not C_{log}^{0A0B}|_{∂M_{i}} = 0). From (3.54) one therefore obtains

C_{log}^{0101}|_{∂M_{i}} = 0 .

(3.4) gives

C_{0101} = R_{0101} + \frac{1}{2}(S_{11} − S_{00}) .

On the other hand we have

R_{0101} = −S_{11} − \frac{1}{6}R + R_{i1i1}

⟹

C_{0101} = −\frac{1}{2}(S_{00} + S_{11} + \frac{1}{3}R) + R_{i1i1} .

From (A.12) one obtains

C_{0101}^{log}|_{∂M_{i}} = −(\frac{1}{2}h^{AB}S_{AB}^{log} + S_{11}^{log})|_{∂M_{i}} ,

and (3.65)–(3.66) below give

0 = C_{0101}^{log}|_{∂M_{i}} = [\frac{|Dω|_{g}^{2}}{2} (K_{11}^{log} + K^{log} + \frac{4}{3} ω_{0}^{log} + \frac{2}{3} ω^{log})] |_{∂M_{i}} .  (3.55)

(3.52) follows from (3.53) and (3.55). □

Recall that the choice of ω_{0}^{log} is arbitrary, so that for fixed K̃^{log} the function K^{log} is arbitrary. It follows that (3.52) imposes a relation between the functions ω^{log} and K_{11}^{log} (determined by the seed fields if the conformal construction has been used) modulo choices of gauges. If, e.g., we have K̃^{log} = 0, we can choose ω_{0}^{log} = 0 so that K^{log} = 0 and then eq. (3.55) reads

ω^{log}|_{∂M_{i}} = −\frac{3}{2}K_{11}^{log}|_{∂M_{i}} .  (3.56)

This last equation shows in particular, that under the conditions of Section 2 and with this choice of gauges the vanishing of all log terms in K_{ij} implies no log terms in ω as well.

Theorem 3.2 Under the hypotheses of Proposition 3.3 let (V, γ̃) be a Cauchy development of (M, g̃, K̃) such that γ̃ ∈ C_{loc}^{3}(V); suppose moreover that g_{ij} ∈ C^{3}(M̄) and

lim_{M∋p→∂M_{i}} C_{abcd} = 0 ,  (3.57)

in an orthonormal frame such that e_{0} ⊥ M. Then the function ω can be extended from M to a function Ω on V (locally of the same differentiability class as ω) such that

C_{abcd}^{log}|_{∂M_{i}} = 0  (3.58)

if and only if

K̃^{log}|_{∂M_{i}} = 0 ,  (3.59)
K_{1A}^{log}|_{∂M_{i}} = 0 ,  (3.60)
(K_{AB}^{log} − \frac{1}{2}h^{CD}K_{CD}^{log} h_{AB}) |_{∂M_{i}} = 0 .  (3.61)

Moreover if (3.58) holds then we also have

R_{abcd}^{log}|_{∂M_{i}} = 0 .  (3.62)

Remark: When the topology of the boundary is spherical and (g̃, K̃) are constructed by the conformal method starting from smooth “seed fields”, as described in Section 2, then the proof below shows that Ω^{−1}C_{abcd}^{log} ∈ C_{0}(M̄) and is polyhomogeneous if and only if (3.60)–(3.61) hold.

Proof: Regardless of the topology of ∂M it follows from equation (3.30) that C_{01A0}^{log}|_{∂M_{i}} =

S_{1A}^{log}|_{∂M_{i}} = 0 whenever (3.57) holds, so that if we choose ω to be of the form (3.48) and ω_{0} to be of the form (3.47) then (3.60) will hold by (3.51). If we then choose ω_{00} so that

ω_{00} = ω̂_{00} + \frac{1}{2!}ω_{0}^{2} ω^{2} log ω ω_{00}^{log} ,  (3.63)

with ω̂_{00} ∈ C^{2}(M̄), ω_{00}^{log} satisfying the 2–condition, then the decomposition (3.49) inserted in (3.5) implies

S_{0i}^{log}|_{∂M_{i}} = − [|Dω|_{g}ω_{0}(K_{i1}^{log} + ω_{0}^{log}n_{i})] |_{∂M_{i}} ,  (3.64)
S_{00}^{log}|_{∂M_{i}} = − [|Dω|_{g}^{2}\{ω_{00}^{log} + \frac{1}{3}(ω^{log} − ω_{0}^{log})\}] |_{∂M_{i}} ,  (3.65)
S_{ij}^{log}|_{∂M_{i}} = [|Dω|_{g}^{2}\{ − K_{ij}^{log} − ω^{log}n_{i}n_{j} + \frac{1}{3}(ω^{log} − ω_{0}^{log})g_{ij}\}] |_{∂M_{i}} .  (3.66)

Let us consider the 2 + 2 decomposed form of (3.58). From (3.4) and (A.12) one obtains

R_{1A1B} = C_{1A1B} + S_{AB} + S_{11}h_{AB} ⟹ C_{1A1B}^{log}|_{∂M_{i}} = − (S_{AB}^{log} + S_{11}^{log}h_{AB}) |_{∂M_{i}} .

Equation (3.66) yields

C_{1A1B}^{log}|_{∂M_{i}} = [|Dω|_{g}^{2} (K_{AB}^{log} + (K_{11}^{log} + \frac{1}{3}ω^{log} + \frac{2}{3}ω_{0}^{log}) h_{AB})] |_{∂M_{i}} ,  (3.67)

and (3.55) shows that (3.58) with abcd = 1A1B is equivalent to (3.61). From (3.4), (A.11) and (3.66) with a little work one finds

C_{1A0B}^{log}|_{∂M_{i}} = [|Dω|_{g}ω_{0} (K_{AB}^{log} + \frac{1}{2}(K_{11}^{log} + ω_{0}^{log}) h_{AB})] |_{∂M_{i}} ,

so that (3.61) and (3.53) imply that C_{1A0B}^{log}|_{∂M_{i}} will vanish if and only if (3.59) holds. Manipulations with the Bianchi identities and eq. (3.6) similar to those outlined at the end of the proof of Lemma 3.1 can be used to derive the following

[|Dω|_{g}(S_{Ab}^{log} + 2C_{1Ab1}^{log})]|_{∂M_{i}} = 2[C_{1Ab0}^{log}ω_{0}]|_{∂M_{i}}  ⟹  C_{1A01}^{log}|_{∂M_{i}} = −\frac{1}{2} S_{A0}^{log}|_{∂M_{i}} ;
[C_{ABbd}^{log}∇^{d}Ω]|_{∂M_{i}} = 0  ⟹  C_{ABC0}^{log}|_{∂M_{i}} = ϵ C_{ABC1}^{log}|_{∂M_{i}}, C_{AB01}^{log}|_{∂M_{i}} = 0 .

Calculations similar to the ones at the beginning of this proof show that all the remaining components of C_{abcd}^{log}|_{∂M_{i}} vanish when (3.59)–(3.61) hold. Then eq. (3.64) shows that S_{i0}^{log}|_{∂M_{i}} = 0 provided we choose

ω_{0}^{log}|_{∂M_{i}} = −K_{11}^{log}|_{∂M_{i}} .  (3.68)

Similarly it follows from (3.64) that S_{00}^{log}|_{∂M_{i}} = 0 will hold if we set

ω_{00}^{log}|_{∂M_{i}} = \frac{1}{3} (ω_{0}^{log} − ω^{log}) |_{∂M_{i}} .  (3.69)

Equations (3.61) and (3.55) show now that the right hand side of (3.66) vanishes identically, and the proof is completed. □

4 A direct approach

Throughout this section, in contrast to section 3, all indices are coordinate indices, and not tetrad indices. In this section we shall derive, by a direct analysis of the constraint equations, the obstructions to smoothness of the solutions of the constraint equations. The relationship between the “log terms” and the boundary values of the space–time Weyl tensor derived in the previous Section will also be rederived here by a straightforward 2 + 2 decomposition calculation. We will assume that the fields g_{ij}, L_{ij}, ω have the regularity described in Proposition 3.3 and that

K̃ ≡ 3ϵ,  ϵ = ± .  (4.1)

(If K̃ ≡ constant ≠ ±3 the results described below go through with trivial modifications.) As in [2] (cf. Lemma 2.1 of [2]) it is possible to prove that the following gauge conditions can be imposed:

Proposition 4.1 Assume that g ∈ C^{∞}(M̄), that (4.1) holds, and that the hypotheses of Proposition 3.3 are satisfied. We may without loss of generality choose the coordinate system and the conformal factor so that

1. the coordinates (x, y^{2}, y^{3}) form a Gaussian coordinate system near ∂M_{i}, and

2. trλ = 0 on ∂M_{i}.

The above and the scalar constraint equation imply then

ω = x + O(x^{3}) near ∂M_{i} .

Remark: It is worthwile pointing out that with the above conditions we have |Dx|_{g} = 1. Moreover we have the following useful identity on ∂M_{i}:

λ_{AB}λ^{B}_{C} = \frac{1}{2}λ : λh_{AC} ,  (4.2)

where we have used the notation

ν : μ = ν_{AB}μ^{AB} ,

where ν, μ are covariant 2-tensors on ∂M_{i}. The identity (4.2) is easily proved using the fact that λ_{AB} is symmetric and therefore may be diagonalized. Equation (4.2) implies in particular

μ^{AB}μ_{BC}ν^{C}_{A} = 0 .  (4.3)

for any symmetric trace free 2-tensors μ, ν on ∂M_{i}. Similarly we will use the notation λ : μ : ν = λ_{A}^{B}μ_{B}^{C}ν_{C}^{A}, etc.

We shall start our analysis with the vector constraint equation.

4.1 The vector constraint equation

Consider the momentum constraint equation for the physical data,

D̃^{i}(K̃_{ij} − K̃g̃_{ij}) = 0.

Setting

K̃_{ij} = ω^{−1}L_{ij} + \frac{1}{3}K̃g̃_{ij}  (4.4)

with K̃ ≡constant, the momentum constraint equation takes the form

ωD^{i}L_{ij} − 2ω^{,i}L_{ij} = 0 .  (4.5)

A 2+1 decomposition calculation yields

(x∂_{x} − 2 − xλ_{F}^{F})L_{11} + xD^{A}L_{1A} + xλ^{AB}L_{AB} = o(x^{2})  (4.6)
(x∂_{x} − 2 − xλ_{F}^{F})L_{1B} + xD^{A}L_{AB} = o(x^{2})  (4.7)

We have: Proposition 4.2 Let the hypotheses and the gauge conditions of Proposition 4.1 hold. According to Proposition 3.3 we can write

L_{ij} = L̂_{ij} + x^{2} log xL̂_{ij}^{log} ,  (4.8)

with L̂_{ij} ∈ C^{2}(M̄) and L̂_{ij}^{log} satisfying the 2–condition at ∂M_{i}. Suppose moreover that

L̂_{AB}^{log}|_{∂M_{i}} = −\frac{1}{2}L̂_{11}^{log}h_{AB} .  (4.9)

Then

L_{1j}|_{∂M_{i}} = 0 ,  (4.10)
∂_{x}L_{11}|_{∂M_{i}} = λ^{AC}L_{AC} ,  (4.11)
∂_{x}L_{1B}|_{∂M_{i}} = D^{C}L_{CB} ,  (4.12)

and the terms L̂_{1j}^{log} in the decomposition of L_{ij} take the following values on ∂M_{i}:

L̂_{11}^{log}|_{∂M_{i}} = −(D^{A}D^{B}L_{AB} + ∂_{x}λ_{AB}L^{AB} + λ^{AB}∂_{x}L_{AB})
= −(D^{A}D^{B}L_{AB} + R_{AB}L^{AB} + λ^{AB}∂_{x}L_{AB}) ,  (4.13)
L̂_{1A}^{log}|_{∂M_{i}} = −∂_{x}(D^{B}L_{AB})
= − [D^{B}(∂_{x}L_{AB}) + 2λ^{BC}D_{B}L_{AC} + 2L_{AB}D^{C}λ_{CB}
+L^{BC}D_{A}λ_{BC}] ,  (4.14)

where R_{AB} denotes the pull-back of the Ricci tensor of g_{ij} to ∂M_{i}.

Remarks:

1. The tensor L̂_{ij}^{log}|_{∂M_{i}} defined above in eq. (4.8) differs from the trace–free part of K_{ij}^{log}|_{∂M_{i}}

defined in eq. (3.49) by a normalization factor of ϵ/2.

2. The hypothesis (4.9) holds for solutions of the constraint equations constructed by the conformal method starting from seed fields which are smooth up to the boundary.

Proof: The result is obtained by inserting the decomposition (4.8) in (4.6)–(4.7) and comparing terms with appropriate powers of x. To get (4.13) one uses the identity

R_{AB} = ∂_{x}λ_{AB} + r_{AB} + 2λ_{AC}λ^{C}_{B} − λ_{AB}λ^{C}_{C}.  (4.15)
□

Note that L̂_{AB}^{log} is expressed in terms of L̂_{11}^{log}. This means that the vanishing of L̂_{ij}^{log} is equivalent to the vanishing of the three functions L̂_{11}^{log}, L̂_{1A}^{log} on ∂M_{i}.

In the remainder of this subsection we shall consider those solutions of the constraint equations which have been constructed by the conformal method starting from seed fields g, A, x ∈ C^{∞}(M̄), as described in Section 2. For simplicity we shall only consider those A_{ij}’s for which we have

A_{ij}D^{i}x|_{∂M_{i}} = 0.  (4.16)

((4.10) shows that any tensor L_{ij} satisfying (4.5) necessarily satisfies (4.16). This observation and a simple argument using uniqueness of solutions of (2.4) shows that the space of transverse traceless tensors constructed as described in Section 2 is exhausted by those tensors which are obtained from tensors satisfying (4.16).) Under (4.16) it follows by comparing terms with appropriate powers of x in (2.4) that the desired solution X of (2.4) satisfies

X = O(x^{3}) .  (4.17)

Equations (2.5) and (4.17) show that we have

(L_{AB} − \frac{1}{2}h^{CD}L_{CD}h_{AB})|_{∂M_{i}} = (A_{AB} − \frac{1}{2}h^{CD}A_{CD}h_{AB})|_{∂M_{i}} ,  (4.18)

(∂_{x}L_{AB} − \frac{1}{2}h^{CD}∂_{x}L_{CD}h_{AB})|_{∂M_{i}} = (∂_{x}A_{AB} − \frac{1}{2}h^{CD}∂_{x}A_{CD}h_{AB})|_{∂M_{i}} .  (4.19)

It is natural to ask the question, for which seed fields will one obtain solutions of the constraint equations such that L_{ij} ∈ C^{∞}(M̄) – as discussed in Section 2, this will occur if L̂_{ij}^{log}|_{∂M_{i}} = 0. The appropriate conditions on A_{ij} can be derived by inserting (4.18)–(4.19) into (4.13)–(4.14), they do not, however, seem to be very enlightening in general. Some simplifications occur if in addition to (4.16) we assume that “Scri is shear free”, in the sense that

L_{AB}|_{∂M_{i}} = ϵλ_{AB} ,  (4.20)

with ϵ = ±1. Under these conditions the equation L̂_{11}^{log}|_{∂M_{i}} = 0 can be rewritten as

L̂_{11}^{log}|_{∂M_{i}} = − [λ^{AB}∂_{x}A_{AB} + ϵ(D^{A}D^{B}λ_{AB} + R_{AB}λ^{AB})] |_{∂M_{i}} = 0 .  (4.21)

With some work from (4.14) one finds that

L̂_{1A}^{log}|_{∂M_{i}} = −D^{B}p_{AB} ,  (4.22)

where

p_{AB} ≡ ∂_{x}A_{AB} + \frac{3}{2}ϵλ_{CD}λ^{CD}h_{AB} .  (4.23)

From L^{i}_{i} = 0 and from (4.20) it is simple to show that p_{AB} is trace–free, so that p_{AB} is a symmetric traceless tensor on the two dimensional manifold ∂M_{i}. If ∂M_{i} is a sphere, L̂_{1A}^{log}|_{∂M_{i}} = 0 implies p_{AB} = 0; for other topologies this will, however, not be true in general.

4.2 The scalar constraint equation

Under the conditions of Proposition 4.1 we can write

ω = x + x^{3}ω_{3} + x^{4} log(x)ω_{4,1} + l.o. ,  (4.24)

where l.o. denotes lower order terms. Inserting the expansion (4.24) in equation (2.6) one obtains

ω_{3}|_{∂M_{i}} = \frac{1}{12}[−4∂_{x}(λ_{F}^{F}) − L_{AB}L^{AB} + R(g)]  (4.25)
ω_{4,1}|_{∂M_{i}} = \frac{1}{16}[2∂_{x}^{2}(λ_{F}^{F}) + 2L^{AB}∂_{x}L_{AB} + 4L_{A}^{C}L_{C}^{B}λ^{A}_{B} − ∂_{x}R]
= \frac{1}{16}[2∂_{x}^{2}(λ_{F}^{F}) + 2L^{AB}∂_{x}L_{AB} − ∂_{x}R] .  (4.26)

where we have used (4.3) and the fact that L_{1j}|_{∂M_{i}} = 0 which implies that h^{AB}L_{AB}|_{∂M_{i}} = 0. A 2+1 decomposition of the second Bianchi identity D_{i}R = 2D^{j}R_{ij} using λ_{F}^{F} = 0 on ∂M_{i} gives the relation

∂_{x}R|_{∂M_{i}} = 2(∂_{x}R_{11} − D^{A}D^{B}λ_{AB} + λ^{AB}R_{AB}) .  (4.27)

Note that since r_{AB} = \frac{1}{2}rh_{AB}, λ_{F}^{F} |_{∂M_{i}} = 0 and λ_{A}^{B}λ_{B}^{C}λ_{C}^{A}|_{∂M_{i}} = 0 , one finds from the identity (4.15) that λ^{AB}R_{AB}|_{∂M_{i}} = λ : ∂_{x}λ. Further, using R_{11} = ∂_{x}(λ_{F}^{F}) − λ : λ we find

∂_{x}R_{11}|_{∂M_{i}} = ∂_{x}^{2}(λ_{F}^{F}) − 2λ^{AB}∂_{x}λ_{AB} ,

where we have used (4.3). Thus we get

ω_{4,1} = \frac{1}{8}[R_{AB}λ^{AB} + D^{A}D^{B}λ_{AB} + ∂_{x}L : L]|_{∂M_{i}} ,  (4.28)

where in the last step we have used the fact that g^{ij}L_{ij} ≡ 0 and L_{1j}|_{∂M_{i}} = 0, which implies h^{AB}L_{AB}|_{∂M_{i}} = 0 and therefore L_{AC}L^{C}_{B}|_{∂M_{i}} = \frac{1}{2}L_{CD}L^{CD}h_{AB}. In particular, if “Scri is shear free” in the sense that (4.20) holds, then

ω_{4,1} = −\frac{ϵ}{8}L_{11}^{log}|_{∂M_{i}} .  (4.29)

This gives an alternative proof of Lemma 3.2, namely that under the present conditions the vanishing of L_{11}^{log}|_{∂M_{i}} implies that the boundary value of the log coefficient in ω vanishes as well.

4.3 The Weyl tensor

Throughout this section we shall assume that the hypotheses and the gauge conditions of Proposition 4.1 are satisfied. By a transformation of the form (3.16) with an appropriate ϕ_{0} we can also achieve

K = 0 ,  (4.30)

and we shall also assume in this section that (4.30) holds. Due to Ein(γ̃) = 0, C_{i0k}^{0} is the negative of the Ricci trace of the spatial part of Riem(γ̃):

−C_{i0k}^{0} = −C̃_{i0k}^{0} = −R̃_{i0k}^{0}
= −γ̃_{β}^{α}R̃_{iαk}^{β} + g̃_{l}^{j}R̃_{ijk}^{l}
= g̃_{l}^{j}R̃_{ijk}^{l} ,

where C_{abc}^{d}, respectively C̃_{abc}^{d}, denotes the Weyl tensor of the metric γ, respectively γ̃. Using the conformal transformation rules and the Gauss equation (A.12) we get the formula

C_{ij} ≡ C_{0i0j} = R_{ij}(g) − \frac{1}{3}R(g)g_{ij} + \frac{1}{ω} (D_{i}D_{j}ω − \frac{1}{3}D_{k}D^{k}ωg_{ij} + ϵL_{ij})
−L_{i}^{k}L_{kj} + \frac{1}{3}L^{kl}L_{kl}g_{ij} .  (4.31)

By l’Hospital’s rule we have

[\frac{1}{ω} (D_{A}D_{B}ω + ϵL_{AB})] |_{∂M_{i}} = −∂_{x}λ_{AB} + ϵ∂_{x}L_{AB}

so that, using λ_{A}^{A}|_{∂M_{i}} = 0, L_{A}^{A}|_{∂M_{i}} = 0 and the fact that the square of a symmetric traceless two dimensional tensor is proportional to the metric one is led to

C_{AB}|_{∂M_{i}} = R_{AB} − ∂_{x}λ_{AB} + ϵ∂_{x}L_{AB} + κh_{AB} ,  (4.32)

for some function κ. Now we have C_{0}^{i}_{0i} = C_{0}^{α}_{0α} = 0, and since C_{0101}|_{∂M_{i}} ≡ 0 by Section 3 it follows^{5} that κ can be determined by the requirement that the right hand side of (4.32) is trace–free. The embedding equation (4.15) allows us to rewrite eq. (4.32) as

C_{AB}|_{∂M_{i}} = ϵ (∂_{x}L_{AB} − \frac{1}{2}h^{CD}∂_{x}L_{CD}h_{AB}) .  (4.33)

Thus, the value of C_{AB} on ∂M_{i} is just the trace free part of ∂_{x}L_{AB}. Equations (4.33) and (4.22)–(4.23) give an alternative proof of the assertion, that under the shear–free condition (4.20) the boundary values of the log terms in L_{1A} are determined by C_{abcd}|_{∂M_{i}}.

Let us finally mention that with some work it follows from (4.31) that Ĉ_{ij}^{log} takes the form

Ĉ_{11}^{log}|_{∂M_{i}} = 8ω_{4,1} + ϵL̂_{11}^{log} ,
Ĉ_{1A}^{log}|_{∂M_{i}} = ϵL̂_{1A}^{log} ,
Ĉ_{AB}^{log}|_{∂M_{i}} = 4ω_{4,1}h_{AB} − ϵL̂_{AB}^{log}

5This identity can of course be derived directly by a somewhat lengthy 2 + 2 decomposition calculation.

(cf. also (3.50)). We see from (4.29) and from Proposition 4.2 that under the hypotheses of this section the fields Ĉ_{11}^{log} and Ĉ_{AB}^{log} vanish identically on ∂M_{i} (cf. also the proof of Lemma 3.2).

5 The evolution problem

The aim of this Section is to discuss the question of existence of developments (V, γ̃) of the “hyperboloidal” Cauchy data (g̃, K̃) which admit some kind of conformal completion (V̄ , γ). Let us start with smooth up to boundary initial data:

Theorem 5.1 Let ∂M_{i} ≈ S^{2}, the two dimensional sphere, and let (g̃, K̃) be a solution of the vacuum constraint equations for which there exist smooth functions ω and ω_{0}, with ω/x ∈ C^{∞}(M̄), such that the fields

g_{ij} = ω^{2}g̃_{ij} ,  (5.1)
K^{ij} = ω^{−3}(K̃^{ij} − \frac{1}{3}g̃_{kl}K̃^{kl}g̃^{ij}) + \frac{g̃_{kl}K̃^{kl} − 3 ω_{0}}{3 ω} g^{ij} ,  (5.2)

are in C^{∞}(M̄). Suppose moreover that there exists ϵ = ±1 such that

(λ_{AB} − \frac{h^{CD}λ_{CD}}{2} h_{AB})|_{∂M_{i}} = ϵ(K_{AB} − \frac{h^{CD}K_{CD}}{2} h_{AB})|_{∂M_{i}} .  (5.3)

Then

1. there exists a Lorentzian manifold with boundary (V̄ , γ) and a function Ω ∈ C^{∞}(V̄) (defining function for ∂V) such that (int V̄ , Ω^{−2}γ) is a development of (M, g̃, K̃), with ∂M_{i} being a smooth submanifold of V̄ \ V.

2. If moreover M = B(1) (the unit ball in IR^{3}), then there exists ϵ_{0} > 0 such that if

||ω − ω̌||_{C^{8}(M̄)} + ||g_{ij} − ǧ_{ij}||_{C^{7}(M̄)} + ||ω_{0}||_{C^{7}(M̄)} + ||K^{ij}||_{C^{6}(M̄)} < ϵ_{0} ,  (5.4)

where ǧ_{ij} is the standard flat metric on B(1) and ω̌ = (1 − r^{2})/2, then (int V̄ , Ω^{−2}γ) is timelike and null future geodesically complete.

Remarks:

1. The norm in (5.4) can be considerably weakened at the price, however, of a rather more complicated statement of the corresponding result, cf. [9] for details.

2. Part 1 above still holds when ∂M_{i} ≉ S^{2} provided one adds the assumption that C_{0A0B}|_{∂M_{i}} = 0.

Proof: The calculations of Section 3 show that ∇_{0}Ω|_{M} ∈ C^{∞}(M̄) and ∇_{0}∇_{0}Ω|_{M} ∈ C^{∞}(M̄) can be chosen so that S_{ab}|_{M} ∈ C^{∞}(M̄), Ω^{−1}C^{a}_{bcd}|_{M} ∈ C^{∞}(M̄). Point 1 follows then from Theorem 6.5 of [8], point 2 follows from Theorem 3.5 of [9] and from the results of [11]. □

The condition (5.3) can be rewritten in terms of the seed fields (g, A), so that from (4.21)– (4.23), (3.56) and Theorem 5.1 one obtains:

Corollary 5.1 Let ∂M ≈ S^{2}, the two dimensional sphere, and let (g̃, K̃) be constructed as described in Section 2 starting with smooth seed fields (g, A). Suppose moreover that there exists ϵ = ±1 such that

(λ_{AB} − \frac{λ}{2}h_{AB})|_{∂M} = ϵ [A_{AB} − \frac{1}{2}h^{CD}A_{CD}h_{AB}] |_{∂M} .  (5.5)

Without loss of generality we may choose the “gauge”

λ|_{∂M_{i}} = 0

with x, v^{A} — Gauss coordinates near the boundary. Suppose that moreover in this gauge we have

A_{1j}|_{∂M} = 0 ,

[D^{A}D^{B}λ_{AB} + R_{AB}λ^{AB}] |_{∂M} = 0 ,

(∂_{x}A_{AB} − \frac{1}{2}h^{CD}∂_{x}A_{CD}h_{AB}) |_{∂M} = 0 .

Then the fields ω, and K constructed by the conformal method are in C^{∞}(M̄). Further, ω^{−1}C^{a}_{bcd} is in C^{∞}(M̄) and the conclusions of Theorem 5.1 hold.

Remark: It is clear from the methods of [1] that under the hypotheses of Corollary 5.1 the condition (5.4) can be replaced by the condition that (g_{ij}, A^{ij}) are sufficiently close to (ǧ_{ij}, 0) in some possibly higher norm.

We shall now consider the case of initial data sets for which the log terms occur. Note first, that if K^{ij} has log terms in its expansion at ∂M, then even if we start with g_{ij} ∈ C^{∞}(M̄) then by evolution g_{ij} will pick up log terms at the next moment of time. It follows that the appropriate setup to discuss the evolution problem is that of polyhomogeneous space–time metrics γ. Before proceeding any further, let us introduce some terminology. Let M̄ be an n–dimensional connected, paracompact, Hausdorff manifold with boundary, thus M ≡ intM̄ and ∂M ≡ ∂M̄ ≡ M̄ \ M are smooth manifolds. A semi–Riemannian metric f on M̄ will be called polyhomogeneous if f ∈ C_{loc}^{∞}(M) ∩ C^{0}(M̄) and if in local coordinates y = (x, v) near ∂M, where x is a defining function for ∂M and v are coordinates on ∂M, the functions f_{μν} ≡ f(∂/∂y^{μ}, ∂/∂y^{ν}) are polyhomogeneous. Given a polyhomogeneous metric f on M it is natural to enlarge the atlas on M to include all those local coordinate transformations which preserve the polyhomogeneous character of the metric near ∂M.

We shall say that the initial data (M, g̃, K̃) are polyhomogeneous if M is as described above and if there exists polyhomogeneous functions ω and ω_{0}, with ω/x ∈ C^{0}(M̄), such that the fields (g, K) given by (5.1)–(5.1) are polyhomogeneous.

We shall say that a development (V, γ̃) of polyhomogeneous initial data (M, g̃, K̃) admits a polyhomogeneous Scri if

1. there exists a manifold with boundary V̄ such that V = intV̄, and such that ∂M is a smooth submanifold of ∂V; 2. there exist a polyhomogeneous function Ω ∈ C^{1}(V̄) satisfying Ω|_{∂M} = 0, ∇Ω(p) ≠ 0 for

p ∈ ∂V, such that the metric Ω^{2}γ̃ is polyhomogeneous;

3. there exists a time function t defined in a neighbourhood of M̄ in V̄ such that t|_{M} = 0, compatible with the polyhomogeneous structure, with the associated lapse function being uniformly bounded away from zero.

It should be pointed out, that all the results described in Section 2 go through under polyhomogeneity conditions on the seed fields (g, A). More precisely, if g is any polyhomogeneous metric on M̄ and A^{ij} is any uniformy bounded polyhomogeneous symmetric tensor field on M̄, then there exist solutions (X, ω) of (2.4) and (2.6) such that L^{ij} given by (2.5) is polyhomogeneous and uniformly bounded on M̄, and ω/x is polyhomogeneous, uniformly bounded, and uniformly bounded away from zero on M̄.

Given a polyhomogeneous initial data set it is natural to ask the question whether or not there exists an obstruction to the existence of a development of the initial data with a polyhomogeneous Scri. We have the following:

Proposition 5.1 Let (M, g̃, K̃) be a polyhomogeneous initial data set with g_{ij} ∈ C^{1}(M̄) and K^{ij} ∈ C^{0}(M̄). If both for ϵ = 1 and for ϵ = −1 we have

(λ_{AB} − \frac{λ}{2}h_{AB})|_{∂M_{i}} ≠ ϵ(K_{AB} − \frac{κ}{2}h_{AB})|_{∂M_{i}} ,  (5.6)

then there exists no development (V, γ̃) of (M, g̃, K̃) which admits a polyhomogeneous Scri.

Proof: Suppose that a polyhomogeneous Scri exists; therefore one can find a time function t defined in a neighbourhood of M̄ in V̄ such that t|_{M} = 0, and the associated lapse function α and the shift vector β are, in local coordinates, polyhomogeneous, with α uniformly bounded away from zero in a neighbourhood of M̄. We have the evolution equations (we use the notation of [19], in particular indices are coordinate indices)

\frac{∂g_{ij}}{∂t} − L_{β}g_{ij} = −2αK_{ij} ,  (5.7)

\frac{∂K_{ij}}{∂t} − L_{β}K_{ij} = −D_{i}D_{j}α + 2\frac{α}{Ω} (D_{i}D_{j}Ω∇_{0}ΩK_{ij})

+α [−\frac{3|∇Ω|_{γ}^{2} − Ω□ Ω}{Ω^{2}}g_{ij} +^{3}R_{ij} + KK_{ij} − 2K_{i}_{k}K^{k}_{j}] ,  (5.8)

where α is the lapse function, β is the shift vector, L_{β} denotes a Lie derivative and ^{3}R_{ij} is the Ricci tensor of g_{ij}. Note that by hypothesis all the functions α, β, g_{ij}, Ω and K^{ij} are polyhomogeneous and uniformly bounded. If (3.19) does not hold, then the terms D_{i}D_{j}Ω in the trace free part of eq. (5.8) with ij = AB will give a contribution which blows up as Ω^{−1} near ∂M, while all the remaining terms in the trace free part of the ij = AB equation are at most blowing up as some power of log x, which gives a contradiction. □

Proposition 5.1 suggests very strongly, that there exists no polyhomogeneous Scri’s which are not shear free, even without the extra regularity conditions g ∈ C^{1}(M̄), K^{ij} ∈ C^{0}(M̄) which we have imposed above. Such a result has been observed in [7] in a Bondi–Sachs type setting. Recall that the condition (5.3) has been derived in Section 3 precisely under the condition g ∈ C^{1}(M̄), K^{ij} ∈ C^{0}(M̄). It is rather clear that an analysis of (5.8) under general polyhomogeneity conditions on (g, K) will lead to an analogous “no–go” theorem, with possibly a somewhat modified version of condition (5.6).

Proposition 5.1 shows in particular that generic initial data constructed as in Section 2 with smooth seed fields (g, A) will not admit a development which admits a polyhomogeneous Scri (this will hold even when ω and L are smooth up to boundary as long as (5.6) holds). The above “no–go” result does of course not exclude the possibility of existence of some non– polyhomogeneous conformal completions of (V, γ̃).

It may be of some interest to stress that if V is a spacetime which is the maximal devlopment of initial data for which (5.3) holds, then in the class of polyhomogeneous conformal completions one obtains a conformal completion V̄ for which V̄ \ V is a two dimensional smooth manifold, namely ∂M. The physical significance of such space–times and completions remains to be investigated.

Consider now a polyhomogeneous initial data set for which (5.5) holds — if this condition were not preserved by evolution via Einstein equations, it would follow from Proposition 5.1 that the resulting space–time could not admit a polyhomogeneous Scri. It is therefore natural to ask the question, whether (5.5) is preserved by evolution. We shall show that this is the case for initial data satisfying g_{ij} ∈ A^{phg} ∩ C^{1}(M̄), K^{ij} ∈ A^{phg} ∩ C^{0}(M̄), and satisfying moreover the conditions

K̃ = 3ϵ,  K = 0,  λ^{A}_{A}|_{∂M_{i}} = 0.

It can be shown [7] that for any polyhomogeneous metric Gauss coordinates can be constructed, the coefficients of the transformed metric being again polyhomogeneous in the Gauss coordinate system. In what follows we shall use this coordinate system on M in a neighbourhood of ∂M_{i}. Let n^{i} be the three dimensional normal to the sets {Ω = c_{1}} ∩ {t = c_{2}}, for some constants c_{1}, c_{2}, thus n^{i} = D^{i}Ω/|DΩ| and the extrinsic curvature of those sets is given by λ_{AB} = −L_{n}g_{AB}, where L is a Lie derivative. From the evolution equations (5.7)–(5.8) with a little work one finds, assuming (5.5),

\frac{∂}{∂t}(λ_{AB} −ϵK_{AB})|_{∂M_{i},t=0} = ϵL_{αn+ϵβ}(λ_{AB} −ϵK_{AB})+ϵD_{A}D_{B}α − \frac{1}{2}L_{[ \frac{∂}{∂t} −β,n]}g_{AB} +κg_{AB} , (5.9)

with some function κ. (Fortunately, as explained below, we do not need to use (5.7)–(5.8) to evaluate κ, which would require quite a messy calculation.) From K_{ij}n^{j}|_{∂M_{i}} = 0 it follows that

[\frac{∂}{∂t}, n] |_{∂M_{i},t=0} = \frac{∂n}{∂t}|_{∂M_{i},t=0} = 0 .

It should be noted that in (5.9) the lapse function α and the shift vector β are arbitrary. We can therefore require that they are polyhomogeneous, and that

α|_{∂M_{i},t=0} = 1 ,  (5.10)

n(α)|_{∂M_{i},t=0} = 0 ,  (5.11)

(β + ϵn)|_{∂M_{i},t=0} = 0 .  (5.12)

This leads to

\frac{∂}{∂t}(λ_{AB} − ϵK_{AB})|_{∂M_{i},t=0} = κg_{AB} .  (5.13)

(It might be of some interest to point out that with the choice (5.12) we have

\frac{∂}{∂t}|_{∂M_{i},t=0} = (αη + β)|_{∂M_{i},t=0} = − (\frac{ϵα}{|Dω|_{g}} ∇Ω) |_{∂M_{i},t=0} ,

so that at t = 0 the vector field \frac{∂}{∂t}|_{∂M_{i},t=0} is proportional to ∇Ω on ∂M_{i}, as should be if we want the integral curves of \frac{∂}{∂t} to follow the level set Ω = 0.) Consider now the tensor field

σ_{AB} ≡ λ_{AB} − \frac{1}{2}h^{CD}λ_{CD}h_{AB} − ϵ (K_{AB} − \frac{1}{2}h^{CD}K_{CD}h_{AB}) .

By definition we have h^{AB}σ_{AB} ≡ 0, so that

h^{AB} \frac{∂σ_{AB}}{∂t} = −\frac{∂h^{AB}}{∂t} σ_{AB} .

Since σ_{AB}|_{∂M_{i},t=0} ≡ 0 it follows that h^{AB} \frac{∂σ_{AB}}{∂t} |_{∂M_{i},t=0} = 0, which together with the definition of σ_{AB} and (5.13) yields

κ = −\frac{1}{2} \frac{∂(h^{CD}λ_{CD} − ϵh^{CD}K_{CD})}{∂t}|_{∂M_{i},t=0} .  (5.14)

Under the conditions (5.10)–(5.12) one finally finds from (5.13)–(5.14) that

\frac{∂σ_{AB}}{∂t} |_{∂M_{i},t=0} = 0 .  (5.15)

(5.15) shows that the requirement of preservation of the vanishing of the shear of the conformal boundary does not lead to any supplementary conditions on the initial data.

A Appendix

For p ∈ M let e_{a} = e_{a}^{μ}∂_{μ}, a = 1, …, 4, be an orthonormal tetrad, with

e_{0}^{μ} = η^{μ} , e_{1}^{μ} = n^{μ} ,

where η^{μ}∂_{μ} is a unit timelike vector field orthogonal to M in (V, γ), and n = n^{μ}∂_{μ} is any unit vector field tangent to M such that for p ∈ ∂M_{i}, n is the inward pointing vector orthogonal to ∂M_{i} (such tetrads can easily be constructed by e.g. a Gramm–Schmidt procedure, with the differentiability class of e_{i}^{μ}∂_{μ} = e_{i}^{k}∂_{k}, i, k = 1, 2, 3, being that of g_{ij} = γ_{ij}). By definition we have

γ_{ab} = γ_{μν}e_{a}^{μ}e_{b}^{ν} = diag(−1, 1, 1, 1) ,

γ^{ab} = γ_{ab}; set e^{a} = γ^{ab}e_{b}. Define

ω^{a}_{bc} = ⟨∇_{e_{c}}e_{b}, e^{a}⟩_{γ} ⟺ ∇_{e_{a}}e_{b} = ω^{c}_{ba}e_{c} ,  (A.1)

where ⟨ , ⟩_{f} is the scalar product in the metric f, and ∇ is the Levi–Civita derivative operator of γ. It follows

∇_{a}(X^{b}e_{b}) ≡ ∇_{e_{a}}(X^{b}e_{b}) = (e_{a}(X^{b}) + ω^{b}_{ca}X^{c})e_{b}
=: (∇_{a}X^{b})e_{b} .  (A.2)

Let g be the metric induced from (V, γ) on M, let h be the metric induced from (M, g) on ∂M_{i}, we shall denote by D_{i}, respectively by D_{A}, the Levi–Civita derivative operator of g, respectively of h. As is well known (cf. e.g. [15]), the tetrad formalism is most convenient in 3 + 1 and 2 + 1 decomposition calculations because we have

D_{i}(X^{j}e_{j}) ≡ D_{e_{i}}(X^{j}e_{j})
= [e_{i}(X^{j}) + ω^{j}_{ki}X^{k}]e_{j}  (A.3)
=: (D_{i}X^{j})e_{j} ,
D_{A}(X^{B}e_{B}) ≡ D_{e_{A}}(X^{B}e_{B})
= [e_{A}(X^{B}) + ω^{B}_{CA}X^{C}]e_{B}  (A.4)
=: (D_{A}X^{B})e_{b} ,

where the ω_{··}^{·} coefficients in (A.3)–(A.4) are the same as those appearing in (A.2). From (A.1) we have

K_{ij} = ω_{0ij} = ω_{0ji} = ⟨∇_{e_{i}}e_{j}, η⟩_{γ} ,  (A.5)

where K_{ij} is the extrinsic curvature tensor of M in (V, γ) (the sign conventions here agree with those of [40] when η is chosen to be future pointing), and

λ_{AB} = ω_{1AB} = ω_{1BA} = ⟨∇_{e_{A}}e_{B}, n⟩_{γ}
= ⟨D_{e_{A}}e_{B}, n⟩_{g} ,  (A.6)

where λ_{AB} is the extrinsic curvature of ∂M_{i} in (M, g) (note the unconventional sign in (A.6) due to the fact that n is the inward pointing normal to ∂M_{i}. This choice of n is convenient for our purposes because if x is a defining function for ∂M, then n = Dx/|Dx|_{g}.) The following equations are often used:

∇_{i}X_{j} = D_{i}X_{j} + X_{0}K_{ij} ,  (A.7)
D_{A}X_{B} = D_{A}X_{B} − X_{1}λ_{AB} ,  (A.8)
∇_{i}X_{0} = D_{i}X_{0} + K_{i}^{j}X_{j} ,  (A.9)
□ Ω = −Ω_{00} + Δ_{g}Ω + KΩ_{0} ,  (A.10)
Ω_{00} ≡ ∇_{0}∇_{0}Ω , Ω_{0} ≡ e_{0}(Ω) , K ≡ K_{i}^{i} ,
□ ≡ ∇^{a}∇_{a} , Δ_{g} ≡ D^{i}D_{i} .

We have the following embedding equations (cf. e.g. [19])

R_{ijk0}(γ) = D_{j}K_{ik} − D_{i}K_{jk} ,  (A.11)
R_{ijkl}(γ) = R_{ijkl}(g) + K_{ik}K_{jl} − K_{il}K_{jk} ,  (A.12)

R_{abcd}(f) being the Riemann tensor of a metric f. We shall need the following well known result:

Proposition A.1 Let γ be a conformally vacuum metric on a four dimensional manifold. Then the Weyl tensor C_{abcd} of γ satisfies

C_{ijkl} = 2(γ_{i[k}C_{l]j} + γ_{j[l}C_{k]i}) ,  (A.13)

where

C_{ij} ≡ C_{0i0j} .

Proof: Since (A.13) is conformally invariant without loss of generality we may assume that γ is vacuum. A 3 + 1 decomposition calculation gives (cf. e.g. [19])

C_{ij} = R_{ij}(g) + \frac{K}{3} L_{ij} − L_{ik}L^{k}_{j} + \frac{2}{9}K^{2}g_{ij} ,

L_{ij} ≡ K_{ij} − \frac{K}{3} g_{ij}

(g_{ij} = γ_{ij}), while from (A.12) one obtains

R_{ijkl}(g) = 2g_{i[k}C_{l]j} + 2g_{j[l}C_{k]i} + A_{ijkl} ,
A_{ijkl} = 2g_{i[k}L_{l]e}L^{e}_{j} + 2g_{j[l}L_{k]e}L^{e}_{i} + 2L_{i[k}L_{l]j} − |L|_{g}^{2} g_{i[k}g_{l]j} .

Tracelessness and symmetry of L_{ij} together with the identity

δ_{[k}^{i}δ_{l}^{j}L_{e}^{e}L_{f]}^{f} = 0  (A.14)

imply A_{ijkl} ≡ 0, and the result follows. □

B Asymptotically Minkowskian coordinates

The object of this section is to show that polyhomogeneous initial data sets for which (3.19) holds are asymptotically flat, in the sense that the Cauchy data set can be realised by an embedding of the initial data surface in an asymptotically Minkowskian (not necessarily vacuum) space–time. The notion of asymptotic flatness used here is rather rough, and it is clear from the formulae below that the metrics considered posses finer properties than (B.3), we shall however not attempt to formalize that.

Throughout this Section, for simplicity of calculations, we shall assume that changing the time orientation if necessary we have

K̃ = −3,  K = 0 .  (B.1)

(There is little doubt that the results below still hold without these conditions.)

We shall say that (V, γ̃) is asymptotically Minkowskian in lightlike directions if there exist constants R, C_{1}, C_{2} and a coordinate system on V covering the set

O ≡ O_{R,C_{1},C_{2}} ≡ {(t, x) : r ≥ R, C_{1} < t − r < C_{2}} ,  (B.2)

and constants C and N such that on O we have

|γ̃_{μν} − η_{μν}| + |∂_{σ}γ̃_{μν}| ≤ Cr^{−1} log^{N} r .  (B.3)

We have the following:

Proposition B.1 Let (M, g̃, K̃) be a polyhomogeneous data set (as defined in Section 5) with g ∈ C^{1}(M̄) and K ∈ C^{0}(M̄). Suppose moreover that we have ∂M_{i} ≈ S^{2}, and that (B.1) holds, and that (changing time orientation if necessary) we have

(λ_{AB} − \frac{h^{CD}λ_{CD}}{2} h_{AB} + K_{AB} − \frac{h^{CD}K_{CD}}{2} h_{AB})|_{∂M_{i}} = 0 .  (B.4)

Then there exists a polyhomogeneous space–time (V, γ̃) which is asymptotically Minkowskian in lightlike directions and an embedding i : M → V such that i^{∗}γ̃ = g̃, and K̃ is the extrinsic curvature tensor of i(M) in (V, γ̃).

Remarks:

1. We do not claim that (V, γ̃) can be chosen vacuum, though we believe that this is indeed the case.

2. The method of proof of Proposition B.1 works without the conditions g ∈ C^{1}(M̄), K ∈ C^{0}(M̄) as long as g and K are polyhomogeneous provided that the coefficients of the log^{i} x terms in K_{AB} are related to some of the coefficients of x log^{i} x in g_{ij}.

Proof: Let, as usual, h_{AB} = g_{AB}|_{∂M_{i}}. As is well known, there exists a conformal diffeomorphism Φ : S^{2} → S^{2} such that Φ^{∗}h_{AB} = ϕ̆^{2}h̆_{AB}, where h̆_{AB} is the standard round metric on S^{2},

h̆_{AB} = dθ^{2} + sin^{2} θ dϕ^{2} .

Let us still denote by g_{ij} the metric obtained by pull back with the map (x, v) → (x, Φ(v)), we thus have g_{AB}|_{∂M_{i}} = ϕ̆^{2}h̆_{AB}. Let ϕ denote any function on M̄ such that ϕ|_{∂M_{i}} = ϕ̆, with ∂ϕ/∂x|_{∂M_{i}} so chosen that the extrinsic curvature of ∂M_{i} in the metric ϕ^{−2}g is pure trace. Using once again the symbol g_{ij} for the metric ϕ^{−2}g_{ij} we have thus obtained g_{AB}|_{∂M_{i}} = h̆_{AB}. As has been shown in [7], for any polyhomogeneous metric Gauss coordinates near the boundary can be introduced, so that we have g^{xA} = g(dx, dx^{A}) ≡ 0, g^{xx} = g(dx, dx) ≡ 1 for 0 ≤ x ≤ x_{0}, for some x_{0}. We thus have

g_{ij}dx^{i}dx^{j} = dx^{2} + g_{AB}dx^{A}dx^{B} ,

with

g_{AB}|_{∂M_{i}} = h̆_{AB}, λ|_{∂M_{i}} = −\frac{1}{2}h̆^{AB} \frac{∂g_{AB}}{∂x} |_{∂M_{i}} = 0 .

It now follows from (2.6) that

f ≡ \frac{ω}{x} = 1 + O(x^{2}) .

Introducing r = 1/x we have

g̃_{ij}dx^{i}dx^{j} = ω^{−2}g_{ij}dx^{i}dx^{j} = f^{−2}r^{−2}dr^{2} + f^{−2}r^{2}g_{AB}dx^{A}dx^{B} .  (B.5)

Define

τ ≡ t − \sqrt{1 + r^{2}} ,

and set

γ̃_{tt} ≡ −1,  γ̃_{ti} ≡ 0,  γ̃_{ij}(τ = 0) = χ_{ij} ,  (B.6)

with

χ_{rr} = \frac{r^{2}}{1 + r^{2}} + g̃_{rr} = \frac{r^{2}}{1 + r^{2}} + \frac{1}{f^{2}r^{2}} ,  (B.7)
χ_{rA} = 0 ,  (B.8)
χ_{AB} = g̃_{AB} = \frac{r^{2}}{f^{2}} g_{AB} .  (B.9)

It follows that the embedding (r, x^{A}) →^{i} (t = \sqrt{1 + r^{2}}, r, x^{A}) has the desired property that i^{∗}γ̃ = g̃, for any γ̃ which on i(M) takes the values given by (B.6)–(B.9). To analyze the extrinsic curvature of the slice τ = 0 it is useful to pass to the coordinates (τ, r, x^{A}) so that the metric

ds̃^{2} = γ̃_{μν}dx^{μ}dx^{ν} = −dt^{2} + γ̃_{ij}dx^{i}dx^{j}  (B.10)

takes the form

ds̃^{2} = −dτ^{2} − 2\frac{r}{\sqrt{1 + r^{2}}} dτdr + g̃_{ij}dx^{i}dx^{j} .  (B.11)

The extrinsic curvature tensor of i(M) will take the desired values if we set

\frac{∂γ̃_{ij}}{∂τ} (τ = 0) = ψ_{ij} ,

with

ψ_{ij} = −2\sqrt{χ_{rr}}frK̃_{ij} + β^{k} \frac{∂g̃_{ij}}{∂x^{k}} + \frac{∂β^{k}}{∂x^{i}} g̃_{kj} + \frac{∂β^{k}}{∂x^{j}} g̃_{ki}

= 2\sqrt{χ_{rr}}fr [g̃_{ij} − \frac{r}{f} (K_{ij} − \frac{1}{3}g^{kl}K_{kl}g_{ij})] − \frac{f^{2}r^{3}}{\sqrt{1 + r^{2}}} \frac{∂g̃_{ij}}{∂r}

+\frac{∂β^{k}}{∂x^{i}} g̃_{kj} + \frac{∂β^{k}}{∂x^{j}} g̃_{ki} ,  (B.12)
β^{A} = 0 ,  (B.13)
β^{r} = −\frac{f^{2}r^{3}}{\sqrt{1 + r^{2}}} = −r^{2} + O(1) ,  (B.14)

where g̃^{ij} is the matrix inverse to g̃_{ij}. One easily finds

ψ_{rr} = 2\sqrt{χ_{rr}}frg̃_{rr} + 2β^{r} \frac{∂g̃_{rr}}{∂r} + \frac{∂β^{r}}{∂r} g̃_{rr} + O(1/r^{2}) = O(1/r^{2}) ,  (B.15)
ψ_{rA} = O(1/r) ,  (B.16)
ψ_{AB} = −2r^{2}(K_{AB} − \frac{1}{2} \frac{∂h_{AB}}{∂x}) + O(r log^{N} r) = O(r log^{N} r)  (B.17)

for some constant N (note that ψ_{AB} = O(r log^{N} r) wouldn’t hold if (B.4) weren’t satisfied). Let finally ϕ ∈ C^{∞}(IR) be any function satisfying suppϕ ⊂ (−ϵ, ϵ), 0 ≤ ϕ ≤ 1, ϕ|_{[−ϵ/2,ϵ/2]} = 1, set ϕ̃(t, r) = ϕ(t − \sqrt{1 + r^{2}}). Define

γ̃_{μν}dx^{μ}dx^{ν} = −dt^{2} + (1 − ϕ̃) [dr^{2} + r^{2}(dθ^{2} + sin^{2} θ dϕ^{2})] + ϕ̃ [χ_{ij} + (t − \sqrt{1 + r^{2}})ψ_{ij}] dx^{i}dx^{j} .

It is easily seen that ϵ can be chosen small enough so that the metric γ̃ satisfies (B.3) in the quasi–Minkowskian coordinates (t, x = r sin θ sin ϕ, y = r sin θ cos ϕ, z = r cos θ) on the set {r ≥ 1/x_{0}, t ∈ (−∞, ∞)}. □

C The extrinsic curvature of a null hypersurface

Let (V, γ) be a Lorentzian manifold, let N be a null differentiable hypersurface in V, i.e., N is a differentiable submanifold of codimension 1 such that for all p ∈ N there exists a neighbourhood O_{p} of p and a vector field k defined on O_{p} such that ∀q ∈ O_{p} we have 0 ≠ k_{q} ∈ T_{q}N and γ(k, k) = 0. It follows that k_{q} is defined up to the multiplication by a non-zero number. For X, Y ∈ T_{p}N define

N(X, Y) = γ(X, ∇_{Y} k) .  (C.1)

It is easily seen that N is a symmetric tensor field on N satisfying

∀X ∈ T_{p}N  N(k, X) = 0 .  (C.2)

In the space of 2–covariant symmetric tensors let us define the equivalence relation

N_{1} ∼_{1} N_{2}  iff  ∀ X, Y ∈ T_{p}N N_{1}(X, Y)_{p} = α_{p}N_{1}(X, Y)_{p} , α_{p} ∈ IR .  (C.3)

It follows that the equivalence class [N]_{1}, N given by (C.1), is an object well defined by the geometry of V and by N. [N]_{1} will be called the extrinsic curvature tensor of N.

Consider next a Lorentzian metric γ_{1} on V such that γ_{1} = ϕ^{2}γ, where ϕ is a differentiable function on V; set

X, Y ∈ T_{p}N  N_{1}(X, Y) = γ_{1}(X, ∇_{Y}^{1} k) ,  (C.4)

where ∇^{1} is the Levi–Civita covariant derivative of the metric γ_{1}. One finds

N_{1}(X, Y) = ϕ^{2}N(X, Y) + βγ(X, Y) ,  (C.5)
β = −ϕ k^{μ}∂_{μ}ϕ .

In the space of 2–covariant symmetric tensors let us define a second equivalence relation

N_{1} ∼_{2} N_{2}  iff  ∀ X, Y ∈ T_{p}N N_{2}(X, Y)_{p} = α_{p}N_{1}(X, Y)_{p} + β_{p}γ(X, Y) , α_{p}, β_{p} ∈ IR .

(C.6)

From what has been said it follows that the equivalence class [N]_{2}, N given by (C.1), is is an object well defined by the conformal geometry of V and by N. [N]_{2} will be called the conformal extrinsic curvature tensor of N.

D Proof of Proposition 3.3

Consider the equation

S_{ab} = \frac{|DΩ|_{g}^{2} − (∇_{0}Ω)^{2}}{Ω^{2}} γ_{ab} − \frac{2∇_{a}∇_{b}Ω}{Ω} ;  (D.1)

the equations

lim_{M∋p→∂M_{i}} [(∇_{0}Ω)^{2} − |DΩ|_{g}^{2}] = 0 ,  (D.2)
lim_{M∋p→∂M_{i}} [\frac{(∇_{0}Ω)^{2} − |DΩ|_{g}^{2}}{Ω} γ_{ab} − 2 ∇_{a}∇_{b}Ω] = 0 ,  (D.3)

are clearly necessary conditions for

S_{ab}|_{M} = Ŝ_{ab} + Ω log Ω S_{ab}^{log} ,  (D.4)

with Ŝ_{ab} ∈ C^{1}(M̄), and S_{ab}^{log} satisfying the 1–condition (we will show that they are also sufficient). (D.2) is easily satisfied by setting

∇_{0}Ω|_{M} = ω_{0} ,  ω_{0}|_{∂M_{i}} = ϵ|Dω|_{g}|_{∂M_{i}} ,  (D.5)
ϵ = ±1 ,

ω_{0} otherwise arbitrary, with an ϵ which shall be determined later. A 3 + 1 decomposition of the trace–free part of (D.3),

lim_{M∋p→∂M_{i}} (∇_{a}∇_{b}Ω − \frac{1}{4}□Ω γ_{ab}) = 0 ,

gives the equations

lim_{M∋p→∂M_{i}} ∇_{0}∇_{0}Ω = −\frac{1}{3} lim_{M∋p→∂M_{i}} (Δ_{g}ω + Kω_{0}) ,  (D.6)
lim_{M∋p→∂M_{i}} D_{i}ω_{0} = − lim_{M∋p→∂M_{i}} (K_{i}^{j}D_{j}ω) ,  (D.7)
lim_{M∋p→∂M_{i}} (D_{i}D_{j}ω − \frac{1}{3}Δ_{g}ω g_{ij}) = − lim_{M∋p→∂M_{i}} [ω_{0}(K_{ij} − \frac{1}{3}Kg_{ij})] .  (D.8)

(D.6) will be satisfied if we choose

∇_{0}∇_{0}Ω|_{M} = ω_{00} ,  ω_{00}|_{∂M_{i}} = −\frac{1}{3}(Δ_{g}ω + Kϵ|Dω|_{g})|_{∂M_{i}} ,  (D.9)

ω_{00} otherwise arbitrary. To analyze (D.7), recall that we have

K̃^{ij} = ω^{3}K^{ij} + ω_{0}g̃^{ij} ,  (D.10)

in particular

K̃ = ωK + 3ω_{0} ⇒ K̃|_{∂M_{i}} = 3ω_{0}|_{∂M_{i}} .  (D.11)

(D.10) gives

D̃_{i}(K̃^{ij} − K̃g̃^{ij}) = 0 ⟺ ωD_{i}(K^{ij} − Kg^{ij}) = 2(K^{ij}D_{i}ω + D^{j}ω_{0}) ,  (D.12)

which shows that (D.7) is automatically satisfied. A (2+1) decomposition of (D.8) with i = A, j = 1 gives

D_{A}D_{1}ω|_{∂M_{i}} = −(ω_{0}K_{1A})|_{∂M_{i}} .  (D.13)

where D_{A} is the Levi–Civita connection of the metric h_{AB} induced on ∂M_{i} by g. Equation (D.13) is satisfied beacuse of (D.7) and (D.11). To show that (D.8) with i = 1, j = 1 holds, recall that we have the scalar constraint equation,

ωΔ_{g}ω − \frac{3}{2}|Dω|_{g}^{2} + \frac{1}{4}ω^{2}(R(g) − |L|_{g}^{2}) + \frac{1}{6}K̃^{2} = 0 ,  (D.14)

from which it follows that

|Dω|_{g}|_{∂M_{i}} = \frac{1}{3}|K̃||_{∂M_{i}}  (D.15)

(which is consistent with (D.11)). Applying e^{1} to (D.14) one obtains

(2D_{1}D_{1}ω + λD_{1}ω − \frac{1}{3}K̃D_{1}K̃)|_{∂M_{i}} = 0 ,  (D.16)

where

λ ≡ h^{AB}λ_{AB} ,

with λ_{AB} – the extrinsic curvature of ∂M_{i} in (M, g). With a little work from (D.16) one finds that

D_{1}D_{1}ω|_{∂M_{i}} = −\frac{1}{2}[λD_{1}ω + (3K_{11} − K)ω_{0}]|_{∂M_{i}} ,  (D.17)

which is precisely (D.8) with i = 1, j = 1. It follows that the only non–trivial condition imposed by (D.8) is the equation

[ω_{0}(K_{AB} − \frac{κ}{2}h_{AB})]|_{∂M_{i}} = [|Dω|_{g}(λ_{AB} − \frac{λ}{2}h_{AB})]|_{∂M_{i}} ,  (D.18)
κ ≡ h^{AB}K_{AB} ,

which will be satisfied if we choose ϵ according to the sign occuring in (3.20). Now the trace of (D.3) gives

lim_{M∋p→∂M_{i}} (□ Ω − 2 \frac{|Dω|_{g}^{2} − (ω_{0})^{2}}{ω} ) = 0 ,

which with a little effort can be seen to hold if (D.9) holds.

We have thus shown that (D.8) will hold whenever (D.18) holds provided we choose ω_{0} and ω_{00} to satisfy (D.5) and (D.9). It is then elementary to show that (D.4) will be satisfied provided ω_{0} and ω_{00} are moreover required to satisfy

ω_{0} = ω̂_{0} + \frac{1}{3!}x^{3} log x ω̂_{0}^{log} ,

ω_{00} = ω̂_{00} + \frac{1}{2!}x^{2} log x ω̂_{00}^{log} ,

with ω̂_{0} ∈ C^{3}(M̄), ω̂_{00} ∈ C^{2}(M̄), ω̂_{0}^{log} satisfying the 3–condition, ω̂_{00}^{log} satisfying the 2– condition. [This is easily satisfied as we have a complete freedom of choice of the extensions of ω off M. We can e.g. choose ω_{0} to be any function in C_{loc}^{∞}(M) ∩ C^{3}(M̄) satisfying (D.5), and choose ω_{00} to be any function in C_{loc}^{∞}(M) ∩ C^{2}(M̄) satisfying (D.9) (with such a choice we will have ω̂_{0}^{log} ≡ 0, ω̂_{00}^{log} ≡ 0; some other choices might be more convenient for some other purposes).] Now the embedding equations (A.11) and (A.12) together with the conditions on g and K show that

R_{ijk0}|_{M} = R̂_{ijk0} + x log x R̂_{ijk0}^{log} ,  (D.19)
R_{ijkl}|_{M} = R̂_{ijkl} + x log x R̂_{ijkl}^{log} ,  (D.20)

with R̂_{ijk0}, R̂_{ijk0}^{log}, R̂_{ijkl}, R̂_{ijkl}^{log} satisfying the appropriate conditions. Since

R_{i0j0} = −R_{ij} + g^{kl}R_{ikjl} ,

equation (3.21) follows from (3.23) (which has already been established) and from (D.19)– (D.20). □

E Conventions, function spaces

The summation convention is used throughout this paper. Conventions on signature, covariant derivatives and extrinsic curvatures can be found in Appendix A.

Let M̄ be a compact, connected Hausdorff manifold with boundary. Thus M ≡ intM̄ is a smooth, paracompact, Hausdorff manifold of dimension 3, and ∂M ≡ ∂M̄ is a smooth manifold which is a (necessarily finite) sum of its connected components ∂M_{i}. Throughout this paper x will denote a defining function for ∂M, i.e. a function satisfying x|_{∂M} = 0, x ≥ 0, |dx||_{∂M} > 0, and the implication x(p) = 0 ⇒ p ∈ ∂M holds.

We can always choose a finite number of coordinate charts ϕ_{j} : O_{j} → IR^{n,+} ≡ {y ∈ IR^{n} : y^{1} ≥ 0}, j = 1, …, J, covering a neighbourhood of ∂M such that y^{1} = x. When referring to local coordinates we shall implicitly assume that y^{1} = x, and we shall use the letter v to denote the coordinates y^{2}, y^{3};

v^{A} = y^{A} , A = 2, 3 .

Thus

y = (x, v) .

The standard Schwartz multi-index notation is used throughout, thus if α = (α_{1}, … α_{n}), then

∂^{α} = ∂_{y}^{α} = ∂_{y^{1}}^{α_{1}} ⋯ ∂_{y^{n}}^{α_{n}} = ∂_{x}^{α_{1}}∂_{y^{2}}^{α_{2}} ⋯ ∂_{y^{n}}^{α_{n}} = ∂_{x}^{α_{1}}∂_{v}^{β} ,

where β = (α_{2}, …, α_{n}).

For k ∈ IN_{0} ≡ IN ∪ {0} the spaces C_{loc}^{k}(M) are the spaces of functions k–times differentiable on M. The spaces C^{k}(M̄) are the Banach spaces of functions differentiable k-times on M such that f and its derivatives can be extended to continuous functions on M̄, and equipped with the supremum norm.

Let f_{i} be a sequence of functions, f_{i} ∈ C_{loc}^{∞}(M), such that for every N ∈ IN and for all |α| ≤ N we have

|∂_{y}^{α}f_{i}| ≤ C_{i,N}x^{s_{i,N}} ,

for some sequence s_{i,N} →_{i→∞} ∞ and some constants C_{i,N}. We shall write

f ∼ ∑_{i=0}^{∞} f_{i}

if for every n, M ∈ IN there exists N ∈ IN and a constant C(n, M) such that for all |α| ≤ M

|∂_{y}^{α}(f − ∑_{i=0}^{N} f_{i})| ≤ C(n, M) x^{n} .

f will be said to be polyhomogeneous, f ∈ A_{phg}, if f ∈ C_{loc}^{∞}(M) and if there exists a sequence {N_{j}}_{j=0}^{∞}, N_{j} ∈ IN_{0}, and a sequence of functions f_{jk} ∈ C^{∞}(M̄) such that

f ∼ ∑_{j=0}^{∞} ∑_{k=0}^{N_{j}} f_{jk} x^{j} ln^{k} x .  (E.1)

Let F be a function space over M. A tensor field X = (X^{A}_{β}), where A, β are some multiindices, |A| = r, |β| = s, will be said to belong to F if in local coordinates as described at the beginning of this section the components X^{A}_{β} of X are in F. We shall write that X ∈ x^{α}F when x^{−α}X ∈ F, etc. We shall write that X ∈ F_{1} + F_{2}, where F_{a}, a = 1, 2 are two function spaces, if there exist X_{a} ∈ F_{a} such that X = X_{1} + X_{2}.

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