The Taylor expansion at past time-like infinity
aa r X i v : . [ g r- q c ] J un The Taylor expansion at past time-like infinity.
Helmut FriedrichMax-Planck-Institut f¨ur GravitationsphysikAm M¨uhlenberg 114476 Golm, GermanyAugust 26, 2018
Abstract
We study the initial value problem for the conformal field equationswith data given on a cone N p with vertex p so that in a suitableconformal extension the point p will represent past time-like infinity i − , the set N p \ { p } will represent past null infinity J − , and the freelyprescribed (suitably smooth) data will acquire the meaning of theincoming radiation field for the prospective vacuum space-time. It isshown that: (i) On some coordinate neighbourhood of p there existsmooth fields which satisfy the conformal vacuum field equations andinduce the given data at all orders at p . The Taylor coefficients ofthese fields at p are uniquely determined by the free data. (ii) On N p there exists a unique set of fields which induce the given free dataand satisfy the transport equations and the inner constraints inducedon N p by the conformal field equations. These fields and the fieldswhich are obtained by restricting the functions considered in (i) to N p coincide at all orders at p . Introduction
A purely radiative, asymptotically flat space-time should be generated solely by gravi-tational radiation coming in from past null infinity, extraneous information entering thespace-time at past time-like infinity should be excluded. A natural problem to study isthen the asymptotic characteristic initial value problem for the conformal vacuum fieldequations where data are prescribed on a cone N p with vertex p , similar to the cone { x µ x µ = 0 , x ≥ } in Minkowski space with vertex at x µ = 0. It is to be arranged suchthat the prospective vacuum solution admits a smooth conformal extension in which thepoint p acquires the meaning of past time-like infinity i − and the set N p \ { p } , swept outby the future directed null geodesics through p , represents past null infinity J − .As in any other initial value problem for Einstein’s field equations two different sub-problems must be analysed here: (i) one needs to analyse which part of the initial datacan be prescribed freely and how the remaining data are determined on the initial set bythe field equations, (ii) for suitably given data one has to show the existence of a smoothsolution inducing these data on the initial set. In the situation indicated above both tasksare complicated by the fact that the initial set N p is a smooth hypersurface only away fromthe vertex p . The notion of smoothness and the way data are given thus require particularconsiderations. The present article will be concerned with problem (i), the second problemwill be dealt with in a forthcoming article by Chru´sciel and Paetz ([2]).From the point of view of the physical/geometrical interpretation one would like toconstruct the space-times from a minimal set of data on N p which admit a physical inter-pretation. There are various ways to prescribe data for Einstein’s field equations in char-acteristic initial value problems (cf. [1]), the specific choice usually depending on technicalconsiderations and the particular situation at hand. A natural datum to prescribe at nullinfinity is the radiation field , a complex-valued function that encodes information on thetwo components of the conformal Weyl tensor with the slowest fall-off behaviour at pastnull infinity, thought to represent the two polarization states of the incoming gravitationalradiation.That the radiation field is convenient from the technical point of view has been shownin the proof of J. K´ann´ar’s existence results on the characteristic asymptotic initial valueproblem where data are prescribed on an incoming null hypersurface C which intersectspast null infinity in a space-like slice Σ = C ∩J − and on the future J ′− of that slice in pastnull infinity [7]. A basic step in that proof consists in showing that given the radiationfield on J ′− , the solution and its derivatives of any order can be determined on J ′− bysolving ODE’s along the null generators of J ′− , where the initial data for the integrationare derived from the data prescribed on C and Σ.In the problem to be considered here the analysis is complicated by the fact that theinitial hypersurface tends to loop back onto itself near past time-like infinity, forcing anyanalogue of Σ to shrink to a point and leaving no space for a hypersurface like C . Theinformation for the integration of the solution along the null generators of N p has thusto be extracted completely from the radiation field on N p . Together with the need fora careful discussion of smoothness requirements near the vertex p this leads to variousalgebraic subtleties. first study of this problem was made in [5], where it was shown that for a suitablysmooth prescribed radiation field on N p and a gauge involving a null coordinate adaptedto N p the prospective solution to the conformal field equations is determined uniquely atall orders along the cone N p . However, even under the most convenient assumptions such anull coordinate is singular at and near p . To show that any smooth solution is determineduniquely in the future of the cone N p by its radiation field , there has been performed in [5]a transformation into a gauge which is regular up to an order sufficient for the argument.An existence result for smooth solutions would require, however, a smooth gauge and thus,due to the quasi-linearity of the equations, a transformation which enters the solutions atall orders.To simplify this tedious problem (in section 9 it will be seen that the analysis ofthe transport equations on N p requires a discussion of singular equations in any case) theanalysis in the present article will be based on a smooth gauge right from the outset. Afterintroducing and discussing the field equations and suitable gauge conditions in sections 2- 4 the normal expansion at the point p representing space-like infinity and the propertiesof the radiation field on N p are discussed in section 5.In section 6 an argument by Penrose ([8], [9]) is adapted to the present situation andit is shown in Lemma 6.1 that the covariant derivatives at p of the curvature fields, theconformal factor, and a further scalar field are determined on a formal level uniquely atall orders by the radiation field and that the latter is not subject to any restriction. Torelate these data to a space-time metric we consider in section 7 the structural equations,written as equations for the metric coefficients and connection coefficients. It turns outthat already a subset of the equations suffices to determine the formal Taylor expansionsof these fields and that the expansion coefficients so obtained encode the information onthe chosen gauge (Lemma 7.1).By Borel’s theorem there exist then smooth fields near p whose Taylor expansioncoefficients at p are given precisely by the (symmetric parts of the) coefficients determinedin the formal calculations above. Because only the symmetric parts of the covariantderivatives enter the definition of these functions and only a subset of the structuralequation has been considered in the formal calculations, it remains to be shown that thefunctions so defined do indeed satisfy the conformal fields equations at all orders at p . Asomewhat involved induction argument shows that this is the case (Proposition 8.1).The conformal field equations induce a set of inner equations on N p which splitsnaturally into two subsets. The equations in the first set, referred to as transport equations,determine all unknown fields entering the conformal field equations once the radiation fieldis given (Proposition 9.1). The equations in the second set are inner constraints on thefields so determined. It turns out that they are satisfied by a solution to the transportequations without imposing restrictions on the prescribed radiation field.To identify and analyse the inner equations on N p one needs to express the equationsin terms of a frame adapted to the cone, which is necessarily singular at p . If the resultingequations are solved and the fields are then transformed back into the regular gaugeunderlying Proposition 8.1 they coincide with the field discussed in that Proposition at allorders at p and thus satisfy a necessary smoothness requirement.The fields so obtained, which constitute a complete set of initial data on N p for theconformal field equations, can be considered as a starting point for an existence proof in he category of smooth functions.As pointed out at various places, the analysis presented in this paper also applies tothe characteristic initial value problem where data are given on a finite cone N p whichis thought of as being generated by the (future directed) null geodesics through a point p which is considered as an inner point of a smooth vacuum space-time. In fact, theanalogues of the arguments used in sections 2 - 8 considerably simplify in that case. Insection 9, however, we take advantage of the fact that the conformal Weyl tensor vanishesat null infinity. This allows us to obtain explicit expression for various fields. The analogueof Proposition 9.1 has to be established in the finite problem by an abstract discussion ofthe transport equations, which will not be given here. Let g denote a Lorentzian metric on a four dimensional manifold and ∇ a connectionwhich is metric compatible so that ∇ g = 0. In the following we shall make use of aframe { e k } k =0 ,..., which is orthonormal so that g ij = g ( e i , e j ) = η ij . With the directionalcovariant derivative operators ∇ i ≡ ∇ e i the connection coefficients Γ i k j are define by theequation ∇ i e j = Γ i k j e k . The relation ∇ g = 0 is then equivalent to the anti-symmetryΓ i l j = − Γ i j l , where Γ i l j = Γ i k j g kl . All tensors (except the frame fields) will be givenin the following in terms of the frame e k .For a vector field Z the commutator of the covariant derivatives satisfies( ∇ i ∇ j − ∇ j ∇ i ) Z l = r l kij Z k − t i k j ∇ k Z l , (2.1)where t k i l denotes the torsion tensor, given in terms of coordinates x µ and the framecoefficients e µ k = < e k , dx µ > by the relation t k i l e µ i = e µ k, ν e ν l − e µ l, ν e ν k − (Γ l i k − Γ k i l ) e µ i , (2.2)and r i jkl is the curvature tensor, given by r i jkl ≡ Γ l i j, µ e µ k − Γ k i j, µ e µ l + Γ k i p Γ l p j − Γ l i p Γ k p j (2.3) − (Γ k p l − Γ l p k − t k p l ) Γ p i j . The last term on the right hand side of the equation above can also be expressed in termsof the commutator of the frame fields because [ e k , e l ] = (Γ k p l − Γ l p k − t k p l ) e p by (2.2).The metric is torsion free if and only if the torsion tensor vanishes, which is the case ifand only if ( ∇ j ∇ k − ∇ k ∇ j ) f = 0 , (2.4)for any C -function f .The torsion and the curvature tensor satisfy in general the Bianchi identities X cycl ( ijl ) ∇ i t j k l = X cycl ( ijl ) ( r k ijl − t i m j t m k l ) , (2.5) cycl ( ijl ) ∇ i r h kjl = X cycl ( ijl ) t j m i r h kml , (2.6)where the sums are performed after a cyclic permutation of the indices i, j, l .Assume now that the metric g is torsion free and related by a conformal rescaling g = Ω ˜ g with a conformal factor Ω to a ‘physical’ metric ˜ g which satisfies Einstein’svacuum field equations. These equations can then be expressed in terms of g and Ω andderived fields as follows. We write R ijkl = C ijkl + 2 { g i [ k L l ] j + L i [ k g l ] j } , where C ijkl is the conformal Weyl tensor and L ij = 12 ( S ij + 112 R g ij ) with S ij = R ij − R g ij , denotes the Schouten tensor of g with Ricci tensor R kl and Ricci scalar R . In terms of thetensor fieldsΩ , g ij = η ij , L ij , W i jkl = Ω − C i jkl , Π = 14 ∇ i ∇ i Ω + 124 R Ω , the (metric) conformal field equations read ([3], [4])6 Ω Π − ∇ i Ω ∇ i Ω = 0 , ∇ j ∇ k Ω = − Ω L jk + Π g jk , ∇ l Π = −∇ k Ω L kl , ∇ i L jk − ∇ j L ik = ∇ l Ω W l kij , ∇ i W i jkl = 0 . These equations must be complemented by the structural equations, namely the torsion-free condition t k i l = 0 , (2.7)and the equation r i jkl = R i jkl , (2.8)which will be referred to as the Ricci identity .We note that with the choice Ω ≡ e µ k , Γ i j k , and W i jkl = C i jkl andthe only non-trivial equations are the vacuum Bianchi identity ∇ i W i jkl = 0 and thestructural equations.In the case of a more general conformal factor the equation 6 Ω Π − ∇ i Ω ∇ i Ω = 0will be satisfied on the connected component C q of a point q if it holds at q and the otherequations are satisfied on C q . This is a consequence of the fact that the other equationsimply the relation ∇ k (6 Ω Π − ∇ j Ω ∇ j Ω) = 0 . In the situations considered here, in which either Ω = 0, ∇ i Ω = 0 or Ω ≡ p , the equation 6 Ω Π − ∇ i Ω ∇ i Ω = 0 need not be considered any longer. The 2-index spinor representation
The 2-index spin frame formalism is well adapted to the null geometry and will simplify ouralgebraic task considerably. It amounts essentially to taking complex linear combinationsof various expressions in terms of maps of the form T ijk... → T AA ′ BB ′ CC ′ ... = T ijk... α i AA ′ α j BB ′ α k CC ′ . . . , (3.1)where the α ’s denote the constant van der Waerden symbols α i AA ′ = 1 √ (cid:18) δ i + δ i δ i − i δ i δ i + i δ i δ i − δ i (cid:19) , which are hermitian matrices so that α k AB ′ = α k BA ′ . Frame indices k, l, . . . are thusreplaced by pairs of indices AA ′ , BB ′ , . . . , where A, B, . . . , A ′ , B ′ , . . . take values 0 and 1.None of the operations applied in the following to spinor fields mix primed and unprimedindices. Therefore we shall write T ABC...A ′ B ′ C ′ ... instead of T AA ′ BB ′ CC ′ ... if convenient.There is an operation of complex conjugation under which unprimed indices are convertedinto primed indices and vice versa. Because of the hermiticity of the α ’s the reality of atensor T ijk... is then expressed by the relation T AA ′ BB ′ CC ′ ... = T AA ′ BB ′ CC ′ ... . These tensor fields are considered as members of a tensor algebra which is generated bya 2-dimensional complex vector space and its primed version, both being related to eachother by an operation of complex conjugation. The members of these spaces are calledspinors. For more details (not in all cases employing the same curvature conventions asused here) we refer to [9].The e k are also replaced by e AA ′ = α k AA ′ e k so that the indices A, A ′ specify in thiscase the frame vector fields. Then e ′ , e ′ are real and e ′ , e ′ are complex (conjugate)null vector fields with scalar products g ( e AA ′ , e BB ′ ) = η jk α j AA ′ α k BB ′ = ǫ AB ǫ A ′ B ′ , (3.2)where ǫ AC , ǫ A ′ C ′ , ǫ AC , ǫ A ′ C ′ denote the anti-symmetric spinor fields with ǫ = ǫ ′ ′ = ǫ = ǫ ′ ′ = 1, so that, assuming the summation rule for primed and unprimed indicesseparately, ǫ A B = ǫ AC ǫ BC and ǫ A ′ B ′ = ǫ A ′ C ′ ǫ B ′ C ′ denote Kronecker spinors. The ǫ ’s areuse to raise and lower indices according to the rules κ A = ǫ AB κ B , κ B = κ A ǫ AB , and similar rules apply to primed indices. Upper frame indices can be converted intospinor indices by the van der Waerden symbols α i AA ′ = η ij ǫ AB ǫ A ′ B ′ α j BB ′ Though it will occasionally be convenient to go back to the standard frame notation(or to employ a hybrid notation as discussed below), we shall assume most of the time the elds (except the frame and the spin frame) to be given by their components with respecta suitably chosen spin frame field { ι A } A =0 , which is normalized such that ǫ ( ι A , ι B ) = ǫ AB , (3.3)where ǫ denotes the antisymmetric form on spinor space. As discussed in detail in [9], thefields e ′ = ι ¯ ι ′ and e ′ = ι ¯ ι ′ correspond to real null vector fields while e ′ = ι ¯ ι ′ and e ′ = ι ¯ ι ′ correspond to complex (conjugate) null vector fields which have the scalarproducts (3.2) as a consequence of (3.3).We set Γ AA ′ BB ′ CC ′ = Γ i j k α i AA ′ α BB ′ j α k CC ′ . As a consequence of the anti-symmetryΓ ijk = − Γ ikj these connection coefficients can be decomposed in the formΓ AA ′ BB ′ CC ′ = Γ AA ′ B C ǫ C ′ B ′ + ¯Γ AA ′ B ′ C ′ ǫ C B , with spin connection coefficients Γ AA ′ B C = Γ AA ′ BE ′ CE ′ that satisfy Γ AA ′ BC = Γ AA ′ ( BC ) .Covariant derivatives of spinor fields κ A resp. π A ′ are then defined by ∇ AA ′ κ B = e µAA ′ ∂ µ κ B + Γ AA ′ B C κ C , ∇ AA ′ π B = e µAA ′ ∂ µ π B ′ + ¯Γ AA ′ B ′ C ′ π C ′ , and the definition of the covariant derivative is extended to arbitrary spinor fields byrequiring the Leibniz rule for spinor products. For the commutators of covariant derivativeswe get ( ∇ CC ′ ∇ DD ′ − ∇ DD ′ ∇ CC ′ ) κ A = R A BCC ′ DD ′ κ B , (3.4)and its complex conjugate, where R ABCC ′ DD ′ = R ( AB ) CC ′ DD ′ denotes the curvaturespinor. The usual curvature tensor describing the commutator of covariant derivativesacting of vector field is then given by R AA ′ BB ′ CC ′ DD ′ = R i jkl α i AA ′ α j BB ′ α k CC ′ α l DD ′ (3.5)= R A BCC ′ DD ′ ǫ B ′ A ′ + ¯ R A ′ B ′ CC ′ DD ′ ǫ B A . The curvature spinor admits a decomposition of the form R ABCC ′ DD ′ = Ψ ABCD ǫ C ′ D ′ + Φ ABC ′ D ′ ǫ CD + 2 Λ ǫ A ( C ǫ D ) B ǫ C ′ D ′ . (3.6)The different components are the Weyl spinorΨ ABCD = Ψ ( ABCD ) = − C ijkl α i AE ′ α j B E ′ α k CF ′ α l D F ′ , which contains the information on the conformal Weyl tensor, given by C AA ′ BB ′ CC ′ DD ′ = − Ψ ABCD ǫ A ′ B ′ ǫ C ′ D ′ − ¯Ψ A ′ B ′ C ′ D ′ ǫ AB ǫ CD , and the spinorΦ ABA ′ B ′ = Φ ( AB )( A ′ B ′ ) = ¯Φ ABA ′ B ′ = 12 ( R jk − R η jk ) α j AA ′ α k BB ′ , hich represents the trace free part of the Ricci tensor, andΛ = ¯Λ = 124 R. It holds then L ABA ′ B ′ = Φ ABA ′ B ′ + Λ ǫ AB ǫ A ′ B ′ , and the rescaled conformal Weyl tensor W i jkl = Ω − C i jkl is represented by the rescaledWeyl spinor ψ ABCD = Ω − Ψ ABCD . With this notation the conformal field equations read ∇ AA ′ Π = −∇ BB ′ Ω (Φ
ABA ′ B ′ + Λ ǫ AB ǫ A ′ B ′ ) , ∇ AA ′ ∇ BB ′ Ω = − Ω (Φ
ABA ′ B ′ + Λ ǫ AB ǫ A ′ B ′ ) + Π ǫ AB ǫ A ′ B ′ , ∇ A D ′ Φ BCB ′ D ′ + 2 ǫ A ( B ∇ C ) B ′ Λ = ψ ABCD ∇ D B ′ Ω , ∇ D B ′ ψ ABCD = 0 . and the structural equations take the form0 = e µ AA ′ , ν e ν BB ′ − e µ BB ′ , ν e ν AA ′ − (Γ BB ′ CC ′ AA ′ − Γ AA ′ CC ′ BB ′ ) e µ CC ′ ,r A BCC ′ DD ′ = Ω ψ A BCD ǫ C ′ D ′ + Φ A BC ′ D ′ ǫ CD + 2 Λ ǫ A ( C ǫ D ) B ǫ C ′ D ′ . where r A BCC ′ DD ′ = (3.7)Γ DD ′ A B, µ e µ CC ′ − Γ CC ′ A B, µ e µ DD ′ + Γ CC ′ A F Γ DD ′ F B − Γ DD ′ A F Γ CC ′ F B − (Γ CC ′ F F ′ DD ′ − Γ DD ′ F F ′ CC ′ ) Γ F F ′ A B . In the case of the vacuum field equations, in which Ω ≡
1, the non-trivial unknowns aregiven by e µ AA ′ , Γ AA ′ B C , ψ ABCD and the field equations reduce to ∇ D B ′ ψ ABCD = 0and the structural equations.The following observations will become important later. Forget the meaning of thefields considered above and let the spinor field R ABCC ′ DD ′ in (3.6) be given by spinorfields Ψ ABCD , Φ
ABC ′ D ′ ǫ CD , and Λ which satisfy the symmetries and reality conditionsstated above. The tensor R AA ′ BB ′ CC ′ DD ′ defined by (3.5) then satisfies the analogueof the first Bianchi identity R i [ jkl ] = 0 as a consequence of the symmetries and realityconditions. In fact, the anti-symmetric tensor ǫ ijkl = ǫ [ ijkl ] with ǫ = 1 has the spinorrepresentation ǫ AA ′ BB ′ CC ′ DD ′ = i ( ǫ AC ǫ BD ǫ A ′ D ′ ǫ B ′ C ′ − ǫ AD ǫ BC ǫ A ′ C ′ ǫ B ′ D ′ ) , hich implies R AA ′ BB ′ CC ′ DD ′ ǫ EE ′ BB ′ CC ′ DD ′ = 2 i ( R AH EA ′ H E ′ − ¯ R A ′ H ′ AE ′ E H ′ ) = 0 , (3.8)because R AH EA ′ H E ′ = Φ AEA ′ E ′ − ǫ AE ǫ A ′ E ′ = ¯ R A ′ H ′ AE ′ E H ′ . An analogue of the second Bianchi identity ∇ [ m R ij kl ] = 0 follows under suitable assump-tions. It holds ∇ EE ′ R AA ′ BB ′ CC ′ DD ′ ǫ EE ′ F F ′ CC ′ DD ′ (3.9)= 2 i ( ǫ B ′ A ′ ∇ CD ′ R ABCF ′ F D ′ + ǫ BA ∇ CD ′ ¯ R A ′ B ′ CF ′ F D ′ ) , and, with Ψ ABCD = Ω ψ ABCD , ∇ CD ′ R ABCF ′ F D ′ = − Ω ∇ C F ′ ψ ABCF (3.10)+ n ∇ F D ′ Φ ABF ′ D ′ + 2 ǫ F ( A ∇ B ) F ′ Λ − ∇ C F ′ Ω ψ ABCF o , which will vanish if the conformal field equations are satisfied. These relations are notsurprising, because the Bianchi identities have in fact been used to derive the symmetryproperties of the curvature spinors and also the conformal field equations. Later on weshall need to consider the last two relations, however, under circumstances in which it isnot clear, whether the conformal field equations hold.To shorten the following expressions it will be convenient to introduce some additionalnotation. In the case of spinor fields which carry pairs of spinor indices like AA ′ whichcorrespond to a standard frame indices j we shall occasionally employ a hybrid notationby using the index j , so that equation (3.7) takes for instance the form r A B ij =Γ j A B, µ e µ i − Γ i A B, µ e µ j + Γ i A F Γ j F B − Γ j A F Γ i F B − (Γ i k j − Γ j k i ) Γ k A B . The symmetric part of a spinor field S AB...EF is denoted by S ( AB...EF ) . The totallysymmetric part of a spinor field T A ... A k B ′ ...B ′ j is then given by T ( A ... A k ) ( B ′ ...B ′ j ) . If T is a spinor field and n = ( i , . . . , i n ) a multi-index of order | n | = n we write ∇ n T = ∇ i . . . ∇ i n T and ∇ ( n ) T = ∇ ( i . . . ∇ i n ) T . If X i is a vector field we set X n = X i . . . X i n and write X i . . . X i n ∇ i . . . ∇ i n T = X n ∇ n T = X n ∇ ( n ) T . Unless stated otherwise the connection ∇ will be assumed in the following to be g -compatible and torsion free. We need to restrict the gauge freedom for the conformalfactor, the frame, the coordinates. The conformal gauge near i − . he data for the conformal field equations are to be prescribed on the cone N p = J − ∪ { i − } . The vertex p = i − is to represent past time-like infinity and N p is thought tobe generated by the future directed null geodesics starting at p . Thus one must assumethat Ω = 0 , ∇ AA ′ Ω = 0 , Π = 0 at p. The equations ∇ j ∇ k Ω = − Ω L jk + Π g jk and ∇ l Π = −∇ k Ω L kl suitably transvected withthe geodesic null vectors tangent to the null generators of N p imply then thatΩ = 0 and Π = 0 on N p , ∇ j Ω = 0 on N p \ { p } . (Note that the assumption Π | p = 0 would imply that ∇ j Ω = 0 on N p ).The sign of Π depends on the signature of g . The equation ∇ µ ∇ ν Ω = − Ω L µν + Π g µν implies for a future directed time-like geodesics γ starting at p the relation Π g ( γ ′ , γ ′ ) | p = ∇ γ ′ ∇ γ ′ Ω | p . If we want this to be positive we must assume that sign (Π) = sign ( g ( γ ′ , γ ′ )) = sign ( η ) at i − . This discussion shows that with the assumptions above on Ω and Π at p the field equations themselves will take care for the conformal factor Ω to evolve so thatit will show near p the desired behaviour on N p and on the physical space-time region I + ( N p ).Under a rescaling g µν → ˆ g µν = θ g µν , Ω → ˆΩ = θ Ω with some function θ > | p → ˆΠ | p = (Π θ − ) | p . The transformation laws R µν [ g ] → R µν [ˆ g ] = R µν [ g ] − θ − ∇ µ ∇ ν θ + 4 θ − ∇ µ θ ∇ ν θ −{ θ − ∇ λ ∇ λ θ + θ − ∇ λ θ ∇ λ θ } g µν , and R [ g ] → R [ˆ g ] = θ − { R [ g ] − θ − ∇ λ ∇ λ θ } , (4.1)of the Ricci tensor and the Ricci scalar imply the transformation behaviour S µν [ g ] → S µν [ˆ g ] = S µν [ g ] − θ − ∇ µ ∇ ν θ + 4 θ − ∇ µ θ ∇ ν θ + 12 { θ − ∇ λ ∇ λ θ − θ − ∇ λ θ ∇ λ θ } g µν . Let l µ = 0 denote the tangent vector of a future directed null geodesics γ ( τ ) on J p with γ (0) = p , so that ∇ l l = 0. Then ˆ l = θ − l satisfies ˆ g (ˆ l, ˆ l ) = 0, ˆ ∇ ˆ l ˆ l = 0. This gives θ ˆ l µ ˆ l ν S µν [ˆ g ] = l µ l ν S µν [ g ] − θ − ( l µ ∇ µ ) θ + 4 θ − ( l µ ∇ µ θ ) , or equivalently ˆ l µ ˆ l ν S µν [ˆ g ] θ = l µ l ν S µν [ g ] θ − + 2 ( l µ ∇ µ ) ( θ − ) . (4.2)For prescribed value of ˆ l µ ˆ l ν S µν [ˆ g ] this represents an ODE for θ along the null generatortangent to l . While the value of θ can be fixed at p by specifying there the value of | Π | , here remains the freedom to specify the value of ∇ µ θ at p . The equations above suggestthat a convenient conformal gauge can be defined in a neighbourhood of p in J + ( N p ) byrequiring Ω = 0 , ∇ µ Ω = 0 , Π = 2 η at p, (4.3)and l µ l ν S µν [ g ] = 0 on N p near p, R [ g ] = 0 on J + ( N p ) near p. (4.4)This conformal gauge will be assumed in the following without any problem. When thistype of conformal gauge is used in a wider context, however, it is important to know thatfor a given smooth background g equation (4.2) with ˆ l µ ˆ l ν S µν [ˆ g ] = 0 yields a rescalingfactor θ on N p which has the appropriate smoothness behaviour on N p near the vertex p so that the wave equation obtained on the right hand side of (4.1) by setting R [ˆ g ] = 0can be solved with these data on N p for a smooth function θ near p . This question willbe discussed in the article [2]. The choice of the coordinates near i − . We shall consider p -centered g -normal coordinates x µ near p . These are determinedby the requirements that x µ ( p ) = 0, that g µν (0) = η µν and that for given x µ = 0 and areal parameter τ with | τ | small enough the curve γ : τ → τ x µ is a geodesic through thepoint p . If g µν and Γ µ ρ ν denote the metric coefficients and the Christoffel symbols in thecoordinates x µ the latter condition is equivalent to0 = 2 g µρ ( ∇ γ ′ γ ′ ) ρ = 2 g µρ x ν Γ ν ρ λ ( τ x ) x λ = 2 x ν g νµ,λ ( τ x ) x λ − x ν x λ g νλ,µ ( τ, x ) , which gives in particular that x ν Γ ν ρ λ ( τ x ) x λ = 0 , (4.5)for small enough | τ | . The first equation above implies further 0 = x ν x µ g νµ,λ ( τ x ) x λ = ddτ ( x ν x µ g νµ ( τ x )) and thus x ν x µ g νµ ( τ x ) = x ν x µ g νµ (0), whence2 x ν g νµ ( τ x ) + τ x ν x λ g νλ,µ ( τ x ) = 2 x ν g νµ (0) . With the first equations it follows then0 = τ x ν g νµ,λ ( τ x ) x λ − τ x ν x λ g νλ,µ ( τ, x ) = 2 ddτ { τ ( x ν g νµ ( τ x ) − x ν g νµ (0)) } , and thus x ν g νµ ( τ x ) = x ν g νµ (0) . (4.6)This equation implies in turn x ν x µ g νµ ( τ x ) = x ν x µ g νµ (0) which gives by differentiation τ x ν x λ g νλ,µ ( τ x ) = − x ν g νµ ( τ x ) + 2 x ν g νµ (0) = 0. Because differentiation of (4.6) withrespect to τ gives 0 = x ν g νµ,λ ( τ x ) x λ we see that (4.6) implies that the curves γ consid-ered above are in fact geodesics. The relation (4.6) thus completely characterizes normalcoordinates in terms of algebraic conditions on the metric coefficients. It follows from theequations above that g µν, ρ ( p ) = 0, Γ µ ρ ν ( p ) = 0. n this gauge N p is now given by the set { x µ ∈ R | η µν x µ x ν , x ≥ } . The choice of the frame near i − . Assume now that p -centered g -normal coordinates x µ are given on a convex normalneighbourhood U ′ of p and take their values in a neighbourhood U of the origin of R . Aframe { e k } k =0 , , , is called a normal frame centered at p if it satisfies on U ′ g ( e j , e k ) = η jk , and ∇ γ ′ e k = 0 , for any geodesic γ passing through p . The frame coefficients satisfying e k = e µ k ∂ µ areassumed to satisfy e µ k (0) = δ µk . The 1-forms dual to e k will be denoted by σ j . Then σ j = σ j ν dx ν with σ j µ e µ k = δ jk .That the frame field depends in fact smoothly on the coordinates x µ follows by argumentsknown from the discussion of the exponential function.The equation x ν g νµ ( τ x ) e µ k ( τ x ) = x ν η νµ δ µk expresses that the scalar product g ( γ ′ , e k )is constant along the geodesic γ . The representation g µν = η ij σ i µ σ j ν allows us to rewriteit in the form x µ σ j µ ( τ x ) = x µ δ jµ resp. x µ δ jµ e ν j ( τ x ) = x ν . (4.7)With this relation equation (4.6) implies x µ η µρ δ ρj σ j ν ( τ x ) = x µ η µν resp. x µ η µν e ν k ( τ x ) = x µ η µν δ ν k . (4.8)If the fields σ j µ and the coordinates x µ satisfy the last two relations it follows withoutfurther assumptions that the metric g µν = η ij σ i µ σ j ν satisfies (4.6). In terms of the framefield the information that the x µ are normal coordinates is thus encoded in (4.7), (4.8).Writing ∇ i ≡ ∇ e i , the connection coefficients Γ i j k with respect to the frame e j aredefined by the relations ∇ i e k = Γ i j k e j . They satisfy Γ ijk = − Γ ikj , where Γ ijk = Γ i l k η lj .The tensor field X ( x ) = x µ ∂ µ tangential to the geodesics through p is characterizeduniquely by the conditions X ( p ) = 0 , ∇ µ X ν ( p ) = g µ ν ( p ) , ∇ X X = X. (4.9)By (4.7) it can be written X = X k e k with X k ( x ) = δ kν x ν . The relation ∇ X e j = 0 isequivalent to X k ( x ) Γ k i j ( x ) = δ kν x ν Γ k i j ( x ) = 0 , x µ ∈ U, (4.10)or X AA ′ ( x ) Γ AA ′ B C ( x ) = 0 , x µ ∈ U. (4.11)This is the characterizing property of the normal frame.In the following we shall refer to coordinates x µ and a frame e k (resp. e AA ′ ) whichsatisfy the conditions above as to a normal gauge . We shall always assume this to besupplemented by a normalized spin-frame { ι A } A =0 , which satisfies e AA ′ = ι A ¯ ι A ′ and ∇ X ι A = 0. All spinor fields will be assumed to be given in this frame. Normal expansions
Let x µ and e AA ′ be given in a normal gauge and let X be the vector field defined by (4.9) sothat X = X i e i = X AA ′ e AA ′ with X AA ′ ( x ) = x µ α µ AA ′ , where we set α AA ′ µ = δ i µ α i AA ′ .Let T denote a smooth spinor field and T A ...A j B ′ ...B ′ k its components in the normalframe. If x µ ∗ = 0, then we get with (4.7) and (4.11) along the geodesic γ : τ → τ x µ ∗ ddτ T A ...A j B ′ ...B ′ k ( τ x ∗ ) = T A ...A j B ′ ...B ′ k , µ ( τ x ∗ ) x µ ∗ = x CC ′ ∗ n T A ...A j B ′ ...B ′ k , µ ( τ x ∗ ) e µ CC ′ ( τ x ∗ ) − Γ CC ′ D A T D ...A j B ′ ...B ′ k ( τ x ∗ ) . . . − ¯Γ CC ′ E ′ k B ′ k T A ...A j B ′ ...B ′ k ( τ x ∗ ) o = x CC ′ ∗ ∇ CC ′ T A ...A j B ′ ...B ′ k ( τ x ∗ )with x CC ′ ∗ = x µ ∗ δ i µ α i CC ′ . Applying the argument repeatedly gives d n dτ n T A ...A j B ′ ...B ′ k ( τ x ∗ ) = x C C ′ ∗ . . . x C n C ′ n ∗ ∇ C C ′ . . . ∇ C n C ′ n T A ...A j B ′ ...B ′ k ( τ x ∗ ) . Setting x µ = τ x µ ∗ in the Taylor expansion T A ...A j B ′ ...B ′ k ( τ x ∗ ) = N X n =0 n ! τ n d n dτ n T A ...A j B ′ ...B ′ k (0) + O ( | τ | N +1 ) , the Taylor expansion of T A ...A j B ′ ...B ′ k at p is obtained in the form T A ...A j B ′ ...B ′ k ( x ) = N X | n | =0 | n | ! X n ∇ n T A ...A j B ′ ...B ′ k (0) + O ( | x | N +1 ) (5.1)= N X | n | =0 | n | ! X n ∇ ( n ) T A ...A j B ′ ...B ′ k (0) + O ( | x | N +1 ) . This will be referred to as the normal expansion of T at p . It will be known once the symmetrized covariant derivatives ∇ ( n ) T A ...A j B ′ ...B ′ k ( p ), | n | >
0, are given.
The set C p ∼ S of future directed null vectors at p satisfying g ( l, l ) = 0 and g ( l, e ) = η / √ N p , which are given in thenormal gauge by the curves τ → τ l µ , l µ ∈ C p , 0 ≤ τ < a for some suitable a >
0. Denoteby W p the subset of N p which is generated by the null generators parametrized by a properopen subset W of C p . et κ A ( x ) be a smooth spinor field on W p \{ p } which is parallely propagated along thenull generators and such that κ A ¯ κ A ′ is tangent to the null generators of W p . Because thecomponents κ A are given in the normal frame they are constant along the null generators.Thus, κ A assumes a limit as τ → τ → τ l µ and it can be assumed that κ A ¯ κ A ′ = l AA ′ along that curve. The field κ A is then determined uniquely up to phasetransformations κ A → e i φ κ A with smooth phase factors which are constant along the nullgenerators.For a given tensor field T with spin frame components T A ... A j B ′ ... B ′ k we define its null datum on W p as the spin weighted function T ( x ) = κ A ( x ) . . . κ A j ( x )¯ κ B ′ ( x ) . . . ¯ κ B ′ k ( x ) T A ... A j B ′ ... B ′ k ( x ) , x µ ∈ W p \ { p } . With the normal expansion for T given above this gives at p the asymptotic representation T ( τ x ) = N X n =0 τ n n ! κ C . . . ¯ κ C ′ n κ A . . . ¯ κ B ′ k ∇ C C ′ . . . ∇ C n C ′ n T A ...A j B ′ ...B ′ k (0) + O ( | τ | N +1 ) . for τ >
0. The sum is determined uniquely by the coefficients˜ T n ( κ ) = κ C . . . ¯ κ C ′ n κ A . . . ¯ κ B ′ k ∇ C C ′ . . . ∇ C n C ′ n T A ...A j B ′ ...B ′ k (0) . Because the directions κ A ¯ κ A ′ = l AA ′ are allowed to vary in the open subset W of C p ,knowing these coefficients is equivalent to knowing the symmetrized derivatives T ( A ...A j ) ( B ′ ...B ′ k ) (0) , ∇ ( C ( C ′ . . . ∇ C n C ′ n T A ...A j ) B ′ ...B ′ k ) (0) , n = 1 , , . . . . (5.2)In fact, let S A ...A p A ′ ...A ′ q = S ( A ...A p ) ( A ′ ...A ′ q ) be a symmetric spinor. It will be knownonce its ‘essential components’, denoted by S ij = S ( A ...A p ) i ( A ′ ...A ′ q ) j , are known, whichare obtained by setting for given integers i, j , with 0 ≤ i ≤ p , 0 ≤ j ≤ q , precisely i unprimed resp. j primed indices to equal to one. Choose ( κ , κ ) = β (1 , z ) with z ∈ C and the factor β = (1 + | z | ) − / which ensures the normalization condition on l µ . If thefunction S ( κ ) = κ A . . . ¯ κ A ′ q S A ...A p A ′ ...A ′ q is known then also the function β − p − q S ( κ ) = p X i =0 q X j =0 (cid:18) pi (cid:19)(cid:18) qj (cid:19) S ij z i ¯ z j , and the essential components are given by S ij = ( p − i )! p ! ( q − j )! q ! ∂ iz ∂ j ¯ z ( β − p − q S ( κ )) | z =0 .While the null datum on W p is a spin weighted function which depends on the choice of κ A , the spinors (5.2) at p are given with respect to the spin-frame ι A and are independentof any phase factors. They will be referred to as to the null data of T at p .Of particular importance will be for us the null datum ψ = κ A κ B κ C κ D ψ ABCD , (5.3)associated with the rescaled conformal Weyl spinor ψ ABCD . It is referred to as the radia-tion field . o illustrate some of its properties it will be convenient to proceed as follows. Let SU (2 , C ) denote the subgroup of transformations ( s A B ) A,B =0 , ∈ sl (2 , C ) which satisfy ǫ AC s A B s C D = ǫ BD and s A B ¯ s A ′ B ′ α BB ′ = α AA ′ . Then the null vectors l µ = l µ ( s ) at p with spinor components l AA ′ = s A ¯ s A ′ ′ sweep out the null directions at p and the m AA ′ = m AA ′ ( s ) = s A ¯ s A ′ ′ are complex null vectors orthogonal to l AA ′ . By requiringthem to be constant along the null generators tangent to l AA ′ they will be parallelytransported and tangent to N p along the generators.The information on the radiation field is equivalent to the information containedin the pull back of the tensor W ijkl l i l k to N p . In fact, the latter can be specifiedby the contractions of the symmetric tensor W ijkl l i l k with the field m and ¯ m . Be-cause W ijkl l i m j l k m l and W ijkl l i ¯ m j l k ¯ m l are complex conjugates of each other and thetrace-freeness of W ijkl implies that W ijkl l i m j l k ¯ m l = 0, the information is stored in W ijkl l i m j l k m l = s A s B s C s D ψ ABCD = ψ . Note that this description includesthe complete freedom to perform phase transformations. If this is to be removed, one hasto restrict the choice of s to a local section of the Hopf map SU (2) ∋ s → l AA ′ ( s ) ∈ S ,where S is identified with the set of future directed null directions at p .The null data of ψ at p can be extracted from the null datum ψ on N p as follows. Bytaking derivatives with respect to τ at τ = 0 one gets from the null datum the quantities˜ ψ n ( s ) = s C ¯ s C ′ ′ . . . s C n ¯ s C ′ n ′ s A s B s C s D ∇ C C ′ . . . ∇ C n C ′ n ψ ABCD (0) . As discussed in detail in [5], these functions on SU (2 , C ) translate naturally into ex-pansions in terms of the coefficients T m i j ( s ) of certain finite unitary representations ofthe group SU (2 , C ). With this understanding the essential components of the null data ∇ ( C ( C ′ . . . ∇ C n C ′ n ) ψ ABCD ) (0) can be obtained by performing integrals of ˜ ψ n ( s ) ¯ T m i j ( s )with respect to the Haar measure on SU (2 , C ). Any ambiguities related to choices of phasefactors as indicated above are cancelled out by the integration.To prescribe the null datum in a way which ensures the necessary smoothness proper-ties we start with some symmetric spinor field ψ ∗ ABCD = ψ ∗ ABCD ( x µ ) which is defined andsmooth in a suitable neighbourhood of the origin p of R (so that x µ ( p ) = 0). This fieldwill be thought as being given in a conformal and normal gauge as described in section 4.Assuming s A B as above, one can then consider on the cone N p = { η µν x µ x ν = 0 , x ≥ } (or more precisely on the bundle ˜ N p ∼ R +0 × SU (2) over N p , see section 9) the complex-valued function ψ ( τ, s ) = s A s B s C s D ψ ∗ ABCD ( τ α µEE ′ s E ¯ s E ′ ′ ) , (5.4)as a ‘smooth’ radiation field.The gauge conditions give control on the null data at p for some of the unknowns inthe conformal field equations. It follows immediately from the discussion above and thefirst of conditions (4.4) that the conformal gauge impliesΦ AB A ′ B ′ (0) = 0 , ∇ ( C ( C ′ . . . ∇ C n C ′ n Φ AB ) A ′ B ′ ) (0) = 0 , n = 1 , , . . . . (5.5) Formal expansions at i − . In a conformal gauge satisfying (4.4) the conformal field equations read ∇ AA ′ ∇ BB ′ Ω = − Ω Φ
ABA ′ B ′ + Π ǫ AB ǫ A ′ B ′ , (6.1) ∇ AA ′ Π = −∇ BB ′ Ω Φ
ABA ′ B ′ , (6.2) ∇ A D ′ Φ BCB ′ D ′ = ψ ABCD ∇ D B ′ Ω , (6.3) ∇ D B ′ ψ ABCD = 0 , (6.4)and the curvature spinor (3.6) takes the form R ABCC ′ DD ′ = Ω ψ ABCD ǫ C ′ D ′ + Φ ABC ′ D ′ ǫ CD . (6.5)The following algebraic considerations will be simplified by rewriting equations (6.3) and(6.4). The symmetry of ψ ABCD and the fact that vanishing spinor contractions indicateindex symmetries imply that equation (6.4) is equivalent to ∇ E E ′ ψ ABCD = ∇ ( E E ′ ψ ABCD ) . (6.6)If (6.4) holds, equation (6.3) and its complex conjugate are equivalent to the equations ∇ A A ′ Φ BC B ′ C ′ − ∇ B A ′ Φ AC B ′ C ′ = − ǫ AB ∇ C H ′ Ω ¯ ψ A ′ B ′ C ′ H ′ , (6.7) ∇ A A ′ Φ BC B ′ C ′ − ∇ A B ′ Φ BC A ′ C ′ = − ǫ A ′ B ′ ∇ HC ′ Ω ψ ABCH . (6.8)With the identity ∇ A A ′ Φ BC B ′ C ′ = ∇ ( A ( A ′ Φ BC ) B ′ C ′ ) + 23 ∇ ( A H ′ Φ BC ) H ′ ( B ′ ǫ C ′ ) A ′ − ǫ A ( B ∇ H ( A ′ Φ C ) H B ′ C ′ ) − ǫ A ( B ∇ HH ′ Φ C ) HH ′ ( B ′ ǫ C ′ ) A ′ , these two equations are seen to be equivalent to the equation ∇ A A ′ Φ BC B ′ C ′ = ∇ ( A ( A ′ Φ BC ) B ′ C ′ ) (6.9)+ 23 ψ ABCH ∇ H ( B ′ Ω ǫ C ′ ) A ′ + 23 ǫ A ( B ∇ C ) H ′ Ω ¯ ψ A ′ B ′ C ′ H ′ . We note that ψ ABCD (0) , ∇ E E ′ ψ ABCD (0) = ∇ ( E E ′ ψ ABCD ) (0) , (6.10)represent null data of ψ ABCD and that the conformal gauge (4.3), (4.4) implies by (5.5)and (6.9) thatΦ
BC B ′ C ′ (0) = 0 , ∇ A A ′ Φ BC B ′ C ′ (0) = ∇ ( A ( A ′ Φ BC ) B ′ C ′ ) (0) = 0 . (6.11) ith this it follows from equations (6.2), (6.1) and the gauge conditions that ∇ AA ′ ∇ BB ′ Ω(0) = Π(0) ǫ AB ǫ A ′ B ′ , ∇ k Ω(0) = 0 for | k | = 0 , , , , , (6.12) ∇ k Π(0) = 0 for | k | = 1 , , . (6.13)The relations above imply furthermore that R ABCC ′ DD ′ (0) = 0 , ∇ EE ′ R ABCC ′ DD ′ (0) = 0 . (6.14)The following result, which relates the formal expansion of the curvature fields at agiven point p to the null data of ψ ABCD at p , applies and extends arguments of the theoryof exact sets of fields discussed in [8], [9]. Lemma 6.1
In a neighbourhood of the point p let the fields Ω , Π , Φ ABA ′ B ′ , ψ ABCD , e µ AA ′ , Γ AA ′ B C be smooth and be given in a p -centered normal gauge for the coordinatesand the frame and in a conformal gauge satisfying (4.3), (4.4). Then, if they satisfythe structural equations and the conformal field equations the covariant derivatives of thefields Ω , Π , Φ ABA ′ B ′ , ψ ABCD at all orders are determined uniquely at p by the null data ∇ ( E ( E ′ . . . ∇ E n E ′ n ) ψ ABCD ) ( p ) , n ∈ N , at p .The resulting map which relates to the null data of ψ at p the covariant derivatives ofthe fields Ω , Π , Φ ABA ′ B ′ , ψ ABCD at p extends in a unique way so that it associates withany freely specified sequence of totally symmetric spinors ξ ABCD , ξ E ′ ...E ′ n E ...E n ABCD , n = 1 , , , . . . at p formally ‘covariant derivatives’ the of fields Ω , Π , Φ ABA ′ B ′ , ψ ABCD of any order at p such that ψ ABCD ( p ) = ξ ABCD , ∇ ( E ( E ′ . . . ∇ E n E ′ n ) ψ ABCD ) ( p ) = ξ E ′ ...E ′ n E ...E n ABCD . (6.15) Remark : The coefficients e µ AA ′ and Γ AA ′ B C have been listened in the first statementbecause the field equations involve covariant derivatives of tensor fields and thus requirethe frame and connection coefficients for their formulation. The following argument will,however, never make use of explicit expressions of covariant derivatives in terms of thesecoefficients and partial derivatives of the fields. It only uses formal expressions of covariantderivatives and the standard rules for covariant derivatives such as commutation relationsand the Leibniz rule. Therefore the coefficients are not mentioned in the second part ofthe Lemma. How they are determined will be discussed in the following section.
Proof : At lowest order the first assertion of the Lemma follows from (6.10), (6.11), (6.12)and (6.13). That it is true at higher orders will be shown by an induction argument. Inthis we shall repeatedly make use of (3.4) and (3.6) with Λ = 0. With the identity ∇ CC ′ ∇ DD ′ − ∇ DD ′ ∇ CC ′ = ǫ CD ∇ H ( C ′ ∇ H D ′ ) + ǫ C ′ D ′ ∇ ( C | H ′ | ∇ D ) H ′ , t is seen that (3.4) and its complex conjugate are with our assumptions equivalent to therelations ǫ C ′ D ′ ∇ ( C C ′ ∇ D ) D ′ κ A = Ω ψ ABCD κ B , ǫ C ′ D ′ ∇ ( C C ′ ∇ D ) D ′ ¯ κ A ′ = Φ CDA ′ B ′ κ B ′ ,ǫ CD ∇ C ( C ′ ∇ D D ′ ) κ A = Φ AB C ′ D ′ κ B , ǫ CD ∇ C ( C ′ ∇ D D ′ ) ¯ κ A ′ = Ω ¯ ψ A ′ B ′ C ′ D ′ ¯ κ B ′ . While the induction argument is fairly obvious for the fields Ω, Π, it is more involvedin the case of Φ
ABA ′ B ′ and ψ ABCD . The following observations are important. Considerthe quantities ∇ E E ′ . . . ∇ E n E ′ n ψ ABCD with n ≥
2. If the covariant derivatives wouldcommute it would follow that ∇ E E ′ . . . ∇ E n E ′ n ψ ABCD = ∇ ( E ( E ′ . . . ∇ E n E ′ n ) ψ ABCD ) . (6.16)In fact, any order of the upper indices can be achieved by commuting the covariant deriva-tives. If can be shown that the lower indices can be brought into any order without chang-ing the position of the upper indices, the assertion will follow. Consider, for instance,the index positions given on the left hand side of the equation above. To interchange theindices E k and A (say) we commute ∇ E k E ′ k to the right until we can use (6.6) to swap E k and A , then we commute again to bring ∇ A E ′ k back to the k -th position. To show thatindices E k , E j can be interchanged we operate with ∇ E k E ′ k as before to get ∇ A E ′ k , thencommute ∇ E j E ′ j to the right and use (6.6) again to get ∇ E k E ′ j , then commute ∇ A E ′ k to the right to get ∇ E j E ′ k by using again (6.6). Finally, commute ∇ E j E ′ k and ∇ E k E ′ j into the k -th and j -th position respectively so that the order of the upper indices remainsunchanged.If the covariant derivatives do not commute one can still operate as above but use (3.4)and (3.6) with Λ = 0 each time we commute derivatives. By this procedure the curvaturespinor R A BCC ′ DD ′ and its derivatives enter the expressions and (6.16) is replaced by anequation of the form ∇ E E ′ . . . ∇ E n E ′ n ψ ABCD = ∇ ( E ( E ′ . . . ∇ E n E ′ n ) ψ ABCD ) + . . . , (6.17)where the dots indicate terms which depend on the curvature tensor and its derivativesand thus via the field equations on the fields Ω, s , Φ ABA ′ B ′ , ψ ABCD and their covariantderivatives of order ≤ n −
2. Restriction to p then implies with the induction hypothesisthat the ∇ n ψ ABCD (0)with | n | ≥ s (0) and the null data of ψ ABCD of order ≤ n .Using (6.7) and (6.8) to interchange unprimed as well as primed indices we concludeby similar arguments that for n ≥ ∇ E E ′ . . . ∇ E n E ′ n Φ BC B ′ C ′ = ∇ ( E ( E ′ . . . ∇ E n E ′ n Φ BC ) B ′ C ′ ) + . . . , (6.18)where the dots indicate the terms of order ≤ n −
2, which are generated by commutatingcovariant derivatives and the terms which arise from the right hand sides of equations (6.7)and (6.8). These terms and the commutators contain expressions ∇ k ψ ABCD , ∇ j ¯ ψ A ′ B ′ C ′ D ′ with | k | , | j | ≤ n − ∇ l Ω with | l | ≤ n . Equation (6.1) allows us to expressthe latter in terms of ∇ m Ω, ∇ p s and ∇ q Φ ABCD with | m | , | p | , | q | ≤ n −
2. Restricting to µ = 0 and observing that the right hand side of (6.7), (6.8) vanish at p , we conclude withour induction hypothesis that ∇ n Φ BCB ′ C ′ (0) is obtained as an expression of s (0) and thenull data of ψ ABCD of order ≤ n − ∇ n Ω(0) the induction step follows immediately from (6.1) and forthe quantities ∇ n s (0) it follows with (6.2) by using (6.1) again.This proves the first part of the Lemma. The second statement follows because equa-tion (6.17) shows that no restrictions are imposed by the field equations on the quantities ∇ ( E ( E ′ . . . ∇ E n E ′ n ) ψ ABCD ) (0). By the argument given above all formal covariant deriva-tives are given by algebraic expressions of the null data of ψ at p and these expressionimpose no restrictions on the null data. (cid:3) By (5.1) the symmetric parts of the covariant derivatives determined in Lemma 6.1can be regarded as Taylor coefficients of corresponding tensor fields. By Borel’s theo-rem ([6]) we can then find smooth fields ˆΩ, ˆΠ, ˆ ψ ABCD , ˆΦ
ABA ′ B ′ near p whose Taylorcoefficients at p coincide with the Taylor coefficients determined by the procedure above(but fairly arbitrary away from p ). We can assume that these fields satisfy near p thesymmetry and the reality properties discussed in section 3. With these fields we setˆ R ABCC ′ DD ′ = ˆΩ ˆ ψ ABCD ǫ C ′ D ′ + ˆΦ ABC ′ D ′ ǫ CD , which corresponds to the curvature spinorwhose Taylor coefficients entered the discussion above, and define the ‘curvature tensor’ˆ R i jkl by following (3.5).To decide whether these smooth fields do in fact satisfy the field equations at allorders at p we first need to determine frame and connection coefficients consistent withthe curvature tensor. The frame and the connection coefficients which we want to satisfy the structural equa-tions with the ‘curvature spinor’ ˆ R ABCC ′ DD ′ will be denoted in the following by ˆ e µ i andˆΓ AA ′ C B . It turns out that these functions are determined already by the subsystemˆ t k i l ˆ e µ i X l = 0 , (ˆ r A B kl − ˆ R A B kl ) X k = 0 , (7.1)of the structural equations, where the fields ˆ t k i l and ˆ r A B kl are given by the right handsides of (2.2), (2.3) with e and Γ replaced by ˆ e and ˆΓ and where X i = δ i µ x µ . Assuming(4.7) and (4.10) to be satisfied by ˆ e µ i and ˆΓ AA ′ C B , these equations can be writtenˆ e µ k, ν x ν + ˆ e µ l ( δ lν ˆ e ν k − δ lk ) + ˆΓ k i l X l ˆ e µ i = 0 , (7.2)ˆΓ l A B, µ x µ + ˆΓ k A B δ kµ ˆ e µ l + ˆΓ l j k X k ˆΓ j A B = ˆ R A B kl X k , (7.3)where the ˆΓ k i l are given in spinor notation byˆΓ AA ′ CC ′ BB ′ = ˆΓ AA ′ C B ǫ B ′ C ′ + ¯ˆΓ AA ′ C ′ B ′ ǫ B C , so that they are real and satisfy ˆΓ k i l = − ˆΓ k l i as a consequence of ˆΓ l AB = ˆΓ l ( AB ) .Equations (7.2), (7.3) imply that a smooth solution ˆ e µ i ( x µ ), ˆΓ AA ′ C B ( x µ ) near x µ = 0 ith det(ˆ e µ i ) = 0 must satisfyˆ e µ k (0) = δ µ k , ˆΓ l A B (0) = 0 . (7.4)Equations (7.2), (7.3) can be discussed by analysing the ODE’s which are implied bythem along the curves τ → τ x µ ∗ , x µ ∗ = 0. These ODE’s will be considered in section 9, forour present purpose a more direct approach will be sufficient. To simplify the algebra werewrite the equations in terms of the unknownsˆ c µ k ≡ ˆ e µ k − δ µ k , ˆΓ AA ′ C B , to obtain them in the formˆ c µ k, ν x ν + ˆ c µ k + ˆ c µ l δ lν ˆ c ν k + ˆΓ k i l X l ˆ c µ i + ˆΓ k i l X l δ µ i = 0 , (7.5)ˆΓ l A B, ν x ν + ˆΓ l A B + ˆΓ k A B δ kµ ˆ c µ l + ˆΓ l j k X k ˆΓ j A B − ˆ R A B kl X k = 0 . (7.6)By taking formally partial derivatives, observing (7.4), and evaluating at x µ = 0 oneobtains unique sequences of derivativesˆ c µ k, ν ... ν k (0) , ˆΓ l A B, ν ... ν k (0) , k ∈ N , which are symmetric in the indices ν . . . ν k and are determined by the partial derivativesof the field ˆ R A B kl at the origin. By Borel’s theorem ([6]) we can then find smooth fieldsˆ c µ k and ˆΓ l A B near x µ = 0 whose Taylor coefficients coincide with the coefficients givenabove. Because of ˆ R AB kl = ˆ R ( AB ) kl and the structure of the equations, these fields canbe chosen such that ˆ c µ k is real and ˆΓ l AB = ˆΓ l ( AB ) . While the choice of the fields is ratherarbitrary away from x µ = 0 they satisfy the structural equations at all orders at x µ = 0so that ˆ c µ k, ν x ν + ˆ c µ k + ˆ c µ l δ lν ˆ c ν k + ˆΓ k i l X l ˆ c µ i + ˆΓ k i l X l δ µ i = O ( | x | ∞ ) , (7.7)ˆΓ l A B, ν x ν + ˆΓ l A B + ˆΓ k A B δ kµ ˆ c µ l + ˆΓ l j k X k ˆΓ j A B − ˆ R A B kl X k = O ( | x | ∞ ) , (7.8)where the symbols O ( | x | ∞ ) on the right hand sides indicate that the quantities on the lefthand side are for all n ∈ N of the order O ( | x | n ) as x µ → c µ k (0) = 0 , ˆ c µ k,ν (0) = 0 , ˆΓ l A B (0) = 0 . (7.9)We restrict the following discussion to some neighbourhood of the origin on which thesmooth field ˆ e µ k ≡ δ µ k + ˆ c µ k satisfies det(ˆ e µ k ) = 0. It is there orthonormal for themetric ˆ g µν ≡ η ij ˆ σ i µ ˆ σ j ν , where the ˆ σ i µ denote the 1-forms dual to the ˆ e µ k . BecauseˆΓ i AB = ˆΓ i BA , whence ˆΓ i j k = − ˆΓ i k j , the connection ˆ ∇ defined by ˆ e µ k and ˆΓ i j k resp.ˆΓ i A B , which satisfies for instance ˆ ∇ i ˆ e k = ˆΓ i j k ˆ e k with ˆ ∇ i ≡ ˆ ∇ ˆ e i , is ˆ g -metric compatiblein the sense that ˆ ∇ ˆ g = 0.The symmetries of the fields ˆΓ k AB and ˆ R AB jk imply the following results. emma 7.1 (i) The coordinates x µ and the frame coefficients ˆ e µ k satisfy the require-ments (4.7), (4.8), (4.10) of a normal gauge at all orders at x µ = 0 , so that (ˆ e ν j ( x ) − δ ν j ) δ jµ x µ = O ( | x | ∞ ) , (7.10) x µ η µν (ˆ e ν k ( x ) − δ ν k ) = O ( | x | ∞ ) , (7.11) δ kν x ν ˆΓ k i j ( x ) = O ( | x | ∞ ) . (7.12) (ii) Consider the curve τ → x µ ( τ ) = τ x µ ∗ , x µ ∗ = 0 , through the origin. The components ofits tangent vectors ˙ x µ = x µ ∗ in the frame ˆ e µ k , given by z k ( τ ) = x µ ∗ ˆ σ k µ ( τ x ∗ ) , satisfies z k ( τ ) − δ k µ x µ ∗ = O ( | τ x ∗ | ∞ ) , (7.13) the curve satisfies the geodesic equation at all orders at τ = 0 , ˆ ∇ ˙ x ˙ x = O ( | τ x ∗ | ∞ ) , (7.14) and the frame ˆ e k = ˆ e µ k ∂ x µ satisfies the equation of parallel transport along these curvesat all orders at τ = 0 , ˆ ∇ ˙ x ˆ e k = O ( | τ x ∗ | ∞ ) . (7.15) Proof : To obtain the relations (7.10), (7.11), (7.12) we contract (7.7) and (7.8) with δ kµ x µ and δ lµ x µ respectively to obtain the relationsˆ c µ , ν x ν + ˆ c µ l δ lν ˆ c ν + ˆΓ i l X l (ˆ c µ i + δ µ i ) = O ( | x | ∞ ) , (7.16)ˆΓ A B, ν x ν + ˆΓ k A B δ kµ ˆ c µ + ˆΓ j k X k ˆΓ j A B = O ( | x | ∞ ) . (7.17)for the quantitiesˆ c ν ≡ ˆ c ν j ( x ) δ jµ x µ , ˆ c ν k ≡ x µ η µν ˆ c ν k ( x ) , ˆΓ A B ≡ δ kν x ν ˆΓ k A B ( x ) , ˆΓ i j ≡ δ kν x ν ˆΓ k i j ( x ) . If ˆ c µ = O ( | x | p ) and ˆΓ A B = O ( | x | q ) with some p, q ∈ N , these relations imply with (7.9)relations of the formˆ c µ , ν x ν = O ( | x | p +2 ) + O ( | x | q +1 ) , ˆΓ A B, ν x ν = O ( | x | p +1 ) + O ( | x | q +2 ) . Because ˆ c µ = O ( | x | ) and ˆΓ A B = O ( | x | ) by (7.9), the second relation implies thatˆΓ A B, ν x ν = O ( | x | ) whence also ˆΓ A B = O ( | x | ) and the first relation gives then ˆ c µ , ν x ν = O ( | x | ) whence ˆ c µ = O ( | x | ). Repeating the argument we conclude that ˆ c µ = O ( | x | ∞ )and ˆΓ A B = O ( | x | ∞ ), which are the relations (7.10) and (7.12).Observing thatˆΓ k i l X l δ µ i x λ η λµ = ˆΓ k j l X l η ji δ µ i x λ η λµ = ˆΓ k j l X j X l = 0 , the contraction of (7.7) with x λ η λµ givesˆ c k, ν x ν + ˆ c l δ lν ˆ c ν k + ˆΓ k i l X l ˆ c i + ˆΓ k i l X l δ µ i = O ( | x | ∞ ) , (7.18) hich implies with the previous result that ˆ c k = O ( | x | ∞ ), which is in fact (7.11).Contraction of the relation (ˆ e ν j ( τ x ∗ ) − δ ν j ) δ jµ x µ ∗ = O ( | τ x ∗ | ∞ ), which holds by (7.10),with − ˆ σ k ν ( τ x ∗ ) gives (7.13). In terms of the frame one has( ˆ ∇ ˙ x ˙ x ) k = ddτ z k + z j ˆΓ j k l ( τ x ∗ ) z l = O ( | τ x ∗ | ∞ ) , and ˆ ∇ ˙ x ˆ e k = z j ˆΓ j l k ( τ x ∗ ) ˆ e l = O ( | τ x ∗ | ∞ ) , by (7.13) and (7.12). (cid:3) The subsystem (7.1) of the structural equations determines the functions ˆ e µ k and ˆΓ i j k uniquely and implies thatˆ t i j k = O ( | x | n ) , ˆ r h kjl − ˆ R h kjl = O ( | x | n ) (8.1)with n = 1. Moreover, direct calculations involving (7.4), (6.10), (6.11), (6.12), (6.13)show that ˆ ∇ k ˆΩ(0) = ∇ k Ω(0) , ˆ ∇ k ˆΠ(0) = ∇ k Π(0) , (8.2)for | k | ≤ ∇ k ˆΦ ABA ′ B ′ (0) = ∇ k Φ ABA ′ B ′ (0) , ˆ ∇ k ˆ ψ ABCD (0) = ∇ k ψ ABCD (0) , (8.3)whence ˆ ∇ k ˆ R i jkl (0) = ∇ k R i jkl (0) , (8.4)for | k | ≤ s , ˆΦ ABA ′ B ′ , ˆ ψ ABCD , ˆ R i jkl at the point x µ = 0 with respect to the connectionˆ ∇ . These relations imply thatˆ ∇ AA ′ ˆ ∇ BB ′ ˆΩ + ˆΩ ˆΦ ABA ′ B ′ − ˆΠ ǫ AB ǫ A ′ B ′ = O ( | x | n ) , (8.5)ˆ ∇ AA ′ ˆΠ + ˆ ∇ BB ′ ˆΩ ˆΦ ABA ′ B ′ = O ( | x | n ) , (8.6)ˆ ∇ F D ′ ˆΦ ABF ′ D ′ − ˆ ∇ C F ′ ˆΩ ˆ ψ ABCF = O ( | x | n ) , (8.7)ˆ ∇ C F ′ ˆ ψ ABCF = O ( | x | n ) , (8.8)hold with n = 1. Because the quantities ∇ k R i jkl (0) have been determined by invokingthe Bianchi identities (see the discussion of (3.9), (3.10)) it follows from (8.4) that X cycl ( ijl ) ˆ ∇ i ˆ R h kjl = O ( | x | n ) , (8.9)with n = 1. The purpose of this section is to derive the following result. roposition 8.1 Relations (8.1) to (8.9) hold true for all integers n ∈ N resp. multi-indices k . Remark 8.2
The following argument covers in particular the vacuum case in which Ω isset equal to and the only non-trivial fields are given by e µ k , Γ i j k and ψ ABCD . Before we begin with the proof we need to make a few observations. Because onlya subsystem of the structural equations has been used so far, it is not clear whether theorder relations (8.1) hold for all n ∈ N . The following result shows in particular how thisquestion is related to the Bianchi identity (8.9). Lemma 8.3
Denote by ˆ γ k i l the connection coefficients of the Levi-Civita connection ofthe metric ˆ g µν = η jk ˆ σ j µ ˆ σ k ν with respect to the frame ˆ e k . If the torsion tensor ˆ t i j k ofthe connection ˆ ∇ behaves as ˆ t i j k = O ( | x | N ) for some N ∈ N , N ≥ , then ˆΓ k i l − ˆ γ k i l = O ( | x | N ) .If N ∈ N , N ≥ , and X cycl ( ijl ) ˆ ∇ i ˆ R h kjl = O ( | x | N ) , (8.10) then ˆ t j k l = O ( | x | N +2 ) , ˆ r h kjl − ˆ R h kjl = O ( | x | N +1 ) . Proof : Denote by ˆ c l j k the commutator coefficients satisfying [ˆ e l ˆ e k ] = ˆ c l j k ˆ e j . Withˆ c l i k = ˆ c l j k η ji and ˆ t l i k = ˆ t l j k η ji the torsion free relation can be written ˆΓ l i k − ˆΓ k i l − ˆ c l i k = ˆ t k i l . It is well known that this implies2 ˆΓ kli − { ˆ c l i k + ˆ c k l i − ˆ c i k l } = ˆ t l i k + ˆ t k l i − ˆ t i k l . The same relations hold with ˆ t l i k = 0 if ˆΓ l i k is replaced by ˆ γ l i k . This gives2 (ˆΓ kli − ˆ γ kli ) = ˆ t l i k + ˆ t k l i − ˆ t i k l , which implies the desired result.The connection ˆ ∇ defined by ˆ e k and ˆΓ i j k is metric compatible but at this stage notknown to be torsion free. As pointed out in section 2, the Bianchi identities for the torsiontensor ˆ t i j k and the curvature tensor ˆ r i jkl then take the form X cycl ( ijl ) ˆ ∇ i ˆ t j k l = X cycl ( ijl ) (ˆ r k ijl − ˆ t i m j ˆ t m k l ) , (8.11) X cycl ( ijl ) ˆ ∇ i ˆ r h kjl = X cycl ( ijl ) ˆ t j m i ˆ r h kml . (8.12)By the symmetries and reality conditions of the fields defining ˆ R k ijl the arguments whichled to (3.8) imply P cycl ( ijl ) ˆ R k ijl = 0 near p . Equation (8.11) can thus be written X cycl ( ijl ) ˆ ∇ i ˆ t j k l = X cycl ( ijl ) (ˆ r k ijl − ˆ R k ijl − ˆ t i m j ˆ t m k l ) . ransvecting this equation with X i , observing (7.1) and the anti-symmetry of the torsiontensor gives X i ˆ ∇ i ˆ t j k l + ˆ t j k i ˆ ∇ l X i + ˆ ∇ j X i ˆ t i k l = x i (ˆ r k ijl − ˆ R k ijl ) . Similarly, transvecting the rewrite X cycl ( ijl ) ˆ ∇ i (ˆ r h kjl − ˆ R h kjl ) = X cycl ( ijl ) ˆ t j m i ˆ r h kml − X cycl ( ijl ) ˆ ∇ i ˆ R h kjl , of (8.12) with X i gives X i ˆ ∇ i (ˆ r h kjl − ˆ R h kjl ) + (ˆ r h kji − ˆ R h kji ) ˆ ∇ l X i + ˆ ∇ j X i (ˆ r h kil − ˆ R h kil )= ˆ t l m j ˆ r h kmi X i − X i X cycl ( ijl ) ˆ ∇ i ˆ R h kjl . The result follows now with (4.9) by taking derivatives and evaluating at x µ = 0. (cid:3) Assume that there exist a smooth solution to the field equations in the given gaugewhich induces the prescribed null data at p . By the arguments given above the ∞ -jetof the solution at p must then coincide with the expressions on the right hand sides of(8.2), (8.3), (8.4). It is not obvious, however, that it must also coincide with the ∞ -jetsof the functions ˆ e µ k , ˆΓ i j k , ˆΩ, ˆΠ, ˆΦ ABA ′ B ′ , ˆ ψ ABCD at p . The reason is, that, following(5.1), these functions have been defined so that their Taylor coefficients at x µ = 0 coincidewith the symmetrized derivatives ∇ ( k ) Ω(0), ∇ ( k ) Π(0), ∇ ( k ) Φ ABA ′ B ′ (0), ∇ ( k ) ψ ABCD (0)and it is not clear how much of the information encoded in the unsymmetrized derivativesis transported by the symmetrized derivatives. In particular, while the Bianchi identitiesare by (3.9), (3.10) part of the conformal field equations and the coefficients on the righthand sides of (8.2), (8.3), (8.4) have been determined so as to satisfy these identities, it isnot obvious at this stage that relation (8.10) should be satisfied for integers
N > Proof of Proposition 8.1 : The induction argument to be given below will make use ofthe following general considerations. Let T A ...A j B ′ ...B ′ k denote a smooth spinor field and ∇ a metric compatible connection with curvature tensor r i jkl and torsion tensor t i j k .To begin with assume that t i j k = 0. If the derivatives on the right hand side of thesymmetrization formula ∇ ( i . . . ∇ i n ) T A ...A j B ′ ...B ′ k = 1 n ! X π ∈S n ∇ i π (1) . . . ∇ i π ( n ) T A ...A j B ′ ...B ′ k are then commuted to bring them into their natural order, one obtains an equation of theform ∇ n T A ...A j B ′ ...B ′ k = ∇ ( n ) T A ...A j B ′ ...B ′ k + C ∗ n A ...A j B ′ ...B ′ k , where the spinor field C ∗ is a sum of terms which depend on the covariant derivativesof T and r i jkl of order ≤ | n | −
2. Using these formulas to substitute successively in the ormulas for n = 3 , , . . . the covariant derivatives of T of lower order by their symmetricparts one obtains formulas ∇ n T A ...A j B ′ ...B ′ k = ∇ ( n ) T A ...A j B ′ ...B ′ k + C n A ...A j B ′ ...B ′ k , | n | ≥ , (8.13)with spinor valued functions C n = C n ( ∇ ( p ) T, ∇ q r ) where | p | , | q | ≤ | n | − , which satisfy C ( n ) = 0. These formulas show how the covariant derivatives of T at thepoint x = 0 are determined from the Taylor coefficients in (5.1) and the derivatives of thecurvature tensor at x = 0.Formulas (8.13) represent universal relations. The functions C n depend on the con-nection ∇ only via the derivatives ∇ q r of its curvature tensor. (We ignore the fact thatthe explicit dependence of C n on the ∇ q r may be written in different forms by using thesymmetries and the differential identities satisfied by the curvature tensor). The full indexnotation of (8.13) emphasizes that the explicit structure of the functions C n does dependon the index type of the spinor field T and in following equations we shall write out theappropriate indices.With the notation of Section 6 the unknowns in the field equations must have repre-sentations of the form ∇ n Ω = ∇ ( n ) Ω + C n ( ∇ ( p ) Ω , ∇ q R ) , (8.14) ∇ n Π = ∇ ( n ) Π + C n ( ∇ ( p ) Π , ∇ q R ) , (8.15) ∇ n Φ ABA ′ B ′ = ∇ ( n ) Φ ABA ′ B ′ + C n ABA ′ B ′ ( ∇ ( p ) Φ , ∇ q R ) , (8.16) ∇ n ψ ABCD = ∇ ( n ) ψ ABCD + C n ABCD ( ∇ ( p ) ψ, ∇ q R ) , (8.17)with | p | , | q | ≤ | n | − ∇ q R which are understood as derivatives of thecurvature defined by ∇ . Using these as a starting point we can impose equations (6.1)to (6.5) and proceed as in Section 6 to derive for all multi-indices k expressions for thequantities ∇ k Ω(0), ∇ k Π(0), ∇ k Φ ABA ′ B ′ (0), ∇ k ψ ABCD (0), and thus also for ∇ k R i jkl (0)in terms of the null data, which are given by the totally symmetric part of ˆ ∇ n ˆ ψ ABCD (0).These expressions could be inserted into the equations above but for the sake of comparisonit will be better not to do this here.Formulas (8.13) do not immediately apply to the functions ˆΩ, ˆΠ, ˆΦ
ABA ′ B ′ , ˆ ψ ABCD with the connection ˆ ∇ and the curvature tensor ˆ r i jkl . They can be generalized, however,to the case where the connection ∇ is not torsion free by observing (2.1). The functions C n will then depend on the symmetrized derivatives ∇ ( k ) T A ...A j B ′ ...B ′ k of order | k | ≤ | n | − p with x µ ( p ) = 0 and assume that t i j k ( x µ ) = O ( | x | N ′ ) withsome integer N ′ ≥ x µ →
0. It follows then with (2.1) that the restriction of (8.13)to the point x µ = 0 is valid as it stands if | n | ≤ N ′ + 1. At that point we thus get for | n | ≤ N ′ + 1 the relations ˆ ∇ n ˆΩ = ˆ ∇ ( n ) ˆΩ + C n ( ˆ ∇ ( p ) ˆΩ , ˆ ∇ q ˆ r ) , (8.18) ∇ n ˆΠ = ˆ ∇ ( n ) ˆΠ + C n ( ˆ ∇ ( p ) ˆΠ , ˆ ∇ q ˆ r ) , (8.19)ˆ ∇ n ˆΦ ABA ′ B ′ = ˆ ∇ ( n ) ˆΦ ABA ′ B ′ + C n ABA ′ B ′ ( ˆ ∇ ( p ) ˆΦ , ∇ q ˆ r ) , (8.20)ˆ ∇ n ˆ ψ ABCD = ˆ ∇ ( n ) ˆ ψ ABCD + C n ABCD ( ˆ ∇ ( p ) ˆ ψ, ˆ ∇ q ˆ r ) , (8.21)with | p | , | q | ≤ | n | − C n which are identical with those appearing in thecorresponding equation in (8.14) to (8.17).To compare these two sets of equations we observe that only the properties (4.7) and(4.10) of the frame and the connection coefficients have been used to derive the normalexpansion (5.1). Because these are satisfied by Lemma 7.1 also by the coefficients ˆ e µ k andˆΓ i j k , the normal expansions of the fields ˆΩ, ˆ s , ˆΦ ABA ′ B ′ , ˆ ψ ABCD can thus be expressed interms of the derivatives with respect to the connection ˆ ∇ . This implies ˆ ∇ ( k ) ˆΩ = ∇ ( k ) Ω,ˆ ∇ ( k ) ˆΠ = ∇ ( k ) Π, ˆ ∇ ( k ) ˆΦ ABA ′ B ′ = ∇ ( k ) Φ ABA ′ B ′ , ˆ ∇ ( k ) ˆ ψ ABCD = ∇ ( k ) ψ ABCD for all multi-indices k (here and in the following all spinors are thought to be taken at the point x µ = 0).It follows that the right hand sides of the two sets of equations are distinguished now onlyby the occurrence of the spinors ∇ q R in the first set and the spinors ˆ ∇ q ˆ r in the secondset. Consider now as a induction hypothesis the relations (8.2), (8.3) (8.4) with multi-indices k such that | k | ≤ N ′ . Because the formal derivatives of the tensor R i jkl have beendetermined such that the Bianchi identities are satisfied at all orders, relations (8.4) implythat (8.10) holds with N = N ′ . It follows then from Lemma 8.3 that the assumptionabove on the torsion tensor is satisfied and (8.4) implies with the Lemma that ˆ ∇ q ˆ r h kjl =ˆ ∇ q ˆ R h kjl = ∇ q R h kjl with | q | ≤ N ′ . Comparing the two sets of equations above we canobtain relations (8.2), (8.3) (8.4) with multi-indices k such that | k | = N ′ + 1.With the properties noted in the beginning of this section this implies that (8.2), (8.3)(8.4) hold true for multi-indices k of all orders. It follows that the order relations (8.1)and (8.5) to (8.9) are true for all integers n ∈ N . (cid:3) We have prescribed the radiation field, read off the null data at the vertex p , and con-structed sequences of expansion coefficients at p which can be realized as ∞ -jets at p ofsmooth fields which satisfy the (conformal) field equations at all orders at p . We want todiscuss now which information can be derived from the radiation field in some neighbour-hood of p on N p .By definition, the characteristics of any hyperbolic system of first order are thosehypersurfaces on which the system induces inner equations on (combinations of) the de-pendent variables. On the other hand, the (conformal) Einstein equations induce as aconsequence of their gauge freedom constraints on their Cauchy data on any hypersurface.On null hypersurfaces, which represent the characteristics of the (conformal) Einsteinequations, these facts combine and result in a particular set of inner equations. This setsplits into two subsets. There are equations which involve in particular derivatives inthe direction of the null generators of the null hypersurface. These will be referred to as transport equations . The remaining equations only involve derivatives in directions which re still tangent to the null hypersurface but transverse to the null generators. These willbe referred to as inner constraints .At most points of N p none of the frame vectors e k in the normal gauge is tangent to N p .To derive from the complete set of equations subsystems which only contain derivativesin directions tangent to N p , one thus needs to take (point dependent) linear combinationsof the equations and the dependent variables. Whatever one does to obtain the maximalnumber of transport equations will amount in the end to expressing the equations in termsof a new frame field on N p \ { p } which is such that three of the new frame vectors will betangent to N p \ { p } .We shall describe the procedure and the resulting equations and derive the informationwhich will be needed to construct the desired fields on N p near p . The following discussion,which works out some of the considerations at the end of section 5 in a systematic way,makes use of the analysis in [5], to which we refer for more details. Let { κ a } a =0 , denotethe new spin frame field. If it is chosen such that the null vector κ ¯ κ ′ is tangent to thenull generators on N p \ { p } , the vectors κ ¯ κ ′ and κ ¯ κ ′ will be tangent to N p \ { p } aswell. Because such a frame field cannot have a direction independent limit at p , particularcare has to be taken to construct this frame so near p that the resulting equations willstill admit a convenient analysis near p . It will be required that the frame assumes regularlimits at the point p if p is approached along the null generators of N p \ { p } . Let κ a denote such a limit frame at p . It can be expanded in terms of the normal spin frame ι A underlying our earlier analysis in the form κ a = κ A a ι A . It will be convenient andimplies no restriction to assume the spinors κ A a , a = 0 , κ A a ) A,a =0 , to be normalized such that κ A a ǫ AB κ B b = ǫ ab , κ A a τ AB ′ ¯ κ B ′ b ′ = τ ab ′ . (9.1)Here τ AB ′ = √ α AA ′ = ǫ A ǫ A ′ ′ + ǫ A ǫ A ′ ′ , the quantities ǫ ab , τ ab ′ and α µ aa ′ referringto the new frame take the same numerical values as ǫ AB , τ AB ′ and α µ AA ′ , and the smallletter indices are treated in the same way as the large letter indices.Because we did not specify the null generator along which the limit was taken, theconditions above characterize in fact a family of frames at p . To describe them in detail,denote by SU (2) the Lie group given by the set of complex 2 × s a b ) a,b =0 , satisfying the conditions s a c ǫ ab s b d = ǫ bd , s a c τ ab ′ ¯ s b ′ d ′ = τ cd ′ . (9.2)Any s ∈ SU (2) can be written in the form s = (cid:18) α − ¯ ββ ¯ α (cid:19) , α, β ∈ C , | α | + | β | = 1 , (9.3)and a basis of its Lie-algebra is given by the matrices h = 12 (cid:18) i − i (cid:19) , u = 12 (cid:18) ii (cid:19) , u = 12 (cid:18) −
11 0 (cid:19) . (9.4)The subgroup of SU (2) consisting of the matrices exp ( φ h ) = 12 e i φ e − i φ ! , φ ∈ R , (9.5) ill be denoted by U (1). Comparing (9.1) with (9.2) shows that a complete parametriza-tion of the transformation matrices κ A a is obtained by setting κ A a ( s ) = δ A b s b a with s ∈ SU (2). The corresponding frame spinors will be denote by κ a ( s ).We shall make use of the left invariant vector fields Z u , Z u , Z h generated by u , u , h and define the operators Z + = − ( Z u + i Z u ) , Z − = − ( Z u − i Z u ) , which satisfy the commutation relation [ Z + , Z − ] = 2 i Z h . It should be noted that SU (2)is a real but not a complex analytic Lie group and Z u , Z u must be considered as realvector fields while Z + and Z − take values in the complexifications of the tangent spacesof SU (2) and are complex conjugate to each other. If f is a complex-valued function on SU (2) with complex conjugate ¯ f it holds thus Z ± ¯ f = Z ∓ f . In particular, if the κ A a areconsidered as complex-valued functions on SU (2) as indicated above we get Z + κ A = 0 , Z + κ A = κ A , Z − κ A = − κ a , Z − κ A = 0 , (9.6)and if ¯ κ A ′ a ′ is its spinor complex conjugate we find with the rule above Z + ¯ κ A ′ ′ = − ¯ κ A ′ ′ , Z + ¯ κ A ′ ′ = 0 , Z − ¯ κ A ′ ′ = 0 , Z − ¯ κ A ′ ′ = ¯ κ A ′ ′ . (9.7)Let c µ aa ′ ( s ) = e µ AA ′ κ A a ( s ) ¯ κ A ′ a ′ ( s ) be the frame field associated with κ a at p anddenote by S the sphere { x µ ∈ T p M/x µ x µ = 0 , √ x = 1 } in the tangent space of p . Itholds x µ ∗ ( s ) ≡ c µ ′ ( s ) = α µaa ′ s a ¯ s a ′ ′ (9.8)= 1 √ δ µ + 2 Re ( α ¯ β ) δ µ + 2 Im ( α ¯ β ) δ µ + ( | α | − | β | ) δ µ ) , and x µ ∗ ( s · t ) = x µ ∗ ( s ) for all t ∈ U (1). The Hopf map S ∼ SU (2) ∋ s → x µ ∗ ( s ) ∈ S , thus associates with the left cosets s · U (1), s ∈ SU (2) the null directions x µ ∗ ( s ). It willbe assumed that the frame κ a ( s ) (resp. c aa ′ ( s )) is parallelly propagated along the nullgeodesic τ → τ x µ ∗ ( s ), τ ≥
0, of N p . Because ι A (resp. e AA ′ ) is a p -centered normalframe, it is related to the frame κ a (resp. c aa ′ ( s )) along this curve by the τ -independenttransformation κ A a ( s ) (resp. κ A a ¯ κ A ′ a ′ ( s ), which corresponds to a rotation in SO (3 , R ) ∼ SU (2)) / { , − } that leaves the direction e invariant). While the null directions x µ ∗ ( s ) areinvariant under the action of U (1) the frames ι a resp. c aa ′ are not and our prescriptiondefines in fact a smooth bundle of frames ι a ( τ, s ) (resp. c aa ′ ( τ, s )) over N p \ { p } withprojection π : ι a ( τ, s ) → τ x µ ∗ ( s ) (resp. c aa ′ ( τ, s ) → τ x µ ∗ ( s )) and structure group U (1)(resp. U (1) / { , − } ). For simplicity we will concentrate in the following on the bundle ofspin frames, the discussion of the bundle of vector frames being very similar. The paralleltransport of the frames defines lifts of the null geodesics τ → τ x µ ∗ ( s ) to this bundle(‘horizontal curves’). The tangent vector field defined by the lifts will be denoted by ∂ τ and τ will be considered as a coordinate on ˜ N p . In the limit as τ → SU (2) (in this ense the limit is even preserving the bundle structure). However, while the projection π has rank three over points of N p \ { p } , its rank drops to one in the limit to π − ( p ). Inthe new setting this fact will be reflected by the singular behaviour at π − ( p ) of the frameand the connection coefficients defined below. We denote the bundle in the following by˜ N p and consider it as a four dimensional smooth manifold with boundary π − ( p ), the setof frames c aa ′ ( s ) at p , diffeomorphic to R +0 × SU (2).To discuss the field equations one could choose a local section of the Hopf fibrationat p and push it forward with the flow of ∂ τ to generate a section of ˜ N p . Because therestriction of the projection π will then be a 1 : 1 map away from π − ( p ), it will then beobvious how to lift the frame field. However, apart from a subtlety which will be discussedin the proof of the second part of Proposition 9.1 it will in fact be more convenient toformulate the transport equations as equations on ˜ N p , as has been done in [5].A suitable lift of the frame field can conveniently be discussed by introducing on ˜ N p besides ∂ τ vector fields X ± and S . Because the set π − ( p ) is parametrized by SU (2), thefield Z ± transfer naturally to this set. We set X ± = Z ± , S = − i Z h on π − ( p ) , and extend these fields to ˜ N p by Lie transport so that[ ∂ τ , X ± ] = 0 , [ ∂ τ , S ] = 0 . It follows then that S is tangent to the fibers of ˜ N p and X ± τ = 0 , [ X + , X − ] = − S. In fact, the first result follows from 0 = [ ∂ τ , X ± ] τ = ∂ τ ( X ± τ ) − X ± ∂ τ ( X ± τ ) andthe observation that lim τ → X ± τ = 0 because the fields X ± become in this limit tangentto the set π − ( p ) on which τ vanishes. The second result follows because it is satisfiedin the limit as τ → ∂ τ , [ X + , X − ] + S ] = 0 on˜ N p . Because the images of the fields Z ± under the Hopf map are linearly independent,the images of the fields X ± under the projection π will be linearly independent for τ > s lift from N p to ˜ N p by simple pull-back under the projectionmap. The fields ψ ABCD and Φ
ABA ′ B ′ are in addition subject to a frame transformation sothat they are related to the lifted fields by ψ abcd ( τ, s ) = ψ ABCD ( τ x µ ∗ ( s )) κ A a ( s ) . . . κ D d ( s )and Φ aba ′ b ′ ( τ, s ) = ψ ABA ′ B ′ ( τ x µ ∗ ( s )) κ A a ( s ) . . . ¯ κ B ′ b ′ ( s ).Only the fields c aa ′ = e AA ′ κ A a κ A ′ a ′ with aa ′ = 11 ′ are tangent to N p at the points τ x µ ∗ ( s ) with τ >
0. Lifts of these tangent vector fields on N p to points of ˜ N p are notimmediately well defined because the kernel of the projection π is one-dimensional. For τ > c aa ′ for aa ′ = 11 ′ , i.e. fields satisfying T π (˜ c aa ′ ) = c aa ′ , that can be expanded in terms of the vector fields ∂ τ , X + , X − . Because c ′ istangent to the null geodesics of N p , it follows then immediately that ˜ c aa ′ = ∂ τ . Toanalyse the precise behaviour of ˜ c aa ′ as τ →
0, we observe that by our earlier discussions e µ AA ′ = α µ AA ′ + O ( | x | ) as x µ →
0, which gives c µ aa ′ = α µ AA ′ κ A a κ A ′ a ′ + O ( | τ | ) in thislimit. For any smooth function f = f ( x µ ) we find thus with x µ = τ x µ ∗ ( s ) and (9.3) f ,µ c µ aa ′ = f ,µ α µ AA ′ κ A a κ A ′ a ′ + O ( | τ | ) for aa ′ = 11 ′ . o see how this is related to the action of the vector field X + on the lift of this function to˜ N p , we observe that the vector fields X ± inherit properties of the fields Z ± such as (9.6),(9.7) and find with (9.8) X + f = τ f ,µ X + ( α µ AA ′ κ A ¯ κ A ′ ′ ) = − τ f ,µ α µ AA ′ κ A ¯ κ A ′ ′ , and similarly X − f = − τ f ,µ α µ AA ′ κ A ¯ κ A ′ ′ , so that we can write f ,µ c µ ′ = − τ X + f + O ( | τ | ) , f ,µ c µ ′ = − τ X − f + O ( | τ | ) . It follows that the lifted fields with aa ′ = 11 ′ must have expansions of the form˜ c aa ′ = ǫ a ǫ a ′ ′ ∂ τ − τ ( ǫ a ǫ a ′ ′ X + + ǫ a ǫ a ′ ′ X − ) + c ∗ aa ′ , (9.9)with c ∗ aa ′ = b aa ′ X + + ¯ b aa ′ X − + r aa ′ ∂ τ , (9.10)and complex fields b aa ′ and r aa ′ satisfying¯ r aa ′ = r aa ′ , b ′ = 0 , r ′ = 0 , b aa ′ = O ( | τ | ) , r aa ′ = O ( | τ | ) . (9.11)Because there has not been specified a rule how to extend the new coordinates and thefields ˜ c aa ′ off ˜ N p , there cannot be given an explicit coordinate expression for the field ˜ c ′ .It should be noted, however, that the field ˜ c ′ is determined on ˜ N p once the fields ˜ c aa ′ , aa ′ = 11 ′ , are known there.If it is assumed that the relation c aa ′ = e AA ′ κ A a κ A ′ a ′ holds in a full neighbour-hood of the point p with an x µ -dependent transformation matrix κ A a and it is used that κ a A ≡ ǫ ab κ B b ǫ BA satisfies κ A a κ a B = − ǫ B A , the well known transformation law whichrelates the connection coefficients ˜Γ aa ′ bc with respect to the frame c aa ′ to the connectioncoefficients Γ AA ′ BC with respect to the frame e AA ′ is obtained in the form˜Γ aa ′ bc = − κ B b ǫ BC κ C c,µ c µ aa ′ + Γ AA ′ BC κ A a κ A ′ a ′ κ B b κ C c . Under our assumptions the derivatives κ C c,µ c µ aa ′ are defined on ˜ N p only for aa ′ = 11 ′ so that the formula above can only be used under this restriction. With (9.6) and (9.9) itfollows then that˜Γ aa ′ bc = − τ ( ǫ a ǫ a ′ ′ ǫ b ǫ c + ǫ a ǫ a ′ ′ ǫ b ǫ c ) + Γ aa ′ bc for aa ′ = 11 ′ , (9.12)with a complex-valued field Γ aa ′ bc that satisfiesΓ ′ bc = 0 , Γ aa ′ bc = O ( | τ | ) , (9.13)so that ˜Γ ′ bc = 0 . n this form the coefficients lift to ˜ N p . As discussed in [5], the coefficients ˜Γ aa ′ bc are infact obtained by contracting the connection form on the bundle of frames with the framefield ˜ c aa ′ .On ˜ N p the covariant derivative in the direction of ˜ c aa ′ , aa ′ = 11 ′ , which will be denotedby ˜ ∇ aa ′ , is now given with (9.9), (9.12) by the same rule as known on the base space sothat e.g. ˜ ∇ d ′ ψ abcd = ǫ de (˜ c e ′ ( ψ abcd ) − ˜Γ e ′ f ( a ψ bcd ) f ) . It will be convenient to introduce Σ AA ′ = ∇ AA ′ Ω as an additional unknown tensorfield. Because no rule has been specified to extend the new coordinates and the fields ˜ c aa ′ away from ˜ N p , there cannot be given an explicit coordinate expression for the derivativeof Ω in the direction of ˜ c ′ . Because the field ˜ c ′ is determined on ˜ N p once the fields ˜ c aa ′ , aa ′ = 11 ′ , are known there, the field Σ aa ′ ( τ, s ) = Σ AA ′ κ A a ¯ κ B ′ a ′ can still be discussed asa tensor field on ˜ N p .We are in a position now to obtain the expressions for the transport equations inducedon ˜ N p in the new gauge and to prove the following result. Proposition 9.1
In the conformal gauge (4.3), (4.4) the transport equations induced on ˜ N p by the conformal field equations and the structural equations uniquely determine thefields Ω , Π , Φ aba ′ b ′ and ψ abcd on ˜ N p once the radiation field ψ ( τ, s ) = κ A κ B κ C κ D ψ ABCD | x µ = τ α µEE ′ κ E ¯ κ E ′ ′ , (9.14) is prescribed there.The fields so obtained also satisfy the inner constraint equations on ˜ N p . Remark 9.2
A similar result can be obtained in the vacuum case
Ω = 1 . The discussionof that case is more complicated than the one below because then the conformal Weyl tensordoes not necessarily vanish on ˜ N p . We do not work out the details here. Proof : The gauge conditions (4.3), (4.4) read in the present settingΩ = 0 , Σ aa ′ = 0 , Π = Π ∗ ≡ η on π − ( p ) , (9.15)Φ ′ ′ = 0 , Λ = 0 on ˜ N p . (9.16)The transport equations induced by (6.1), i.e. the equations which involve the directionalderivative ˜ c ′ imply in particular ∂ τ Ω = Σ ′ , ∂ τ Σ ′ = 0 , and thus Ω = 0, Σ ′ = 0 on ˜ N p . With this it follows further ∂ τ Σ ′ = 0 , ∂ τ Σ ′ = 0 , whence Σ ′ = 0, Σ ′ = 0 on ˜ N p . The transport equations induced by (6.1), (6.2) thenfinally imply ∂ τ Σ ′ = Π , ∂ τ Π = 0 , nd thus Σ ′ = τ Π ∗ , Π = Π ∗ on ˜ N p . Collecting results we findΩ = 0 , Σ aa ′ = τ Π ∗ ǫ a ǫ a ′ ′ , Π = Π ∗ on ˜ N p . (9.17)The transport equations induced by the torsion free conditions are given by0 = ˜ t bb ′ aa ′ = [˜ c bb ′ , ˜ c aa ′ ] − (˜Γ bb ′ ee ′ aa ′ − ˜Γ aa ′ ee ′ bb ′ ) ˜ c ee ′ , with bb ′ = 00 ′ and aa ′ = 11 ′ . Inserting here expressions (9.9), (9.12) and setting thefactors of ∂ τ , X + , X − in the resulting equation separately equal to zero shows that thecontent of this equation is equivalent to the conditions ∂ τ b aa ′ + 1 τ b aa ′ + 1 τ ¯Γ aa ′ ′ ′ = Γ aa ′ b ′ + ¯Γ aa ′ ′ ′ b ′ , (9.18) ∂ τ r aa ′ + 1 τ r aa ′ = Γ aa ′ r ′ + ¯Γ aa ′ ′ ′ r ′ − Γ aa ′ − ¯Γ aa ′ ′ ′ , (9.19)(which are satisfied identically for aa ′ = 00 ′ ).The Ricci identity is given for cc ′ , dd ′ = 11 ′ on ˜ N p by˜ c cc ′ (˜Γ dd ′ ab ) − ˜ c dd ′ (˜Γ cc ′ ab ) + ˜Γ cc ′ af ˜Γ dd ′ f b − ˜Γ dd ′ af ˜Γ cc ′ f b − (˜Γ cc ′ ff ′ dd ′ − ˜Γ dd ′ ff ′ cc ′ ) ˜Γ ff ′ ab = Ω ψ abcd ǫ c ′ d ′ + Φ abc ′ d ′ ǫ cd . With (9.17), ˜Γ ′ ab = 0 and ˜ c ′ = ∂ τ it follows ∂ τ ˜Γ ′ ab − ˜Γ ′ ˜Γ ′ ab − ¯˜Γ ′ ′ ′ ˜Γ ′ ab = Φ ab ′ ′ ,∂ τ ˜Γ ′ ab − ˜Γ ′ ˜Γ ′ ab − ¯˜Γ ′ ′ ′ ˜Γ ′ ab = 0 , and thus with (9.9), (9.12) ∂ τ Γ ′ ab + 1 τ (cid:8) Γ ′ ab − Γ ′ ǫ a ǫ b + ¯Γ ′ ′ ′ ǫ a ǫ b (cid:9) (9.20)= Γ ′ Γ ′ ab + ¯Γ ′ ′ ′ Γ ′ ab + Φ ab ′ ′ ,∂ τ Γ ′ ab + 1 τ (cid:8) Γ ′ ab + Γ ′ ǫ a ǫ b + ¯Γ ′ ′ ′ ǫ a ǫ b (cid:9) (9.21)= Γ ′ Γ ′ ab + ¯Γ ′ ′ ′ Γ ′ ab . The transport equations induced by (6.3) are ˜ ∇ c ′ Φ bcb ′ c ′ = ψ bcd Σ d b ′ or, more explicitly, ∂ τ Φ bcb ′ ′ + 1 τ n X + Φ bcb ′ ′ − ǫ ( b Φ c )0 b ′ ′ + ǫ b ′ ′ Φ bc ′ ′ + Φ bcb ′ ′ o − c ∗ ′ (Φ bcb ′ ′ ) (9.22)= − ′ f ( b Φ c ) fb ′ ′ − ¯Γ ′ f ′ b ′ Φ bcf ′ ′ − ¯Γ ′ f ′ ′ Φ bcb ′ f ′ − τ Π ∗ ψ bc ǫ b ′ ′ , hile the transport equations induced by (6.4) are ˜ ∇ d ′ ψ abcd = 0, or, more explicitly, ∂ τ ψ abc + 1 τ (cid:8) X − ψ abc + 3 ǫ ( a ψ bc )01 + ψ abc (cid:9) − c ∗ ′ ( ψ abc ) (9.23)= − ′ f ( a ψ bc ) f − Γ ′ f ψ abcf . While the initial data at τ = 0 are given for b aa ′ , r aa ′ , Γ aa ′ bc by (9.11) and (9.13),they still have to be specified for Φ aba ′ b ′ , ψ abcd . In principle they can be read off from theformal expansions determined earlier but we give a different argument because it shedssome light on the content of the equations. It is convenient here to use the ‘essentialcomponents’ ψ k = κ A ( a κ B b κ C c κ D d ) k ψ ABCD (0)) which are obtained by setting k of thelower indices in brackets equal to 1 and the remaining ones equal to 0. Because the vectorfields X ± approach in the limit τ → Z ± , it follows with (9.6) and (9.14)lim τ → X − ψ = Z − ( κ A κ B κ C κ D ) ψ ABCD (0) = − τ → ψ , and, more generally,lim τ → X − ψ k = − (4 − k ) lim τ → ψ k +1 , k = 0 , . . . , . In the notation of (9.23) this is precisely the relationlim τ → ( X − ψ abc + 3 ǫ ( a ψ bc )01 + ψ abc ) = 0 . It allows one to determine the initial data ψ abcd (0) from the radiation field and at the sametime ensures that the formally singular term in (9.23) admits a limit as τ → N p . Similarly one can determine by X + and X − operationsthe values of lim τ → Φ aba ′ b ′ from Φ ′ ′ with the result thatlim τ → ( X + Φ bcb ′ ′ − ǫ ( b Φ c )0 b ′ ′ + ǫ b ′ ′ Φ bc ′ ′ + Φ bcb ′ ′ ) = 0 , so that the formally singular term in (9.22) admits a limit along a fixed null generator.However, because Φ ′ ′ = 0 on ˜ N p by (9.16), it follows thatlim τ → Φ aba ′ b ′ = 0 . The gauge condition (9.16) and the vanishing of the Weyl tensor on ˜ N p lead to sim-plifications. With this (9.22) implies ∂ τ Φ ′ ′ + 2 τ Φ ′ ′ = 2 Γ ′ Φ ′ ′ + 2 ¯Γ ′ ′ ′ Φ ′ ′ . Because Φ ′ ′ is by assumption the complex conjugate of Φ ′ ′ it follows thatΦ ′ ′ = 0 , Φ ′ ′ = 0 on ˜ N p . (9.24) quation (9.21) implies the coupled system ∂ τ Γ ′ + 2 τ Γ ′ = (Γ ′ + ¯Γ ′ ′ ′ ) Γ ′ ,∂ τ Γ ′ + 1 τ Γ ′ = Γ ′ Γ ′ + ¯Γ ′ ′ ′ Γ ′ , for Γ ′ and Γ ′ whenceΓ ′ = 0 , Γ ′ = 0 on ˜ N p . (9.25)With (9.16), (9.24), (9.25) equation (9.20) implies ∂ τ Γ ′ = Γ ′ Γ ′ ,∂ τ Γ ′ + 1 τ Γ ′ = Γ ′ Γ ′ , from which we conclude thatΓ ′ = 0 , Γ ′ = 0 on ˜ N p . (9.26)With this the remaining equations of (9.21) and (9.20) read ∂ τ Γ ′ + 1 τ Γ ′ = 0 ,∂ τ Γ ′ + 1 τ Γ ′ = Φ ′ ′ , (9.27)which give Γ ′ = 0 , Γ ′ = 1 τ Z τ τ ′ Φ ′ ′ dτ ′ on ˜ N p . (9.28)With these results it follows from (9.18), (9.19) that b aa ′ = 0 , r aa ′ = 0 , c ∗ aa ′ = 0 on ˜ N p for aa ′ = 11 ′ . (9.29)With the resulting simplifications equations (9.22) read ∂ τ Φ ′ ′ + 2 τ Φ ′ ′ = 0 ,∂ τ Φ ′ ′ + 1 τ Φ ′ ′ = − τ Π ∗ ψ ,∂ τ Φ ′ ′ + 1 τ { X + Φ ′ ′ + Φ ′ ′ } = − τ Π ∗ ψ ,∂ τ Φ ′ ′ + 1 τ { X + Φ ′ ′ + 2 Φ ′ ′ } = 0 ,∂ τ Φ ′ ′ + 1 τ { X + Φ ′ ′ + Φ ′ ′ } = − ′ Φ ′ ′ − ¯Γ ′ ′ ′ Φ ′ ′ − τ Π ∗ ψ . he first three of these equations implyΦ ′ ′ = 0 , Φ ′ ′ = − Π ∗ τ Z τ τ ′ ψ dτ ′ , Φ ′ ′ = − Π ∗ τ Z τ τ ′ ψ dτ ′ on ˜ N p . (9.30)Explicit expressions can also be obtained for the solutions of the remaining equations.In particular, imposing the reality conditions, using in the forth equation the expressionfor Φ ′ ′ given by (9.30), and observing that X ± τ = 0 gives for Φ ′ ′ the alternativeexpression Φ ′ ′ = Π ∗ τ Z τ Z τ ′ τ ′′ X − ψ dτ ′′ ! dτ ′ . (9.31)Comparing this with the expression in (9.30), it is seen that consistency requires ∂ τ ψ + 1 τ { X − ψ + 4 ψ } = 0 , which is in fact the first of the equations which follow.With the results obtained so far the transport equations (9.23) read ∂ τ ψ + 1 τ { X − ψ + 4 ψ } = 0 , (9.32) ∂ τ ψ + 1 τ { X − ψ + 3 ψ } = − Γ ′ ψ , (9.33) ∂ τ ψ + 1 τ { X − ψ + 2 ψ } = − ′ ψ , (9.34) ∂ τ ψ + 1 τ { X − ψ + ψ } = − ′ ψ . (9.35)Equation (9.32) has the regular solution ψ = − τ Z τ τ ′ X − ψ dτ ′ . With (9.28), (9.30) one obtainsΓ ′ = − Π ∗ τ Z τ Z τ ′ τ ′′ ¯ ψ ′ ′ ′ ′ dτ ′′ ! dτ ′ , (9.36)which allows one to obtain successively integral expressions for the remaining componentsof ψ abcd on ˜ N p . This completes the proof of the first part of the Proposition.Equations (6.1) and (6.2) imply the inner constraints0 = ˜ c ′ (Σ bb ′ ) − ˜Γ ′ f b Σ fb ′ − ¯˜Γ ′ f ′ b ′ Σ bf ′ + Ω Φ b ′ b ′ − Π ǫ b ǫ ′ b ′ , = ˜ c ′ (Π) + Σ bb ′ Φ b ′ b ′ , and their complex conjugates. A direct calculation using (9.17), (9.25), (9.26) shows thatthey are indeed satisfied on ˜ N p .There do not arise inner constraints from (6.3), (6.4). Those which have not beendiscussed yet contain the operator ˜ c ′ and thus differentiations in directions transverseto N p .Inner constraints are implied by the torsion-free condition and the Ricci identity.Formula (2.2) suggests that the torsion free condition should read on ˜ N p n [˜ c ′ , ˜ c ′ ] − (˜Γ ′ ee ′ ′ − ˜Γ ′ ee ′ ′ ) ˜ c ee ′ o (9.37)There arises, however, a subtlety because the commutator of the fields ˜ c ′ and ˜ c ′ con-tributes a component which is tangential to the fibers of ˜ N p . One way to deal this problemis to follow the torsion-free condition in the form (2.4) and test whether the operator aboveapplied to a function f vanishes if this function is the lift of a scalar function on N p , whenceconstant on the fibres. For reasons which become clear when we discuss the Ricci identitywe prefer a different procedure. If the operator (2.2) is lifted according to our rules, itshould not contain a vertical part and therefore the formula above should be corrected bysubtracting the vertical part supplied by the commutator. By (9.29) the commutator is,however, totally vertical, [˜ c ′ , ˜ c ′ ] = 1 τ [ X + , X − ] = − τ S, and thus drops out after the correction altogether (as it does if applied to the lift of ascalar function). A second subtlety arises because the relation above appears to involve theoperator ˜ c ′ which suggests that it is not an inner condition on ˜ N p . With (9.25), (9.26) andwith (9.28), which states that Γ ′ as well as its complex conjugate ¯Γ ′ ′ ′ vanishes, itfollows, however that not only the factor of ˜ c ′ vanishes but that (˜Γ ′ ee ′ ′ − ˜Γ ′ ee ′ ′ ) =0 for arbitrary indices ee ′ . The inner constraint induced by the torsion free conditions isthus indeed satisfied on ˜ N p .The problem arising from the commutator of ˜ c ′ and ˜ c ′ also affects the discussion ofthe inner constraints induced by the Ricci identity. If one calculates the spinor analogueof (2.1), which reads for the components of interest here( ˜ ∇ ′ ˜ ∇ ′ − ˜ ∇ ′ ˜ ∇ ′ ) λ a = r a b ′ ′ λ b − t ′ ee ′ ′ ∇ ee ′ λ a , one finds that the second term on the right hand side contains a term of the form[˜ c ′ , ˜ c ′ ]( λ a ) − (˜Γ ′ ee ′ ′ − ˜Γ ′ ee ′ ′ ) ˜ c ee ′ ( λ a ) . Performing here the replacement [˜ c ′ , ˜ c ′ ] → [˜ c ′ , ˜ c ′ ]( λ a ) + τ S ( λ a ) and then ignoringthe torsion term as suggested above, has to be compensated by the replacement r a b ′ ′ λ b → r a b ′ ′ λ b − τ S λ a , f the curvature term. To show that the inner constraint induced by the Ricci identityvanishes, we have to take into account the corrected curvature term.Under the action of the group U (1) the frame κ a transforms as κ a → κ b ( exp ( φ h )) b a and the components of a spinor field λ = λ a κ a transform thus as λ a → ( exp ( − φ h )) a b λ b .This implies that S λ a = − i ddφ (( exp ( − φ h ) a b λ b ) | φ =0 = 2 i h a b λ b , with ( h a b ) a,b =0 , denoting the matrix h in (9.4). The equation which should be checkedthus reads 0 = ˜ c ′ (˜Γ ′ ab ) − ˜ c ′ (˜Γ ′ ab ) + ˜Γ ′ af ˜Γ ′ f b − ˜Γ ′ af ˜Γ ′ f b − (˜Γ ′ ff ′ ′ − ˜Γ ′ ff ′ ′ ) ˜Γ ff ′ ab − iτ h ab − Ω ψ ab ǫ ′ ′ − Φ ab ′ ′ ǫ , where we set h ab = h c b ǫ ca . In the cases ab = 00 and ab = 01 a direct calculation usingthe results obtained above shows that this condition is indeed satisfied on ˜ N p . The case ab = 11 is slightly more difficult. With the given results it readily reduces to the condition0 = − τ X + ˜Γ ′ − Φ ′ . Observing (9.36), taking the complex conjugate, and using (9.31) shows that the conditionis indeed satisfied. This proves the second assertion of the Proposition. (cid:3) N p in the normal gauge. In the first part of this section has been shown that there is associated with the radiationfield (5.4), which reads in the present notation ψ ( τ, s ) = κ A κ B κ C κ D ψ ∗ ABCD ( τ α µEE ′ κ E ¯ κ E ′ ′ ) , a unique set of fieldsΩ , Σ aa ′ , Π , Φ aba ′ b ′ , ψ abcd , and ˜ c aa ′ , ˜Γ aa ′ bc , aa ′ = 11 ′ , (9.38)on ˜ N p which satisfy the transport equations and the inner constraints induced by theconformal field equations so that the 0000 components of ψ abcd coincides with ψ ( τ, s ).Apart from the explicitly described singular terms of ˜ c aa ′ and ˜Γ aa ′ bc these fields are smoothfunctions of τ and s ∈ SU (2). On the other hand, it has been shown in sections 6 to 8that with the null data derived from ψ at p can be associated fieldsˆΩ , ˆΣ AA ′ , ˆΠ , ˆΦ ABA ′ B ′ , ˆ ψ ABCD , ˆ e µ AA ′ , ˆΓ AA ′ BC , (9.39)which are defined and smooth on a neighbourhood of p , satisfy at p the conformal fieldequations at all orders, and which have ∞ -jets at p which are uniquely determined by this roperty and the requirement that null data derived from ψ at p coincide with null dataat p derived fromˆ ψ ( τ, s ) = κ A κ B κ C κ D ˆ ψ ABCD ( τ α µEE ′ κ E ¯ κ E ′ ′ ) . While the Taylor expansions of these functions at p are fixed uniquely, they are fairlyarbitrary away from p .To understand the relations between these two sets of fields, we consider the fields(9.39) at the points x µ = τ α µEE ′ κ E ¯ κ E ′ ′ of N p and use the τ -independent frame trans-formation κ A a employed in section 9 to express the fields (9.39) in terms of the adaptedframe to obtain on R +0 × SU (2) ∼ ˜ N p the fieldsˆΩ( τ, s ) = ˆΩ( τ α µEE ′ κ E ¯ κ E ′ ′ ) , ˆΠ( τ, s ) = ˆΠ( τ α µEE ′ κ E ¯ κ E ′ ′ ) , (9.40)ˆΣ aa ′ ( τ, s ) = ˆΣ AA ′ ( τ α µEE ′ κ E ¯ κ E ′ ′ ) κ A a ¯ κ A ′ a ′ , (9.41)ˆΦ aba ′ b ′ ( τ, s ) = ˆΦ ABA ′ B ′ ( τ α µEE ′ κ E ¯ κ E ′ ′ ) κ A a κ B b ¯ κ A ′ a ′ ¯ κ B ′ b ′ , (9.42)ˆ ψ abcd ( τ, s ) = ˆ ψ ABCD ( τ α µEE ′ κ E ¯ κ E ′ ′ ) κ A a κ B b κ C c κ D d . (9.43)Further, we use the considerations of section 9 to derive fields ˆ c aa ′ , ˆΓ aa ′ bc , aa ′ = 11 ′ , on R +0 × SU (2) from ˆ e µ AA ′ , ˆΓ AA ′ BC which have the meaning and the singularity/regularitystructure described in (9.9), (9.12).Because the fields (9.39) satisfy the field equations at all orders at p and have only beensubject to a coordinate and frame transformation, the new fields (9.40) - (9.43) must satisfytogether with the transformed frame and connection coefficients the transport equationsand inner constraints induced on ˜ N p at all orders at p . The uniqueness property statedin Proposition 9.1 thus implies that the Taylor expansion of the fields (9.40) - (9.43) interms of τ at τ = 0 must coincide with the corresponding Taylor expansion of the fields(9.38) at τ = 0.This fact can be expressed in the following way. If the curvature fields given by (9.38)are transformed into the normal gauge of section 4 by settingΦ ABA ′ B ′ = Φ aba ′ b ′ κ a A κ b B ¯ κ a ′ A ′ ¯ κ b ′ B ′ , ψ ABCD = ψ abcd κ a A κ b B κ c C κ d D , (9.44)on N p , thenΦ ABA ′ B ′ = N X n =0 n ! τ n κ E ¯ κ E ′ ′ . . . κ E n ¯ κ E ′ n ′ ∇ E E ′ . . . ∇ E n E ′ n Φ ABA ′ B ′ (0)+ O ( | τ | N +1 ) ,ψ ABCD = N X n =0 n ! τ n κ E ¯ κ E ′ ′ . . . κ E n ¯ κ E ′ n ′ ∇ E E ′ . . . ∇ E n E ′ n ψ ABCD (0) + O ( | τ | N +1 ) , for given N ∈ N , where the coefficients on the right hand sides are the expansion coefficientsassociated with the null data derived from φ at p as described in sections 5 and 6.One can also transform the frame vector fields and the connection coefficients givenby (9.38) into the normal gauge but more complete information is obtained by using thecurvature spinor R ABCC ′ DD ′ = Ω ψ ABCD ǫ C ′ D ′ + Φ ABC ′ D ′ ǫ CD , upplied on N p by (9.44) to integrate the analogues of equations (7.5) and (7.6) on N p along the curves τ → x µ ( τ ) = τ x µ ∗ , where x µ ∗ = α µ AA ′ κ A ¯ κ A ′ ′ is constant along thesecurves. Let e µ k and Γ i A B denote the frame and connection coefficients which constitutein the normal gauge together with the fields Ω, Π Φ ABC ′ D ′ , ψ ABCD supplied by (9.38)initial data on N p for the conformal vacuum equations and set c µ k = e µ k − δ µ k . Therestriction of equations (7.5) and (7.6) to the curves x µ ( τ ) can then be written in the form τ ddτ c µ k + c µ k + c µ l δ lν c ν k + Γ k i l τ X l ∗ ( c µ i + δ µ i ) = 0 , (9.45) τ ddτ Γ k A B + Γ k A B + Γ l A B δ lµ c µ k + Γ k i l τ X l ∗ Γ i A B − R A B ik τ X i ∗ = 0 , (9.46)with X l ∗ = δ l µ x µ ∗ . We are interested here in the solutions which are C in τ and satisfy c µ k | τ =0 = 0 , Γ k A B | τ =0 = 0 . If the left hand sides of the equations are contracted with X l ∗ , the curvature term dropsout and one gets for c µ ≡ c µ k X k ∗ and Γ A B ≡ X k ∗ Γ k A B equations which can be written τ ddτ ( τ c µ ) + ( τ − c µ l ) δ lν ( τ c ν ) + ( τ Γ i l ) X l ∗ ( c µ i + δ µ i ) = 0 ,τ ddτ ( τ Γ A B ) + ( τ − Γ l A B ) δ lµ ( τ c µ ) + ( τ Γ i l ) X l ∗ Γ i A B = 0 . Because of the smoothness assumption and the initial conditions we can assume that τ − c µ l and τ − Γ l A B extend as continuous functions to τ = 0. This allows us to concludethat ( e µ k − δ µ k ) δ k µ x µ = 0 , δ k µ x µ Γ k A B = 0 along x µ ( τ ) . By contracting (9.45) with x ν ∗ η νµ and observing that Γ k i l X l ∗ δ µ i x ν ∗ η νµ = Γ k i l X i ∗ X l ∗ =0, one gets for c k = x ν ∗ η νµ c µ k the equation ddτ ( τ c k ) + ( τ c l ) δ lν ( τ − c ν k ) + Γ k i l X l ∗ ( τ c i ) = 0 , which implies x ν η νµ ( e µ k − δ µ k ) = 0 along x µ ( τ ) . This shows that the gauge conditions (4.7), (4.8), (4.11) will be satisfied on N p by any C solution to (9.45), (9.46).We know from the explicit calculations above that Φ ADA ′ D ′ κ A a κ D ¯ κ A ′ ¯ κ D ′ d ′ =Φ a ′ d ′ = 0 on N p . This implies that R A B CC ′ DD ′ κ B X CC ′ ∗ = Φ A B C ′ D ′ κ B ¯ κ C ′ κ D = 0 along x µ ( τ ) . The contraction of (9.46) with κ B thus gives ddτ ( τ Γ k A B κ B ) + ( τ Γ l A B κ B ) δ lµ ( τ − c µ k ) + Γ k i l X l ∗ ( τ Γ i A B κ B ) = 0 , hence Γ k A B κ B = 0 along x µ ( τ ) . Consequently, Γ k CC ′ DD ′ X DD ′ ∗ = Γ k C D κ D ¯ κ C ′ ′ + ¯Γ k C ′ D ′ κ C ¯ κ D ′ ′ = 0 along x µ ( τ )and equation (9.45) reduces to τ ddτ c µ k + c µ k + c µ l δ lν c ν k + Γ k i l X l ∗ τ c µ i = 0The only C solution vanishing at τ = 0 is given by c µ k = 0 and thus e µ k = δ µ k whence g µν = η µν along x µ ( τ ) . ACKNOWLEDGEMENTS: The author would like to thank Piotr Chru´sciel and Tim Paetzfor discussions and the Erwin Schr¨odinger Institut for financial support.
References [1] P.T. Chru´sciel, T. Paetz. The many ways of the characteristic initial value problem.
Class. Quantum Grav.
29 (2012) 145006 (27 pp), arXiv:1203.4534 [gr-qc][2] P.T. Chru´sciel, T. Paetz. Solutions of the vacuum Einstein equations with initial dataon past null infinity. In preparation[3] H. Friedrich. On the regular and the asymptotic characteristic initial value problemfor Einstein’s field equations.
Proc. Roy. Soc. Lond. A
375 (1981) 169-184.[4] H. Friedrich. The asymptotic characteristic initial value problem for Einstein’s vac-uum field equations as an initial value problem for a first-order quasilinear symmetrichyperbolic system.
Proc. Roy. Soc. Lond. A
378 (1981) 401-421.[5] H. Friedrich. On purely radiative space-times.
Commun. Math. Phys.
103 (1986) 35-65.[6] J. Dieudonn´e.
Foundations of modern analysis . Academic Press, New York, 1969.[7] J. K´ann´ar. On the existence of C ∞ solutions to the asymptotic characteristic initialvalue problem in general relativity. Proc. Roy. Soc.
A 452 (1996) 945 - 952.[8] R. Penrose. Null hypersurface initial data for classical fields of arbitrary spin and forGeneral Relativity.
Gen. Rel. Grav.
12 (1980) 225 - 264.[9] R. Penrose, W. Rindler.
Spinors and Space-Time