KIDs like cones

We analyze vacuum Killing Initial Data on characteristic Cauchy surfaces. A general theorem on existence of Killing vectors in the domain of dependence is proved, and some special cases are analyzed in detail, including the case of bifurcate Killing horizons.


Introduction
Killing Initial Data (KIDs) are defined as initial data on a Cauchy surface for a spacetime Killing vector field. Vacuum KIDs on spacelike hypersurfaces are well understood (see [1,7] and references therein). In the spacelike case they play a significant role by providing an obstruction to gluing initial data sets [3,5].
The question of KIDs on light-cones has been recently raised in [11]. The object of this note is to analyze this, as well as KIDs on characteristic surfaces intersecting transversally. It turns out that the situation in the light-cone case is considerably simpler than for the spacelike Cauchy problem, which explains our title.
For definiteness we assume the Einstein vacuum equations, in dimensions n + 1, n ≥ 3, possibly with a cosmological constant, Similar results can be proved for Einstein equations with matter fields satisfying well-behaved evolution equations.

Light-cone
Consider the (future) light-cone C O issued from a point O in an (n + 1)dimensional spacetime (M , g), n ≥ 3; by this we mean the subset of M covered by future-directed null geodesics issued from O. (We expect that our results remain true for n = 2; this requires a more careful analysis of some of the equations, which we have not attempted to do.) Let (x µ ) = (x 0 , x i ) = (x 0 , r, x A ) be a coordinate system such that x 0 is constant on C O . In the theorem that follows the initial data for the sought-for Killing vector field are provided by a spacetime vector field Y which is defined on C O only. We will need to differentiate Y , for this we need a covariant derivative operator which involves only derivatives tangent to the characteristic hypersurfaces. In a coordinate system such that the hypersurface under consideration is given by the equation x 0 = 0, for the first derivatives the usual spacetime covariant derivative ∇ i Y µ applies. However, the tensor of second spacetime covariant derivatives involves the undefined fields ∇ 0 Y µ . To avoid this we set, on the hypersurface {x 0 = 0}, with an obvious similar formula for D i D j Y µ . When the restriction ∇ 0 Y µ to the hypersurface {x 0 = 0} of the x 0 -derivative is defined we have Clearly, D i D j Y 0 coincides with ∇ i ∇ j Y 0 when Y µ is the restriction to the hypersurface {x 0 = 0} of a Killing vector field Y µ , as then ∇ 0 Y 0 = 0. This is of key importance for our equations below.
Our main result is: Furthermore, (1.5) is not needed on the closure of the set on which the divergence τ of C O is non-zero.
As such, the analysis of KIDs on light-cones can be split into two cases: The first is concerned with the region sufficiently close to the tip of the cone where the expansion τ has no zeros. Once a spacetime with Killing field has been constructed near the vertex, the initial value problem for the remaining part of the cone can be reduced to a characteristic initial value problem with two transversally intersecting null hypersurfaces, which will be adressed in Theorem 1.2. From this point of view the key restriction for light-cones is (1.4).
The proof of Theorem 1.1 can be found in Section 2.5. In order to prove it we will first establish some intermediate results, Theorems 2.1 and 2.5 below, which require further hypotheses. It is somewhat surprising that these additional conditions turn out to be automatically satisfied.
Equations (1.4) provide thus necessary-and-sufficient conditions for the existence, to the future of O, of a Killing vector field. They can be viewed as the light-cone equivalent of the spacelike KID equations, keeping in mind that (1.5) should be added when C O contains open subsets on which τ vanishes. As made clear by the definitions, (1.4)-(1.5) involve only the derivatives of Y in directions tangent to C O .
We shall see in Section 2.6 that some of the equations (1.4) can be integrated to determine Y in terms of data at O. Once this has been done, we are left with the trace-free part of D A Y B + D B Y A = 0 as the "reduced KID equations".
It should be kept in mind that a Killing vector field satisfies an overdetermined system of second-order ODEs which can be integrated along geodesics starting from O, see (2.56) below. This provides both X, and its restriction X to C O , in a neighborhood of O, given the free data X α | O and ∇ [α X β] | O . We will see in the course of the proof of Theorem 2.1 how such a scheme ties-in with the statement of the theorem, cf. in particular Section 2.3 In Section 2.6 we give a more explicit form of the KID equations (1.4) on a cone and discuss some special cases.

Two intersecting null hypersurfaces
Throughout, we employ the symbol "· " to denote the trace-free part of the field " · " with respect tog = g AB dx A dx B . Further, an overbar denotes restriction to the initial surface.
In what follows the coordinates x A are assumed to be constant on the generators of the null hypersurfaces. The analogue of Theorem 1.1 for two intersecting hypersurfaces reads as follows: Theorem 1.2 Consider two smooth null hypersurfaces N 1 = {x 1 = 0} and N 2 = {x 2 = 0} in an n + 1-dimensional vacuum spacetime (M , g), with transverse intersection along a smooth submanifold S. Let Y be a vector field defined on N 1 ∪ N 2 . There exists a smooth vector field X satisfying the Killing equations on J + (S) and coinciding with Y on N 1 ∪ N 2 if and only if on N 1 it holds that where D is the analogue on N 1 of the derivative operator (1.2)-(1.3); similarly on N 2 ; while on S one needs further to assume that (1.14) Similarly to Theorem 1.1, (1.9) can be replaced by the requirement that g AB D A Y B = 0 on regions where the divergence of N 1 is non-zero. An identical statement applies to N 2 . Theorem 1.2 is proved in Section 3. As before, (1.6)-(1.14) provide necessary-and-sufficient conditions for the existence, to the future of S, of a Killing vector field. Hence they provide the equivalent of the spacelike KID equations in the current setting. Note that in (1.6)-(1.13) the derivative D coincides with ∇; it is only in (1.14) that the operators differ.
2 The light-cone case

Adapted null coordinates
We use coordinates (x 0 ≡ u, x 1 ≡ r, x A ) adapted to the light-cone as in [2], in the sense that the cone is given by C O = {x 0 = 0} . Further, the coordinate x 1 parameterizes the null geodesics emanating to the future from the vertex of the cone, while the x A 's are local coordinates on the level sets {x 0 = 0, x 1 = const} ∼ = S n−1 , and are constant along the generators. Then the metric takes the following form on C O : (2.1) We stress that we do not assume that this form of the metric is preserved under differentiation in the x 0 -direction, i.e. we do not impose any gauge condition off the cone. On C O the inverse metric reads

A weaker result
We start with a weaker version of Theorem 2.5 which, moreover, assumes that the vector field Y there is the restriction to the light-cone of some smooth vector field Y :

7)
are satisfied by the restriction Y of Y to C O .
Proof of Theorem 2.1: To prove necessity, let X be a smooth vector field satisfying the Killing equations on J + (O): the tangent components of which give (2.4). It further easily follows from and (2.5)-(2.7) similarly follow, (2.7) since A 00 = 0. To prove sufficiency, by contracting (2.9) one finds (which equals −λX σ under (1.1)). So, should a solution X of our problem exist, it will necessarily satisfy the wave equation (2.10). Now, it follows from e.g. [6] that for any smooth vector field Y defined on M there exists a smooth vector field X on M solving (2.10) to the future of O, such that, Here, and elsewhere, overlining denotes restriction to As such, for any solution of (2.10) the tensor field A µν defined in (2.8) satisfies Note that under (1.1) the last term −2L X R µν equals −2λA µν (and, in fact, cancels with the before-last one, but this is irrelevant for the current discussion), which shows that A µν satisfies a homogeneous linear wave equation. It follows from uniqueness of solutions of (2.12) that a solution X of (2.10) will satisfy the Killing equation on J + (O) if and only if But by (2.4) we already have so it remains to show that the equations A 0µ = 0 hold. (Annoyingly, these equations involve the derivatives ∂ 0 X µ which cannot be expressed as local expressions involving only the initial data X = Y .) The theorem follows now directly from Lemma 2.4 below. 2 Definition 2.2 It is convenient to introduce, for a given vector field X, the tensor field Whenever it is clear from the context which vector field is meant we will suppress its appearance and simply write S µνσ .
Using the algebraic symmetries of the Riemman tensor we find: 3 It holds that:

If (2.5) and (2.6) hold, then (2.7) is equivalent to
Proof: It turns out to be convenient to consider the identity Thus, it holds that In adapted null coordinates (2.16) implies Due to Lemma 2.3 we have When (ij) = (11) that yields Inserting into (2.17) with ν = 1, after some simplifications one obtains Using the vanishing of the Γ 0 i1 's [2, Appendix A] and the A ij 's, this becomes a linear homogeneous ODE for A 01 ; in the notation of the last reference: 1 (2.21) If A 01 = 0, the vanishing of S 110 immediately follows.
To prove the reverse implication, for definiteness we assume here and in what follows a coordinate system as in [2,Section 4.5]. In this coordinate system τ behaves as (n − 1)/r for small r, ν 0 satisfies ν 0 = 1 + O(r 2 ), and (2.21) is a Fuchsian ODE with the property that every solution which is o(r −(n−1)/2 ) for small r is identically zero. As A 01 is bounded, when S 110 vanishes we conclude that This proves point 1 of the lemma. Next, (2.17) with ν = D reads to eliminate ∇ 0 A 1D from (2.23), and invoking (2.22), on C O one obtains a system of Fuchsian radial ODEs for A 0D , with zero being the unique solution with the required behavior at r = 0 when S B10 = 0: This proves point 2 of the lemma. Let us finally turn attention to (2.17) with ν = 0: Throughout we shall make extensively use of the formulae for the Christoffel symbols in adapted null coordinates computed in [2, Appendix A]. Apart from the vanishing of the Γ 0 i1 's the expressions for Γ 0 01 , Γ 1 11 = κ, Γ A 1B and Γ 0 AB will be often used.
The transverse derivatives ∇ 0 A 11 and ∇ 0 A 1A can be eliminated using (2.19) and (2.24), The remaining one, g AB ∇ 0 A AB , fulfills the following equation on C O , which follows from (2.18), where we have setS Note thatS is the negative of the left-hand side of (2.7), and we want to show that the vanishing ofS is equivalent to that of A 00 . Equation (2.28) with A ij = 0 and A 0i = 0 (i.e. S i10 = 0) yields (2.30) ForS = 0 this is again a Fuchsian radial ODE for A 00 , with the only regular solution A 00 = 0, and the lemma is proved. 2

The free data for X
Let us explore the nature of (1.4). Making extensive use of [2, Appendix A], and of the notation there (thus κ ≡ Γ 1 denotes the null second fundamental form of C O ), we find

32)
For definiteness, in the discussion that follows we continue to assume a coordinate system as in [2,Section 4.5], in particular κ = 0 and Under (1.4) the left-hand sides of (2.31)-(2.33) vanish. Hence, we can determine X 1 by integrating (2.31), for some function of the angles. We continue by integrating (2.32). This is a Fuchsian ODE for X B , the solutions of which are of the form whereD is the covariant derivative operator of the unit round metric s on S n−1 .
In a neighborhood of O, where τ does not vanish, the component X 1 can be algebraically determined from the equation A A A = 0, leading to where ∆ s is the Laplace operator of the metric s. As before, letȂ AB denote theg-trace free part of A AB . The equation We wish, now, to relate the values of c and f A to the values of the vector field X at the vertex, under the supplementary assumption that X is the restriction to C O of a differentiable vector field defined in spacetime. Following [2], we denote by y µ normal coordinates centered at O. Given the coordinates y µ the coordinates x α can be obtained by setting with x A local coordinates on S n−1 , and We underline the components of tensor fields in the y α -coordinates, thus (2.44) In particular, for vector fields such that X µ is continuous, we obtain Thus, for such vector fields, X 1 (0) is a linear-combination of ℓ = 0 and ℓ = 1 spherical harmonics, and contains the whole information about X α (0). We conclude that (2.46) Equation (2.40) will be satisfied if and only if c is of the form (2.46), which can be seen by noting that (2.46) provides a family of solutions of (2.40) with the maximal possible dimension.
To determine f A when X µ is differentiable at the origin we Taylor expand X there, so that

A second intermediate result
Proof of Theorem 2.5: We wish to apply Theorem 2.1. The crucial step is to construct the vector field Y needed there. For further reference we note that (2.51) will not be needed for this construction.
In the argument that follows we shall ignore the distinction between X and Y whenever it does not matter.
By hypothesis it holds that We define an antisymmetric tensor F µν via Moreover, With (2.52) that gives Further, To sum it up, (2.52) and (2.53) imply that the equations (2.56) Note that the first four equations above determine uniquely the initial data X µ on C O needed to obtain a unique solution of the wave equation for X µ . Now, we claim that there exists a smooth vector field Y µ defined near O so that X µ is the restriction of Y µ to the light-cone. To see this, letl µ be given and define (x µ (s), Z µ (s), F αβ (s)) as the unique solution of the problem (2.57) For initial values such that It follows from smooth dependence of solutions of ODEs upon initial data that Y µ is smooth in all initial variables, in particular inl µ . If the x µ 's are normal coordinates centered at O, then x µ (s = 1) =l µ , which implies that (2.58) defines a smooth vector field in a neighborhood of O. It then easily follows that the restriction of Y µ to C O equals X µ , as defined by the first four equations in (2.56). The hypotheses of Theorem 2.1 are now satisfied, and Theorem 2.5 is proved. 2

Proof of Theorem 1.1
To prove Theorem 1.1 we will use Theorem 2.1, together with some of the ideas of the proof of Theorem 2.5. More precisely, we need to show that (1.4) together with the Einstein equations imply both the existence of a smooth extension Y of Y , and that (2.5)-(2.7) hold.

Properties of S µνσ
Recall the definition and Lemma 2.3. In the context of Theorem 1.1, only those components of the tensor field S αβγ which do not involve ∂ 0 -derivatives of X are a-priori known. One easily checks: of the restriction to C O of S µνσ can be algebraically calculated from X ≡ Y .
We wish, next, to calculate ∇ α S αβγ and ∇ γ S αβγ . This requires the knowledge of ∇ 0 X µ , of ∇ 0 ∇ 0 X µ , and even of ∇ 0 ∇ 0 ∇ 0 X µ in some equations. For this, let X be any extension of X from the light-cone to a punctured neighborhood O \ {O} of O, so that the transverse derivatives appearing in the following equations are defined. X is assumed to be smooth on its domain of definition, and we emphasise that we do not make any hypotheses on the behavior of the extension X as the tip O of the light-cone is approached. As will be seen, the transverse derivatives of X on C O drop out from those final formulae which are relevant for us.
We will make use several times of which is a standard consequence of the second Bianchi identity when the Ricci tensor is proportional to the metric. We start with ∇ α S αβγ . Two commutations of derivatives allow us to rewrite the first term in the divergence of S αβγ over the first index as Hence, since R αβ = λg αβ , and using the first Bianchi identity in the second line Similarly, Now, on C O and in coordinates adapted to the cone In order to handle undesirable terms such as ∇ 0 S βγ1 we write (2.66)

Analysis of condition (2.5)
Lemma 2.7 Assume that A 1i = 0 andȂ AB = 0. Then, in vacuum, Consider (2.63) with (βγ) = (11). Setting a := g AB A AB we find Due to Lemma 2.3 we have Using (2.65) with (βγ) = (11), as well as the last equation, and employing again Lemma 2.3 we obtain, on C O , Using (2.66) we find Proof: From (2.66) we obtain This allows us to rewrite (2.65) with (βγ) = (A1) on C O as Combining with (2.63), which reads with (βγ) = (A1) we obtain on the initial surface This allows us to rewrite the right-hand side of (2.73) as Now, using in addition that S 110 = 0, Hence, and thus, again due to S 110 = 0 and Lemma 2.3, But zero is the only solution of this equation which is o(r −(n−1) ), and to be able to conclude that we need to check the behavior of S A10 at the vertex. For definiteness we assume a coordinate system as in [2,Section 4.5]. Now, by definition, From (2.37)-(2.39) we find Using the formulae from [2, Appendix A] one obtains which implies that (2.74) holds, and Lemma 2.9 is proved.
Proof: Equation (2.62) yields with A iµ = 0, 2X + λX = 0 and in vacuum On the other hand, (2.64) gives with Lemma 2.3 and 2.10 on C O , Moreover, from (2.66) and Lemma 2.3 we deduce that, on C O , Hence, Using the vacuum constraint [2] We employ (2.30), which holds since all the hypotheses of Lemma 2.4 are fulfilled, to end up with Regularity at O in coordinates as in [2,Section 4.5] givesS = O(r −1 ), which implies thatS = 0 is the only possibility. together with κ describes the free data on the light-cone, is chosen in such a way that, in the region where τ has no zeros, the KID equations A ij = 0 admit a non-trivial solution X. Written as equations for the vector field X, they read (we use the formulae from [2, Appendix A]) where σ A B denotes the trace-free part of χ A B , ξ A := g AB ξ B and ζ := 2g AB Γ 1 AB + τ g 11 , (2.82) The analysis of these equations is identical to that of their covariant counterpart, already discussed in Section 2.3. The first three equations, arising from A 1i = 0 and g AB A AB = 0 determine a class of candidate fields (depending on the integration functions c(x A ) and f A (x B ), withD A c and f A being conformal Killing fields on (S n−1 , s). Note that it is crucial for the expansion τ to be non-vanishing in order for g AB A AB = 0 to provide an algebraic equation for X 1 . Regardless of whether τ has zeros or not, we can determine X 1 by integrating radially (1.5), compare Remark 3.2 below.

Killing vector fields tangent to spheres
Let us consider the special case where the spacetime admits a Killing field X with the property that X 0 = X 1 = 0 on C O . The KID equations (2.78)-(2.81) then reduce to which leads us to the following corollary of Theorem admits an r-independent Killing field f A = f A (x B ).

Killing vector fields tangent to the light-cone
Let us now restrict attention to those Killing fields which are tangent to the cone C O , i.e. we assume X 0 = 0 . (2.84) We start by noting that in the coordinates of (2.42) we have for an anti-symmetric matrix ω µν . Hence, quite generally, where we have set f A := g AB f B . Equations (2.91)-(2.93) provide thus a relatively simple form of the necessary-and-sufficient conditions for existence of Killing vectors initially tangent to C O . If we choose a gauge where τ = (n − 1)/r (cf. e.g. [4]), the last three equations become (2.94)

6)
where D is the analogue on N 1 of the derivative operator (1.2)-(1.3); similarly on N 2 ; while on S one needs further to assume that Proof: The proof is essentially identical to the proof of Theorem 2.1. The candidate field is constructed as a solution of the wave equation (2.10); the delicate question of regularity of Y needed at the vertex in the cone case does not arise. Existence of the solution in J + (N 1 ∪ N 2 ) follows from [8].
The main difference is that one cannot invoke regularity at the vertex to deduce the vanishing of, say on N 2 , A 2µ | N 2 from the equations which correspond to (2.21), (2.25) and (2.30). Instead, one needs further to require (3.7) as well as (∇ 2 Y A + ∇ A Y 2 )| S = 0 and ∇ 2 Y 2 | S = 0 .
However, the last two conditions follow from (3.1) and (3.2) on N 1 . 2
A straigthforward adaptation of Lemma 2.11 to the current setting shows that, say on N 2 , g AB S AB2 − 1 2 τ g 12 A 22 vanishes, supposing that it vanishes initially, as guaranteed by (1.14). Hence also (3.6) is fulfilled. 2 Remark 3.2 In the case of two transversally intersecting null hypersurfaces the expansion τ may vanish. In that case the trace of (3.3) on, say, N 1 will fail to provide an algebraic equation for X 2 . Also, Corollary 2.8 cannot be applied to deduce the vanishing of S 221 , equivalently, the validity of (3.4), in the regions where τ vanishes. Instead one can use the second-order ODE (3.4) to find a candidate for X 2 , and then Lemma 2.7 guarantees that the trace of the left-hand side of (3.3) vanishes when g AB A AB | S = 0 = ∂ 2 (g AB A AB )| S .