On free general relativistic initial data on the light cone
aa r X i v : . [ g r- q c ] O c t On free general relativistic initial data on thelight cone
Piotr T. Chru´scielUniversit¨at Wien Jacek JezierskiUniwersytet WarszawskiKMMF, Ho ˙za 69, 00-682 WarszawaNovember 6, 2018
Abstract
We provide a simple explicit parameterization of free general rela-tivistic data on the light cone.
Contents Introduction
In a series of recent papers [1–4], solutions of the vacuum Einstein equationsdefined to the future of a light cone, say C O , issued from a point O , have beencharacterized in terms of data on a light cone. Part of those data is providedby a symmetric degenerate tensor on C O , and the approach there requiresthis degenerate tensor to be induced on C O by some smooth Lorentzianmetric C = C µν dx µ dx ν . The question then arises, how to usefully describethe induced tensors having this property. Now, tensor fields on (0 , R ) × S with vanishing r –components, where r parameterizes (0 , R ), can always bewritten in the form (see, e.g., [7, Appendix E]) r (cid:2) (1 + γ )˚ s AB + 2 α || AB − ˚ s AB ˚ s CD α || CD + ˚ ǫ AC β || CB + ˚ ǫ BC β || CA (cid:3) dx A dx B , (1.1)where ˚ s ≡ ˚ s AB dx A dx B is the round unit metric on S , and || denotes co-variant differentiation on ( S , ˚ s ). Further, ˚ s AB is the inverse metric to ˚ s AB ,˚ ǫ AB := ˚ s AC ˚ ǫ CB , and ˚ ǫ AB is the alternating tensor on ( S , ˚ s ). This shifts theextendibility question to that of the properties of the functions α , β and γ .The aim of this note is to prove the following (see Section 2.1 for terminologyand Section 2.2 for the proof): Theorem 1.1
A tensor field on (0 , R ) × S of the form (1.1) is the restrictionof a smooth metric in normal coordinates to its light cone if and only if thefunctions α , β and γ are C O –smooth, except possibly for the ℓ = 0 and ℓ = 1 spherical harmonics of α and β which give zero contribution to (1.1) . Consider the vacuum general relativistic characteristic constraint equa-tion in the affinely parameterized gauge (see, e.g., [3]): ∂ τ + τ n − | σ | = 0 , (1.2)where τ is the divergence of C O and σ its shear. Given α and β , Equa-tion (1.2) can be viewed as a non-linear ODE for γ , and thus the functions α and β can be thought of as representing unconstrained degrees of freedomof the gravitational field.Theorem 1.1 invokes normal coordinates for the metric C , and its proofrequires a useful description of the components of a metric tensor in normalcoordinates. This is provided by the following result, proved in Section 2.1,which has some interest of its own: 2 heorem 1.2 The coordinates w µ are normal for a metric C µν if and onlyif there exists a tensor field Ω αβγδ satisfying Ω αβγδ = Ω γδαβ = − Ω βαγδ (1.3) such that g αγ = η αγ + Ω αβγδ w β w δ , (1.4) where underlined tensor components denote coordinate components in thecoordinate system w µ , and where η is the Minkowski metric. Remark 1.3
While we are mainly interested in Lorentzian metrics, we notethat Theorem 1.2 has a direct counterpart in all signatures.The main issue of our work is the understanding of the behaviour of theobjects at hand near the vertex of the cone. Many of the considerationsbelow are valid only within the domain of definition of normal coordinatescentered at the vertex of the light cone, which is sufficient for the purpose.
Consider a smooth metric C in normal coordinates w µ . As already pointedout, we write C βγ for the coordinate components of the metric tensor in thiscoordinate system. We reserve the notation C µν for the components of C inthe coordinate system ( x ≡ u, x ≡ r, x A ), defined as w = x − x , w i = x Θ i ( x A ) with n X i =1 (cid:2) Θ i ( x A ) (cid:3) = 1 . (2.1)Thus ∂ u = − ∂ w , ∂ r = ∂ w + w i r ∂ w i , and η = − ( dx ) + 2 dx dx + r ˚ s AB dx A dx B , η ♯ = ∂ r + 2 ∂ u ∂ r + r − ˚ s AB ∂ A ∂ B . The explicit form of the transformation formulae for a symmetric tensor T µν reads T ≡ T , T ≡ − T − T i Θ i , T A ≡ − T i r ∂ Θ i ∂x A , (2.2)3 ≡ T + 2 T i Θ i + T ij Θ i Θ j , T A ≡ T i r ∂ Θ i ∂x A + T ji r Θ j ∂ Θ i ∂x A , (2.3) T AB ≡ T ij r ∂ Θ i ∂x A ∂ Θ j ∂x B . (2.4)Conversely, T λµ = ∂x α ∂w λ ∂x β ∂w µ T αβ gives T ≡ T , T i ≡ − ( T + T )Θ i − T A ∂x A ∂w i , (2.5) T ij = ( T +2 T + T )Θ i Θ j +( T A + T A ) (cid:18) Θ i ∂x A ∂w j + Θ j ∂x A ∂w i (cid:19) + T AB ∂x A ∂w i ∂x B ∂w j . (2.6)An overline over a function f denotes restriction of the function to thelight cone C O = { w = | ~w |} : if we parameterize the cone by ~w ≡ ( w i ), wehave f ( ~w ) := f ( w = | ~w | , ~w ) , where | ~w | := P i ( w i ) .Note that the domain of definition of normal coordinates for a generalmetric is rarely global, and that our considerations apply only within thisdomain.Since C O can be coordinatised by ~w , functions on C O can be identifiedwith functions of ~w . A function ϕ on C O will be said to belong to C k ( C O ) if ϕ can be written as ˆ ϕ + r ˇ ϕ , where ˆ ϕ and ˇ ϕ are C k functions of ~w . A functionon C O will be called C O –smooth if it can be written as ˆ ϕ + r ˇ ϕ , where ˆ ϕ andˇ ϕ are smooth functions of ~w . A similar definition is used for real-analyticfunctions. It is not too difficult to show that a function ϕ is C O –smooth ifand only if there exists a smooth function ϕ on space-time such that ϕ = f .In other words: Proposition 2.1
A function ϕ defined on C O := (cid:26) w µ ∈ R n +1 : w = sX i ( w i ) (cid:27) can be extended to a C k , respectively smooth, respectively analytic, functionon R n +1 if and only if ϕ is C k ( C O ) , respectively C O –smooth, respectively C O –analytic. The proof of Proposition 2.1 for real-analytic functions can be found in [2];the remaining cases are covered in Appendix A.4 .1 Normal coordinates
Recall that (local) coordinates w µ are normal for the metric C if and only ifit holds that [10] C µν w µ = η µν w µ . (2.7)For completeness, and because of restricted accessibility of [10], we give aproof of this in Appendix B.It follows from (2.2) and (2.7) that C = 1 r C µν w µ w ν = 1 r η µν w µ w ν = 0 , (2.8) C = − r C ν w ν = − r η ν w ν = 1 , (2.9) C A = C iµ w µ ∂ Θ i ∂x A = η iµ w µ ∂ Θ i ∂x A = r X i Θ i ∂ Θ i ∂x A = 12 X i r ∂ (Θ i Θ i ) ∂x A = 0 (2.10)(note that the only information, that does not immediately follow from thefact that C O is the future light cone for the metric C , is provided by(2.9);the remaining equations can serve as consistency checks).We set h µν := C µν − η µν , and we will lower and raise all indices with the metric η . Hence the coordi-nates w α are normal for C = C µν dw µ dw ν if and only if h µν w µ = 0 . (2.11)Note that, from (2.8)-(2.10), h µ = 0 ⇐⇒ h µ := η α η µβ h αβ = 0 . (2.12)The question arises, how to describe exhaustively, and in a useful way,the set of tensors satisfying (2.11). One obvious way of doing this is to use aprojection operator: indeed, for any smooth symmetric tensor φ µν , the tensorfield P αµ P βν ( η ρσ w ρ w σ ) φ µν , where P αβ = δ βα − η αµ w µ w β η ρσ w ρ w σ
5s a smooth tensor field satisfying (2.11). This leads to a restricted class oftensors because of the multiplicative factor ( η ρσ w ρ w σ ) above (in particularthe resulting tensor induced on the light cone has vanishing AB components),and it is not clear how to guarantee smoothness of the final result withoutthe multiplicative factor. Variations on the above using a space projector δ ij − r − x i x j lead to similar difficulties.Note, however, that solutions of (2.11) can be constructed as follows: letΩ αβγδ be any smooth tensor field satisfying (1.3). Then the tensor field h αγ = Ω αβγδ w β w δ (2.13)is symmetric, and satisfies (2.11). Theorem 1.2 follows now immediatelyfrom: Proposition 2.2
A tensor field h µν satisfies (2.11) if and only if there existsa tensor field Ω αβγδ satisfying (1.3) such that (2.13) holds. Proof.
We work in a given smooth coordinate system x µ . The sufficiencyhas already been established. To show necessity recall, first, that any smoothtensor field satisfying A µ x µ = 0 (2.14)can be represented as A µ = Ω µν x ν , with Ω µν = − Ω νµ . To see this, note first that differentiation of (2.14) shows that A µ (0) = 0;then A µ ( x σ ) = Z dds [ sA µ ( sx σ )] ds = Z [ A µ ( sx σ ) + sx ν ∂ ν A µ ( sx σ )] ds = x ν Z s ( ∂ ν A µ − ∂ µ A ν )( sx σ ) ds | {z } =:Ω µν , where we have used ∂ µ ( x ν A ν ) = 0 = ⇒ A µ ( sx σ ) = − sx ν ∂ µ A ν ( sx σ ) . Applying this to h µν at fixed ν we find that there exists a field Ω αβν , anti-symmetric in α and β , such that h µν ( x ρ ) = Ω µαν ( x ρ ) x α . µ and α weconclude that Ω µαν ( x ρ ) = Ω µανβ ( x ρ ) x β for some field Ω αβγδ , anti-symmetric in γ and δ . This is of the desired form,but the pair-interchange symmetry is not completely clear. However, theabove prescription givesΩ µσνλ ( x ρ ) = Z s Z t ( ∂ λ ∂ σ h µν − ∂ λ ∂ µ h σν + ∂ ν ∂ µ h σλ − ∂ ν ∂ σ h µλ ) ( stx ρ ) dt ds , (2.15)which makes manifest all the symmetries claimed. This equation defines thecomponents of the tensor field Ω µσνλ ( x ρ ) in the coordinate system x µ . ✷ One should bear in mind that Ω αβγδ is not uniquely defined by (2.13).However, (2.15) can be used as a canonical choice, if needed.It would be of interest to provide an answer to the corresponding questionfor tensor fields satisfying (2.11) on the light cone only: h µν w µ = 0 . (2.16)We return to this question in Section 3, where some partial results are given,but we have not attempted an exhaustive study. In any case, on the lightcone (2.11) gives the following: h = Ω i j w i w j , (2.17) h i = (cid:0) − Ω j i r + Ω jik w k (cid:1) w j , (2.18) h ij = Ω i j r + (cid:2) − Ω ijk r − Ω jik r + Ω ikjℓ w ℓ (cid:3) w k . (2.19)In coordinates adapted to the light cone (2.17)-(2.19) translate to h = Ω i j w i w j , (2.20) h µ = 0 , (2.21) h A = r (cid:0) Ω j i r − Ω jik w k (cid:1) w j ∂ Θ i ∂x A , (2.22) h AB = r (cid:18) Ω i j r + (cid:0) − Ω ijk r − Ω jik r + Ω ikjℓ w ℓ (cid:1) w k (cid:19) ∂ Θ i ∂x A ∂ Θ j ∂x B . (2.23)7n particular h A factors out through r and is O ( r ), while h AB factors outthrough r and is O ( r ).For further use we note h AB ∂x A ∂w p ∂x B ∂w q = Ω i j ( rδ ip − w i Θ p )( rδ jq − w j Θ q )+ (cid:2) − Ω iqk ( rδ ip − w i Θ p ) − Ω jpk ( rδ jq − w j Θ q ) + Ω pkqℓ w ℓ (cid:3) w k = Ω i j w i w j Θ p Θ q + w k w i (cid:0) Ω iqk Θ p + Ω ipk Θ q (cid:1) (2.24)+ r Ω p q − Ω i q w i w p − Ω p i w i w q − (cid:0) r Ω pqk + r Ω qpk − Ω pkqℓ w ℓ (cid:1) w k . This equation has been derived under the assumption that the coordinates w µ are normal; however, h AB dx A dx B is intrinsic to the light cone, and hencethis equation provides the most general form of a tensor field h AB dx A dx B arising from some smooth metric C µν in coordinates which coincide with thenormal ones on the light cone C O .Note that given the specific structure of the terms containing Θ i above,it is clear how to extract Ω i j and Ω ijk from h AB .We shall say that a tensor field h is C O –smooth if there exists a coordinatesystem w µ in which the components of h are C O –smooth. We conclude that(keeping in mind the local character of normal coordinates): Proposition 2.3
A tensor field ϕ AB dx A dx B on C O arises from the restric-tion to the light cone of a metric in normal coordinates if and only if there ex-ist C O –smooth tensor fields A ij , symmetric in its indices, A ijk , anti-symmetricin the last two indices, and A ijkl , satisfying A ijkl = A klij = − A jikl , such that (cid:0) ϕ AB − r ˚ s AB (cid:1) ∂x A ∂w p ∂x B ∂w q = A ij w i w j Θ p Θ q + w k (cid:0) A iqk w i Θ p + A jpk w j Θ q (cid:1) + r A pq − A iq w i w p − A pi w i w q − (cid:0) rA pqk + rA qpk − A pkqℓ w ℓ (cid:1) w k . (2.25) Proof.
The necessity is clear from (2.24). To show sufficiency, suppose thata tensor field satisfying (2.25) is given. Let Ω µνρσ be any smooth tensor fieldsatisfying Ω µνρσ = − Ω νµρσ = Ω ρσµν such thatΩ i j = A ij , Ω ijk = A ijk , Ω ijkl = A ijkl ;8xistence of Ω αβγδ follows from Proposition 2.1. Then ϕ AB is the restrictionto the light cone of the smooth tensor field η µν + Ω µρνσ w ρ w σ for which thecoordinates w µ are normal. ✷ Recall [3] (compare [8, 9]) that solutions of the Cauchy problem for thevacuum Einstein equations with initial data on an affinely-parameterizedlight cone are uniquely determined by the conformal class of C AB dx A dx B .The remaining components of C µν are thus irrelevant for that purpose, andfor the sake of computations it is convenient to choose them as simple aspossible. It is therefore of interest to enquire whether any C AB can be realizedby a smooth metric satisfying C = − , C i = 0 , C ij w j = w i . (2.26)Our equations above show that this is only possible for C AB ’s which, incoordinates which coincide with the normal ones on C O , are of the form h AB ∂x A ∂w p ∂x B ∂w q = Ω pkqℓ w ℓ w k . Equivalently, all the functions h AB ∂x A ∂w p ∂x B ∂w q are C O –smooth.We finish this section by the following curious observation, which showsthat normal coordinates can be induced from one-dimension-up: Proposition 2.4
The coordinates w i | w =0 are normal for the metric g ij | w =0 dw i dw j . Proof.
From h µν w µ = 0 one finds h ij | w =0 w i = 0, and the result followsfrom the Riemannian counterpart of the equivalence (2.11). ✷ So far we have been using general space dimension n . For n = 3, using astandard decomposition (cf., e.g., [7]) of symmetric tensors on S n − = S wecan write C AB = r (cid:2) (1 + γ )˚ s AB + 2 α || AB − ˚ s AB ˚ s CD α || CD + ˚ ǫ AC β || CB + ˚ ǫ BC β || CA (cid:3) , (2.27)9e wish to find necessary and sufficient conditions on the functions α , β and γ so that C AB arises from a smooth metric on space-time.For reasons that will become apparent shortly, we want to calculate η αµ η βν ˚ ∇ α ˚ ∇ β C µν and η σρ T α w β ǫ αβγδ ˚ ∇ ρ ˚ ∇ γ C δσ , where ˚ ∇ is the covariant derivative of the metric η , while T α := η α = − δ α , w α = η αβ w β . The calculation of η αµ η βν ˚ ∇ α ˚ ∇ β C µν can, and will, be done without as-suming n = 3; we will use the symbol ˚ s to denote the unit round metric on S n − . Writing ( x a ) = ( x , x ), from˚Γ ArB = 1 r δ AB , ˚Γ uAB = − r η AB = ˚Γ rAB , we find˚ ∇ µ h µν := η µσ η νβ ˚ ∇ σ h αβ = ∂ A h Aν + ∂ a h aν + 2 h rB ˚Γ νBr + n − r h νr + h νA ˚Γ BAB + h AB ˚Γ νAB . Hence ˚ ∇ µ h µb = h Ab || A + ∂ a h ab + n − r h br − r h AB η AB , (2.28)where || denotes covariant differentiation on ( S n − , ˚ s ). Further, using ˚ ∇ µ X µ = | det η | − / ∂ µ ( | det η | / X µ ), η αµ η βν ˚ ∇ α ˚ ∇ β h µν = h AB || AB + h ab,ab + 2 ∂ a h aA || A + n + 3 r h rA || A + n + 1 r ∂ a h ra − r ( ∂ u H + ∂ r H ) − r H + n − r h rr , (2.29)and H := η AB h AB , hence H = η µν h µν . To analyze the right-hand side of (2.29) the following formulae are useful: h rr = h uu + 2 h ur + h rr = h ij Θ i Θ j , (2.30) h ur = h ur + h rr = h i Θ i + h ij Θ i Θ j , (2.31) h uA = h rA = h i r ∂ Θ i ∂x A + h ji w j ∂ Θ i ∂x A , (2.32) h uu = h rr = h + 2 h i Θ i + h ij Θ i Θ j , (2.33) H ≡ η AB h AB = η µν h µν + h − h ij Θ i Θ j . (2.34)10unctions of the form r − ( µ + rν ), where µ and ν are restrictions to the lightcone of smooth functions on space-time, will be called mildly singular . Inwhat follows one should keep in mind that any function ϕ can be written as r ϕ/r , and is thus mildly singular if ϕ is C O –smooth. In particular, all h ab ’sand h ab ’s are mildly singular if the metric C is smooth.Denoting by “m.s.” the sum of all mildly singular terms that might occur,one finds ∂ a ∂ b h ab = Θ i Θ j Θ k Θ ℓ ∂ w k ∂ w ℓ h ij + m.s. , ∂ a h aB || B = − i Θ j Θ k Θ ℓ ∂ w i ∂ w j h ij − nr Θ i Θ j Θ k ∂ w k h ij + 2 nr Θ i Θ j h ij + m.s. ,n + 3 r h rB || B = − n + 3 r Θ i Θ j Θ k ∂ w k h ij − n ( n + 3) r Θ i Θ j h ij + m.s. ,n + 1 r ∂ a h ar = n + 1 r Θ i Θ j Θ k ∂ w k h ij + m.s. , − r ( ∂ r H + ∂ u H ) = 1 r Θ i Θ j Θ k ∂ w k h ij + m.s. ,n − r h rr − r H = nr Θ i Θ j h ij + m.s. . We conclude that h AB || AB = − ∂ w i ∂ w j h kℓ Θ i Θ j Θ k Θ ℓ − (2 n + 1) r ∂ w i h jk Θ i Θ j Θ k − n r h ij Θ i Θ j + m.s. . (2.35)We emphasize that this formula is independent of the “gauge condition” h µν w µ = 0.We now assume that the space dimension n equals three. From (2.27) wefind γ = H (cid:16) η µν h µν + h − h ij Θ i Θ j (cid:17) , (2.36)which is mildly singular. Let χ AB denote the ˚ s –trace-free part of h AB , then h AB || AB = χ AB || AB + 1 r ˚∆ γ , s . With some work, using˚∆Θ i = − i , we find1 r ˚∆ γ = − ∂ w i ∂ w j h kℓ Θ i Θ j Θ k Θ ℓ − r ∂ w i h jk Θ i Θ j Θ k − r h ij Θ i Θ j + m.s. , (2.37)which shows that χ AB || AB is again of the general form (2.35): χ AB || AB = − ∂ w i ∂ w j h kℓ Θ i Θ j Θ k Θ ℓ − r ∂ w i h jk Θ i Θ j Θ k − r h ij Θ i Θ j + m.s. . (2.38)It turns out that things improve when the normal coordinates conditionis invoked. For then we have h ij w i = − h j w , (2.39) h j w i = − h w , (2.40) h ij w i w j = h ( w ) , (2.41) w k w i w j ∂ k h ij = (cid:0) − h + w k ∂ k h (cid:1) ( w ) , (2.42) w ℓ w k w i w j ∂ ℓ ∂ k h ij = (cid:2) w ℓ ∂ ℓ (cid:0) − h + w k ∂ k h (cid:1) + 3 (cid:0) h − w k ∂ k h (cid:1)(cid:3) ( w ) . On the light cone this gives h ij Θ i = − h j , (2.43) h j Θ i = − h , (2.44)1 r h ij Θ i Θ j = 1 r h , (2.45)1 r Θ k Θ i Θ j ∂ k h ij = − r h + Θ k ∂ k h , (2.46)Θ ℓ Θ k Θ i Θ j ∂ ℓ ∂ k h ij = 1 r w ℓ ∂ ℓ (cid:0) − h + w k ∂ k h (cid:1) + 3 (cid:0) h − w k ∂ k h (cid:1) . (2.47)Since all the right-hand sides are mildly singular, from (2.38) we concludethat˚∆(˚∆ + 2) α = r − ˚ s AC ˚ s BD χ CD || AB = r η AC η BD χ CD || AB = r χ AB || AB = r × m.s. ; (2.48)12quivalently, ˚∆(˚∆ + 2) α is C O –smooth.Up to an element of the kernel of ˚∆(˚∆ + 2), which is irrelevant as it doesnot contribute to (2.27), we find that α is C O –smooth: Indeed, if we let Πdenote the projector, at fixed r , on the space orthogonal to ℓ = 0 and ℓ = 1spherical harmonics, we have Proposition 2.5
Let k ∈ N ∪ {∞} ∪ { ω } , and let ˚∆(˚∆ + 2) α ∈ C k ( C O ) .Then Π α ∈ C k ( C O ) . Proof:
Assume, first, that k < ∞ . Let k X p =2 (cid:0) f i ··· i p Θ i · · · Θ i p + f ′ i ··· i p − Θ i · · · Θ i p − (cid:1) r p + o k ( r k ) (2.49)be the Taylor series of ˚∆(˚∆ + 2) α , as guaranteed by Lemma A.1. (Thefact that the series starts at p = 2 will be justified shortly.) Decomposingthe coefficients f i ...i p and f ′ i ...i p − into trace terms and trace-free parts, andrearranging the result, we can without loss of generality assume that the f i ...i p ’s and f ′ i ...i p − ’s are traceless. It then follows from [5, pp. 201-202] thatthe finite sums X p fixed f i ...i p Θ i · · · Θ i p and X p fixed f ′ i ...i p − Θ i · · · Θ i p − (2.50)are linear combinations of ℓ = p , respectively ℓ = p −
1, spherical harmonics.(This explains why the sum in (2.49) starts with p = 2, as the image of˚∆(˚∆ + 2) is orthogonal to ℓ = 0 and ℓ = 1 spherical harmonics.) Set ϕ := α − k X p =2 p ( p + 1)( p + 2) × (cid:2) p + 3) f i ...i p Θ i · · · Θ i p + 1( p − f ′ i ...i p − Θ i · · · Θ i p − (cid:3) r p . Then ˚∆(˚∆ + 2) ϕ = o k ( r k ) . ∀ ≤ i ≤ k k ∂ ir Π ϕ k H k − i ( S ) = o ( r k − i ) , and our claim easily follows.If k = ω , convergence for small | w | + | ~w | of the series X p =2 p ( p + 1)( p + 2) (cid:2) p + 3) f i ...i p w i · · · w i p + w p − f ′ i ...i p − w i · · · w i p − (cid:3) (2.51)follows immediately from that of X p =2 (cid:0) f i ...i p Θ i · · · Θ i p + f ′ i ...i p − Θ i · · · Θ i p − (cid:1) r p . If k = ∞ we let ˜ α denote the Borel sum, as in Appendix D, associatedwith (2.51). Then ∀ k ˚∆(˚∆ + 2)( α − ˜ α ) = o k ( r k ) , and one concludes as before. ✷ Returning to our main argument, note that it follows from (2.36) and(2.45) that γ = 12 η µν h µν , (2.52)which shows that γ is C O –smooth.We pass now to the term˚ ∇ ρ (cid:16) η σρ T α w β ǫ αβγδ ˚ ∇ γ h δσ (cid:17) . Let ǫ µνρσ be the unique anti-symmetric tensor such that ǫ = 1 , we set ǫ AB = w i r ǫ ijk ∂x A ∂w j ∂x B ∂w k . Here, and in what follows, we use the summation convention on any repeatedindices, regardless of their positions. We have T α w β ǫ αβγδ ˚ ∇ γ h δσ = − w i ǫ ijk ˚ ∇ j h kσ = rǫ AB ˚ ∇ B h Aσ = rǫ AB h Aσ ; B , AC ; B = h AC || B + 1 r η AB ( h uC + h rC ) + 1 r η BC ( h uA + h rA ) ,h Aa ; B = h Aa || B + 1 r η AB ( h ua + h ra ) − δ ra r h AB ǫ AB h AC ; B = ǫ AB h AC || B + 1 r ǫ AC ( h uA + h rA ) ,ǫ AB h Aa ; B = ǫ AB h Aa || B , and finally˚ ∇ ρ (cid:16) η σρ T α w β ǫ αβγδ ˚ ∇ γ h δσ (cid:17) = (cid:0) rǫ AB h Aσ ; B (cid:1) ; σ = 1 r (cid:0) r η ab ǫ AB h Ab ; B (cid:1) ,a + (cid:0) rη CD ǫ AB h AC ; B (cid:1) || D = rǫ AB χ AC || BC + rǫ AB ∂ u h uA || B | {z } h rA || B + 1 r ∂ r (cid:0) r ǫ AB h rA || B | {z } h uA || B + h rA || B (cid:1) , (2.53)where, as before, χ AB is the traceless part of h AB .The left-hand side of the last equation is a smooth function on space-time.Next, rǫ AB h Ar ; B = T α w β ǫ αβγδ ˚ ∇ γ h δσ dw σ ( ∂ r ) = T α w β ǫ αβγδ ˚ ∇ γ h δ + w i r T α w β ǫ αβγδ ˚ ∇ γ h δi , where the right-hand side is the sum of a smooth function and of a smoothfunction divided by r . Hence so is its ∂ u = − ∂ w –derivative, which is thesecond term in the last line of (2.53). We note the identity, rǫ AB h Au ; B = − T α w β ǫ αβγδ ˚ ∇ γ h δ , where the right-hand side is a smooth function on space-time. We concludethat ǫ AB χ AC || BC is mildly singular. (2.54)This implies that ˚∆(˚∆ + 2) β = r − ˚ ǫ AB ˚ s CD χ AD || BC = r ǫ AB η CD χ AD || BC = r ǫ AB χ AC || BC = r × m.s. (2.55)15p to an element of the kernel of ˚∆(˚∆ + 2), which is irrelevant as it does notcontribute to (2.27), we find that β is C O –smooth. We have therefore provednecessity in Theorem 1.1.We wish to show, now, that the conditions of our statement are sufficient: C O –smooth functions α , β , and γ lead to smooth metrics in normal coordi-nates. For this, it is convenient to view tensors on S as tensors on R whichare orthogonal to y i in all indices. For example, the metric ˚ s = ˚ s AB dx A dx B is identified with r − times the projector P ij = δ ij − w i w j r . Indeed,˚ s AB dx A dx B = ˚ s AB ∂x A ∂w i ∂x B ∂w j dw i dw j = r − (cid:18) δ ij − w i w j r (cid:19) dw i dw j . So, if Y i or S ij are tensors satisfying Y i w i = 0 = S ij w j = S ij w i , we have theformulae D i Y j = P ik P j ℓ ∂ k Y ℓ , D i S jm = P ik P j ℓ P mn ∂ k S ℓn , D i S im = P ℓk ∂ k S ℓn . In this formalism we have D i f = P ij ∂ j f , and D i D j f = P ik P jℓ ∂ k ( P ℓm ∂ m f ) = P ik P jℓ ∂ k ∂ ℓ f − r P ij Θ m ∂ m f . Hence P ij D i D j f = P ij ∂ i ∂ j f − r Θ m ∂ m f . (2.56)Let us write α = ˇ α + r ˆ α , where ˇ α and ˆ α are smooth functions of ~w . We note that D i D j α = D i D j ˇ α + r D i D j ˆ α . Equation (2.56) with f replaced by ˇ α gives r (cid:0) D i D j ˇ α − P ij P kℓ D k D ℓ ˇ α (cid:1) = r (cid:0) P ik P j ℓ ∂ k ∂ ℓ ˇ α − P ij P kℓ ∂ k ∂ ℓ ˇ α (cid:1) = Θ i Θ j w k w ℓ ∂ k ∂ ℓ ˇ α − w i w ℓ ∂ j ∂ ℓ ˇ α − w j w ℓ ∂ i ∂ ℓ ˇ α − ( r δ ji − w i w j ) ∂ ℓ ∂ ℓ ˇ α . (2.57)16n identical calculation applies to ˆ α . We conclude that the tensor field (1.1)contains Θ ⊗ Θ terms of the form as in (2.25), with A ij = ∂ i ∂ j ˇ α + r∂ i ∂ j ˆ α . (2.58)Next, we write β = ˇ β + r ˆ β , where ˇ β and ˆ β are smooth functions of ~w . The contribution of ˇ β to the tensorfield (1.1) can be rewritten as rw ℓ (cid:0) ǫ kℓi ∂ w j ∂ w k ˇ β + ǫ kℓj ∂ w i ∂ w k ˇ β (cid:1) + w ℓ w m (cid:0) Θ i ǫ jkℓ + Θ j ǫ ikℓ (cid:1) ∂ w k ∂ w m ˇ β , with a similar formula for ˆ β . The resulting Θ terms are of the right-form w m w ℓ (cid:0) A ℓim Θ j + A ℓjm Θ i (cid:1) as in (2.25) if we set A ℓjm = ǫ mkℓ ( ∂ w j ∂ w k ˇ β + r∂ w j ∂ w k ˆ β ) − ǫ jkℓ ( ∂ w m ∂ w k ˇ β + r∂ w m ∂ w k ˆ β ) . (2.59)To summarize: let Ω i j be a smooth extension of A ij as given by (C.9),and let Ω ijk be a smooth extension of A ijk as given by (2.59), if we setΩ ijkl = 0, then the restrictions to the light cone of the ij components of thetensor field ( η µν + Ω µαρβ w α w β ) dw µ dw ν reproduce the non-manifestly C O –smooth terms in r (cid:2) (1 + γ )˚ s AB + 2 α || AB − ˚ s AB ˚ s CD α || CD + ˚ ǫ AC β || CB + ˚ ǫ BC β || CA (cid:3) ∂x A ∂w i ∂x B ∂w j . So the difference is a C O –smooth tensor field, say f ij = ˆ f ij + r ˇ f ij , with ˆ f ij and ˇ f ij smooth tensors on R , that satisfies f ij w j = 0 . (2.60)Now, it is not directly apparent that we have the desired formula, as inProposition 2.2, f ij = A ikjℓ w k w ℓ (2.61)17or some tensor field A ijkl with the right symmetries, because f ij is not dif-ferentiable. However, one can proceed as follows: Let ˆ f ijk ...k ℓ be the Taylorexpansion coefficients of ˆ f , ∀ m ˆ f ij ( ~w ) = X ≤ ℓ ≤ m ˆ f ijk ...k ℓ w k · · · w k ℓ + o m ( r m ) , similarly for ˇ f ijk ...k ℓ . Then the coefficients in the Taylor expansion of f ij w i have to vanish at every power of r , which implies that for all ℓ ∈ N we have X fixed ℓ (cid:0) ˆ f ijk ...k ℓ Θ k · · · Θ k ℓ + ˇ f ijk ...k ℓ − Θ k · · · Θ k ℓ − (cid:1) Θ i = 0 . Equivalently, X fixed ℓ (cid:0) ˆ f ijk ...k ℓ w k · · · w k ℓ + r ˇ f ijk ...k ℓ − w k · · · w k ℓ − (cid:1) w i = 0 . Comparing this equation with the equation where w k is replaced by − w k weeasily conclude that ˆ f i ( jk ...k ℓ ) = 0 = ˇ f i ( jk ...k ℓ ) . Let e ˇ f ij be obtained by Borel summation of the Taylor series of ˇ f ij , as inAppendix D. Then each partial sum ( e ˇ f ij ) p as defined in (D.1) has vanishingcontraction with w i , and so e ˇ f ij w i = 0 as well by passing to the limit. Sinceˇ f ij and e ˇ f ij have the same Taylor coefficients it holds that ∀ m ˇ f ij − e ˇ f ij = o m ( r m ) , where we write ψ = o m ( r m ) if ψ is m –times differentiable withlim r → ∂ k · · · ∂ k ℓ ψ = 0 for 0 ≤ ℓ ≤ m . This implies that r ( ˇ f ij − e ˇ f ij ) is smooth.Hence ˆ f ij + r ( ˇ f ij − e ˇ f ij )is a smooth tensor field satisfying (cid:2) ˆ f ij + r ( ˇ f ij − e ˇ f ij ) (cid:3) w i = 0 . By Proposition 2.2 we can writeˆ f ij + r ( ˇ f ij − e ˇ f ij ) = ˆ A ikjℓ w k w ℓ , e ˇ f ij = ˇ A ikjℓ w k w ℓ . f ij = (cid:0) ˆ A ikjℓ + r ˇ A ikjℓ | {z } =: A ikjℓ (cid:1) w k w ℓ , as desired.One concludes using Proposition 2.3. ✷ So far we have concentrated on normal coordinates, as these are naturallysingled out by the geometry. However, other (local) coordinate systems y µ in which C O takes the standard form { y = | ~y |} exist, and can be useful forsome purposes. The simplest possibility is provided by coordinate systemsof the form y µ = w µ + η αβ w α w β χ µ , (3.1)for some smooth functions χ µ . It is likely that all coordinate systems forwhich C O = { y = | ~y |} are related to the normal ones in this way, but we arenot aware of a proof of this except in the analytic case in dimension 3 + 1.For sufficiently small | w | + | ~w | the inverse transformation to (2.51) takesa similar form w µ = y µ + η αβ y α y β ψ µ , (3.2)for some smooth functions ψ µ .To avoid ambiguities, let us write g = g y µ y ν dy µ dy ν = g w µ w ν dw µ dw ν ≡ g µν dw µ dw ν ;one finds g y µ y ν = g w µ w ν + 2 g w α w ν χ α y µ + 2 g w α w µ χ α y ν + 4 g w α w β χ α χ β y µ y ν , where y α = η y α y β y β , with η y µ y ν = diag( − , +1 , . . . , +1). Clearly { η y µ y ν y µ y ν =0 } remains a null hypersurface on geometric grounds; a useful consistencycheck in subsequent calculations is to note that the last equation implies g y µ y ν y ν = g w µ w ν y ν = η y µ y ν y ν . (3.3)To avoid a proliferation of notation, we will again use the symbols x α todenote coordinates defined as y = x − x , y i = x Θ i ( x A ) with, as before, n X i =1 (cid:2) Θ i ( x A ) (cid:3) = 1 . (3.4)19t follows from (2.20)-(2.23) that the new h µν = g µν − η µν takes on C O theform h = Ω i j y i y j − rg µ χ µ + 4 r g µν χ µ χ ν , (3.5) h = 2 (cid:0) rg µ χ µ − g µi χ µ y i (cid:1) , (3.6) h A = h = 0 , (3.7) h A = (cid:20) r (cid:0) Ω j i r − Ω jik y k (cid:1) y j + 2 r g µi χ µ (cid:21) ∂ Θ i ∂x A , (3.8) h AB = r (cid:20) Ω i j r + (cid:0) − Ω ijk r − Ω jik r + Ω ikjℓ y ℓ (cid:1) y k (cid:21) ∂ Θ i ∂x A ∂ Θ j ∂x B . (3.9) A Extending functions
Lemma A.1
A function ϕ defined on a light cone C O is the trace f on C O of a C k spacetime function f if and only if ϕ admits an expansion, for small r , of the form ϕ = k X p =0 f p r p + o k ( r k ) , (A.1) with f p ≡ f i ...i p Θ i · · · Θ i p + f ′ i ...i p − Θ i · · · Θ i p − , (A.2) where f i ...i p and f ′ i ...i p − are numbers.The claim remains true with k = ∞ if (A.1) holds for all k . Proof:
The result is trivial away from the origin, so it suffices to considerfunctions defined near the tip of the light cone.Suppose, first, that k < ∞ . To see the necessity, let f be a functionwhich is C k in a neighbourhood of the origin in R n +1 . For any multi-index β = ( β , . . . , β j ) ∈ ( N n +1 ) j , β i ∈ { , , . . . , n } , with length 1 ≤ | β | := j ≤ k set f β := ∂∂y β · · · ∂∂y β j f . f β is C k −| β | in a neighbourhood of the origin, and thus admits a Taylorexpansion f β = k −| β | X p =0 h β ; α ··· α p y α · · · y α p | {z } =: h β + g β |{z} o ( | y | k −| β | ) , (A.3)for some coefficients h β ; α ··· α p ∈ R . Since f β ∈ C k −| β | and h β ∈ C ∞ we have g β = f β − h β ∈ C k −| β | . Similarly f = k X p =0 f α ··· α p y α · · · y α p | {z } =: h + g |{z} o ( | y | k ) , (A.4)with f α ··· α p ∈ R , h ∈ C ∞ and g ∈ C k . The usual formula for the coefficientsof a Taylor expansion implies that h β = ∂∂y β · · · ∂∂y β j h . Hence ∂∂y β · · · ∂∂y β j g = ∂∂y β · · · ∂∂y β j ( f − h )= f β − h β = g β = o ( | y | k − j ) , (A.5)and so g = o k ( | y | k ). Now, f = h + g , and it should be clear that h isof the form (A.2). The estimate g = o k ( r k ) is then straightforward from g = o k ( | y | k ), using ∂∂y i · · · ∂∂y i j g = (cid:18) y i r ∂∂y + ∂∂y i (cid:19) · · · (cid:18) y i j r ∂∂y + ∂∂y i j (cid:19) g . Conversely, let ϕ = ψ + χ be defined on a neighbourhood of O on C O ,where ψ = k X p =0 ( f i ...i p Θ i · · · Θ i p + f ′ i ...i p − Θ i · · · Θ i p − ) r p = k X p =0 ( f i ...i p y i · · · y i p + rf ′ i ...i p − y i · · · y i p − ) , χ = o k ( r k ). Set t = y , ~y = ( y , . . . , y n ), and f ( t, ~y ) = k X p =0 ( f i ...i p y i · · · y i p + tf ′ i ...i p − y i · · · y i p − ) + χ . Then f = ϕ . The function χ ( ~y ), viewed as a function of ( t, ~y ), is trivially o k ( | y | k ), and the proof is completed for finite k .The case k = ∞ is obtained from the above by Borel summation, usingLemma D.1, Appendix D. ✷ B How to recognize that coordinates are nor-mal
In this appendix we prove some simple necessary and sufficient conditionsfor a coordinate system to be normal:
Proposition B.1 (Thomas [10])
Let { x µ } be a local coordinate system de-fined on a star shaped domain containing the origin. The following conditionsare equivalent:1. For every a µ ∈ R n the rays s → sa µ are geodesics;2. Γ µαβ ( x ) x α x β = 0 ;3. ∂g γα ∂x β ( x ) x α x β = 0 ;4. g αβ ( x ) x β = g αβ (0) x β . Proof: . ⇔ . : The rays γ µ ( s ) = sa µ are geodesics if and only if0 = d γ µ ds | {z } =0 +Γ µαβ ( sa σ ) dγ α ds dγ β ds = Γ µαβ ( sa σ ) a α a β , multiplying by s and setting x µ = sa µ the result follows.22 . ⇔ . : g µα ( x σ ) x α = g µα (0) x α ⇐⇒ g µα ( sa σ ) a α = g µα (0) a α (B.1) ⇐⇒ dds ( g µα ( sa σ ) a α ) = 0 (B.2) ⇐⇒ ∂g µα ( x σ ) ∂x β x α x β = 0 . (B.3)2 . ⇒ . : From the formula for the Christoffel symbols in terms of themetric we haveΓ µαβ ( x ) x α x β = 0 ⇐⇒ (cid:18) ∂g µα ∂x β − ∂g αβ ∂x µ (cid:19) x α x β = 0 . (B.4)Multiplying by x µ we obtain ∂g µα ( x σ ) ∂x β x α x β x µ = 0 ⇐⇒ ∂g µα ( sa σ ) ∂x β a α a β a µ = 0 (B.5) ⇐⇒ dds ( g µα ( sa σ ) a α a µ ) = 0 (B.6) ⇐⇒ g µα ( sa σ ) a α a µ = g µα (0) a α a µ (B.7) ⇐⇒ g µα ( x σ ) x α x µ = g µα (0) x α x µ . (B.8)Differentiating it follows that ∂g µα ( x σ ) ∂x γ x α x µ + 2 g γα ( x σ ) x α = 2 g γα (0) x α . Substituting this into the last term in (B.4) one obtains ∂g µα ∂x β ( x σ ) x α x β + g µα ( x σ ) x α − g µα (0) x α = 0 . (B.9)This implies that dds [ g µα ( sa µ ) sa α − g µα (0) sa α ] = 0 , and the result follows by integration.3 . &4 . ⇒ . : Point 4 implies g αβ ( x γ ) x α x β = g αβ (0) x α x β . ∂g αβ ( x γ ) ∂x µ x α x β + 2 g αµ ( x γ ) x α = 2 g αµ (0) x α . The last two terms are equal by point 4 so that ∂g αβ ( x γ ) ∂x µ x α x β = 0 . This shows that the last term in (B.4) vanishes, so does the next-to-last bypoint 3, and the proof is complete. ✷ C Covector fields
The aim of this appendix is to present a simple equivalent of our param-eterization of the metric for covector fields. This can be used for Cauchyproblems on the light cone involving Maxwell fields.We start by noting that every covector field ζ µ on space-time can bewritten as ζ µ = ξ µ + ∂ µ λ , with w µ ξ µ = 0 , for a smooth function λ . This is obtained by setting λ ( w µ ) = w α Z ζ α ( sw µ ) ds . By the arguments in Section 2.1 there exists a smooth anti-symmetric matrixΩ µν such that ξ µ = Ω µν w ν . (C.1)As in the main body of this paper, the restriction to the light cone { w = | ~w |} of ξ µ arises from a smooth vector field on R satisfying w µ ξ µ = 0 if and onlyif the restrictions Ω µν are C O –smooth.An alternative parameterization of ξ is obtained by introducing ξ u = − ξ , γ = Θ i ξ i , ξ A = ξ i w i,A , ξ A = α || A + ǫ AC β || C , (C.2)and we have ξ u = γ in view of the condition ξ µ w µ = 0. We then have:24 heorem C.1 A field of the form (C.2) defined on (0 , R ) × S is the restric-tion to the light cone { w = | ~w |} of a smooth vector field on R satisfying w µ ξ µ = 0 if and only if α = r ˇ α + r ˆ α , β = r ˇ β + r ˆ β , and γ = w i ∂ w i ˇ α + r ˇ γ + r ˆ γ ,where ˇ α , ˆ α , ˇ β , ˆ β , ˇ γ and ˆ γ are smooth functions of ~w ,except for the ℓ = 0 spherical-harmonics components of α and β which donot affect ξ µ . Proof:
Necessity: it follows from the identities r ξ A || A = ˚∆ α = r ∂ w k ξ k − w j w i ∂ w i ξ j − rw i ξ i , (C.3) r ǫ AB ξ A || B = ˚∆ β = rw i ǫ ijk ∂ w k ξ j , (C.4)together with a straightforward generalization of Proposition 2.5 that α and r − β are C O –smooth if ξ µ is smooth, except for their ℓ = 0 components whichare in the kernel of ˚∆. However, the gauge condition 0 = ξ µ w µ = w ξ + w i ξ i implies w j w i ∂ j ξ i = − tw j ∂ j ξ + tξ , and we conclude that Θ i ξ i = − ξ , (C.5) w j w i ∂ j ξ i = − rw j ∂ j ξ + rξ . (C.6)The C O –smoothness of γ follows from (C.5), while that of α/r follows from(C.3) and (C.6).We can write γ = w i ∂ w i ˇ α + ψ , and it remains to show that ψ/r is C O –smooth. The inverse of (C.2) reads ξ k = r∂ w k ˇ α + w i ǫ ikl ∂ w l ˇ β + r ∂ w k ˆ α + rw i ǫ ikl ∂ w l ˆ β − w i ∂ w i ˆ αw k + ψ Θ k . (C.7)Extending ˆ α , ˇ α , etc., to R by requiring the extension to be time-independent,and using the same symbols for this extension, ξ k minus the first line of the25ight-hand side of (C.7) is the restriction to the light cone of the smoothvector field ξ k − (cid:0) t∂ w k ˇ α + w i ǫ ikl ∂ w l ˇ β + r ∂ w k ˆ α + tw i ǫ ikl ∂ w l ˆ β − w i ∂ w i ˆ α w k (cid:1) . (C.8)Hence for every k the function ψ ( ~w )Θ k = ψ ( ~w ) r w k extends to a smooth function on space-time. Choosing k to be one, byProposition 2.1 we can write ψ ( ~w ) r w = ˇ χ ( ~w ) + r ˆ χ ( ~w ) , (C.9)for some smooth functions ˇ χ and ˆ χ . For r = 0 this implies[ ˇ χ ( ~w ) + r ˆ χ ( ~w )] (cid:12)(cid:12) w =0 = 0 ;by continuity this holds for all r . Smoothness of ˇ χ and ˆ γ implies existenceof smooth functions ˇ γ and ˆ γ such thatˇ χ = ˇ χ | w =0 + ˇ γw , ˆ χ = ˆ χ | w =0 + ˆ γw , and (C.9) gives ψ ( ~w ) r w = (cid:2) ˇ γ ( ~w ) + r ˆ γ ( ~w ) (cid:3) w . (C.10)For w = 0 we conclude ψ ( ~w ) = r ˇ γ ( ~w ) + r ˆ γ ( ~w ) , (C.11)and continuity implies that this equation holds everywhere. We concludethat ψ/r is C O –smooth, and the proof of necessity is complete.Sufficiency should be clear from what has been said together with ψ ( ~w )Θ k = (cid:2) ˇ γ ( ~w ) + t ˆ γ ( ~w ) (cid:3) w k . (C.12) ✷ Borel’s summation
In the main body of the paper we will need the details of the following con-struction, which is a straightforward adaption of [6, Volume I, Theorem 1.2.6]:
Lemma D.1 [ Borel summation ] For any sequence { c i ...i k } k ∈ N = { c, c i , c ij , . . . } there exists a smooth function f such that, for all k ∈ N , f − k X p =0 c i ...i p y i · · · y i p = o k ( r k ) . Proof.
Let φ ∈ C ∞ ( R ) be any function such that φ | [0 , / = 1 , φ | [1 , ∞ ) = 0 . Set f = c , and for p > f p = X i ,...,i p φ ( M p | y | ) c i ...i p y i · · · y i p , (D.1)where the constant M p is chosen large enough so that for all p > α satisfying0 ≤ | α | ≤ p − | ∂ α f p | ≤ − p . Then for each α the series ∞ X p =0 ∂ α f p is absolutely convergent. By standard results (see, e.g., [6, Volume I, Theo-rem 1.1.5]), the function f := ∞ X p =0 f p is smooth, and is easily seen to have the required properties. ✷ Acknowledgements:
Supported in part by the Polish Ministry of Scienceand Higher Education grant Nr N N201 372736. PTC acknowledges manyuseful discussions with Y. Choquet-Bruhat and Jose-Maria Martin Garcia onproblems closely related to the ones addressed here.27 eferences [1] Y. Choquet-Bruhat, P.T. Chru´sciel, and J.M. Mart´ın-Garc´ıa,
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