In this section, we will discuss the application of the above framework to the construction of solutions of the ECEs. First, we define the following Banach spaces, for l∈ N,
Xl≡ Hl+2(C (S)) × Hl+2(S T T(S;˚h))× Hl+2(ST T(S;˚h)), Yl≡ Hl+2(S2 0(S;˚h))× H l+2(Λ1(S)) × Hl+2(Λ1(S)) × Hl+3(S2(S)), Zl≡ Hl+1(S2 0(S;˚h))× H l+1(Λ1(S)) × Hl(Λ1(S)) × Hl+1(S2(S)).
The exponents in the definition of Yl andZl are equal to l + t
ν, l− sµ, respectively, where sµ, tν are the Douglis–Nirenberg weights identified in the proof of Proposition 18. In order to establish surjectivity of DvΨ, we shall have to consider the kernel of the adjoint (De vΨ)e ∗ (which is Douglis– Nirenberg elliptic with the same weights as DvΨ) acting one Zl. To be able to apply elliptic regularity, we require l≥ 2 in order that
l + 1≥ t1= 2, l + 1≥ t2= 2, l≥ t3= 2, l + 1≥ t4= 3.
—the left-hand-sides of the inequalities being the exponents appearing inZl. In other words, given
l≥ 2, ker (DvΨ)e ∗∩ Zl⊂ C∞—see also Remark 41. For what follows, define the map
ω : (ϕ, ¯T , T , χ, ¯X, X)7→ χij+13ϕ˚hij ¯ S( ¯X, ¯T )ij S(X, T )ij , as in Chapters 4 and 5. Then we have the following
Proposition 19. Let (S,˚h, ˚K) be a smooth closed initial data set. The map eΨ : Xl× Yl → Zl is continuous. Fixing l≥ 2, ker DvΨ and ker (De vΨ)e ∗ are finite-dimensional and consist of smooth sections. If both kernels trivialise3 then there exist open neighbourhoods U ⊆ Xl, V ⊆ Yl of ( ˚K, ˚T , ˚¯ T ) and (˚χ,˚¯ξ, ˚ξ, ˚h) and a map ν :U → V such that w(ν(u), u) is a solution for the ECEs for
each u∈ U.
6.3. Constructing solutions of the ECEs 113
Proof. The fact that eΨ is a continuous map fromXl× YltoZlis easily verified using the Schauder ring property —see Section 2.3. Finite dimensionality of ker DvΨ and ker (De vΨ)e ∗ follows from Theorem 5 applied to P = DvΨ and P = (De vΨ)e ∗, and smoothness (when l≥ 2) is a consequence of elliptic regularity, as described above. If both kernels trivialise then DvΨ :e Yl → Zl is an isomor- phism by the Fredholm alternative —see Theorem 5. The Implicit Function Theorem guarantees the existence ofU, W and a map ν such that eΨ(ν(u); u) = 0 for each u∈ U. It remains to show that the candidate solutions w(u) ≡ w(u, ν(u)) indeed satisfy the ECEs. First note the zero quantities
constructed from w(u) satisfy (by definition of eΨ)
Jijk= 0, Λi= 0, Vij= 12L(QX)ij. (6.3.1)
Therefore, all that remains to be shown is that ¯
Λi= 0, L(QX)ij= 0.
First consider the latter. Substituting (6.3.1) into the integrability condition (4.1.11b), we find that 0 =−2⟨QX, B◦ L(QX)⟩L2=−2⟨QX, δ◦ L(QX)−12d(trhL(QX))⟩L2
=−2⟨QX, δ◦ L(QX)⟩L2 =∥L(QX)∥2L2.
Since S is closed, we can integrate by parts to find that L(QX)
ij = 0, as required. On the other hand, ¯Λi= 0 follows automatically by virtue of the remaining integrability condition
¯
Λl+12ϵijkDkJijl= 0. Hence, the candidate solutions w(u) indeed solve the ECEs.
Remark 43. The argument to show that L(QX)
ij = 0 can also be applied in the asymptotically- Euclidean setting provided one chooses the functional spaces in such a way that the fields are of the appropriate decay to permit integration by parts.
6.3.1
Application to time symmetric background initial data sets
Let us return to the case of time symmetric background initial data set. On such a background, the linearised auxiliary equations, DvΨ = 0 (see (6.2.1)), reduce toe
˚ D(χ)ijk−˚ϵijl˚L( ¯ξ)kl = 0, (6.3.2a) ˚δ◦ ˚L(ξ) i− ˚SijB(γ)˚ j−12S˚jkD˚iγjk− γjkD˚kS˚ij= 0, (6.3.2b) 1 2P˚Lγij− 1 3(˚S klγ kl)˚hij+13˚δ◦ ˚B(γ)˚hij = 0. (6.3.2c)
Recall that equation (6.3.2a) is equivalent to ˚P(0)(χ, ¯ξ) = 0. The adjoint system, (D vΨ)e ∗= 0 (see (6.2.8)), reduces to ˚ R(ρ)ij+12˚L(ϱ)ij= 0, (6.3.3a) 2˚δ(ρ)i= 0, (6.3.3b) ˚δ◦ ˚L(ς)i = 0, (6.3.3c) 1 2P˚Lηij+ 1 6(2 ˚DiD˚j−˚hij∆)η +˚ 1 2LςS˚ij+ 1 6(˚δ(ς)− 2η)˚Sij− 1 4S˚kl˚L(ς) kl˚h ij = 0. (6.3.3d)
We have the following improvement on Theorem 3 of Chapter 5:
Theorem 6. Fix l≥ 2. Suppose (S,˚h) is a constant scalar curvature Riemannian manifold (˚r = 2λ)
satisfying
(A1) : ∆ + λ˚ injective on C∞(S), (A2) : P˚L injective on Γ(S02(S,˚h))),
(A3) : P˚(0) injective on Γ(S02(S,˚h))⊕ Γ(Λ1(S)).
Then DvΨ :e Yl → Zl is an isomorphism of Banach spaces. There exists an open neighbourhood
U ⊆ Xl of (0, 0, ˚S), an open neighbourhoodW ⊆ Yl of (0, 0, 0,˚h) and a smooth map ν :U → W such that, defining
u≡ (ϕ, ¯T , T ), ν(u)≡(χ(u), ¯X(u), X(u), h(u)),
the following assertions hold: i) for each (ϕ, ¯T , T )∈ U,
w(u)≡(χ(u) +13ϕ˚h, ¯S( ¯X(u), ¯T ), S(X(u), T ), h(u))
is a solution to the extended constraint equations with cosmological constant λ;
ii) the map u 7→ w(u) is injective if we restrict the free datum ϕ to the sub-Banach space
¯
Hl+2(C (S)).
Proof. Injectivity of DvΨ: First note that any solution to De vΨ = 0 must be smooth by elliptice regularity. Equation (6.3.2a) immediately implies that χij = 0 and ¯ξi = 0, condition (A3). Taking the tracefree part of (6.3.2c), one obtains ˚PL¯γij = 0 and so (A2) implies ¯γij = 0. Substituting back into (6.3.2c) and recalling that ˚Sii= 0, we find
(˚∆ + λ)γ = 0
and so condition (A1) implies γ = 0. Hence, γij= 0. Substituting into (6.3.2b), we find ˚δ◦˚L(ξ) = 0, implying ξi = 0 by condition (A3) —recall that ker ˚L⊂ ker ˚P(0)={0}. Hence, DvΨ is injective.e
Surjectivity of DvΨ: By the Fredholm alternative it suffices to verify injectivity of the adjointe map. Note that any solution to (DvΨ)e ∗ = 0 is again automatically smooth by elliptic regularity. The equations (6.3.3a)–(6.3.3b) are precisely ˚P(0)(ρ, ρ), so condition (A3) implies ρ
ij = 0, ρi = 0. Since (A3) implies non-existence of conformal Killing vectors, equation (6.3.3c) implies that ςi = 0. Substituting into (6.3.3d), 1 2∆˚Lηij− 2 3ληij+ 1 6(−˚hij∆η + 2 ˚˚ DiD˚jη) = 0.
6.4. Concluding remarks 115
Taking the trace one obtains
(˚∆ + λ)η = 0,
implying that η = 0 by condition (A1). Substituting back in we find that ˚PLη¯ij = 0, implying ¯
ηij = 0by condition (A2), and hence ηij = 0. Hence, DvΨe is surjective.
Applying Proposition 19 then proves statement i). For statement ii) we need only show that
DuΨ is injective. It is straightforward to compute that, for a time symmetric background,e
DuΨ( ˘e ϕ, ˘T , ˘¯ T ) = 1 3D( ˘ϕ˚˚ h)ijk−˚ϵij lT˘¯ kl ˚δ( ˘T )i − ˘Tij .
Hence, DuΨ( ˘e ϕ, ˘T , ˘¯ T ) = 0 immediately implies that ˘Tij = 0. Decomposing the first equation and using the fact that ˚R(f˚h)ij = 0 for all scalar functions f , one finds that ˘T¯ij = 0 and d ˘ϕi= 0. The latter implies that ˘ϕ is constant, and therefore if we restrict to ¯Hl+2(C (S)) then ˘ϕ = 0 and D
uΨ ise injective.
Remark 44. For umbilical initial data, the KID equations (6.2.6a)–(6.2.6b) can be shown to reduce
to
1
3N Khij+ D(iYj)= 0, (6.3.4a)
DiDjN − N (rij−12rhij) = 0. (6.3.4b) The latter is known in the literature as the static potential equation, and solutions N referred to
as static potentials. In the time-symmetric sub-case, K = 0, the existence of a static potential would imply a non-trivial KID set (take Yi= 0, for example) and therefore a non-trivial element of ker (DvΨ)e ∗. In the case K̸= 0 it is not so clear whether a static potential N can be completed to a KID set, but one still has (
˚ Π 0 0 1 ) DΦ∗ ( N 0 ) = 0,
since the tracefree part of (6.3.4a) decouples from (6.3.4b), and this gives rise to non-trivial elements in ker (DvΨ)e ∗. Note also that, taking the trace of the static potential equation, one obtains
(∆ + 12r)N = 0.
Hence, condition (A1) implies the non-existence of static potentials, but clearly imposes a much stronger restriction on (S,˚h). It would be interesting to see whether (A1) in Theorem 6 can be
weakened to the requirement that (S,˚h) admit no non-trivial static potentials.
The above theorem is an improvement on Theorem 3 in that we do not impose any restrictions on the kernels of ˚∆Y and ˚P(1).
6.4
Concluding remarks
In this chapter an alternative auxiliary extended constraint map was presented which streamlines the approaches of the previous chapters, in the sense of simplifying the analysis of the linearised equations (for time symmetric background initial data) and rendering the sufficiency argument
almost trivial. The two new additions are the use of a new gauge-reduction procedure and the use of an inbuilt mixed-order ellipticity of the ECEs. As an additional pay-off, it was shown that KID sets naturally arise as obstructions to solving the auxiliary system, which is desirable given the role that KID sets play in the problem of linearisation stability. The proposed method was then applied to time symmetric background initial data sets and the conditions (A1)–(A3) were identified as being sufficient conditions for the method’s implementation, thereby improving on Theorem 3 of Chapter 5.
So far we have been unable to give a natural geometric interpretation of the full cokernel, of which the KID sets are only part. It is possible that the obstructions may be given a more geometric characterisation by identifying them with components of some tensor field on the full spacetime manifold, in a way that is analogous to the identification of KID sets with the lapse-shift compo- nents of spacetime Killing vectors —see Section 6.2.2. It is possible that the linearised auxiliary system would be further simplified by using a gauge-reduction which is also adapted to the extrinsic curvature perturbation (in addition to the perturbation of the electric part). At present it is not clear how to do this, particularly in view of the fact that one expects the “correct” approach to leave the linearised scalar curvature unchanged in order that KID sets continue to feature as (potential) obstructions to integrability. However, such a modification may be necessary if the method is to be applied to non-time symmetric background data.
Another direction of study would involve generalising the method to non-compact S —e.g.
asymptotically-Euclidean, or hyperboloidal, initial data. To do so, one would first have to make use of the elliptic machinery for Douglis–Nirenberg systems on non-compact domains, which can be found in [79] for instance. As discussed in Remark 39, certain aspects of the streamlined method presented in this chapter render it unsuitable for generalisation to the full CCEs, suggesting that an approach more in keeping with that of Chapters 4 and 5 is required.
Chapter 7
Extending the Friedrich–Butscher
method to the full CCEs
In this chapter we return to the full Conformal Constraint Equations (CCEs), and the problem of generalising the Friedrich–Butscher method to this context. In Section 7.1, we will describe an elliptic reduction of the CCEs, involving a specification of the free and determined fields that is motivated by the Friedrich–Butscher method. The new feature that arises here is that certain parts of the free data (identified in Section 7.1) are “unphysical” in the sense that they should be thought of as fixing the conformal gauge freedom inherent to solutions of the CCEs —see Section 3.2.2. In Section 7.2, I will describe an application to the construction of non-linear perturbations of the CCEs around the hyperbolic, conformally-rigid background geometries considered in Chapter 4, thought of now as a solution to the full CCEs with trivial conformal factor ˚Ω≡ 1. It will be important to note that the hyperbolic manifolds considered earlier, and which we take as defining our background intrinsic geometry, have vanishing first Betti number —see Remark 14. Hence, by Hodge’s Theorem, the background manifolds admit no non-trivial harmonic 1−forms —i.e. ker ˚∆H = {0}— a fact that will be used explicitly both in the construction of candidate solutions and in the sufficiency argument.