Proof of non-linear stability

The precise form of the linear stability statement proved in §10 is not quite what we need for the proof of non-linear stability. Concretely, in order to realize the linearized Kerr–de Sitter metric gb(b⁰) as a zero-resonant state of the linearized gauged Einstein operatorLb, we needed to add to it a pure gauge termδ^∗_gbω_b^Υ(b⁰) which in general has a non-trivial asymptotic part att∗-frequency zero, since this is the case for the right-hand side in (10.5)—indeed, since our construction of the metricsg_b only ensures the smooth dependence onb, but does not guarantee any gauge condition, the termD_gbΥ(g_b⁰(b⁰)) is in general non-zero and stationary, i.e. hast_∗-frequency zero. (In fact, without a precise analysis of the resonances of the operator family^Υg_b, it is not even clear if ω^Υ_b(b⁰) can be arranged to both depend smoothly on (b, b⁰)∈T B and not be exponentially growing.)

Since the termδ_g^∗

bω^Υ_b(b⁰), while pure gauge and therefore harmless for linear stability considerations, cannot be discarded in a non-linear iteration scheme, we need to treat it differently. The idea is very simple: since this is a gauge term, we take care of it by changing the gauge; the point is that changing the final black hole parameters fromb

to b+b⁰ is incompatible with the gauge Υ(g)−Υ(gb) (with g≈gb+b⁰ being the current approximation of the non-linear solution), but it is compatible with an updated gauge Υ(g)−Υ(gb+b⁰); updating the gauge in this manner will (almost) exactly account for the termδ_g^∗_bω^Υ_b(b⁰).

To motivate the precise formulation, we follow the strategy outlined in§1.1: using the notation of§3.6, let us consider the non-linear differential operator

P0(b, θ,˜g) := (Ric+Λ)(gb₀,b+ ˜g)−˜δ^∗(Υ(gb₀,b+ ˜g)−Υ(gb₀,b)−θ),

withb∈UB, θa modification in some finite-dimensional space which we will determine, and ˜gexponentially decaying. (We reserve the letter ‘P’ for the actual non-linear operator used in the proof of non-linear stability below.) Note that the linearization ofP₀ in ˜g is given by the second-order differential operator

L_b,˜gr:= (D_˜gP₀(b, θ,·))(r) =D_gb

0,b+˜g(Ric+Λ)(r)−˜δ^∗(D_gb

0,b+˜gΥ(r)), (11.3) while a change in the asymptotic Kerr–de Sitter parameterb is infinitesimally given by

(DbP0(·, θ,˜g))(b⁰) =Dg_b0,b+˜g(Ric+Λ)(χg_b⁰(b⁰))−˜δ^∗(Dg_b0,b+˜gΥ(χg_b⁰(b⁰)))

Let us now reconsider the solvability result for Lb₀ described in Proposition 10.2, and use the specific structure of (g_b⁰

0)^Υ(b⁰)=g_b⁰

0(b⁰)+δ_g^∗_b

0ω_b^Υ

0(b⁰) to arrive at a modification of the range which displays the change of the asymptotic gauge advertised above more clearly: namely, instead ofLb₀(χ(g⁰_b

0)^Υ(b⁰)) as in Proposition10.2, we use Lb₀(χg_b⁰₀(b⁰)+δ_g^∗_b

0(χω_b^Υ₀(b⁰)))

as the modification, which is still compactly supported, and thus can be used equally well to eliminate the asymptotic part (g⁰_b

0)^Υ(b⁰) of linear waves. To see the benefit of

=L_b0(χg⁰_b

0(b⁰))+ ˜δ^∗(D_gb

0Υ(χg_b⁰

0(b⁰)))−δ˜^∗θ_χ(b⁰), (11.5) where we introduce the notation

θχ(b⁰) := [Dg_b0Υδ^∗_gb

0, χ]ω_b^Υ₀(b⁰)+[Dg_b0Υ, χ](g⁰_b0(b⁰)). (11.6) The interpretation of the terms in (11.5) is clear: the first gives rise to linearized Kerr–

de Sitter asymptotics, corresponding to the first term in (11.4), the second corrects the gauge accordingly, corresponding to the second term in (11.4) (note that g_b0,b≡ g_b for b=b₀), and the final term patches up the gauge change in the transition region suppdχ; notice that θ_χ(b⁰) is compactly supported in t_∗. We moreover point out that the sum of the first two terms on the right-hand side vanishes for large t_∗, due to the fact thatD_gb

0(Ric+Λ)(g_b⁰

0(b⁰))=0; exponential decay will be the appropriate and stable description, when we discuss perturbations.

In order to put the non-linear stability problem into the framework developed in

§5.2, we define the space

Ge^s={˜g∈H_b^s,α(Ω;S^{2 b}T_Ω^∗M) :k˜gk_H14,α

b < ε}, (11.7)

withε>0 sufficiently small for all our subsequent arguments—which rely mostly on the results of§5.2—to apply; moreover, we choose a trivializationT_UBB∼=B×R⁴. Then, we define the continuous map

z^Υ:UB×Ge^s+2×R⁴−!H_b^s,α(Ω;S^{2 b}T_Ω^∗M)^,−, (b,g, b˜ ⁰)7−!L_b,˜g(χg_b⁰(b⁰))+ ˜δ^∗(D_gb

0,bΥ(χg⁰_b(b⁰))),

(11.8)

fors>14, which is just the linearization ofP0 inb as in (11.4); the range ofz^Υ consists of modifications which take care of changes of the asymptotic gauge. (The map z^Υ is certainly linear inb⁰, asb⁰7!g⁰_b(b⁰) is linear.) Furthermore, we parameterize the space of compactly supported gauge modifications necessitated by these asymptotic gauge changes by

R⁴−!C_c^∞(Ω;^bT_Ω^∗M), b⁰7−!θχ(b⁰),

(11.9) withθ_χ defined in (11.6). (We could make this map depend onb and ˜g, which may be more natural, though it makes no difference, since our setup is stable under perturba-tions.)

We are now prepared to prove the main result of this paper: the non-linear stability of slowly rotating Kerr–de Sitter spacetimes.

Theorem 11.2. Let h, k∈C^∞(Σ0;S²T^∗Σ0) be initial data satisfying the constraint equations (2.2),and suppose that(h, k)is close to the Schwarzschild–de Sitter initial data (hb₀, kb₀) (see (3.38)) in the topology of H²¹(Σ0;S²T^∗Σ0)⊕H²⁰(Σ0;S²T^∗Σ0). Then, there exist Kerr–de Sitter black hole parameters b∈B,a compactly supported gauge mod-ification θ, lying in a fixed finite-dimensional space Θ⊂C_c^∞(Ω;T^∗Ω), and a section

˜g∈H_b^∞,α(Ω;S^{2 b}T_Ω^∗M) such that the 2-tensor g=gb₀,b+ ˜g is a solution of the Einstein vacuum equations

Ric(g)+Λg= 0,

attaining the given initial data(h, k)at Σ₀,in the gaugeΥ(g)−Υ(gb0,b)−θ=0 (see (3.35) for the definition of Υ). More precisely, we obtain (b, θ,g),˜ and thus g=g_b0,b+ ˜g,as the solution of

Ric(g)+Λg−δ˜^∗(Υ(g)−Υ(g_b0,b)−θ) = 0 in Ω,

γ₀(g) =i_b0(h, k) on Σ₀, (11.10)

where i_b0 was defined in Proposition 3.10.

Moreover, the map

C^∞(Σ0;S²T^∗Σ0)²−!UB×Θ×H_b^∞,α(Ω;S^{2 b}T_Ω^∗M), (h, k)7−!(b, θ,˜g),

(11.11) is a smooth map of Fr´echet spaces (in fact,a smooth tame map of tame Fr´echet spaces) for (h, k)in a neighborhood of (h_b0, k_b0)in the topology of H²¹⊕H²⁰.

Here, recall that α>0 is a small fixed number, only depending on the spacetime (M, gb0) we are perturbing. Furthermore, we use any fixed Riemannian fiber metric on S²T^∗Σ0, for instance the one induced byhb0, to define theH^snorm of the initial data.

Proof. Once we have solved (11.10), the fact thatgsolves Einstein’s equations in the stated gauge follows from the general discussion in§2.1. We briefly recall the argument in the present setting: by definition ofib₀, we have Υ(g)|Σ₀=0 (note that Υ(gb₀,b)=0 near Σ0

for allb∈UB), hence Υ(g)−θ=0 at Σ0due to suppθ∩Σ0=∅, and the constraint equations for (h, k) imply thatL∂t∗(Υ(g)−θ)=0 at Σ0as well once we have solved (11.10); but then applying δgGg to (11.10) implies the linear wave equation e^CPg (Υ(g)−Υ(gb0,b)−θ)=0, and hence Υ(g)−Υ(gb0,b)−θ≡0 in Ω and therefore indeed Ric(g)+Λg=0.

In order to solve (11.10), let Θ denote the finite-dimensional space constructed in Proposition10.2, and fix an isomorphismϑ:R^NΘ

∼=

−!Θ, whereN_Θ:=dim Θ. We parame-terize the modification space for the linear equations we will encounter by

z:UB×Ge^s+2×R^4+4+NΘ−! H_b^s,α(Ω;S^{2 b}T_Ω^∗M)^,−  //D^s,α(Ω;S^{2 b}T_Ω^∗M), (b,˜g,(b⁰, b⁰₁,c))7−!z^Υ(b,˜g, b⁰)+ ˜δ^∗(θ_χ(b⁰₁)+ϑ(c)),

(11.12)

using the maps (11.8) and (11.9). Tensors in the range ofz^Υwill be subsumed in changes of the asymptotic gauge condition. The non-linear differential operator we will consider is thus

P(b, b⁰₁,c,g) := (Ric+Λ)(g˜ b₀,b+ ˜g)−δ˜^∗(Υ(gb₀,b+ ˜g)−Υ(gb₀,b)−θχ(b⁰₁)−ϑ(c)), with (b, b⁰₁,c,˜g)∈U_B×R^4+NΘ×Ge^∞, and the non-linear equation we shall solve is

φ(b, b⁰₁,c,˜g) := (P(b, b⁰₁,c,g), γ˜ ₀(˜g)− i_b0(h, k)−γ0(g_b0))) = 0.

To relate this to the abstract Nash–Moser result, Theorem11.1, we define the Banach spaces

B^s:=R⁴×R^4+Nθ×H_b^s,α(Ω;S^{2 b}T_Ω^∗M) and B^s=D^s,α(Ω;S^{2 b}T_Ω^∗M);

we will look for a solution nearu0:=(b0,0,0,0), for whichφ(u0)=(0, γ0(gb₀)−ib₀(h, k)) is small in a Sobolev norm which we shall determine momentarily.

The typical linearized equation we need to study in the Nash–Moser iteration is of the form

D_(b,b⁰

1,c,˜g)φ(b⁰, b⁰⁰₁,c⁰,r) =˜ d= (f, r₀, r₁)∈B^∞; (11.13) withLb,˜g, the linearization of P in ˜g around (b, b⁰₁,c,˜g) (thus Lb,˜g does not depend on b⁰₁ andc), given by the expression (11.3), this is equivalent to

L_b,˜g(˜r) =f−z(b,g,˜ (b⁰, b⁰⁰₁,c⁰)), γ₀(˜r) = (r₀, r₁).

Now, the mapzsatisfies the (surjective) assumptions of Theorem5.14, in particular (5.33) withb₀ taking the place ofw₀; surjectivity holds because of theϑterm in the definition (11.12) of the map z, taking care of pure gauge modes, and the terms involvingb⁰ and b⁰₁ which take care of the linearized Kerr–de Sitter family in view of the computation (11.5). Thus, we do obtain a solution

ψ(b, b⁰₁,c,˜g)(f, r0, r1) := (b⁰, b⁰⁰₁,c⁰,r)˜ ∈B^∞ of (11.13) together with the estimates

|b⁰|+|b⁰⁰₁|+|c⁰|.kdk13 and k˜rks6Cs(kdks+3+(1+k˜gks+6)kdk13),

for s>10; this regularity requirement is the reason we need 2·10=20 derivatives; see below. Two remarks are worth making: first, the norm on ˜g comes from the fact that

for ˜g∈H_b^s+6,α, the non-smooth coefficients of the linearization ofφlie inH_b^s+4,α, corre-sponding to the norm onwein (5.33); see also Remark5.13. Second, the assumption on the skew-adjoint part of the linear operator at the radial set, ˆβ>−1, in the statement of Theorem5.14does hold; indeed, we showed ˆβ>0 in§9.2.

Thus, we obtain (11.2) withd=10. One easily verifies that for this choice ofd, the estimates (11.1) hold as well. (In fact, d=4 would suffice for the latter, see [76, Proof of Theorem 5.10].) Theorem 11.1 now says that we can solveφ(b, b⁰₁,c,g)=0 provided˜ kib0(h, k)−γ0(gb0)kH²¹⊕H²⁰ is small (here, 20=2d), proving the existence of a solution of (11.10) as claimed; the spaceΘ in the statement of the theorem is equal to the sum of the rangesΘ=θχ(R⁴)+ϑ(R^NΘ).

The smoothness of the solution map (11.11) (in fact with tame estimates), or indeed of

(h, k)7−!(b, b⁰₁,c,˜g),

follows from a general argument using the joint continuous dependence of the solution map for the linearized problem on the coefficients and the data, together with the fact thatφitself is a smooth tame map; see, e.g., [68,§III.1.7] for details.

The proof of non-linear stability is complete.

Remark 11.3. We explain in what sense one can see ringdown for the non-linear solution, at least in principle (since no rigorous results on shallow resonances for the linearized gauged Einstein equation are known): assume for the sake of argument that there is exactly one further resonanceσin the strip−α2<Imσ <−α<0, where we assume to have high-energy estimates (5.7) still, with 1-dimensional resonant space spanned by a resonant stateϕ; we assume thatσis purely imaginary andϕis real. The asymptotic expansion of the solution of the first linear equation that one solves in the Nash–Moser iteration then schematically is of the formg⁰_b

0(b⁰)+εg₍₁₎, whereεis the size of the initial data, andg₍₁₎=cϕ+ ˜g₍₁₎, withc∈Rand ˜g₍₁₎∈H_b^∞,α2 of size 1 (in L^∞, say). Proceeding in the iteration scheme, we simply viewcϕ+ ˜g₍₁₎∈H_b^∞,α, so the non-linear solution will beg=g_b0,b+εg₍₁₎+ε²g₍₂₎, withg₍₂₎∈H_b^∞,α of size 1. At the timescale t_∗=C⁻¹log(ε⁻¹), forC >0 large only depending onα,α2andσ, the three components ofgare thus of size

|εϕ| ∼ε^{−(Imσ)/C+1}, |ε˜g₍₁₎| ∼ε^α2/C+1 and |ε²g₍₂₎| ∼ε^α/C+2,

so the termϕcoming from the refined partial expansion dominates by a factor ε^−δ for some smallδ >0; in this sense, one can see the ringdown, embodied byϕhere, even in the non-linear solution. It would be very interesting to understand the asymptotic behavior of the non-linear solution more precisely, possibly obtaining a partial expansion using shallow resonances.