On linearised vacuum constraint equations on Einstein manifolds

We show how to parameterise solutions of the general relativistic vector constraint equation on Einstein manifolds by unconstrained potentials. We provide a similar construction for the trace-free part of tensors satisfying the linearised scalar constraint. We use the potentials to show that one can shield linearised gravitational fields using linearised gravitational fields without imposing the TT gauge (as done in previous work), for any value of $\Lambda \in \mathbb{R}$.


Introduction
One of the key challenges in general relativity is an exhaustive description of solutions of the vacuum constraint equations. As a step towards this, one might wish to obtain such a description in restricted settings, or for simpler related equations.
For instance, consider the general relativistic vacuum vector constraint equation, (1.1) We will provide below an exhaustive description of solutions of this equation on three dimensional Einstein manifolds using unconstrained potentials: We show (see Section 3.1 below) that for an arbitrary symmetric tensor field ψ ij the tensor field K = K ij dx i dx j defined by satisfies (1.1), where the operatorQ is given by and Conversely (see Corollary 3.3 below), if a tensor field K satisfies (1.1) on a three-dimensional simply connected Einstein manifold with vanishing second homology class, then there exists a tensor field ψ such that K is given by (3.49).
We present a similar result for the trace-free part of the linearised scalar constraint equation. On an Einstein manifold, where R ij = λg ij (1.5) (with the sign of λ coinciding with that of the cosmological constant Λ), the linearised scalar constraint becomes the following equation for a symmetric tensor field h ij : For solutions with vanishing trace (e.g., because the trace has been gauged away by an appropriate choice of the initial data surface) this becomes It is not too difficult to check that for any symmetric trace-free tensor field φ = φ ij dx i dx j the tensor field h = h ij dx i dx j defined by the formula is a symmetric trace-free tensor field solving (1.6). Under the same restrictions as for the vector constraint equation, we show (see Theorem 3.1 below) that for any trace-free tensor h solving (1.7) there exists a symmetric trace-free tensor field φ such that (1.8) holds In general gauges, where the trace of h does not vanish, given one solutionh ij of (1.6), the remaining solutions are obtained by adding toh a trace-free solution of (1.7). So our construction gives an exhaustive description of solutions of (1.6) in this sense.
Recall that in [3] (compare [8]) we provided a simple way of shielding gravity linearised at Minkowski space in transverse-traceless gauge (in a sense made precise by Theorem 1.1 below), based on third-order unconstrained potentials for transverse-traceless tensors introduced in [2]. One of the motivations for the current work was to provide a shielding construction for linearised vacuum gravity which applies to initial data with a non-zero cosmological constant Λ, regardless of its sign and of the gauge. The analysis here applies to gravity linearised at Einstein metrics with Λ ∈ R, while that in [3] works on any locally conformally flat manifolds but requires higher-order potentials.
An immediate corollary of our constructions below is the following shielding theorem: be a smooth vacuum initial data set for the linearised gravitational field on a three-dimensional Einstein manifold (M, g). Consider open sets U , U ′ and Ω such that and assume that H 1 (Ω) = H 2 (Ω) = {0}. Then there exists a smooth vacuum initial data set (h ij ,k ij ), solution of the linearised vacuum constraint equations, which coincides with (h ij , k ij ) on U , and such that k and the trace-free part of h vanish outside of U ′ .
In particular, for initial data for which the trace of h can be gauged-away, one obtains the screening of the full initial data set.
In the screening construction of [3] one needs first to transform the metric to a transverse-traceless gauge, which requires solving elliptic equations Neither solving elliptic equations, nor applying preliminary gauge transformations, is needed in the approach taken here.
As discussed in [3], a shielding construction provides immediately a gluing construction, the reader is referred to [3] for details.
The proof of Theorem 1.1 is a repetition of the arguments given in [3, Section 2.3], invoking instead the potentials of Theorems 3.1 and 3.2, and will be omitted.
The hypotheses above on Ω and the metric will clearly be satisfied if Ω is taken to be a three-dimensional sphere with the standard round metric, or R 3 with the flat or with the hyperbolic metric.

Linearised constraint equations
Consider the three-dimensional general relativistic vacuum constraint equations, Denoting by h the linearisation of g and by k the linearisation of K, the linearised version of (2.1)-(2.2) at a solution (g, K ≡ 0) of the above reads where D denotes the covariant derivative of the metric g.
To fix notations, given a symmetric tensor k ij , we denote byk its trace-free part:k (2.5) In this notation, (2.4) is equivalent to The function τ is then uniquely defined byk up to the addition of a constant. It follows that k ij solves (2.4) if and only if there exists a constant c such that with the trace-free symmetric tensork ij solving (2.7), and with τ being a solution of (2.8).
All solutions of (2.7) are parameterised by unconstrained symmetric tensor fields ψ ij in Corollary 3.3 below.
From now on we assume that (M, g) is Einstein.
As already indicated, we will provide in Theorem 3.2 below an exhaustive description of the set of solutions of (2.7) in terms of second-order potentials.
We pass now to a discussion of solutions of the scalar constraint equation While the following will not be assumed in our theorems below, one should keep in mind some situations of particular interest, namely This allows any Λ ∈ R. If we denote by γ the trace of h, we can write The linearised scalar constraint equation (2.3) becomes This equation, viewed as a PDE for γ, has typically a finite dimensional set of solutions of interest. For instance, when the right-hand side vanishes the function τ will be zero on compact manifolds with Λ < 0. Similarly τ will be zero on a flat R 3 and on hyperbolic space when restricting oneself to solutions which tend to zero at infinity, as easily follows afer applying the maximum principle for the Dirichlet problem on larger and larger balls. Compact manifolds with Λ = 0 will lead to constants being the only solutions of the homogeneous equation. A finite dimensional set of non-trivial functions γ might arise on some compact manifolds when Λ > 0. Note, finally, that under gauge transformations that a suitable choice of ξ can bring γ to a constant (possibly, but not necessarily, zero), but gauge transformations is something that we wanted to avoid. By linearity, it remains to describe the set of solutions of this will be done in Theorem 3.1.

Solutions of the linearised constraints on threedimensional Einstein manifolds
Let (M, g ij ) be a Riemannian 3-manifold which is locally conformally flat. Then (as pointed out in [2]) the symmetric, trace-free tensor H ij given by the linearization of the Cotton tensor at the metric g ij in the direction of h ij is 3rdorder linear partial differential operator on h ij , which is divergence-free and vanishes on h ij 's which are conformal Killing forms of vectors X i , i.e.
We will restrict to h ij trace-free. Moreover we assume that (M, It is then possible to write down a concise expression for t ij as follows: and where rot 2 h is the symmetric trace-free tensor given by We note that the operator appearing in (3.1) is eight times the linearization of the trace-free Ricci tensor at g ij in the direction of trace-free tensors h ij , namely Note also that Q coincides withQ defined in (1.3) when restricting to trace-free tensors. Last but not least, this is related to the linearization of the trace-free Ricci tensor at g ij as follows: If one now replaces the variation of the metric h ij by h ij + 1/2tr g hg ij in that last tensor, one obtains 1/8 of the hatted Q.

Complexes
With the standard definition (rot 1 X) i = ǫ i jk D j X k , and with d denoting the differential of a scalar, we have Indeed, (3.3) and (3.4) are direct computations, while (3.5) follows from (3.3) by noting that the formal adjoint L † of L is the negative of div 2 , and rot † 2 = rot 2 . Using that rot 1 d is zero one finds Using formal adjoints we have and a simple computation then gives From this and (3.6) one concludes that it also holds, for all vector fields X, Recall that a complex is called elliptic if the sequence of symbols is exact. We have recovered the elliptic complex, which is a special case of the "conformal complex" derived in [2] (compare [6,8]), which applies to any locally conformally flat metric. Inspection of the identities above leads to the following elliptic complex: de Rham conformal momentum scalar Figure 1: Collected complexes.
where the composition of the fourth and fifth arrow is div 2 H = 0, and the composition of the third and fourth arrow vanishes by taking formal adjoints: Ellipticity of (3.11), and of (3.13) below, follows from the calculations in Appendix A.
Similarly we have a third elliptic complex For this, from (3.5) we find and the composition of the last two arrows vanishes by taking formal adjoints. The complex (3.11) will be called momentum complex, since the condition on h ij to be the trace-free part of a field satisfying the linearized momentum constraints is exactly that rot 1 div 2 [h] be zero. The complex in (3.13) is similarly related to the linearized Hamiltonian constraint, and will be referred to as the Hamiltonian complex.
The complexes just discussed are collected, together with the de Rham complex , in the commutative diagram of Figure 1, where the composition of any two horizontal or vertical arrows vanishes.
The question arises, whether the above complexes are exact at the secondto-last slots. This would provide an exhaustive description of solutions of the equations of interest. We establish this for the hamiltonian complex, cf. Theorem 3.1 below, but have not been able to for the momentum complex. In Theorem 3.2 below we will provide another complete classification of solutions of the momentum constraint. It would be of interest to determine exactness, or lack thereof, at this key slot, of the momentum complex.
Let us mention that the momentum constraint equation on Minkowski spacetime is naturally related to the complex where → 1 is the Killing operator, → 2 is the linearized Schouten tensor, → 3 is the momentum operator. Here S 2 denotes the space of all symmetric twocovariant tensors, without any trace condition. This, however, is not a complex for non-flat Einstein metrics. Note furthermore that when g is the flat metric the above complex is closely related to the "elasticity complex" used in the area of finite elements methods for elasticity (see [1,5], compare [10]).

The potentials
We are interested in the construction of solutions of the momentum and the scalar constraint, as well as of transverse-traceless tensors, on simply connected subsets Ω of three dimensional Einstein manifolds (M, g). If M itself is simply connected and complete, it follows from [11, Corollary 2.4.10] that (M, g) is the three dimensional sphere, hyperbolic space, or Euclidean space. However, we neither assume completeness of Ω, nor that of M , nor simple-connectedness of M .
We start with the scalar constraint. We have: if and only if there exists a symmetric trace-free two-covariant symmetric tracefree tensor field φ on Ω such that Proof. The necessity has been established when proving that (3.13) is a complex. For sufficiency, note first that Eq. (3.15) and our assumptions on Ω imply existence of a field τ i such that Thus the (non-symmetric) tensor given by t ij − ǫ ijk τ k is divergence-free with respect to the index j, i.e. D j (t ij − ǫ ijk τ k ) = 0. It follows that the covector field is divergence-free, and subsequently there exists a tensor field We continue by considering the set of vectors of the form When Ω is connected and simply connected, the set of solutions ξ of this equation forms a four dimensional vector space. The solutions of (3.19) are then determined uniquely throughout Ω by the values of ξ and Dξ at one point. On a flat R 3 the ξ i 's are the parallel vectors, forming a three-dimensional vector space, and are uniquely defined everywhere by the value of ξ i at one point. Both these claims can be established by usual globalisation arguments. (Compare [9] for a proof of a similar statement for the Killing vector equation; identical arguments apply to (3.19), see also [7]. Here it is useful to keep in mind that a Riemannian Einstein metric is real-analytic in harmonic coordinates so that, without loss of generality, we can assume that (M, g) is analytic.) If Ω is a connected subset of the sphere or hyperbolic space, and when the sphere or the hyperbolic space are embedded in a standard way in R 4 , the ξ's are obtained by restricting to Ω the linear functions on R 4 ; on a sphere, these are the ℓ = 1 -spherical harmonics.
Let us show that any solution H ij of (3.18) can be written in the form where N is some arbitrarily chosen point on M . Likewise there exist smooth fields such that If R = 0, at every point p we can find a function ξ which equals zero at p and has arbitrary gradient there. It follows that Similarly one sees that but we will not use this equation. An analogous argument applies when R = 0. One easily checks that Thus, from (3.25), we obtain and ω ij is antisymmetric. Inserting into (3.28) shows that A ℓ ℓ does not contribute and, using that t ij is trace-free, that where ω ij = 2D [i ν j] . Since L rot 1 = rot 2 L, setting φ := a + Lν provides the desired tensor field. ✷ Next we consider t ij ∈ S 2 0 with rot 1 div 2 t = 0. We will show there exist v ∈ C ∞ and χ ij ∈ S 2 0 so that t = Qχ + Ld v: if and only if there exists a symmetric trace-free two-covariant tensor field χ and a function V on Ω such that Proof. The necessity has been established in Section 3.1. For sufficiency, note first that there exists a function τ so that D i τ = D j t ij . Thus t ij − τ g ij is divergence-free, whence is G i = (t ij − τ g ij )ξ j for every Killing vector ξ. An argument similar to the one leading to (3.21) shows that there exists G ij = G [ij] such that G i = D j G ij and [kℓ] and, of course, D (i ξ j) = 0. As is well known, the last equation implies that Inserting (3.32) into G i = D j G ij and using (3.33) we find the two equations and Using the identity and inserting (3.34) into (3.35) there follows Using the Ricci identity in the next-to-last term we find after some calculation that where V ij = V m(i m j) . Now, we have the trivial decomposition where r imlj is an algebraic Riemann tensor. It follows Now, consider the first two terms (within the round brackets) on the right-hand side of (3.41). They can be rewritten as and give no contribution to (3.38) for no obvious reason. This can be seen as follows: Next, in dimension three, the before-last term in (3.41) vanishes. The last term in (3.41), again in three space dimensions, is of the form In order to determine its contribution to (3.38), we calculate as follows Note that this is symmetric in i and j, which takes care of the symmetrisation occurring in (3.38). Inserting (3.45) into (3.38) and taking a trace yields

A Ellipticity
In this appendix we verify the ellipticity, or lack thereof, of the complexes discussed in Section 3.1. For this we need to calculate the symbols of the operators involved, and determine the relevant images and kernels.
Given an operator A we denote by σ k (A) its symbol, where k = (k i ) is the covector argument. The tensor field h ij is assumed to be symmetric; we will not assume that h ij is traceless here, but tracelessness will be assumed in our applications. We have: This implies σ k ((rot 2 ) 2 )(h) = ǫ iℓm k ℓ (ǫ mnp k n h jp + ǫ j np k n h m p ) + ǫ jℓm k ℓ (ǫ mnp k n h ip + ǫ i np k n h m p ) = k n k ℓ (ǫ iℓm ǫ mnp h jp + ǫ jℓm ǫ mnp h ip ) + k n k ℓ (ǫ iℓm ǫ j np + ǫ jℓm ǫ i np )h m p = k ℓ (k i h jℓ + k j h iℓ ) − 2|k| 2 h ij + (ǫ iℓm ǫ j np + ǫ jℓm ǫ i np )k ℓ k n h m p = 3k ℓ (k i h jℓ + k j h iℓ ) − 4|k| 2 h ij + 2 (|k| 2 g ij − k i k j )h ℓ ℓ − k ℓ k m h ℓm g ij , (A.11) σ k (K)(h) = σ k ((rot 2 ) 2 )(h) + σ k (Ldiv 2 )(h) To check ellipticity of the complexes under consideration we need to calculate the following images Im σ k and kernels Ker σ k , for k = 0. By homogeneity and rotation invariance it suffices to analyse the case where the coordinate differentials dx i form an orthonormal basis with k i dx i = dx 1 . Letting the