The linearized Einstein equations with sources

On vacuum spacetimes of general dimension, we study the linearized Einstein vacuum equations with a spatially compactly supported and (necessarily) divergence-free source. We prove that the vanishing of appropriate charges of the source, defined in terms of Killing vector fields on the spacetime, is necessary and sufficient for solvability within the class of spatially compactly supported metric perturbations. The proof combines classical results by Moncrief with the solvability theory of the linearized constraint equations with control on supports developed by Corvino and Chru\'sciel-Delay.


Introduction
Let (M, g) be a smooth connected globally hyperbolic spacetime, of dimension n + 1 where n ≥ 1, which solves the Einstein vacuum equations Ein(g) = 0, Ein(g) := Ric(g) − 1 2 R g g. (1.1) Here Ric(g) and R g denote the Ricci and scalar curvature, respectively.Let Σ ⊂ M denote a smooth spacelike Cauchy hypersurface; denote its unit normal by ν Σ and the surface measure by dσ.We study linearized perturbations of g sourced by smooth linearized stressenergy-momentum tensors f ∈ C ∞ sc (M ; S 2 T * M ) which are spatially compactly supported (hence the subscript 'sc'): this means that there exists a compact subset K ⊂ Σ so that supp f ⊂ ± J ± (K), with J ± (K) ⊂ M denoting the causal future ('+'), resp.past ('−'), of K. 1 That is, we shall study the equation Recall the second Bianchi identity, which states δ g Ein(g) = 0 for all metrics g (where δ g is the negative divergence operator); linearizing this in g and using (1.1) gives Therefore, a necessary condition for the solvability of (1.2) is δ g f = 0.The main result of this note precisely determines the extent to which this condition is also sufficient.To state it, denote by K (M, g) ⊂ V(M ) = C ∞ (M ; T M ) the (finite-dimensional) space of Killing vector fields on (M, g).
Theorem 1.1 (Main theorem, smooth version).The map (1.5)Here K (M, g) * = L(K (M, g), R) is the dual space.This isomorphism is moreover independent of the choice of Cauchy hypersurface Σ.
In other words, D g Ein(h) = f ∈ C ∞ sc ∩ ker δ g has a solution h ∈ C ∞ sc if and only if all 'charges' (1.4) vanish.(In particular, when K (M, g) = {0}, then δ g f = 0 is sufficient for the solvability of (1.2).)The proof of solvability utilizes the Cauchy problem for a gaugefixed version of this equation.The Cauchy data must be chosen to satisfy the linearized constraint equations on Σ with source ψ = f (ν Σ , •) while being compactly supported; by results of Corvino [Cor00] and Chruściel-Delay [CD03], this is possible if and only if ψ is orthogonal to the cokernel of the linearized constraints map, which can be canonically identified with K (M, g) by a result of Moncrief [Mon75].We recall Moncrief's result in Proposition 2.1 and give a new perspective on its proof.For a finite regularity version of Theorem 1.1, see Theorem 3.3.
In Theorem 3.4, we show that if Σ is noncompact and one drops the support assumptions on h, there are no obstructions to solvability beyond δ g f = 0, the reason being that the aforementioned cokernel on the dual space E (Σ) of C ∞ (Σ) is trivial.When M and Σ have more structure, e.g. if they are asymptotically flat, one can make more precise statements regarding the weights at infinity of f and h depending on which (some of) the additional obstructions given by (1.4) disappear; we shall not discuss this here.
Remark 1.2 (Cosmological constant).Our arguments go through with purely notational modifications if Ein(g) − Λg = 0 where Λ ∈ R is the cosmological constant.In this case, Theorem 1.1 becomes a characterization of those f ∈ C ∞ sc ∩ker δ g for which D g Ein(h)−Λh = f has a solution h ∈ C ∞ sc .
Our motivation for solving D g Ein(h) = f with nontrivial f , and understanding the obstructions to solvability, comes from perturbation theory.To give a concrete example, suppose (M, g) is a vacuum spacetime, and we wish to modify it near a timelike geodesic γ, in local Fermi normal coordinates (t, x) given by γ = {(t, 0) : t ∈ R}, by gluing in a small black hole; in 3 + 1 dimensions this could be a Schwarzschild black hole with mass > 0, given by the metric −dt 2 + dx 2 + 2 r (dt 2 + dr 2 ) + O( 2 r −2 ) where r = |x|.The naive ansatz g = g + χ(r) 2 r (dt 2 + dr 2 ) for the modified spacetime metric leads to Ein(g ) = Ein(g) + f + O( 2 ) where f = O(1) near the gluing region supp χ ⊂ {r > 0} (ignoring the singularity of f at r = 0, which one must deal with separately), and δ g f = 0 by the second Bianchi identity for g = g + O( ).One then wishes to eliminate the error f by replacing g by g + h where h solves D g Ein(h) = −f (while being more regular at r = 0 than 2 r (dt 2 + dr 2 )).For the details of such a gluing procedure, see [Hin23].Prior work on solutions of the linearized Einstein equations has largely focused on solutions of the homogeneous equation D g Ein(h) = 0. Control of solutions modulo pure gauge solutions (i.e.symmetric gradients δ * g ω of 1-forms ω on M , which always satisfy this equation) is one important problem, especially in the context of stability problems [RW57].More pertinent to this work is the problem of linearization stability as introduced by Fischer-Marsden [FM75,FM73], namely whether such an infinitesimal deformation h can be integrated to a nonlinear solution, i.e. whether there exists a family g s of metrics with Ein(g s ) = 0 and h = d ds g s | s=0 .This is the original context of [Mon75], which shows that linearization stability at (M, g) holds for the Einstein equations if and only if K (M, g) = {0}.The necessary conditions for the existence of g s whose linearization is h in the presence of nontrivial Killing vector fields were found in [Mon76,AMM82,FMM80].
In this context, Theorem 1.1 gives a necessary and sufficient condition on f ∈ C ∞ sc ∩ ker δ g so that there exist spacetime metrics g s , s ∈ (−1, 1), with Ein(g s ) = sf + O(s 2 ).(Absent a general natural physical prescription of further lower order terms beyond sf , we do not study the problem of determining when a metric g near g with, say, Ein(g) = sf , s small, exists.)The problem of prescribing the nonlinear Ricci curvature tensor of a Lorentzian metric on a given smooth manifold has been studied by DeTurck [DeT83]; the question of the solvability of the analogue of the constraint equations with source, see [DeT83, Equations (3.1)-(3.2)], is however not discussed there.

Killing vector fields and the linearized constraint equations
We recall classical results by Moncrief [Mon75] and Fischer-Marsden-Moncrief [FMM80].For any Lorentzian metric g for which Σ is spacelike, the 1-form of Σ; this gives rise to the constraints map We write D γ,k P for the linearization of P at (γ, k); thus ). (Here we identify T Σ M = R ⊕ T Σ using the orthogonal projections onto Rν Σ and T Σ, and likewise for the cotangent bundles.)Suppose that Ein(g) = 0.If h ∈ C ∞ (M ; S 2 T * M ), and γ, k ∈ C ∞ (Σ; S 2 T * Σ) denote the linearized initial data at Σ (i.e. the derivatives at s = 0 of the initial data of g + sh), then linearizing (2.1) implies (2.2) Proposition 2.1 (Killing vector fields and the linearized constraints map).(See [Mon75] and also [FMM80, Lemma 2.2].)Let (M, g) be globally hyperbolic with Ein(g) = 0. Then the map Our proof of the surjectivity of (2.3) is based on a distributional characterization of ker(D γ,k P ) * which seems to not have been noted before; see (2.7)-(2.8).Before starting the proof in earnest, we observe that Let (M, g) be a smooth connected N -dimensional pseudo-Riemannian manifold.Then the dimension of the space K (M ) of Killing vector fields on (M, g) Proof.This follows from the well-known fact that X is uniquely determined by X(p) ∈ T p M and ∇X(p) ∈ T * p M ⊗ T p M .(Since the Killing equation imposes 1 2 N (N + 1) linearly independent constraints on X(p), ∇X(p), the space of Killing vector fields has dimension ≤ N + N 2 − 1 2 N (N + 1) = 1 2 N (N + 1).)We recall a prolongation argument for the proof of this statement.The Killing equation implies for Z ∈ V(M ) the following identity, where we write '≡' for equality modulo terms involving only X and ∇X but no higher derivatives: Cyclically permuting the vector fields (Z, V, W ) twice more, we obtain ∇ Z (∇X)(V, W ) ≡ −∇ Z (∇X)(V, W ), and thus ∇ Z (∇X)(V, W ) is an (explicit) expression involving only X and ∇X.Therefore, if α : [0, 1] → M is a smooth curve, then there exists a smooth bundle endomorphism F on the restriction of In particular, if Z(0) = 0, then Z = 0 on α([0, 1]), and thus Z = 0 on M since M is connected.
To prove the final claim, note that if V, W are vector fields on M , then (2.4) gives V X, W + W X, V = 0 at Σ (where X = 0).Fix V to be transversal to Σ at a point p ∈ Σ.For all W which are tangent to Σ, we have W X, V = W (0) = 0 and thus V X, W = 0, while for W = V we obtain V X, V = 0. Therefore, ∇X = 0 at p, from where X = 0 follows from the first part of the proof.

Proof of Proposition
for the linearized initial data at Σ. Since δ g (D g Ein(h)) = 0, the fact that X is Killing implies that also D g Ein(h)(•, X) is divergencefree.Therefore, if Σ t , t ∈ [−1, 1], is a smooth family of spacelike hypersurfaces, with unit normal ν Σt and surface measure dσ t , so that Σ t equals Σ outside a large compact set, then is independent of t and thus equals to be arbitrary at Σ and 0 at Σ 1 , we deduce that I(0) = 0 for all γ, k, and therefore (D γ,k P ) * (X) = 0. Together with Lemma 2.2, we conclude that the map (2.3) is well-defined and injective.(This argument is taken from [FMM80, Lemma 1.5 and Corollary 1.9].) where we use the musical isomorphism for g.Write δ * g for the symmetric gradient on 1-forms; this is related to the Lie derivative via δ * g ω = 1 2 L ω g.We need to show that there exists a solution ω ∈ C ∞ (M ; T * M ) of the Killing equation δ * g ω = 0 which satisfies ω| Σ = ω (0) .At Σ, the requirements ω = ω (0) and 0 = 2(δ * g ω)(ν Σ , V ) = ∇ ν Σ ω(V ) + ∇ V ω(ν Σ ) for V = ν Σ and V ∈ T Σ uniquely determine the 1-jet ω (1) of ω at Σ. Write G g = I − 1 2 g tr g ; then we may extend ω (1) to a 1-form on M by solving the wave equation δ g G g δ * g ω = 0 on M , with the Cauchy data of ω matching ω (1) .Let ) and δ g π = 0 globally.Our aim is to show π = 0. Note that since Ein(g) = 0, we have where R g is a 0-th order operator involving the curvature of g.
In view of this hyperbolic equation for G g π, we only need to show that the 1-jet of G g π (or, equivalently, that of π) vanishes at Σ (2.6) to conclude the proof.This is where the equation satisfied by ω (0) will enter.

Proof of the main result; variants
The independence of the map (1.4) of the choice of Cauchy hypersurface Σ in (1.4) follows from the divergence theorem, since δ g f = 0 and X ∈ K (M, g) implies that f (X, •) is divergence-free.The vanishing of Σ f (ν Σ , X) dσ for f = D g Ein(h) was shown in the first step of the proof of Proposition 2.1.Thus, (1.5) is well-defined.
For the proof of Theorem 1.1, we require the following (well-known) result: Theorem 3.1 (Solvability of the linearized constraints).Let (Σ, γ) be a smooth connected Riemannian manifold, and let Proof.The existence of regular solutions with controlled support is due to Corvino [Cor00], with the existence of smooth solutions proved by Chruściel-Delay [CD03]; see [Del12] for general results of this flavor.The key point is that (D γ,k P ) * is an overdetermined (Douglis-Nirenberg) elliptic partial differential operator for which a priori estimates hold on spaces of tensors defined on a smoothly bounded open precompact subset U ⊂ Σ, containing supp ψ, which allow for exponential growth at ∂U.By duality, this gives the solvability of D γ,k P ( γ, k) = ψ, provided ψ is orthogonal to the cokernel, with γ, k exponentially decaying at ∂U, whence their extension by 0 to Σ \ U furnishes the desired solution.
Proof of Theorem 1.1.• Injectivity of (1.5).Suppose f ∈ C ∞ sc (M ; S 2 T * M ) satisfies δ g f = 0 and Σ f (ν Σ , X) dσ = 0 for all X ∈ K (M, g).By Proposition 2.1, ) so that γ, k are the linearized initial data corresponding to any metric perturbation h ∈ C ∞ sc (M ; S 2 T * M ) with Cauchy data We then solve the initial value problem for the gauge-fixed linearized Einstein vacuum equation the solution h satisfies h ∈ C ∞ sc (M ; S 2 T * M ) by finite speed of propagation.If we set η := δ g G g h − θ, then we have by definition of θ.Moreover, applying G g to (3.1) and recalling (2.2), we obtain Together with (3.2), this implies ∇ ν Σ η = 0. Finally, applying δ g G g to (3.1) and using the linearized second Bianchi identity together with δ g f = 0 gives the hyperbolic equation δ g G g δ * g η = 0 for η, whence η = 0 on M and thus D g Ric(h) = G g f .Therefore, f = D g Ein(h) projects to 0 in the quotient space (1.5).
• Surjectivity of (1.5).Given λ ∈ K (M, g) * , there exists a 1-form ψ ∈ C ∞ c (Σ; T * Σ M ) with ψ, X| Σ = λ(X) for all X ∈ K (M, g).Indeed, it suffices to arrange this for X in a basis of K (M, g), in which case it follows from the linear independence of the restrictions of the basis elements to Σ (a consequence of Lemma 2.2).Pick then any f M ; S2 T * M ) denote any compactly supported symmetric 2tensor with f = f (0) at Σ.We claim that there exists ω ∈ C ∞ sc (M ; T * M ) with (ω| Σ , ∇ ν Σ ω) = 0 so that g ω ∈ ker δ g .Indeed, we simply solve the initial value problem for all X ∈ K (M, g) still, and the surjectivity of (1.5) follows.This completes the proof of Theorem 1.1.Remark 3.2 (Equivalence to solving the linearized constraints).Not only did Theorem 3.1 play a key role in the argument; one can conversely deduce Theorem 3.1 from Theorem 1.1.Indeed, given ψ satisfying the assumptions of Theorem 3.1, one constructs ) are the linearized initial data induced by h.