We fix a static patch of de Sitter space by choosing a pointq∈X as the origin of our coordinate system (τ, w1, ..., wn), and homogeneously blowing up q; coordinates on the static patch are then
τ, xi:=wi
τ ,
with the front face given byτ=0. Correspondingly, our frame takes the form e0=τ ∂τ−X
We work on a neighborhood Ω of the causal past ofq; concretely, let us take Ω =
for any fixedεM>0, so Ω is a domain with corners within M:=
Suppose now that r∈C∞(M) is a smooth function on global de Sitter space M. Then, the pull-back ˜rofrto Ω is
˜
r(τ, xi) =r(τ xi) =r(0)+τX
i
xi∂wir(0)+O(τ2); (C.11) continuing the Taylor expansion further, one finds that ˜ris (asymptotically as τ!0) a sum of terms of the formτjtimes a homogeneous polynomial of degreejin thexi. Thus,
one can deduce the resonance expansion in the static patch from the calculations in§C.4 by taking rto be a homogeneous polynomial in the coordinates wi of degree61 (since higher-order terms will giveo(1) contributions, the imaginary part of all resonances being
<2) and reading off the terms in the resulting asymptotic expansions inτ. Indeed, every resonant state on static de Sitter space arises as a term in the asymptotic expansion of the solution of a wave equation with smooth forcing, compactly supported and with support disjoint from X; and conversely, solutions of such equations on static de Sitter spaces admit an expansion into resonances up to terms of any fixed, prescribed rate of decayτC, C∈R; but the asymptotic behavior of waves on static de Sitter space, which are in this way equivalent to knowledge of resonances and resonant states, can simply be read off by restriction from the asymptotics onglobal de Sitter space.
We thus obtain the following result.
Proposition C.1. For the operator L∈Diff2b(Ω;S2 bTΩ∗M),the following is a com-plete list of the resonances σ of L which satisfy Imσ >12(−n+√
n2+8n)−2, and the corresponding resonant states:
(1) σ=σ+=12i(−n+√
n2+8n): resonance of order 1 and rank 1;basis of resonant states
(2) σ=σ+−i: resonance of order 1 and rank n;basis of resonant states
τiσ++1
(3) σ=0: resonance of order 1 and rank 12n(n+1)−1;basis of resonant states
We can now complete the proof that all non-decaying resonant states are pure gauge solutions.
Proposition C.2. All the resonant states corresponding to resonances in Imσ>0, viewed as mode solutions of Lon the spacetime,lie in the range of δ∗g0 acting on 1-forms on the spacetime.
Proof. Atσ+ we compute, in the splittings (C.1),
δg∗0τiσ+ and one can solve away the second term, with the result
δg∗0
Putting f=1 gives the resonant state at σ+, thus proving the result for the resonance σ+, while puttingf=iσ+wj,j=1, ..., n, we find proving the result for the resonanceσ+−i. Finally, for σ=0, we observe
δg∗0
Remark C.3. For any choice of parametersγ1 and γ2, the space of resonances at zero is always non-trivial, and contains the resonant states given in PropositionC.1(3).
There are further modifications one can consider, for instance using a conformally rescaled background metricg0=τγ3g0and considering the gauge Υ(g)−Υ(g0), with
Υ(g) =g(g0)−1δgGgg0,
but this does not affect the previous statement regarding the zero resonance. Thus, if we are restricting ourselves to modifications of Einstein’s equations which are well behaved from the perspective of global de Sitter space, there seems to be no way to eliminate all non-decaying resonances! Choosingγ1, γ2 andγ3 appropriately, one can remove all non-decaying resonances apart from zero, but this is quite delicate.
Denote byNΘ:=n+1+12n(n+1)−1 (soNΘ=9 forn=3) the total dimension of the space of non-decaying resonant states; then, paralleling the proof of Proposition10.2, we let Θ be theNΘ-dimensional space of 1-formsθof the form
θ=−Dg0Υ(δ∗g
0(χω)) =δg0Gg0δg∗
0(χω),
whereχ(τ) is a fixed cutoff, identically 1 nearτ=0 and identically 0 forτ>12, say, and ω is one of the 1-forms used in the proof of PropositionC.2exhibiting the non-decaying modes as pure gauge modes. Thus, we have
L(δ∗g0(χω)) = ˜δ∗θ;
furthermore,θis compactly supported in (0,1)τ, due to
0 =δg0Gg0L(δg∗0ω) =−eCPg0 (Dg0Υ(δ∗g0ω))
and SCP, which givesDg0Υ(δg∗0ω)=0 asδ∗g0ωis a non-decaying mode. One can of course also check directly that the resonant states described in PropositionC.1are annihilated by δg0Gg0; for the zero resonant states, this is straightforward to check, while for the resonant states at σ+ and σ+−i, this follows from the fact that δg0Gg0 applied to the expression in square brackets in (C.10) gives a result of orderτiσ++2. The upshot is that
Θ⊂ Cc∞(Ω;T∗Ω)
can be used as the fixed,NΘ-dimensional space of gauge modifications, using which we can prove the non-linear stability of the static model. That is, modifying the forcing terms of the linearized equations which we need to solve in the course of a non-linear iteration scheme by ˜δ∗θ for suitable θ∈Θ (which are found at each step by the linear solution operators), we can solve the linear equations—and thus the non-linear gauged Einstein equation—in spaces of exponentially decaying 2-tensors.
Theorem C.4. Let Σ0=Ω∩{τ=1} be the Cauchy surface of Ω. Let h, k∈ C∞(Σ0;S2T∗Σ0)
be initial data satisfying the constraint equations (2.2),and suppose (h, k)is close to the data induced by the static de Sitter metric g0 in the topology of
H21(Σ0;S2T∗Σ0)⊕H20(Σ0;S2T∗Σ0).
Then, there exist a compactly supported gauge modification θ∈Θand a section
˜
g∈Hb∞,α(Ω;S2 bTΩ∗M),
with α>0 small and fixed,such that g=g0+ ˜g solves the Einstein equation Ric(g)+ng= 0,
attaining the given initial data atΣ0in the gauge Υ(g)−θ=0 (see(C.4)for the definition of Υ). More precisely,g solves the initial value problem
Ric(g)+ng−δ˜∗(Υ(g)−θ) = 0 in Ω, γ0(g) =i0(h, k) on Σ0,
where i0 constructs correctly gauged (relative to Υ(g)=0) Cauchy data from the given initial data (h, k),analogously to Proposition 3.10.
Proof. Given what we have arranged above, this follows directly from TheoremB.1 if we takez:RNΘ∼=Θ!Cc∞(Ω;T∗Ω) to be the mapθ7!−δ˜∗θ.
The number of derivatives here is rather excessive: in fact, due to the lack of trap-ping, one does not lose derivatives beyond the usual loss of one derivative for hyperbolic equations; thus, one can prove this theorem using a Newton-type iteration method as in [71, §8]. But, since we state this result in order to present a simple analogue of Theorem11.2, we refrain from optimizing it.
While the above arguments prove the stability of the static model of de Sitter space, there is absolutely no direct implication for the initial value problem near a Schwarzschild–de Sitter spacetime: the limit M!0 in which Schwarzschild–de Sitter space becomes de Sitter space is very singular.
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