Exterior Stability of Minkowski Space in Generalized Harmonic Gauge

Abstract

We give a short proof of the existence of a small piece of null infinity for $(3+1)$-dimensional spacetimes evolving from asymptotically flat initial data as solutions of the Einstein vacuum equations. We introduce a modification of the standard wave coordinate gauge in which all non-physical metric degrees of freedom have strong decay at null infinity. Using a formulation of the gauge-fixed Einstein vacuum equations which implements constraint damping, we establish this strong decay regardless of the validity of the constraint equations. On a technical level, we use notions from geometric singular analysis to give a streamlined proof of semiglobal existence for the relevant quasilinear hyperbolic equation.

1 Introduction

The goal of this paper is to introduce a novel generalized wave coordinate gauge on asymptotically flat spacetimes, and to demonstrate its utility by proving the existence of a piece of null infinity for the spacetime evolving from asymptotically flat initial data sets.

Theorem 1.1

(Main theorem, rough version) Let $\Sigma =\{x\in {\mathbb {R}}^3:|x|>R\}$ for some $R>0$, and suppose $\gamma ,k\in \mathcal C^\infty (\Sigma ;S^2 T^*\Sigma )$ are a Riemannian metric, resp. smooth symmetric 2-tensor on $\Sigma$ satisfying the constraint equations,¹ with

where is the standard metric on ${\mathbb {S}}^2$. Let $\ell _0\in (0,1)$. Suppose that ${\tilde{\gamma }}$ and k are small² in the sense that

$$\begin{aligned} \sum _{|\alpha |\le N} \Vert r^{-1/2+\ell _0}(r\nabla )^\alpha ({\tilde{\gamma }},r k) \Vert _{L^2} < \epsilon \end{aligned}$$

(1.1)

with N large, $\epsilon >0$ small. Then there exists a Lorentzian metric g on

$$\begin{aligned} \Omega= & {} \{ (t,r,\omega ) \in [0,\infty )\times (R,\infty )\times {\mathbb {S}}^2 :r_*-t>R+1 \} \\ {}{} & {} \quad \subset {\mathbb {R}}^4 = \{ (z^0=t,z^1,z^2,z^3) \}, \end{aligned}$$

where $r_*=r+2{\mathfrak {m}}\log (r-2{\mathfrak {m}})$, with the following properties:

• (1) g solves the Einstein vacuum equations $\textrm{Ric}(g)=0$;

• (2) identifying $\Sigma \cap \{r_*>R+1\}$ with $t^{-1}(0)\subset \Omega$, the induced metric and second fundamental form of g at $\Sigma$ are given by $\gamma$ and k;

• (3) g is in a modified wave coordinate gauge relative to the Schwarzschild metric $g_{\mathfrak {m}}=-\bigl (1-\tfrac{2{\mathfrak {m}}}{r}\bigr ){\mathrm d}t^2+\gamma _{\mathfrak {m}}$, see (1.5) and (1.3);

• (4) g approaches the Schwarzschild metric in a quantitative manner,

  $$\begin{aligned} g = g_{\mathfrak {m}}+ r^{-1}h, \end{aligned}$$

  where all coefficients $h(\partial _{z^i},\partial _{z^j})$ are uniformly bounded. More precisely, if $L=\partial _t+\partial _{r_*}$ and $\underline{L}=\partial _t-\partial _{r_*}$ denote outgoing and incoming null vector fields for the Schwarzschild metric, and $\Omega$ denotes an arbitrary vector field on ${\mathbb {S}}^2$, then

  

  (1.2)

  while the trace-free part of the restriction of h to $T{\mathbb {S}}^2$ and the component $h(\underline{L},\underline{L})$ have smooth limits as $r\rightarrow \infty$, $|r_*-t|\lesssim 1$, with $(r_*-t)^{-\ell _0}$ decay³.

More generally, we prove a semiglobal existence theorem and the same asymptotics for the solution g of a quasilinear hyperbolic (gauge-fixed) version of the Einstein equations for general (i.e. not necessarily arising from an initial data set) suitably decaying and regular Cauchy data for h; see Corollary 3.36. Through a combination of the new gauge with constraint damping and a simple nonlinear iteration scheme, we are able to obtain these asymptotics in one fell swoop.

We recall that Christodoulou–Klainerman [11] gave the first proof of the nonlinear stability of Minkowski space, with initial data given on all of ${\mathbb {R}}^3$ (requiring stronger decay $\ell _0>{\tfrac{1}{2}}$ but less regularity); the evolving spacetime metric is geodesically complete. Klainerman–Nicolò [27] gave a new proof of the stability of the exterior region (as in Theorem 1.1) using a double null foliation; see [41] for improvements. Earlier work by Friedrich [16] established the global nonlinear stability for special initial data $(\gamma ,k)$ which are equal to $(\gamma _{\mathfrak {m}},0)$ outside a compact set; the existence of such data was proved by Corvino [8, 12]. Bieri [6] lowered the decay assumptions to $\ell _0>-{\tfrac{1}{2}}$ and required only $N=3$ derivatives on the initial data. (There is a vast literature on extensions and variants of the nonlinear stability problem on asymptotically flat spacetimes, including [1, 3, 10, 13,14,15, 19, 24, 28, 29, 31, 37, 44, 45, 47].)

Closely related to the present work is the global stability proof by Lindblad–Rodnianski [33, 34] in the standard wave coordinate gauge $\Box _g z^\mu =0$; due to logarithmic divergences arising in simplistic formal nonlinear iteration arguments (see [34, §1] and also (1.7) below), this gauge condition was considered unsuitable for a proof of global stability until Lindblad–Rodnianski [32] discovered that the Einstein equations in wave coordinates satisfy a weak null condition at null infinity $\mathscr {I}^+$. Lindblad [30] subsequently proved sharp decay at $\mathscr {I}^+$ by using vector field multipliers and commutators adapted to the large scale Schwarzschild geometry (rather than the Minkowski geometry as in [34]). In the wave coordinate gauge, only the first three components in (1.2) have the stated decay, while all other components of h have leading order terms at $\mathscr {I}^+$, with the exception of $h(\underline{L},\underline{L})$ which blows up logarithmically as $r\rightarrow \infty$. This result was extended by the author and Vasy in [22] where it was shown that the geodesically complete spacetime metric g, evolving from initial data close to the trivial data, is polyhomogeneous on a compactification of ${\mathbb {R}}^4$ to a manifold with corners; this result utilizes a wave map gauge relative to the Schwarzschild metric (discussed further below). Furthermore, in this gauge, [22] clarified the nature of the logarithmically divergent leading order term of $h(\underline{L},\underline{L})$ by relating its average over spherical sections of null infinity to the Bondi mass [5, 9]. In a different direction, Keir [25], focusing on the analysis of weak null conditions, proved the global well-posedness of the Einstein equations in harmonic coordinates (in the standard formulation, i.e. without constraint damping) for general small Cauchy data. We also mention the work by Lindblad–Schlue [35, 36] on scattering problems from future null infinity, i.e. backward problems, for (systems of) semilinear equations satisfying the weak null condition; see Wang [46] for such results in $n+1$ dimensions, $n\ge 4$.

By contrast, in the novel gauge introduced here, $h(\underline{L},\underline{L})$ remains bounded, the spherical averages of its limit at $\mathscr {I}^+$ being related to the Bondi mass; and the trace-free spherical part of h directly encodes the Bondi news function and outgoing energy flux, as indicated in Remark 3.38 (following [22, §8]). All other metric components have faster decay; in this sense, our gauge suppresses all ‘non-physical’ degrees of freedom to leading order at null infinity. We discuss this in §1.1. It would be interesting to see if our novel gauge might simplify the analysis of scattering problems for the Einstein vacuum equations by eliminating the need to study weak null conditions as in [36].

In addition to this improved decay, our analysis takes full advantage of notions from geometric singular analysis, concretely the notions of b- and edge-metrics and -operators going back to Melrose [39] and Mazzeo [38]; see §1.2.

1.1 Constraint Damping and Novel Gauge

A natural generalized wave coordinate gauge or generalized harmonic gauge for a spacetime metric g which is a perturbation of $g_{\mathfrak {m}}$ is the wave map gauge relative to $g_{\mathfrak {m}}$: this requires the identity map $(\Omega ,g)\rightarrow (\Omega ,g_{\mathfrak {m}})$ to be a wave map. One can solve the Einstein vacuum equations in this gauge by solving the quasilinear wave equation

$$\begin{aligned}{} & {} \textrm{Ric}(g) - \delta _g^*\Upsilon (g;g_{\mathfrak {m}}) = 0,\qquad \nonumber \\{} & {} \Upsilon (g;g_{\mathfrak {m}})^\kappa :=g^{\mu \nu }(\Gamma (g)_{\mu \nu }^\kappa -\Gamma (g_{\mathfrak {m}})_{\mu \nu }^\kappa )\ \ \text {(gauge 1-form)}, \end{aligned}$$

(1.3)

for g; here, $(\delta _g^*\Upsilon )_{\mu \nu }={\tfrac{1}{2}}(\Upsilon _{\mu ;\nu }+\Upsilon _{\nu ;\mu })$ is the symmetric gradient. Constraint damping amounts to modifying $\delta _g^*$ by zeroth order terms; it was introduced in [4, 17] and used to by Pretorius [40] as a device in numerical evolution schemes to ensure that violations of the gauge condition are damped. It also played a key role in the recent proofs of black hole stability in cosmological spacetimes [20, 21]: when solving linearizations of Eq. (1.3) (with nontrivial right hand side) in a Nash–Moser iteration scheme, constraint damping ensures improved decay of $\Upsilon (g;g_{\mathfrak {m}})$ throughout the iteration. Concretely, we modify $(\delta _g^*\Upsilon )_{\mu \nu }$ in (1.3) by a zeroth order term not involving derivatives of g to

$$\begin{aligned} (\delta _{g,E^\mathcal C}^*\Upsilon )_{\mu \nu }:= & {} {\tfrac{1}{2}}(\Upsilon _{\mu ;\nu }+\Upsilon _{\nu ;\mu }) + 2\gamma ^\mathcal C({\mathfrak {c}}_\mu \Upsilon _\nu +{\mathfrak {c}}_\nu \Upsilon _\mu ) - \gamma ^\mathcal C\Upsilon _\kappa {\mathfrak {c}}^\kappa g_{\mu \nu },\qquad \\ E^\mathcal C= & {} ({\mathfrak {c}},\gamma ^\mathcal C), \end{aligned}$$

where we take ${\mathfrak {c}}=r^{-1}\,{\mathrm d}t$ and $\gamma ^\mathcal C>0$. (This is similar to the definition of ${\tilde{\delta }}^*$ in [22, §3.3].) Usage of $\delta _{g,E^\mathcal C}^*$ instead of $\delta _g^*$ in the modified gauge-fixed Einstein vacuum equations

$$\begin{aligned} \textrm{Ric}(g) - \delta _{g,E^\mathcal C}^*\Upsilon (g;g_{\mathfrak {m}}) = 0 \end{aligned}$$

(1.4)

does not change the gauge in which one solves $\textrm{Ric}(g)=0$. However, it ensures, for general Cauchy data which may violate the constraint equations, that $\Upsilon$ satisfies a modified (here: damped) wave equation $\delta _g{\textsf{G}}_g\delta _{g,E^\mathcal C}^*\Upsilon (g;g_{\mathfrak {m}})=0$ by virtue of the second Bianchi identity. (Here, ${\textsf{G}}_g=1-{\tfrac{1}{2}}g{\text {tr}}_g$ is the trace reversal operator, and $\delta _g$ is the negative divergence.) On Minkowski space and with general Cauchy data, this ensures that $\Upsilon (g;g_{\mathfrak {m}})$ decays faster than $r^{-1}$ at null infinity, which concretely means that certain metric components—4 in number, matching the number of components of the 1-form $\Upsilon (g;g_{\mathfrak {m}})$—of $r^{-1}h=g-g_{\mathfrak {m}}$ (in fact the first three in (1.2), with $\Omega$ accounting for 2 components) similarly have stronger decay; notably, the component h(L, L) controls the deviation of outgoing light cones for the metric g from the Schwarzschildean ones. This improved decay and the resulting fixing of the geometry near null infinity allowed for an application of a global nonlinear iteration scheme for solving (1.4) in [22].

The new gauge we introduce here is a modification of $\Upsilon (g;g_{\mathfrak {m}})$ by a zeroth order term,

$$\begin{aligned} \Upsilon _{E^\Upsilon }(g;g_{\mathfrak {m}})_\mu := \Upsilon (g;g_{\mathfrak {m}})_\mu - 2\gamma ^\Upsilon {\mathfrak {c}}^\nu (g-g_{\mathfrak {m}})_{\mu \nu }; \end{aligned}$$

(1.5)

we again use ${\mathfrak {c}}=r^{-1}{\mathrm d}t$, and $\gamma ^\Upsilon <0$. (See Definitions 2.2 and 3.27, and Remark 2.3 for the duality of gauge modifications and constraint damping.) The gauge-fixed Einstein vacuum equations we shall solve in the proof of Theorem 1.1 are then

$$\begin{aligned} \textrm{Ric}(g) - \delta _{g,E^\mathcal C}^*\Upsilon _{E^\Upsilon }(g;g_{\mathfrak {m}}) = 0. \end{aligned}$$

(1.6)

The coefficients of $r^{-1}h=g-g_{\mathfrak {m}}$ which have improved decay by virtue of $\Upsilon _{E^\Upsilon }(g;g_{\mathfrak {m}})=0$ (or strong decay of $\Upsilon _{E^\Upsilon }(g;g_{\mathfrak {m}})$ at $\mathscr {I}^+$ due to constraint damping) are the same as in the formulation (1.4). On the other hand, upon combining the new gauge with the ungauged Einstein operator, 2 further components of h (the final two in (1.2)) have improved decay, and furthermore $h(\underline{L},\underline{L})$ does not diverge logarithmically anymore; see §3.6. We alert the reader to Appendix A where gauge changes and constraint damping of this sort are discussed in the context of the Maxwell equations; there, we also give a more conceptual explanation for why the gauge modification has the advertised effect.

We substantiate this discussion schematically in terms of the often used model for couplings and semilinear interactions for the Einstein vacuum equations in harmonic gauge,

$$\begin{aligned} \Box _g\phi _1=(\partial _t^2-\partial _x^2)\phi _1=0,\qquad \Box _g\phi _2=(\partial _t\phi _1)^2, \end{aligned}$$

with g the Minkowski metric (see [34, §1]). Here,

• (1) $\phi _1$ encodes gravitational radiation escaping to null infinity and corresponds to the trace-free spherical part of metric perturbations $r^{-1}h$ above;

• (2) $\phi _2\sim h(\underline{L},\underline{L})$ encodes the Bondi mass.

The $\mathcal O(r^{-1})$ decay of $\phi _1$ creates $\mathcal O(r^{-2})$ forcing for $\phi _2$, leading to the logarithmic divergence of $\phi _2=\mathcal O(r^{-1}\log r)$ at $\mathscr {I}^+$. We supplement this by two more equations,

$$\begin{aligned} \Box _g\phi _1 = 0,\qquad \Box _g\phi _2 = (\partial _t\phi _1)^2, \qquad \Box _g\phi _3=0,\qquad \Box _g\phi _4=0, \end{aligned}$$

(1.7)

where we ignore couplings at sub-leading order at $\mathscr {I}^+$. Here,

• (3) $\phi _3$ models the 4 metric coefficients whose leading order behavior at $\mathscr {I}^+$ is constrained by the wave coordinate condition, as discussed after (1.4);

• (4) $\phi _4$ models the remaining 3 metric coefficients which are affected only once one combines the new gauge with the ungauged Einstein equations, and which do not encode any leading order physical degrees of freedom at $\mathscr {I}^+$.

Constraint damping turns the equation for $\phi _3$ into a damped wave equation of the sort

$$\begin{aligned} (\Box _g + 2\gamma ^\mathcal Cr^{-1}\partial _t)\phi _3 = 0; \end{aligned}$$

this leads to $\phi _3=\mathcal O(r^{-1-\gamma ^\mathcal C})$ decay at null infinity. Since a main effect of constraint damping is of quasilinear nature (namely, it fixes the geometry near null infinity), a more precise model than (1.7) replaces all occurrences of g by “$g+\phi _3$”; this makes apparent the advantage of ensuring better decay for $\phi _3$.

The improvement afforded by the gauge change leads to the schematic model

$$\begin{aligned} \begin{aligned} \Box _{g+\phi _3}\phi _1&=0,&\quad (\Box _{g+\phi _3}-2\gamma ^\Upsilon r^{-1}\partial _t)\phi _2&= (\partial _t\phi _1)^2, \\ (\Box _{g+\phi _3}+2\gamma ^\mathcal Cr^{-1}\partial _t)\phi _3&=0,&\quad (\Box _{g+\phi _3}-2\gamma ^\Upsilon r^{-1}\partial _t)\phi _4&=0. \end{aligned} \end{aligned}$$

(1.8)

Thus, $\phi _3$ and $\phi _4$ have better-than-$r^{-1}$ decay at $\mathscr {I}^+$, and the $\mathcal O(r^{-2})$ forcing term for $\phi _2$ is no longer borderline, and hence $\phi _2=\mathcal O(r^{-1})$. This leaves $\phi _1,\phi _2$ as the only components with nontrivial radiation fields; the other components ($\phi _3$ and $\phi _4$) decay faster. The relationship between the model (1.8) and the gauge-fixed Einstein equations is further discussed at the end of §2, after Definition 3.20, and after the statement of Corollary 3.31.

1.2 Energy Estimates and Edge-b-Metrics

Our analysis here is based on energy estimates. The rough ‘background’ estimate uses the vector field multiplier

$$\begin{aligned} W=\Bigl (\frac{r}{r_*-t}\Bigr )^{2\alpha _{\!\mathscr {I}}}(r_*-t)^{2\alpha _0}\bigl (r L + (r_*-t)\partial _t\bigr ),\qquad L=\partial _t+\partial _{r_*}, \end{aligned}$$

for suitable weights $\alpha _0,\alpha _{\!\mathscr {I}}\in {\mathbb {R}}$; this is stronger than $\partial _t$ and weaker than the conformal Morawetz vector field, while still being compatible with the types of metric perturbations one encounters in the stability problem. Concretely, usage of W allows one to control the derivatives of the metric perturbation along

$$\begin{aligned}{} & {} r(\partial _t+\partial _{r_*}),\quad (r_*{-}t)(\partial _t{-}\partial _{r_*}) \ \text {(weighted approx. outgoing/incoming derivative)}, \nonumber \\{} & {} \Bigl (\frac{r_*-t}{r}\Bigr )^{1/2}\Omega \ \ (\text {spherical vector} \text {fields with } r^{-1/2} \text {decay at null infinity}), \end{aligned}$$

(1.9)

in a weighted spacetime $L^2$-space. (One can replace the incoming null vector field by the scaling vector field $t\partial _t+r_*\partial _{r_*}$.) Higher regularity is proved by commuting stronger vector fields (see Remark 3.9) through the equation; a minor simplification is that due to our strong background estimate we can relax the requirements on these commutator vector fields, cf. Lemma 3.16. By contrast, in [34], the background estimate is weaker than the edge-b-estimate, and thus the commutator vector fields need to be chosen more carefully, much as in [26].

Besides proving the improved asymptotics in the new gauge, a secondary goal of this paper is to contribute to the development of the global analytic point of view for nonelliptic PDE using techniques from geometric singular analysis. Concretely, as discussed in detail in §3.2, the Schwarzschild metric is a weighted edge-metric (with Lorentzian signature) at null infinity in the sense of Mazzeo [38]; see Corollary 3.14. Indeed, pulling back $g_{\mathfrak {m}}$ to the interior of

$$\begin{aligned} {\mathbb {R}}_{t_*} \times [0,\infty )_{x_{\!\mathscr {I}}} \times {\mathbb {S}}^2,\qquad t_*=t-r_*,\quad x_{\!\mathscr {I}}=r^{-1/2}, \end{aligned}$$

(1.10)

with $x_{\!\mathscr {I}}^{-1}(0)$ being (the interior of) null infinity, one computes

where $x^2,x^3$ are local coordinates on ${\mathbb {S}}^2$. Dual to the 1-forms ${\mathrm d}t_*$, $\tfrac{{\mathrm d}x_{\!\mathscr {I}}}{x_{\!\mathscr {I}}}$, $\tfrac{{\mathrm d}x^a}{x_{\!\mathscr {I}}}$ appearing here are the vector fields $\partial _{t_*}$, $x_{\!\mathscr {I}}\partial _{x_{\!\mathscr {I}}}$, $x_{\!\mathscr {I}}\partial _{x^a}$, which are precisely those smooth vector fields on the manifold (1.10) which are tangent to the fibers of the fibration $(t_*,\omega )\mapsto \omega$ of the boundary $x_{\!\mathscr {I}}^{-1}(0)$; and linear combinations of these vector fields are precisely those listed in (1.9). The compactification of the domain $\Omega$ in Theorem 1.1, as shown on the right in Fig. 1, has a second boundary hypersurface $I^0$ where $g_{\mathfrak {m}}$ is a weighted b-metric [39]; globally, $g_{\mathfrak {m}}$ is a weighted edge-b-metric, or eb-metric for short. This observation is key for the streamlining of the functional analytic setup in the present paper as compared to [22].⁴

Fig. 1

Illustration of Theorem 1.1, on the left in a Penrose-diagrammatic fashion, and on the right in the blow-up of the Penrose diagram at spacelike infinity $i^0$

The metric perturbations arising in Theorem 1.1 are lower order perturbations of $g_{\mathfrak {m}}$ as symmetric edge-b-2-tensors; see Lemma 3.23. Thus, regularity with respect to the vector fields (1.9) is a very natural notion. Unlike in Riemannian geometry, there are typically many different types of rescaled vector bundles and boundary fibration structures with respect to which a given Lorentzian metric is nondegenerate down to boundaries at infinity, such as $x_{\!\mathscr {I}}^{-1}(0)$ in (1.10). And indeed, while the edge-b point of view is convenient for the purpose of proving estimates, controlling the geometry of metric perturbations is more conveniently done in terms of the standard vector fields $\partial _{z^\mu }$ on ${\mathbb {R}}^4$ or linear combinations thereof as used in (1.2) and discussed in detail around Definition 3.10; cf. the significance of h(L, L) for controlling outgoing light cones. Since the Einstein equations are quasilinear, it is important to understand the relationship between the two points of view (Lemma 3.13).

1.3 Structure of the Paper

In §2, we present calculations for the linearized gauge-fixed Einstein vacuum equations on Minkowski space, with constraint damping and the (linearization of the) novel gauge, which lend support to the claims made in §1.1. In §3, we prove Theorem 1.1. We first introduce edge-b-structures in §3.1; the partial compactification of the spacetime on which we shall work and a basic edge-b-energy estimate are presented in §3.2. The stability proof starts in §3.3 where we define the class of metric perturbations arising in the stability problem in our new gauge. In §3.4, we define the modified gauge-fixed Einstein operator and describe its structure as an edge-b-differential operator. This is used in §3.5 to prove (tame) energy estimates and in §3.6 to obtain sharp decay for metric perturbations using a Nash–Moser iteration. Appendix A illustrates the choice of gauge and constraint damping in the simpler setting of the Maxwell equations.

2 Motivation; Calculations on Minkowski Space

Fix a background metric $g^0$. Then the operator

$$\begin{aligned} P(g) = \textrm{Ric}(g) - \delta _g^*\Upsilon (g;g^0),\qquad \Upsilon (g;g^0)=g(g^0)^{-1}\delta _g{\textsf{G}}_g g^0, \end{aligned}$$

(2.1)

is a quasilinear wave operator when g is a Lorentzian metric; that is, its linearization is principally scalar, and its principal part is equal to ${\tfrac{1}{2}}\Box _g$ (see below).

Lemma 2.1

(Linearizations) The linearization of the Ricci tensor is

$$\begin{aligned} D_g\textrm{Ric}= & {} {\tfrac{1}{2}}\Box _g - \delta _g^*\delta _g{\textsf{G}}_g + \mathscr {R}_g,\quad (\mathscr {R}_g u)_{\mu \nu }=R^\kappa {}_{\mu \nu }{}^\lambda u_{\kappa \lambda }\nonumber \\ {}{} & {} \quad +{\tfrac{1}{2}}(\textrm{Ric}(g)_\mu {}^\kappa u_{\kappa \nu }+\textrm{Ric}(g)^\kappa {}_\nu u_{\mu \kappa }), \end{aligned}$$

(2.2)

where R is the Riemann curvature tensor of g, and $\Box _g=-{\text {tr}}_g\nabla ^2$. Moreover,

$$\begin{aligned}{} & {} \Upsilon (g;g^0)_\mu =g_{\mu \nu }g^{\kappa \lambda }(\Gamma (g)_{\kappa \lambda }^\nu -\Gamma (g^0)_{\kappa \lambda }^\nu ), \nonumber \\{} & {} D_g\Upsilon (-;g^0) = -\delta _g{\textsf{G}}_g - \mathscr {C}_g + \mathscr {Y}_g, \nonumber \\{} & {} \mathscr {C}_g(u)_\kappa = g_{\kappa \lambda }C_{\mu \nu }^\lambda u^{\mu \nu },\quad C_{\mu \nu }^\lambda = \Gamma (g)_{\mu \nu }^\lambda - \Gamma (g^0)_{\mu \nu }^\lambda ,\qquad \nonumber \\{} & {} \mathscr {Y}_g(u)_\kappa = \Upsilon (g;g^0)^\lambda u_{\kappa \lambda }. \end{aligned}$$

(2.3)

Proof

See [18, 22, §3.3]. $\quad \square$

In the linearization of P around $g=g^0$, given by $D_g P = D_g\textrm{Ric}+ \delta _g^*\delta _g{\textsf{G}}_g$, we now generalize $\delta _g^*$, resp. $\delta _g$, for the purpose of (linearized) constraint damping, resp. gauge change, as follows:

Definition 2.2

(Modifications) Let $E^\mathcal C=({\mathfrak {c}}^\mathcal C,\gamma ^\mathcal C)$, where ${\mathfrak {c}}^\mathcal C$ is a 1-form on spacetime, and $\gamma ^\mathcal C\in {\mathbb {R}}$. The modified symmetric gradient is then defined as

$$\begin{aligned} \delta _{g,E^\mathcal C}^*:= \delta _g^* + \gamma ^\mathcal C\bigl (2{\mathfrak {c}}^\mathcal C\otimes _s(-) - g \iota _{g^{-1}({\mathfrak {c}}^\mathcal C)}\bigr ). \end{aligned}$$

(2.4)

For a pair $E^\Upsilon =({\mathfrak {c}}^\Upsilon ,\gamma ^\Upsilon )$, we define the modified divergence by

$$\begin{aligned} \delta _{g,E^\Upsilon } = (\delta _{g,E^\Upsilon }^*)^* = \delta _g + \gamma ^\Upsilon \bigl (2\iota _{g^{-1}({\mathfrak {c}}^\Upsilon )} - {\mathfrak {c}}^\Upsilon {\text {tr}}_g\bigr ). \end{aligned}$$

(2.5)

Finally, the linearized modified gauge-fixed Einstein operator is

$$\begin{aligned} P'_{g,E^\mathcal C,E^\Upsilon }:= D_g\textrm{Ric}+ \delta _{g,E^\mathcal C}^*\delta _{g,E^\Upsilon }{\textsf{G}}_g. \end{aligned}$$

(2.6)

Consider now the Minkowski metric

(2.7)

In this section, we study the asymptotic behavior of solutions of $P'_{\underline{g},E^\mathcal C,E^\Upsilon }(r^{-1}u)=0$ at null infinity $\mathscr {I}^+$, i.e. for bounded $x^1$ when $x^0\rightarrow \infty$. In this region, we fix

$$\begin{aligned} {\mathfrak {c}}^\mathcal C={\mathfrak {c}}^\Upsilon =r^{-1}\,{\mathrm d}t,\qquad \gamma ^\mathcal C\in (0,1),\quad \gamma ^\Upsilon \in (-1,0). \end{aligned}$$

(2.8)

We work with the bundle splittings

(2.9)

Here and in the rest of the paper, slashed quantities and operators are those on the unit sphere ${\mathbb {S}}^2$. Thus, we rescale the spherical part of the cotangent bundle, recording e.g. the covector with as . We shall only record the ‘main’ terms of

$$\begin{aligned} 2 P'_{\underline{g},E^\mathcal C,E^\Upsilon } = \Box _{\underline{g}} + 2(\delta _{\underline{g},E^\mathcal C}^*\delta _{\underline{g},E^\Upsilon }-\delta _{\underline{g}}^*\delta _{\underline{g}}){\textsf{G}}_{\underline{g}} + 2\mathscr {R}_{\underline{g}} \end{aligned}$$

and drop all ‘error’ terms (writing ‘$\equiv$’ for an equality up to error terms). Concretely, we assign the weights 1, $-1$, 0, 0 to r, $\partial _0$, $\partial _1$, $\mathcal V({\mathbb {S}}^2)$ (thus regarding $r\partial _0\sim r(\partial _t+\partial _r)$ $\partial _1\sim \partial _t-\partial _r$, $\mathcal V({\mathbb {S}}^2)$ as unweighted vector fields), and only record terms of total weight $\le 0$. In the proof of Proposition 3.29, we shall find $r^2\Box _{\underline{g}} r^{-1} \equiv 4 \partial _1 r\partial _0$ and expressions for $\delta _{\underline{g}}$, $\delta _{\underline{g}}^*$, and ${\textsf{G}}_{\underline{g}}$ (the first terms in (3.39), (3.42), and (3.40), respectively), and for $\delta _{\underline{g},E^\mathcal C}^*-\delta _{\underline{g}}^*$, resp. $\delta _{\underline{g},E^\Upsilon }-\delta _{\underline{g}}$ (the first terms in (3.36), resp. (3.43)). They give

$$\begin{aligned} L_{\underline{g},E^\mathcal C,E^\Upsilon }:= 2 r^2 P'_{\underline{g},E^\mathcal C,E^\Upsilon } r^{-1} \equiv 2\partial _1(2 r\partial _0+A_{E^\mathcal C,E^\Upsilon }), \end{aligned}$$

(2.10)

where the endomorphism $A_{E^\mathcal C,E^\Upsilon }$ of $S^2 T^*{\mathbb {R}}^4$ is given by

(2.11)

Passing to $\rho _{\!\mathscr {I}}=(x^0)^{-1}$ (which, in the region of bounded $x^1=t-r$, vanishes at future null infinity $\mathscr {I}^+$), we note that $2 r=x^0-x^1=\rho _{\!\mathscr {I}}^{-1}(1-\rho _{\!\mathscr {I}} x^1)$ and $\partial _0=-\rho _{\!\mathscr {I}}^2\partial _{\rho _{\!\mathscr {I}}}$; thus,

$$\begin{aligned} r^2 P'_{\underline{g},E^\mathcal C,E^\Upsilon } r^{-1}\equiv -2\partial _1(\rho _{\!\mathscr {I}}\partial _{\rho _{\!\mathscr {I}}}-A_{E^\mathcal C,E^\Upsilon }). \end{aligned}$$

By standard regular-singular ODE analysis (and as previously shown rigorously in [22, §3.3]), we can read off the decay at $\mathscr {I}^+$ of a metric perturbation u solving

$$\begin{aligned} P'_{\underline{g},E^\mathcal C,E^\Upsilon }u=0 \end{aligned}$$

(2.12)

from the spectral decomposition of $A_{E^\mathcal C,E^\Upsilon }$.

Remark 2.3

(Duality of constraint damping and gauge change) By (2.2), the adjoint of $D_g\textrm{Ric}$ is $(D_g\textrm{Ric})^*={\textsf{G}}_g\circ D_g\textrm{Ric}\circ {\textsf{G}}_g$ (i.e. ${\textsf{G}}_g\circ D_g\textrm{Ric}$ is formally self-adjoint); thus,

$$\begin{aligned} {\textsf{G}}_g (P'_{g,E^\mathcal C,E^\Upsilon })^*{\textsf{G}}_g = P'_{g,E^\Upsilon ,E^\mathcal C}, \end{aligned}$$

(2.13)

demonstrating a duality between constraint damping and gauge changes. Equation (2.13) also implies ${\textsf{G}}_{\underline{g}} A_{E^\mathcal C,E^\Upsilon }^*{\textsf{G}}_{\underline{g}}=-A_{E^\Upsilon ,E^\mathcal C}$. Since we would like as many eigenvalues as possible of $A_{E^\mathcal C,E^\Upsilon }$ to be positive, this suggests taking $\gamma ^\mathcal C$ and $\gamma ^\Upsilon$ to have opposite signs. In view of (2.13), this forces the endomorphism $A_{E^\Upsilon ,E^\mathcal C}$ corresponding to $(P'_{g,E^\mathcal C,E^\Upsilon })^*$ to have many negative eigenvalues. See Appendix A for a discussion of this point in a simpler context.

To study $A_{E^\mathcal C,E^\Upsilon }$, we introduce the bundle projections (respecting the splitting (2.9))

(2.14)

Then $\pi ^\mathcal CA_{E^\mathcal C,E^\Upsilon }(1-\pi ^\mathcal C)=0$; in the splitting ${\text {ran}}\pi ^\mathcal C\oplus {\text {ran}}(1-\pi ^\mathcal C)$, the top left block of $A_{E^\mathcal C,E^\Upsilon }$ (capturing rows and columns 1, 3, 6) is then

$$\begin{aligned} \pi ^\mathcal CA_{E^\mathcal C,E^\Upsilon }\pi ^\mathcal C= \begin{pmatrix} 2\gamma ^\mathcal C&{} \quad 0 &{} \quad 0 \\ 0 &{} \quad \gamma ^\mathcal C&{} \quad 0 \\ 2\gamma ^\mathcal C&{} \quad 0 &{} \quad \gamma ^\mathcal C\end{pmatrix} \end{aligned}$$

(2.15a)

Thus, $\pi ^\mathcal Cu$ is expected to have components of size $\mathcal O(\rho _{\!\mathscr {I}}^{1+\lambda })$ at $\mathscr {I}^+$, where $\lambda =\gamma ^\mathcal C,2\gamma ^\mathcal C$. Next, the bottom right block (capturing rows 2, 4, 5, 7) is

$$\begin{aligned} (1-\pi ^\mathcal C)A_{E^\mathcal C,E^\Upsilon }(1-\pi ^\mathcal C) = \begin{pmatrix} -\gamma ^\Upsilon &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ -2\gamma ^\Upsilon &{} \quad -2\gamma ^\Upsilon &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad -\gamma ^\Upsilon &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix}, \end{aligned}$$

(2.15b)

with eigenvalues $-\gamma ^\Upsilon$, $-2\gamma ^\Upsilon$, and 0. It is thus natural to further split off the trace-free spherical part (the final row) using , with . In §3.4, we will see that when solving the nonlinear Einstein equations via an iteration scheme, the part will be a source term for the (1, 1) component in the subsequent iteration step; this is why we further split the bundle ${\text {ran}}(\pi ^\Upsilon +\pi _{1 1})$ into the ranges of $\pi ^\Upsilon$ (rows 2 and 5 of $A_{E^\mathcal C,E^\Upsilon }$), $\pi _{1 1}$ (row 4).

Altogether then, the solution u of (2.12) can be analyzed step by step for bounded $t-r$ as follows (we omit error terms throughout):

• (1) $\pi ^\mathcal Cu$ satisfies a decoupled equation (to leading order), thus $\pi ^\mathcal Cu=\mathcal O(\rho _{\!\mathscr {I}}^{1+\gamma ^\mathcal C})$.

• (2) $\pi ^\Upsilon u$ satisfies an equation with source terms given by $\pi ^\mathcal Cu$. Choosing our parameters so that $-\gamma ^\Upsilon <\gamma ^\mathcal C$, we then have $\pi _\Upsilon u=\mathcal O(\rho _{\!\mathscr {I}}^{1-\gamma ^\Upsilon })$.

• (3) has a radiation field, i.e. a leading order term of size $\mathcal O(\rho _{\!\mathscr {I}})=\mathcal O(r^{-1})$, and has lower order terms of size $\rho _{\!\mathscr {I}}^{1-\gamma ^\Upsilon }$ from coupling to the remaining metric components.

• (4) $\pi _{1 1} u$ has the same decay as $\pi ^\Upsilon u$.

These improved decay rates (compared to the $\mathcal O(\rho _{\!\mathscr {I}})$ decay of typical scalar waves on Minkowski space) will persist for the nonlinear gauge-fixed Einstein vacuum equations, except for the decay of $\pi _{1 1}u$ (which is replaced by a $\mathcal O(\rho _{\!\mathscr {I}})$-leading order term), as already indicated before. In terms of our model (1.8), thus correspond to $\phi _1,\phi _2,\phi _3,\phi _4$, respectively. Leaving the model (1.8) behind, one can simplify the above scheme by solving at once for $(\pi ^\mathcal C+\pi ^\Upsilon )u$, which in itself satisfies a decoupled equation leading to improved decay. This is the path we will take in §3.6.

3 Nonlinear Stability

3.1 Differential Operators and Function Spaces

Consider an n-dimensional manifold M with corners which has exactly two embedded boundary hypersurfaces $H_0$, $H_1$. Assume furthermore that $H_1$ is equipped with a fibration $\phi :H_1\rightarrow Y$ with a typical fiber F.

Definition 3.1

(b- and edge-b-vector fields)

• (1) The Lie algebra $\mathcal V_{\textrm{b}}(M)\subset \mathcal V(M)$ of b-vector fields [39] consists of all smooth vector fields $V\in \mathcal V(M)$ which are tangent to $H_0$ and $H_1$.

• (2) The Lie algebra $\mathcal V_{\textrm{e,b}}(M)\subset \mathcal V_{\textrm{b}}(M)$ of edge-b-vector fields consists of all b-vector fields $V\in \mathcal V_{\textrm{b}}(M)$ for which $V|_{H_1}\in \mathcal V_{\textrm{b}}(H_1)$ is tangent to the fibers of $H_1\rightarrow Y$.

On manifolds with a single embedded boundary hypersurface, edge vector fields were introduced by Mazzeo [38]. See [2] for iterated structures giving rise to generalizations of $\mathcal V_{\textrm{e,b}}(M)$.

We discuss here only the case of interest for us: $H_0$ and $H_1$ have nonempty intersection, and a neighborhood of $H_0\cap H_1\subset M$ is diffeomorphic to

$$\begin{aligned} {[}0,\infty )_{\rho _0} \times [0,\infty )_{\rho _1} \times {\mathbb {R}}^{n-2}_y,\qquad y=(y^2,\ldots ,y^{n-1}), \end{aligned}$$

(3.1)

where $\rho _0,\rho _1$ are defining functions of $H_0,H_1$, respectively, and with the fibration of $H_1$ given by $(\rho _0,y)\mapsto y$; thus, the fibers F are 1-dimensional. In this case, elements of $\mathcal V_{\textrm{b}}(M)$, resp. $\mathcal V_{\textrm{e,b}}(M)$ are linear combinations, with $\mathcal C^\infty (M)$ coefficients, of

$$\begin{aligned} \rho _0\partial _{\rho _0},\ \rho _1\partial _{\rho _1},\ \partial _{y^2},\dots ,\partial _{y^{n-1}},\qquad \text {resp.}\qquad \rho _0\partial _{\rho _0},\ \rho _1\partial _{\rho _1},\ \rho _1\partial _{y^2},\dots ,\rho _1\partial _{y^{n-1}}.\nonumber \\ \end{aligned}$$

(3.2)

The b-tangent bundle and eb-tangent bundle

$$\begin{aligned} {}^{{\textrm{b}}}TM\rightarrow M,\qquad {}^{{\textrm{e,b}}}TM\rightarrow M, \end{aligned}$$

are then the rank n vector bundles with local frames given by the respective sets of vector fields (3.2); over the interior $M^\circ$, these are naturally isomorphic to the standard tangent bundle. By continuous extension from $M^\circ$, one can thus regard smooth sections of ${}^{{\textrm{b}}}TM$ as vector fields on M, and in this sense, we have $\mathcal V_{\textrm{b}}(M)=\mathcal C^\infty (M;{}^{{\textrm{b}}}TM)$, likewise $\mathcal V_{\textrm{e,b}}(M)=\mathcal C^\infty (M;{}^{{\textrm{e,b}}}TM)$.⁵ The dual bundles ${}^{{\textrm{b}}}T^*M\rightarrow M$ and ${}^{{\textrm{e,b}}}T^*M\rightarrow M$ are called b-cotangent bundle and eb-cotangent bundle, respectively. Their smooth sections are linear combinations with $\mathcal C^\infty (M)$ coefficients of

$$\begin{aligned} \frac{{\mathrm d}\rho _0}{\rho _0},\ \frac{{\mathrm d}\rho _1}{\rho _1},\ {\mathrm d}y^2,\ldots ,{\mathrm d}y^{n-1}, \qquad \text {resp.}\qquad \frac{{\mathrm d}\rho _0}{\rho _0},\ \frac{{\mathrm d}\rho _1}{\rho _1},\ \frac{{\mathrm d}y^2}{\rho _1},\ldots ,\frac{{\mathrm d}y^{n-1}}{\rho _1}. \end{aligned}$$

Definition 3.2

(b- and eb-differential operators) Let $k\in {\mathbb {N}}_0$, $\alpha _0,\alpha _1\in {\mathbb {R}}$.

• (1) The space $\textrm{Diff}_{\textrm{b}}^{k,(\alpha _0,\alpha _1)}(M)=\rho _0^{-\alpha _0}\rho _1^{-\alpha _1}\textrm{Diff}_{\textrm{b}}^k(M)$ consists of all differential operators P on $M^\circ$ of the form $P=\rho _0^{-\alpha _0}\rho _1^{-\alpha _1}P_0$, where $P_0\in \textrm{Diff}_{\textrm{b}}^k(M)$ is a locally finite sum of compositions of up to k b-vector fields.

• (2) The space $\textrm{Diff}_{\textrm{e,b}}^{k,(\alpha _0,\alpha _1)}(M)=\rho _0^{-\alpha _0}\rho _1^{-\alpha _1}\textrm{Diff}_{\textrm{e,b}}^k(M)$ is defined analogously, with eb-vector fields replacing b-vector fields.

• (3) The space $\textrm{Diff}_{{\textrm{e,b}};{\textrm{b}}}^{m;k}(M)$ consists of all locally finite sums of operators of the form $P^\flat P^\sharp$ where $P^\flat \in \textrm{Diff}_{\textrm{e,b}}^m(M)$, $P^\sharp \in \textrm{Diff}_{\textrm{b}}^k(M)$.

Assume for the moment that M is compact. Fix a smooth positive b-density on M; in local coordinates as above, this takes the form $a(\rho _0,\rho _1,y)|\frac{{\mathrm d}\rho _0}{\rho _0}\frac{{\mathrm d}\rho _1}{\rho _1}{\mathrm d}y|$ with $a>0$ smooth. We then denote the $L^2$ space on M by $L^2_{\textrm{b}}(M)$.

Definition 3.3

(b- and eb-Sobolev spaces) Let $k\in {\mathbb {N}}_0$, $\alpha _0,\alpha _1\in {\mathbb {R}}$.

• (1) The weighted b-Sobolev space

  $$\begin{aligned} H_{{\textrm{b}}}^{k,(\alpha _0,\alpha _1)}(M) = \rho _0^{\alpha _0}\rho _1^{\alpha _1}H_{{\textrm{b}}}^k(M) \end{aligned}$$

  consists of all functions u of the form $u=\rho _0^{\alpha _0}\rho _1^{\alpha _1}u_0$ with $u_0\in L^2_{\textrm{b}}(M)$ and $P u_0\in L^2_{\textrm{b}}(M)$ for all $P\in \textrm{Diff}_{\textrm{b}}^k(M)$. Equivalently, $P u\in L^2_{\textrm{b}}(M)$ for all $P\in \textrm{Diff}_{\textrm{b}}^{k,(\alpha _0,\alpha _1)}(M)$.

• (2) The weighted eb-Sobolev space $H_{{\textrm{e,b}}}^{k,(\alpha _0,\alpha _1)}(M)=\rho _0^{\alpha _0}\rho _1^{\alpha _1}H_{{\textrm{e,b}}}^k(M)$ is defined analogously, with $\textrm{Diff}_{\textrm{e,b}}$ replacing $\textrm{Diff}_{\textrm{b}}$.⁶

• (3) Let $m\in {\mathbb {N}}_0$. The mixed eb-b-Sobolev space $H_{{\textrm{e,b}};{\textrm{b}}}^{(m;k),(\alpha _0,\alpha _1)}(M) = \rho _0^{\alpha _0}\rho _1^{\alpha _1}H_{{\textrm{e,b}};{\textrm{b}}}^{(m;k)}(M)$ consists of all functions u for which $P u\in H_{{\textrm{e,b}}}^m(M)$ for all $P\in \textrm{Diff}_{\textrm{b}}^{k,(\alpha _0,\alpha _1)}(M)$. (Equivalently, $P u\in L^2_{\textrm{b}}(M)$ for all $P\in \rho _0^{-\alpha _0}\rho _1^{-\alpha _1}\textrm{Diff}_{{\textrm{e,b}};{\textrm{b}}}^{m;k}(M)$.)

All these spaces can be given the structure of Hilbert spaces; for instance, we can equip $H_{{\textrm{e,b}}}^1(M)$ with the squared norm $\Vert u\Vert _{H_{{\textrm{e,b}}}^1}^2=\Vert u\Vert _{L^2_{\textrm{b}}}^2+\sum \Vert V_i u\Vert _{L^2_{\textrm{b}}}^2$, where $\{V_i\}$ is a finite set of edge-b-vector fields spanning $\mathcal V_{\textrm{e,b}}(M)$ over $\mathcal C^\infty (M)$. We also note the $L^\infty$ estimate

$$\begin{aligned} H_{{\textrm{b}}}^{s,(\alpha _0,\alpha _1)}(M) \hookrightarrow \rho _0^{\alpha _0}\rho _1^{\alpha _1}L^\infty (M),\qquad s>\frac{n}{2}. \end{aligned}$$

(3.3)

This follows from the standard Sobolev embedding after the change of variables $z_0=\log \rho _0$, $z_1=\log \rho _1$, which transforms $\rho _j\partial _{\rho _j}$ into $\partial _{z_j}$ and the b-density $|\tfrac{{\mathrm d}\rho _0}{\rho _0}\tfrac{{\mathrm d}\rho _1}{\rho _1}{\mathrm d}y|$ into $|{\mathrm d}z_0\,{\mathrm d}z_1\,{\mathrm d}y|$.

If M is a manifold with boundary and $\rho \in \mathcal C^\infty (M)$ denotes a boundary defining function, then $H_{{\textrm{b}}}^{k,\alpha }(M)=\rho ^\alpha H_{{\textrm{b}}}^k(M)$ is defined completely analogously (with respect to a smooth b-density). In the setting of Definition 3.3, this allows us to define spaces such as $H_{{\textrm{b}}}^{k,\alpha }(H_1)$.

Lastly, if M is noncompact and equipped with a smooth positive b-density, the spaces $H_{{\textrm{b}},{\textrm{loc}}}^k(M)$ consist of distributions which upon multiplication with elements of $\mathcal C^\infty _{\textrm{c}}(M)$ lie in $L^2_{\textrm{b}}(M)$ together with all their derivatives along all $P\in \textrm{Diff}_{\textrm{b}}^k(M)$; weighted spaces, eb-Sobolev spaces, and mixed edge-b;b-Sobolev spaces are defined analogously. If $\Omega \Subset M$ is an open set with compact closure, then we define

$$\begin{aligned} H_{\textrm{e,b}}^k(\Omega ) = \{ u|_\Omega :u\in H_{{\textrm{e,b}},{\textrm{loc}}}^k(M) \}; \end{aligned}$$

it can be given the structure of a Hilbert space as before. We analogously define

$$\begin{aligned} H_{{\textrm{e,b}};{\textrm{b}}}^{(m;k),(\alpha _0,\alpha _1)}(\Omega ). \end{aligned}$$

Finally, we introduce the following general notation:

Definition 3.4

(Operators with generalized coefficients) If $\mathcal F\subset \mathscr {D}'(M^\circ )$ is a linear subspace of the space of distributions on $M^\circ$ and $\mathcal D$ denotes a space of differential operators on M with smooth coefficients, then $\mathcal F\mathcal D$ is the space of all operators $\mathcal C^\infty (M^\circ )\rightarrow \mathscr {D}'(M^\circ )$ of locally finite linear combinations $\sum a_i P_i$, where $a_i\in \mathcal F$ and $P_i\in \mathcal D$.

Examples of interest in the present paper are spaces such as $H_{{\textrm{b}}}^{k,(\alpha _0,\alpha _1)}\textrm{Diff}_{\textrm{e,b}}^2(M)$.

3.2 Spacetime Manifold; Basic Energy Estimate

Definition 3.5

(Schwarzschild spacetime) Let ${\mathfrak {m}}\in {\mathbb {R}}$. The Schwarzschild spacetime is

The Regge–Wheeler tortoise coordinate $r_*$ and the null coordinates $x^0,x^1$ are defined by

$$\begin{aligned} r_*:= r + 2{\mathfrak {m}}\log (r-2{\mathfrak {m}}),\qquad x^0=t+r_*,\quad x^1=t-r_*. \end{aligned}$$

(3.4)

Lemma 3.6

(Compactification of the far field of the Schwarzschild spacetime) Define

$$\begin{aligned} \rho _0:= \frac{1}{r_*-t},\quad \rho _{\!\mathscr {I}}:= \frac{r_*-t}{r},\quad x_{\!\mathscr {I}}:=\rho _{\!\mathscr {I}}^{1/2},\qquad \rho :=\rho _0\rho _{\!\mathscr {I}}=r^{-1}. \end{aligned}$$

(3.5)

• (1) Put $\bar{x}_{\!\mathscr {I}}:=\min \bigl (\sqrt{\frac{3}{2}},\tfrac{1}{\sqrt{8{\mathfrak {m}}}}\bigr )$ when ${\mathfrak {m}}>0$ and ${\bar{x}}_{\!\mathscr {I}}:=\sqrt{3/2}$ when ${\mathfrak {m}}\le 0$. Then the map $(\rho _0,x_{\!\mathscr {I}},\omega )\rightarrow (t,r_*,\omega )$ with domain $M^\circ =(0,2)_{\rho _0}\times (0,\bar{x}_{\!\mathscr {I}})_{x_{\!\mathscr {I}}}\times {\mathbb {S}}^2$, where

  $$\begin{aligned} M:= [0,2)_{\rho _0}\times [0,{\bar{x}}_{\!\mathscr {I}})_{x_{\!\mathscr {I}}}\times {\mathbb {S}}^2, \end{aligned}$$

  (3.6)

  is a diffeomorphism onto its image.

• (2) Denoting the pullback of $g_{\mathfrak {m}}$ to M by $g_{\mathfrak {m}}$ still, the hypersurface $x_{\!\mathscr {I}}^{-1}(c)\subset M$ is spacelike for all $c\in (0,{\bar{x}}_{\!\mathscr {I}})$, and the hypersurface $\rho _0^{-1}(c)$ is lightlike for all $c\in (0,1)$.

Proof

We have $r=\rho _0^{-1}\rho _{\!\mathscr {I}}^{-1}=\rho _0^{-1}x_{\!\mathscr {I}}^{-2}>{\tfrac{1}{2}}x_{\!\mathscr {I}}^{-2}>4{\mathfrak {m}}$, hence $r_*=r+2{\mathfrak {m}}\log (r-2{\mathfrak {m}})$ is well-defined, and we then have $t=r_*-\rho _0^{-1}$. This proves the first part. For the second part, we record that in the coordinates (3.4), the Schwarzschild metric reads as

(3.7)

Thus, $\rho _0^{-2}\,{\mathrm d}\rho _0=-{\mathrm d}(r_*-t)={\mathrm d}x^1$ is null indeed. Furthermore,

$$\begin{aligned} r\,{\mathrm d}\rho _{\!\mathscr {I}}&= -r\,{\mathrm d}\Bigl (\frac{x^1}{r}\Bigr ) = -{\mathrm d}x^1+\frac{x^1}{r}{\mathrm d}r = -{\mathrm d}x^1+\frac{x^1}{r}\Bigl (1-\frac{2{\mathfrak {m}}}{r}\Bigr ){\mathrm d}r_* \\&= \frac{x^1}{2 r}\Bigl (1-\frac{2{\mathfrak {m}}}{r}\Bigr ){\mathrm d}x^0 - \Bigl (1+\frac{x^1}{2 r}\Bigl (1-\frac{2{\mathfrak {m}}}{r}\Bigr )\Bigr ){\mathrm d}x^1, \end{aligned}$$

the inner product of which with itself is

$$\begin{aligned} g_{\mathfrak {m}}^{-1}(r\,{\mathrm d}\rho _{\!\mathscr {I}},r\,{\mathrm d}\rho _{\!\mathscr {I}}) = \frac{2 x^1}{r}\Bigl (1+\frac{x^1}{2 r}\Bigl (1-\frac{2{\mathfrak {m}}}{r}\Bigr )\Bigr ) < 0. \end{aligned}$$

(3.8)

Indeed, $\tfrac{x^1}{r}=-x_{\!\mathscr {I}}^2<0$; and the second factor is positive, too, since $|\tfrac{x^1}{2 r}|={\tfrac{1}{2}}x_{\!\mathscr {I}}^2<\tfrac{3}{4}<1$. $\quad \square$

The upper bound $\rho _0<2$ can be increased arbitrarily; choosing a larger upper merely places a stronger restriction on $\bar{x}_{\!\mathscr {I}}$.⁷

Definition 3.7

(Ideal boundaries) The boundary hypersurfaces of the spacetime manifold M defined by (3.6) are denoted $I^0:=\rho _0^{-1}(0)$ (blown-up spacelike infinity) and $\mathscr {I}^+:= x_{\!\mathscr {I}}^{-1}(0)$ (null infinity). Moreover, $\mathscr {I}^+$ is fibered by the projection $\mathscr {I}^+=[0,\infty )_{\rho _0}\times {\mathbb {S}}^2\rightarrow {\mathbb {S}}^2$.

In the context of Definition 3.1, the boundary hypersurfaces of M are $H_0:=I^0$ and $H_1:=\mathscr {I}^+$, with defining functions $\rho _0$ and $\rho _1:=x_{\!\mathscr {I}}=\rho _{\!\mathscr {I}}^{1/2}$, respectively. Thus, the space $\mathcal V_{\textrm{e,b}}(M)$ is spanned over $\mathcal C^\infty (M)$ by

$$\begin{aligned} \rho _0\partial _{\rho _0},\qquad x_{\!\mathscr {I}}\partial _{x_{\!\mathscr {I}}} = 2\rho _{\!\mathscr {I}}\partial _{\rho _{\!\mathscr {I}}},\qquad x_{\!\mathscr {I}}\Omega = \rho _{\!\mathscr {I}}^{1/2}\Omega , \end{aligned}$$

(3.9)

where $\Omega$ ranges over all vector fields on ${\mathbb {S}}^2$. (These are, roughly, the spacetime scaling vector field, the weighted outgoing null vector field, and weighted spherical vector field.)

Remark 3.8

(Comparison of function spaces) The Sobolev space $H_\mathscr {I}^1$ of [22, Definition 4.1] is the same as $H_{{\textrm{e,b}}}^1(M)$ (upon restricting to functions with compact support in M) in view of (3.9); moreover $H_{\mathscr {I},{\textrm{b}}}^{1,k}=H_{{\textrm{e,b}};{\textrm{b}}}^{(1,k)}(M)$. Moreover, in the notation of [22], if we adjoin $\rho _{\!\mathscr {I}}^{1/2}=x_{\!\mathscr {I}}$ to the smooth structure of the spacetime manifold M there, smooth sections of the bundle $S^2\,{}^\beta T M+\rho _{\!\mathscr {I}} S^2\,{}^{{\textrm{b}}}TM$ in [22, Equation (4.17)] are the same as smooth sections of $S^2\,{}^{{\textrm{e,b}}}TM$ for the manifold M in (3.6).

For later use, we compute

$$\begin{aligned}&\begin{aligned} \partial _0\equiv \partial _{x^0}&= \frac{1}{2}(\partial _t+\partial _{r_*}) = -\frac{1}{2}\rho _0\rho _{\!\mathscr {I}}^2\Bigl (1-\frac{2{\mathfrak {m}}}{r}\Bigr )\partial _{\rho _{\!\mathscr {I}}} \in \rho _0\rho _{\!\mathscr {I}}\mathcal V_{\textrm{e,b}}(M), \\ \partial _1\equiv \partial _{x^1}&= \frac{1}{2}(\partial _t-\partial _{r_*}) = \rho _0\Bigl (\rho _0\partial _{\rho _0}-\Bigl (1-\frac{1}{2}\rho _{\!\mathscr {I}}\Bigl (1-\frac{2{\mathfrak {m}}}{r}\Bigr )\Bigr )\rho _{\!\mathscr {I}}\partial _{\rho _{\!\mathscr {I}}}\Bigr ) \\ {}&\in \rho _0\mathcal V_{\textrm{e,b}}(M), \end{aligned} \end{aligned}$$

(3.10)

$$\begin{aligned}&\partial _0 r=\frac{1}{2}\partial _{r_*}r=\frac{1}{2}\Bigl (1-\frac{2{\mathfrak {m}}}{r}\Bigr )=-\partial _1 r. \end{aligned}$$

(3.11)

Remark 3.9

(b-regularity) From (3.10), we also obtain

$$\begin{aligned} \rho _{\!\mathscr {I}}\partial _{\rho _{\!\mathscr {I}}} = -r\Bigl (1-\frac{2{\mathfrak {m}}}{r}\Bigr )^{-1}(\partial _t+\partial _{r_*}),\qquad \rho _0\partial _{\rho _0} = (r_*-t)\partial _t + \rho _{\!\mathscr {I}}\partial _{\rho _{\!\mathscr {I}}}. \end{aligned}$$

Membership in $H_{{\textrm{b}}}^k(M)$ is thus equivalent to the condition that up to k derivatives along $r(\partial _t+\partial _{r_*})$, $(r_*-t)(\partial _t-\partial _{r_*})$, $\mathcal V({\mathbb {S}}^2)$ lie in $L^2_{\textrm{b}}(M)$. (These vector fields were already used by Lindblad [30] and in [22].) Thus, unlike in (1.9), the spherical vector fields do not have a decaying weight at $\mathscr {I}^+$ anymore.

Definition 3.10

(Rescaled vector bundle) The vector bundle⁸$\widetilde{T}^*M\rightarrow M$ is defined by

$$\begin{aligned} \widetilde{T}^*M:= \langle {\mathrm d}x^0\rangle \oplus \langle {\mathrm d}x^1\rangle \oplus r T^*{\mathbb {S}}^2. \end{aligned}$$

The prefactor r in front of $T^*{\mathbb {S}}^2$ here means that over the interior $M^\circ$, we identify $\widetilde{T}^*_{M^\circ }M\cong T^*_{M^\circ }M$ by identifying a section of $\widetilde{T}^*M$ with the 1-form on $M^\circ$.

Likewise, we identify sections of $S^2\widetilde{T}^*M$ with symmetric 2-tensors over $M^\circ$. In order to make the scaling of spherical tensors apparent, we thus write the bundle splittings as

(3.12)

by direct analogy with (2.9). Choosing local coordinates $x^2,x^3$ on ${\mathbb {S}}^2$, a smooth section $\omega \in \mathcal C^\infty (M;\widetilde{T}^*M)$ is thus a linear combination $\omega = \omega _0\,{\mathrm d}x^0 + \omega _1\,{\mathrm d}x^1 + \sum _{a=2}^3 \omega _{{\bar{a}}}r\,{\mathrm d}x^a$, $\omega _0,\omega _1,\omega _{{\bar{2}}},\omega _{{\bar{3}}}\in \mathcal C^\infty (M)$. More generally, we use the following index notation:

Definition 3.11

(Weights of spherical indices) For $\mu _1,\ldots ,\mu _p\in \{0,1,2,3\}$, set

$$\begin{aligned} s(\mu _1,\ldots ,\mu _p):= \#\bigl \{ i\in \{1,\ldots ,p\} :\mu _i\in \{2,3\} \bigr \}. \end{aligned}$$

With $x^2,x^3$ denoting coordinates on ${\mathbb {S}}^2$, and for a tensor T on $M^\circ$ of type (p, q), we set

$$\begin{aligned} T_{{\bar{\mu }}_1\dots {\bar{\mu }}_p}^{{\bar{\nu }}_1\dots {\bar{\nu }}_q}:= r^{s(\nu _1,\ldots ,\nu _q)-s(\mu _1,\ldots ,\mu _p)}T_{\mu _1\dots \mu _p}^{\nu _1\dots \nu _q}. \end{aligned}$$

We shall henceforth denote indices in $\{0,1,2,3\}$ by Greek letters $\mu ,\nu ,\kappa ,\ldots$, and spherical indices in $\{2,3\}$ by Roman letters $a,b,c,\ldots$.

Returning to metrics on M, we have, directly from the Definitions (3.7) and (3.12):

Lemma 3.12

(Uniform behavior of $g_{\mathfrak {m}}$) We have $g_{\mathfrak {m}}\in \mathcal C^\infty (M;S^2\widetilde{T}^*M)$, and $g_{\mathfrak {m}}$ is a nondegenerate section with Lorentzian signature down to $I^0\cup \mathscr {I}^+$.

While $S^2\widetilde{T}^*M$ is the appropriate bundle for the unknown in the Einstein equations—the metric—to take values in, the metric also determines the linearized operators we need to study; hence, we need to connect $g_{\mathfrak {m}}$ and its perturbations to the eb-theory in which our (energy) estimates will take place.

Lemma 3.13

(Relationship between $\widetilde{T}^*M$ and ${}^{{\textrm{e,b}}}T^*M$) Let . Then

Writing $\mathcal C^\infty =\mathcal C^\infty (M;S^2\,{}^{{\textrm{e,b}}}T^*M)$ for brevity, we have, for ,

(3.13)

Proof

We only need to prove the first part. It follows by duality from (3.10) and the fact that if $V\in \mathcal V({\mathbb {S}}^2)$, then $r^{-1}V=\rho _0 x_{\!\mathscr {I}}\cdot x_{\!\mathscr {I}} V\in \rho _0 x_{\!\mathscr {I}}\mathcal V_{\textrm{e,b}}(M)$. $\quad \square$

Corollary 3.14

($g_{\mathfrak {m}}$ as an eb-metric) We have $g_{\mathfrak {m}}\in \rho _0^{-2}x_{\!\mathscr {I}}^{-2}\mathcal C^\infty (M;S^2\,{}^{{\textrm{e,b}}}T^*M)$ and $g_{\mathfrak {m}}^{-1}\in \rho _0^2 x_{\!\mathscr {I}}^2\mathcal C^\infty (M;S^2\,{}^{{\textrm{e,b}}}TM)$. Moreover,

Remark 3.15

(Connection of weighted eb-metrics) The Koszul formula, together with the fact that $\mathcal V_{\textrm{e,b}}(M)$ is a Lie algebra (with differentiation along any element of $\mathcal V_{\textrm{e,b}}(M)$ being bounded on $\rho _0^{\ell _0}x_{\!\mathscr {I}}^{\ell _{\!\mathscr {I}}}\mathcal C^\infty (M)$ for any $\ell _0,\ell _{\!\mathscr {I}}\in {\mathbb {R}}$), implies that the Levi-Civita connection of $g_{\mathfrak {m}}\in \rho _0^{-2}x_{\!\mathscr {I}}^{-2}\mathcal C^\infty (M;S^2\,{}^{{\textrm{e,b}}}T^*M)$ satisfies $\nabla \in \textrm{Diff}_{\textrm{e,b}}^1(M;{}^{{\textrm{e,b}}}TM,{}^{{\textrm{e,b}}}T^*M\otimes {}^{{\textrm{e,b}}}TM)$. Writing eb-tensor bundles as ${}^{{\textrm{e,b}}}T^{p,q}M={}^{{\textrm{e,b}}}T^*M^{\otimes p}\otimes {}^{{\textrm{e,b}}}TM^{\otimes q}$, this gives

$$\begin{aligned} \nabla \in \textrm{Diff}_{\textrm{e,b}}^1(M;{}^{{\textrm{e,b}}}T^{p,q}M,{}^{{\textrm{e,b}}}T^{p+1,q}M). \end{aligned}$$

In particular, the tensor wave operator satisfies

$$\begin{aligned} \Box _{g_{\mathfrak {m}}}\in \rho _0^2 x_{\!\mathscr {I}}^2\textrm{Diff}_{\textrm{e,b}}^2(M;{}^{{\textrm{e,b}}}T^{p,q}M),\qquad p,q\in {\mathbb {N}}_0, \end{aligned}$$

(3.14)

Lemma 3.16

($\Box _{g_{\mathfrak {m}}}$ as an eb-operator; commutators) Consider $\Box _{g_{\mathfrak {m}}}$ acting on functions. The operator $L:=\rho _{\!\mathscr {I}}\rho ^{-3}\Box _{g_{\mathfrak {m}}}\rho \in \textrm{Diff}_{\textrm{e,b}}^2(M)$ is equal to

(3.15)

If $\Omega \in \mathcal V({\mathbb {S}}^2)$ is a spherical vector field, then⁹

$$\begin{aligned} {[}L,\rho _0\partial _{\rho _0}],\ [L,\rho _{\!\mathscr {I}}\partial _{\rho _{\!\mathscr {I}}}],\ [L,\Omega ] \in x_{\!\mathscr {I}}\textrm{Diff}_{{\textrm{e,b}};{\textrm{b}}}^{1,1}(M). \end{aligned}$$

(3.16)

Proof

The membership $L\in \textrm{Diff}_{\textrm{e,b}}^2(M)$ is an immediate consequence of (3.14), and will be confirmed here by a direct calculation. The expression for $L\bmod x_{\!\mathscr {I}}\textrm{Diff}_{\textrm{e,b}}^2(M)$ only depends on $g_{\mathfrak {m}}$ modulo $\rho _0^{-2}x_{\!\mathscr {I}}^{-1}\mathcal C^\infty (M;S^2\,{}^{{\textrm{e,b}}}T^*M)$; we may thus replace $g_{\mathfrak {m}}^{-1}$ by the Minkowski dual metric , cf. (2.7), for which, in view of (3.10)–(3.11),

(3.17)

modulo $\rho _0^2 x_{\!\mathscr {I}}^3\textrm{Diff}_{\textrm{e,b}}^2(M)$. Multiplying this on the left by $\rho _{\!\mathscr {I}}\rho ^{-3}=\rho _{\!\mathscr {I}}^{-2}\rho _0^{-3}$ and on the right by $\rho =\rho _0\rho _{\!\mathscr {I}}$ proves (3.15).

The expression (3.15) together with $\rho _0\partial _{\rho _0}\in \mathcal V_{\textrm{e,b}}(M)$ immediately gives $[L,\rho _0\partial _{\rho _0}]=[{\tilde{L}},\rho _0\partial _{\rho _0}]\in x_{\!\mathscr {I}}\textrm{Diff}_{\textrm{e,b}}^2(M)$ since $\mathcal V_{\textrm{e,b}}(M)$ is a Lie algebra. Similarly,

with the commutator lying in $x_{\!\mathscr {I}}\textrm{Diff}_{\textrm{e,b}}^2(M)$; in the first term on the other hand, we can write as a finite sum with spherical vector fields $\Omega _{k,1},\Omega _{k,2},\Omega ^\flat \in \mathcal V({\mathbb {S}}^2)$ and $f\in \mathcal C^\infty ({\mathbb {S}}^2)$; but $\mathcal V({\mathbb {S}}^2)\subset x_{\!\mathscr {I}}^{-1}\mathcal V_{\textrm{e,b}}(M)\cap \mathcal V_{\textrm{b}}(M)$, and hence

(3.18)

Finally, we consider

The term contributes by the same argument as in (3.18). In the second term, we use the fact that lifts of vector fields from the base ${\mathbb {S}}^2$ of the fibration $\mathscr {I}^+\rightarrow {\mathbb {S}}^2$ enjoy improved commutation properties with eb-differential operators, cf. [23, §5.1]. Concretely, in local coordinates (3.1) (with $y^2,y^3$ local coordinates on ${\mathbb {S}}^2$), we have $\Omega =\sum _{j=2}^3\Omega ^j(y)\partial _{y^j}$ where the $\Omega ^j$ are smooth in y (and independent of $\rho _0$); writing any $V\in \mathcal V_{\textrm{e,b}}(M)$ as $V=a^0\rho _0\partial _{\rho _0}+a^1\rho _{\!\mathscr {I}}\partial _{\rho _{\!\mathscr {I}}}+\sum _{j=2}^3 a^j x_{\!\mathscr {I}}\partial _{y^j}$ with $a^\mu \in \mathcal C^\infty (M)$, this gives

$$\begin{aligned} {[}\Omega ,V] = (\Omega a^0)\rho _0\partial _{\rho _0} + (\Omega a^1)\rho _{\!\mathscr {I}}\partial _{\rho _{\!\mathscr {I}}} + x_{\!\mathscr {I}} \sum _j [\Omega ,a^j\partial _{y^j}] \in \mathcal V_{\textrm{e,b}}(M).\nonumber \\ \end{aligned}$$

(3.19a)

(This improves over the naive expectation coming from $\Omega \in x_{\!\mathscr {I}}^{-1}\mathcal V_{\textrm{e,b}}(M)$ that one only has $[\Omega ,V]\in x_{\!\mathscr {I}}^{-1}\mathcal V_{\textrm{e,b}}(M)$.) Using the Leibniz rule, we infer that

$$\begin{aligned} {[}-,\Omega ] :\textrm{Diff}_{\textrm{e,b}}^{2,(\alpha _0,\alpha _1)}(M) \rightarrow \textrm{Diff}_{\textrm{e,b}}^{2,(\alpha _0,\alpha _1)}(M),\qquad \alpha _0,\alpha _1\in {\mathbb {R}}. \end{aligned}$$

(3.19b)

Applying this to ${\tilde{L}}\in x_{\!\mathscr {I}}\textrm{Diff}_{\textrm{e,b}}^2(M)$ gives $[{\tilde{L}},\Omega ]\in x_{\!\mathscr {I}}\textrm{Diff}_{\textrm{e,b}}^2(M)$. This finishes the proof. $\quad \square$

Proposition 3.17

(Energy estimate) In the notation (3.6), let $c<{\bar{x}}_{\!\mathscr {I}}$. Define

$$\begin{aligned} \Omega = \{ x_{\!\mathscr {I}}<c, \rho _0<1 \} \subset M,\quad \Sigma = x_{\!\mathscr {I}}^{-1}(c) \subset M. \end{aligned}$$

Let $k\in {\mathbb {N}}_0$. Let $\alpha _0,\alpha _{\!\mathscr {I}}\in {\mathbb {R}}$ with $\alpha _{\!\mathscr {I}}<\min (\alpha _0,0)$. Suppose $f\in H_{{\textrm{b}}}^{k,(\alpha _0,2\alpha _{\!\mathscr {I}})}(\Omega )=\rho _0^{\alpha _0}x_{\!\mathscr {I}}^{2\alpha _{\!\mathscr {I}}}H_{{\textrm{b}}}^k(\Omega )=\rho _0^{\alpha _0}\rho _{\!\mathscr {I}}^{\alpha _{\!\mathscr {I}}}H_{{\textrm{b}}}^k(\Omega )$ vanishes near $\Sigma$. Then the unique forward solution u (i.e. with vanishing Cauchy data at $\Sigma$) of

$$\begin{aligned} L u=f,\qquad L=\rho _{\!\mathscr {I}}\rho ^{-3}\Box _{g_{\mathfrak {m}}}\rho , \end{aligned}$$

(3.20)

satisfies $u\in H_{{\textrm{e,b}};{\textrm{b}}}^{(1;k),(\alpha _0,2\alpha _{\!\mathscr {I}})}(\Omega )$, with an estimate¹⁰

$$\begin{aligned} \Vert u\Vert _{H_{{\textrm{e,b}};{\textrm{b}}}^{(1;k),(\alpha _0,2\alpha _{\!\mathscr {I}})}(\Omega )}\le C\Vert f\Vert _{H_{{\textrm{b}}}^{k,(\alpha _0,2\alpha _{\!\mathscr {I}})}(\Omega )} = C\Vert f\Vert _{H_{{\textrm{e,b}};{\textrm{b}}}^{(0;k),(\alpha _0,2\alpha _{\!\mathscr {I}})}(\Omega )}. \end{aligned}$$

(3.21)

Proof

We follow the arguments used in the proof of [22, Propositions 4.3 and 4.8] and shall thus be brief. While one can work directly with $\Box _{g_{\mathfrak {m}}}$ (as done in [23, §6]), we work with L in order to simplify the weight arithmetic. Note now that $\Box _{g_{\mathfrak {m}}}$ is symmetric with respect to the volume density

$$\begin{aligned} |{\mathrm d}g_{\mathfrak {m}}|\in \rho _0^{-4}x_{\!\mathscr {I}}^{-4}\mathcal C^\infty (M;|\Lambda ^4\,{}^{{\textrm{e,b}}}T^*M|) = \rho _0^{-4}\rho _{\!\mathscr {I}}^{-3}\mathcal C^\infty (M;|\Lambda ^4\,{}^{{\textrm{b}}}T^*M|), \end{aligned}$$

where we use Lemma 3.13 and the relationship between smooth nonzero eb- and b-densities. Since $\Box _{g_{\mathfrak {m}}}$ is formally self-adjoint on $L^2(M;|{\mathrm d}g_{\mathfrak {m}}|)$, the operator L is formally self-adjoint on $L^2_{\textrm{b}}(M):=L^2(M,\mu _{\textrm{b}})$, where $\mu _{\textrm{b}}=\rho _0^4\rho _{\!\mathscr {I}}^3|{\mathrm d}g_{\mathfrak {m}}|$ is a smooth positive b-density on M.

In order to prove (3.21), one can cut and paste energy estimates using domain of dependence properties. Away from $M^\circ$, the estimate (3.21) estimates the $H^{k+1}$-norm of u by the $H^k$-norm of f; it thus suffices to work near $I^0\cup \mathscr {I}^+$. But away from $\mathscr {I}^+$, L is a b-differential operator, $L\in \textrm{Diff}_{\textrm{b}}^2(M\setminus \mathscr {I}^+)$, for which $x_{\!\mathscr {I}}$ is, near $I^0\setminus \mathscr {I}^+$, a (past directed) time function, the gradient of which can thus be used as a vector field multiplier giving the estimate (3.21) away from $\mathscr {I}^+$—this was discussed in detail in [22, Proposition 4.3].

We thus work in a small neighborhood of $\mathscr {I}^+$, in coordinates $\rho _0,\rho _{\!\mathscr {I}}$. For $k=0$, (3.21) follows from an energy estimate on $\Omega$ with the vector field multiplier

$$\begin{aligned} W=w^2 V,\qquad w:=\rho _0^{-\alpha _0}\rho _{\!\mathscr {I}}^{-\alpha _{\!\mathscr {I}}},\quad V=-(1+c)\rho _{\!\mathscr {I}}\partial _{\rho _{\!\mathscr {I}}}+\rho _0\partial _{\rho _0}\in \mathcal V_{\textrm{e,b}}(M),\nonumber \\ \end{aligned}$$

(3.22)

with $c>0$ (one may take $c=1$)autoedited,;¹¹W is future timelike. Consider the $L^2_{\textrm{b}}(\Omega )$-pairing

$$\begin{aligned} \begin{aligned}&2\langle w L u, w V u\rangle = \langle Q u,u\rangle + \text {[boundary terms]}, \\&\qquad Q:=L^*W+W^*L = [L,W]-({\text {div}}_{\mu _{\textrm{b}}}W)L\in \rho _0^{-2\alpha _0}x_{\!\mathscr {I}}^{-4\alpha _{\!\mathscr {I}}}\textrm{Diff}_{\textrm{e,b}}^2(M). \end{aligned}\nonumber \\ \end{aligned}$$

(3.23)

We compute the principal symbol of Q at $\mathscr {I}^+$. Since

(3.24)

we may replace L by its leading order term in (3.15), and $\mu _{\textrm{b}}$ by $\underline{\mu }_{\textrm{b}}$. Thus,

$$\begin{aligned} {-}{\text {div}}_{\underline{\mu }_{\textrm{b}}}W = w^2 V+(w^2 V)^* = -[V,w^2] = w^2\bigl (-2(1+c)\alpha _{\!\mathscr {I}}+2\alpha _0\bigr ), \end{aligned}$$

and a quick calculation then gives $Q\equiv \underline{Q}\bmod \rho _0^{-2\alpha _0}x_{\!\mathscr {I}}^{-4\alpha _{\!\mathscr {I}}+1}\textrm{Diff}_{\textrm{e,b}}^2(M)$ for

(3.25)

Since $\alpha _{\!\mathscr {I}}<0$ and $c>0$, $\alpha _0-\alpha _{\!\mathscr {I}}>0$, and recalling that , this is a positive elliptic element of $\rho _0^{-2\alpha _0}x_{\!\mathscr {I}}^{-4\alpha _{\!\mathscr {I}}}\textrm{Diff}_{\textrm{e,b}}^2(M)$. Therefore, $\langle \underline{Q} u,u\rangle$ controls one eb-derivative of u in $\rho _0^{\alpha _0}\rho _{\!\mathscr {I}}^{\alpha _{\!\mathscr {I}}}L^2_{\textrm{b}}$. Using a Poincaré inequality to control $\Vert u\Vert _{\rho _0^{\alpha _0}\rho _{\!\mathscr {I}}^{\alpha _{\!\mathscr {I}}}L^2_{\textrm{b}}}$ by $\Vert \rho _{\!\mathscr {I}}\partial _{\rho _{\!\mathscr {I}}}u\Vert _{\rho _0^{\alpha _0}\rho _{\!\mathscr {I}}^{\alpha _{\!\mathscr {I}}}L^2_{\textrm{b}}}$ (using $\alpha _{\!\mathscr {I}}<0$), an application of the Cauchy–Schwarz inequality to (3.23) (and using the fact that the boundary terms vanish at $\Sigma$ and have a good sign at $\rho _0^{-1}(1)$ due to the future causal nature of W and $-{\mathrm d}\rho _0$, and can thus be dropped) implies the estimate (3.21) for $k=0$.

We prove higher b-regularity by commuting the vector fields from Lemma 3.16 through the equation. Suppose we have established (3.21) for $k\in {\mathbb {N}}_0$. Let $f\in H_{{\textrm{b}}}^{k+1,(\alpha _0,2\alpha _{\!\mathscr {I}})}(\Omega )$. Let $X_0=\rho _0\partial _{\rho _0},X_1=\rho _1\partial _{\rho _1}$, and let $X_2,X_3,X_4\in \mathcal V({\mathbb {S}}^2)$ be vector fields spanning $T{\mathbb {S}}^2$ pointwise (e.g. rotation vector fields); put moreover $X_5\equiv 1$. Thus, the $X_k$ span $\textrm{Diff}_{\textrm{b}}^1(M)$ over $\mathcal C^\infty (M)$. By Lemma 3.16, we can write

$$\begin{aligned} {[}L,X_j] = \sum _{k=0}^5 x_{\!\mathscr {I}} Y_{j k}X_k,\qquad Y_{j k}\in \textrm{Diff}_{\textrm{e,b}}^1(M). \end{aligned}$$

Applying the inductive hypothesis to $L(X_j u)=X_j L u+[L,X_j]u$ gives

$$\begin{aligned} \Vert X_j u\Vert _{H_{{\textrm{e,b}};{\textrm{b}}}^{(1;k),(\alpha _0,2\alpha _{\!\mathscr {I}})}} \le C\Bigl (\Vert X_j f\Vert _{H_{{\textrm{b}}}^{k,(\alpha _0,2\alpha _{\!\mathscr {I}})}} + \sum _k \Vert x_{\!\mathscr {I}} X_k u \Vert _{H_{{\textrm{e,b}};{\textrm{b}}}^{(1;k),(\alpha _0,2\alpha _{\!\mathscr {I}})}}\Bigr ) \end{aligned}$$

Summing these estimates over $j=0,\ldots ,5$, the sums over k on the right can be absorbed into the sum over j on the left, provided we localize to a neighborhood of $\mathscr {I}^+$ where $x_{\!\mathscr {I}}$ is small. This gives (3.21) for $k+1$ in place of k, and completes the proof. $\quad \square$

Remark 3.18

(Multiplier in$(t,r_*)$-coordinates) In the coordinates $t<r_*$ and modulo irrelevant lower order terms, the vector field multiplier (3.22) is (using Remark 3.9)

$$\begin{aligned} W = r^{2\alpha _{\!\mathscr {I}}}(r_*-t)^{2(\alpha _0-\alpha _{\!\mathscr {I}})}\bigl ((r_*-t)\partial _t + c r(\partial _t+\partial _{r_*})\bigr ). \end{aligned}$$

3.3 Metric Perturbations

Following the rough discussion of couplings of metric coefficients in §2, we now define the function space for metric perturbations of the Schwarzschild metric $g_{\mathfrak {m}}$ in (3.7) near spacelike and null infinity.

Definition 3.19

(Projections to subbundles) The projections respecting the splitting (3.12) are defined as in (2.14). (If we write for the components of the standard metric on ${\mathbb {S}}^2$ in local coordinates on ${\mathbb {S}}^2$, and similarly for the components of the dual metric, then in the notation of Definition  we have , $\pi ^\Upsilon (h)=(h_{0 1},h_{1{\bar{a}}})$, , and $\pi _{1 1}(h)=(h_{1 1})$.) We set $\pi ^{\mathcal C\Upsilon }:=\pi ^\mathcal C+\pi ^\Upsilon$ (mapping ).

Definition 3.20

(Metric perturbations) Let $\ell _0,\ell _{\!\mathscr {I}}\in {\mathbb {R}}$ with $\ell _{\!\mathscr {I}}<\min (-\gamma ^\Upsilon ,\ell _0,{\tfrac{1}{2}})$, and let $k\in {\mathbb {N}}$, $k\ge 3$. With ${\bar{x}}_{\!\mathscr {I}}$ as in (3.6), fix $c\in (0,{\bar{x}}_{\!\mathscr {I}})$ and put

$$\begin{aligned} \Omega = \Bigl \{ x_{\!\mathscr {I}}<c,\ \rho _0<1+\frac{1}{2}\rho _{\!\mathscr {I}}^{\ell _{\!\mathscr {I}}} \Bigr \}. \end{aligned}$$

(3.26)

Then the space ${\tilde{\mathscr {G}}}^{k,(\ell _0,\ell _{\!\mathscr {I}})}\subset H_{{\textrm{b}}}^{k,(\ell _0,-1)}(\Omega ;S^2\widetilde{T}^*M)$ consists of all h for which there exist

so that $\pi ^{\mathcal C\Upsilon } h$, , $\pi _{1 1}h-h_{1 1}^{(0)} \in H_{{\textrm{b}}}^{k,(\ell _0,2 \ell _{\!\mathscr {I}})}(\Omega ;S^2\widetilde{T}^*M)$. The norm on ${\tilde{\mathscr {G}}}^{k,(\ell _0,\ell _{\!\mathscr {I}})}$ is

where the norms of tensors with spherical components, such as and $\pi ^{\mathcal C\Upsilon }h$, are defined as the sums of the norms of their ${\bar{a}}{\bar{b}}$ or ${\bar{a}}$-components (and moreover summing over a finite cover of ${\mathbb {S}}^2$ by coordinate systems). We finally define the affine space

$$\begin{aligned} \mathscr {G}^{k,(\ell _0,\ell _{\!\mathscr {I}})}:= \{ g_{\mathfrak {m}}+r^{-1}h :h\in {\tilde{\mathscr {G}}}^{k,(\ell _0,\ell _{\!\mathscr {I}})} \}. \end{aligned}$$

Matching the model (1.8) with corresponding to $\phi _1,\phi _2,\phi _3,\phi _4$, the trace-free spherical tensor has a radiation field, and the leading order term of $\pi _{1 1}h$ at $\mathscr {I}^+$ is sourced by it. See Remark 3.38 for an interpretation of these terms.

Remark 3.21

Note that all components of h, modulo the leading order terms of and $\pi _{1 1}h$, have the same decay rates at $\mathscr {I}^+$. This is a significant simplification of [22, Definition 3.1], made possible by the absence of logarithmic terms in $\pi _{1 1}h$ due to our new choice of gauge (cf. by contrast the logarithmic coupling term $B_{h,1 1}$ in [22, Equation (3.26c)]).

Fig. 2

The domain $\Omega$ defined in (3.26) inside the manifold M defined in (3.6). Also shown are the spacelike hypersurfaces $\Sigma ,\Sigma _{\textrm{f}}$ from Lemma 3.24

Notation 3.22

(Remainder space) For $k\in {\mathbb {N}}_0$, $\alpha ,\beta \in {\mathbb {R}}$, we shall use the abbreviation

$$\begin{aligned} \mathcal O_k^{\alpha ,\beta }:= H_{{\textrm{b}}}^{k,(\alpha ,2\beta )}(\Omega ). \end{aligned}$$

When using this notation for tensors, we mean the membership of their components in the splittings (3.12) (thus in particular using the ${\bar{a}}$ or ${\bar{a}}{\bar{b}}$ components for spherical components).

The factor of 2 in the $\mathscr {I}^+$-weight is included so that $\beta$ measures the decay rate in $\rho _{\!\mathscr {I}}$, as $\mathcal O_3^{\alpha ,\beta }\hookrightarrow \rho _0^\alpha x_{\!\mathscr {I}}^{2\beta }L^\infty (\Omega )=\rho _0^\alpha \rho _{\!\mathscr {I}}^\beta L^\infty (\Omega )$. We shall repeatedly use that for $k\ge 3$,

$$\begin{aligned} u_1\in \mathcal O_k^{\alpha _1,\beta _1},&u_2\in \mathcal O_k^{\alpha _2,\beta _2}{} & {} \Longrightarrow \ u_1 u_2\in \mathcal O_k^{\alpha _1+\alpha _2,\beta _1+\beta _2}, \\&u_2\in H_{{\textrm{b}}}^{k,\alpha _2}(\mathscr {I}^+){} & {} \Longrightarrow \ u_1 u_2 \in \mathcal O_k^{\alpha _1+\alpha _2,\beta _1}. \end{aligned}$$

Lemma 3.23

(Metric coefficients) Let $k\ge 3$ and $h\in {\tilde{\mathscr {G}}}^{k,(\ell _0,\ell _{\!\mathscr {I}})}$. Suppose $\Vert h\Vert _{{\tilde{\mathscr {G}}}^{3,(\ell _0,\ell _{\!\mathscr {I}})}}$ is sufficiently small. Then $g=g_{\mathfrak {m}}+r^{-1}h$ is a Lorentzian metric on $\Omega ^\circ$. In the notation of Definition , we have

and the coefficients of the dual metric $g^{-1}$ are

As a symmetric eb-2-tensor, $r^{-1}h$ is a decaying perturbation of $g_{\mathfrak {m}}$ (cf. Corollary ):

$$\begin{aligned} g - g_{\mathfrak {m}}\in \rho _0^{-2}x_{\!\mathscr {I}}^{-2}H_{{\textrm{b}}}^{k,(1+\ell _0,2\ell _{\!\mathscr {I}})}(\Omega ;S^2\,{}^{{\textrm{e,b}}}T^*M). \end{aligned}$$

(3.27)

Proof

Sobolev embedding (3.3) implies the pointwise bound $|h_{{\bar{\mu }}{\bar{\nu }}}| \le C\Vert h\Vert _{{\tilde{\mathscr {G}}}^{3,(\ell _0,\ell _{\!\mathscr {I}})}} \rho _0^{\ell _0}$. The Lorentzian nature of g then follows for small h from the nondegenerate Lorentzian nature of $g_{\mathfrak {m}}$ as a section of $S^2\widetilde{T}^*M$ (see Lemma 3.12). The expressions for the inverse metric follow from (3.7) by working in the bundle $S^2\widetilde{T}^*M$ and writing $g^{-1}=g_{\mathfrak {m}}^{-1}-r^{-1}g_{\mathfrak {m}}^{-1}h g_{\mathfrak {m}}^{-1}+r^{-2}E(h)$, where E(h) vanishes quadratically at $h=0$, so $E(h)\in \mathcal O_k^{2\ell _0,0-}$ since $h\in \mathcal O_k^{\ell _0,0-}$. This gives

$$\begin{aligned} g^{0 0}\in & {} -r^{-1}g_{\mathfrak {m}}^{0 1}h_{1 1}g_{\mathfrak {m}}^{0 1} + \mathcal O_k^{2+2\ell _0,2-} \subset -4 r^{-1}h_{1 1} \\ {}{} & {} \quad + r^{-2}\mathcal C^\infty (M)\cdot \mathcal O_k^{\ell _0,0-} + \mathcal O_k^{2+2\ell _0,2-}, \end{aligned}$$

similarly for the other coefficients.

The statement (3.27) follows from (3.13); for instance, this gives

$$\begin{aligned} r^{-1}h_{0 0}({\mathrm d}x^0)^2&\in \mathcal O_k^{1+\ell _0,1+\ell _{\!\mathscr {I}}} \cdot \rho _0^{-2}\rho _{\!\mathscr {I}}^{-2}\mathcal C^\infty (\Omega ;S^2\,{}^{{\textrm{e,b}}}T^*M) \\&\subset \rho _0^{-2}x_{\!\mathscr {I}}^{-2}H_{{\textrm{b}}}^{k,(1+\ell _0,2\ell _{\!\mathscr {I}})}(\Omega ;S^2\,{}^{{\textrm{e,b}}}T^*M). \end{aligned}$$

$\square$

Lemma 3.24

(Causal nature of $\partial \Omega$) For $h\in {\tilde{\mathscr {G}}}^{3,(\ell _0,\ell _{\!\mathscr {I}})}$ with sufficiently small norm,

$$\begin{aligned} \Sigma = \Bigl \{ x_{\!\mathscr {I}}=c,\ \rho _0<1+\frac{1}{2}\rho _{\!\mathscr {I}}^{\ell _{\!\mathscr {I}}} \Bigr \}\quad \text {and}\quad \Sigma _{\textrm{f}} = \Bigl \{ x_{\!\mathscr {I}}<c,\ \rho _0=1+\frac{1}{2}\rho _{\!\mathscr {I}}^{\ell _{\!\mathscr {I}}} \Bigr \} \end{aligned}$$

are spacelike hypersurfaces for $g=g_{\mathfrak {m}}+r^{-1}h$.

Proof

We recall from the proof of Lemma 3.6 that

$$\begin{aligned} r\,{\mathrm d}\rho _{\!\mathscr {I}}=a_0\,{\mathrm d}x^0-a_1\,{\mathrm d}x^1,\qquad a_0:=\frac{x^1}{2 r}\Bigl (1-\frac{2{\mathfrak {m}}}{r}\Bigr )<-\theta \rho _{\!\mathscr {I}},\quad a_1:=1+a_0,\nonumber \\ \end{aligned}$$

(3.28)

for some $\theta >0$; note here that $\tfrac{x^1}{r}=-\rho _{\!\mathscr {I}}$. The expression (3.8) gives an upper bound $g_{\mathfrak {m}}^{-1}(r\,{\mathrm d}\rho _{\!\mathscr {I}},r\,{\mathrm d}\rho _{\!\mathscr {I}})\le -\theta \rho _{\!\mathscr {I}}$ with $\theta >0$; since $a_0$ and $a_1$ are bounded, we have

$$\begin{aligned} \bigl |(g^{-1}-g_{\mathfrak {m}}^{-1})(r\,{\mathrm d}\rho _{\!\mathscr {I}},r\,{\mathrm d}\rho _{\!\mathscr {I}})\bigr | \le C r^{-1}\Vert h\Vert _{{\tilde{\mathscr {G}}}^{3,(\ell _0,\ell _{\!\mathscr {I}})}} \le \frac{1}{2}\theta \rho _{\!\mathscr {I}} \end{aligned}$$

on $\Omega$ for small h. Therefore, ${\mathrm d}x_{\!\mathscr {I}}$ is (past) timelike for g.

For $\Sigma _{\textrm{f}}$, we compute for the differential of its defining function, using $\rho _0=-(x^1)^{-1}$,

The squared length of the 1-form with respect to $g^{-1}$ is

$$\begin{aligned}{} & {} (4+\mathcal O(r^{-1}))\ell _{\!\mathscr {I}} a_0 (2 \rho _0\rho _{\!\mathscr {I}}^{-\ell _{\!\mathscr {I}}}+\ell _{\!\mathscr {I}} a_1) + \mathcal O(r^{-1}) \\ {}{} & {} \quad \cdot \bigl ( \mathcal O(\rho _{\!\mathscr {I}}^2) \mathcal O(\rho _0^{\ell _0}) + \mathcal O(\rho _0^2\rho _{\!\mathscr {I}}^{-2\ell _{\!\mathscr {I}}}+1)\mathcal O(\rho _0^{\ell _0}\rho _{\!\mathscr {I}}^{\ell _{\!\mathscr {I}}})\bigr ). \end{aligned}$$

For small h, the first term is positive, and in view of (3.28) dominates the second term (collecting the contributions from $h^{0 0}$, $h^{1 1}$) which is of size $\mathcal O(\Vert h\Vert _{{\tilde{\mathscr {G}}}^{3,(\ell _0,\ell _{\!\mathscr {I}})}}\rho _0^{1+\ell _0}\rho _{\!\mathscr {I}}^{1-\ell _{\!\mathscr {I}}})$. $\quad \square$

Lemma 3.25

(Connection coefficients) Let $g\in \mathscr {G}^{k,(\ell _0,\ell _{\!\mathscr {I}})}$, with $g-g_{\mathfrak {m}}\in {\tilde{\mathscr {G}}}^{3,(\ell _0,\ell _{\!\mathscr {I}})}$ small. Then the Christoffel symbols of the first kind $\Gamma _{\kappa \mu \nu }={\tfrac{1}{2}}(\partial _\mu g_{\nu \kappa }+\partial _\nu g_{\mu \kappa }-\partial _\kappa g_{\mu \nu })$ are

where denotes the Christoffel symbols on ${\mathbb {S}}^2$.¹² The Christoffel symbols of the second kind, $\Gamma ^\kappa _{\mu \nu }=g^{\kappa \lambda }\Gamma _{\lambda \mu \nu }$, are

Proof

Direct computation using Lemma 3.23 and Eq. (3.11). $\quad \square$

Corollary 3.26

(Curvature coefficients) Let $g=g_{\mathfrak {m}}+r^{-1}h\in \mathscr {G}^{k,(\ell _0,\ell _{\!\mathscr {I}})}$, $k\ge 4$, with $\Vert h\Vert _{{\tilde{\mathscr {G}}}^{3,(\ell _0,\ell _{\!\mathscr {I}})}}$ small. Define the Riemann curvature tensor by $R^\kappa {}_{\lambda \mu \nu }=\partial _\mu \Gamma ^\kappa _{\lambda \nu }-\partial _\nu \Gamma ^\kappa _{\lambda \mu }+\Gamma ^\kappa _{\mu \rho }\Gamma ^\rho _{\lambda \nu }-\Gamma ^\kappa _{\nu \rho }\Gamma ^\rho _{\lambda \mu }$. Use the notation from Definition . Then, modulo $r^{-3}\mathcal C^\infty +\mathcal O_{k-2}^{3+\ell _0,1+\ell _{\!\mathscr {I}}}$,

$$\begin{aligned} R^0{}_{{\bar{b}} 1{\bar{d}}}&\equiv r^{-1}\partial _1^2 h_{{\bar{b}}{\bar{d}}},&\qquad R^{{\bar{a}}}{}_{1 1{\bar{d}}}&\equiv {\tfrac{1}{2}}r^{-1}\partial _1^2 h_{\bar{d}}{}^{{\bar{a}}}, \end{aligned}$$

while $R^{{\bar{\kappa }}}{}_{{\bar{\lambda }}{\bar{\mu }}{\bar{\nu }}}\equiv 0$ for all other $\kappa ,\lambda ,\mu ,\nu$ with $\mu <\nu$; and $R^{{\bar{\kappa }}}{}_{{\bar{\lambda }}{\bar{\mu }}{\bar{\nu }}}=-R^{{\bar{\kappa }}}{}_{{\bar{\lambda }}{\bar{\nu }}{\bar{\mu }}}$. The Ricci tensor $\textrm{Ric}(g)_{{\bar{\lambda }}{\bar{\nu }}}=R^{{\bar{\kappa }}}{}_{{\bar{\lambda }}{\bar{\kappa }}{\bar{\nu }}}$ satisfies $\textrm{Ric}(g)\in \mathcal O_{k-2}^{3+\ell _0,1+\ell _{\!\mathscr {I}}}$.

Proof

Direct computation. The stated membership of $R^{{\bar{\kappa }}}{}_{{\bar{\lambda }}{\bar{\mu }}{\bar{\nu }}}$ gives $\textrm{Ric}(g)\in r^{-3}\mathcal C^\infty +\mathcal O_{k-2}^{3+\ell _0,1+\ell _{\!\mathscr {I}}}$, with the $r^{-3}\mathcal C^\infty$ term coming from $g_{\mathfrak {m}}$ which satisfies $\textrm{Ric}(g_{\mathfrak {m}})=0$. $\quad \square$

3.4 Gauge-Fixed Einstein Operator

Encouraged by the calculations in §2, we now define the nonlinear gauge-fixed Einstein operator whose linearization will be shown to have the main properties of $L_{\underline{g},E^\mathcal C,E^\Upsilon }$ discussed after (2.10).

Definition 3.27

(Nonlinear modified gauge-fixed Einstein operator) Set ${\mathfrak {c}}^\mathcal C={\mathfrak {c}}^\Upsilon :=r^{-1}\,{\mathrm d}t={\tfrac{1}{2}}r^{-1}({\mathrm d}x^0+{\mathrm d}x^1)$ as in (2.8), and choose $\gamma ^\mathcal C\in (0,1)$, $\gamma ^\Upsilon \in (-1,0)$ with $-\gamma ^\Upsilon <\gamma ^\mathcal C$. Write $E^\bullet =({\mathfrak {c}}^\bullet ,\gamma ^\bullet )$, $\bullet =\mathcal C,\Upsilon$, and define $\delta _{g,E^\mathcal C}^*,\delta _{g,E^\Upsilon }$ by (2.4)–(2.5). Given a Lorentzian metric g, and denoting by $g_{\mathfrak {m}}$ the Schwarzschild metric from Definition 3.5, put

$$\begin{aligned} \Upsilon _{E^\Upsilon }(g;g_{\mathfrak {m}}):= \Upsilon (g;g_{\mathfrak {m}}) - (\delta _{g_{\mathfrak {m}},E^\Upsilon }-\delta _{g_{\mathfrak {m}}}){\textsf{G}}_{g_{\mathfrak {m}}}(g-g_{\mathfrak {m}}), \end{aligned}$$

where $\Upsilon (g;g_{\mathfrak {m}})=g(g_{\mathfrak {m}})^{-1}\delta _g{\textsf{G}}_g g_{\mathfrak {m}}$ as in (2.1). We then define¹³

$$\begin{aligned}&P_{E^\mathcal C,E^\Upsilon }(g) := \textrm{Ric}(g) - \delta _{g_{\mathfrak {m}},E^\mathcal C}^*\Upsilon _{E^\Upsilon }(g;g_{\mathfrak {m}}), \\&\qquad P'_{g,E^\mathcal C,E^\Upsilon } := D_g P_{E^\mathcal C,E^\Upsilon },\qquad L_{g,E^\mathcal C,E^\Upsilon } := 2\rho _{\!\mathscr {I}}\rho ^{-3}P'_{g,E^\mathcal C,E^\Upsilon }\rho . \end{aligned}$$

Lemma 3.28

(Gauge 1-form) For g as in Lemma , we have $\Upsilon _{E^\Upsilon }(g;g_{\mathfrak {m}})\in \mathcal O_{k-1}^{2+\ell _0,1+\ell _{\!\mathscr {I}}}$.

Proof

We have $\Upsilon (g;g_{\mathfrak {m}})^\mu =g^{\kappa \lambda }(\Gamma (g)_{\kappa \lambda }^\mu -\Gamma (g_{\mathfrak {m}})_{\kappa \lambda }^\mu )$; lowering the index using g gives $\Upsilon (g;g_{\mathfrak {m}})_0\equiv -{\tfrac{1}{2}}\Upsilon (g;g_{\mathfrak {m}})^1$ and $\Upsilon (g;g_{\mathfrak {m}})_1\equiv -{\tfrac{1}{2}}\Upsilon (g;g_{\mathfrak {m}})^0$ modulo $\mathcal O_{k-1}^{2+\ell _0,1+\ell _{\!\mathscr {I}}}$. For $E^\Upsilon =(0,0,0)$, the result can now be read off from Lemma 3.25. Likewise,

$$\begin{aligned} \Upsilon _{E^\Upsilon }(g;g_{\mathfrak {m}})-\Upsilon (g;g_{\mathfrak {m}}) = -(\delta _{g_{\mathfrak {m}},E^\Upsilon }-\delta _{g_{\mathfrak {m}}}){\textsf{G}}_{g_{\mathfrak {m}}}(g-g_{\mathfrak {m}}) \in \mathcal O_{k-1}^{2+\ell _0,1+\ell _{\!\mathscr {I}}} \end{aligned}$$

since $\delta _{g_{\mathfrak {m}},E^\Upsilon }-\delta _{g_{\mathfrak {m}}}\in r^{-1}\mathcal C^\infty (M;{\text {Hom}}(S^2\widetilde{T}^*M,\widetilde{T}^*M))$ and $g-g_{\mathfrak {m}}\in \mathcal O_k^{1+\ell _0,1-}$.

$\square$

Proposition 3.29

(Structure of the linearized gauge-fixed Einstein operator) Write symmetric scattering 2-tensors in the splitting (3.12). Let $g=g_{\mathfrak {m}}+r^{-1}h\in \mathscr {G}^{k,(\ell _0,\ell _{\!\mathscr {I}})}$, $k\ge 4$, with $\Vert h\Vert _{{\tilde{\mathscr {G}}}^{3,(\ell _0,\ell _{\!\mathscr {I}})}}$ small. Then the operator $L_{g,E^\mathcal C,E^\Upsilon }$ from Definition  takes the form

(3.29a)

(3.29b)

where the endomorphisms $A_{g,E^\mathcal C,E^\Upsilon }$ and $B_{g,E^\mathcal C,E^\Upsilon }$ of $S^2\widetilde{T}^*M$ are defined by

If $h=0$, then $A_{g,E^\mathcal C,E^\Upsilon }$ equals $A_{E^\mathcal C,E^\Upsilon }$ from (2.11). General h contribute bounded terms at $\mathscr {I}^+$ and do not affect the block triangular structure of $A_{g,E^\mathcal C,E^\Upsilon }$; see §3.6.

Proof of Proposition 3.29

We will analyze the terms in the expression

$$\begin{aligned} \begin{aligned} 2 P'_{g,E^\mathcal C,E^\Upsilon }&= \Box _g + 2(\delta _{g_{\mathfrak {m}},E^\mathcal C}^*-\delta _g^*)\delta _g{\textsf{G}}_g + 2\delta _{g_{\mathfrak {m}},E^\mathcal C}^*(\delta _{g_{\mathfrak {m}},E^\Upsilon }-\delta _{g_{\mathfrak {m}}}){\textsf{G}}_{g_{\mathfrak {m}}} \\&\qquad + 2\delta _{g_{\mathfrak {m}},E^\mathcal C}^*\mathscr {C}_g - 2\delta _{g_{\mathfrak {m}},E^\mathcal C}^*\mathscr {Y}_g + 2\mathscr {R}_g, \end{aligned} \end{aligned}$$

(3.30)

with $\mathscr {C}_g$ and $\mathscr {Y}_g$ defined in (2.3), one by one.

$\bullet {\underline{Tensor\, wave\, operator.}}$ Following Definition 3.11, we set

$$\begin{aligned} \Gamma ^{{\bar{\kappa }}}_{{\bar{\mu }}{\bar{\nu }}}=r^{s(\kappa )-s(\mu ,\nu )}\Gamma ^\kappa _{\mu \nu }, \qquad \Gamma _{{\bar{\kappa }}{\bar{\mu }}{\bar{\nu }}}=r^{-s(\kappa ,\mu ,\nu )}\Gamma _{\kappa \mu \nu }. \end{aligned}$$

By Lemma 3.25, we have

$$\begin{aligned} \Gamma ^{{\bar{\sigma }}}_{0{\bar{\mu }}}&\in r^{-2}\mathcal C^\infty +\mathcal O_{k-1}^{2+\ell _0,1+\ell _{\!\mathscr {I}}},{} & {} s(\sigma ,\nu )<2, \end{aligned}$$

(3.31a)

$$\begin{aligned} \Gamma ^{{\bar{\sigma }}}_{0{\bar{\mu }}}&\in {\tfrac{1}{2}}r^{-1}\delta _\mu ^\sigma +r^{-2}\mathcal C^\infty +\mathcal O_{k-1}^{2+\ell _0,1+\ell _{\!\mathscr {I}}},&\qquad&s(\sigma ,\nu )=2, \end{aligned}$$

(3.31b)

$$\begin{aligned} \Gamma ^{{\bar{\sigma }}}_{{\bar{\kappa }}{\bar{\mu }}}&\in r^{-1}\mathcal C^\infty + \mathcal O_{k-1}^{2+\ell _0,1-}{} & {} \forall \,\sigma ,\kappa ,\mu . \end{aligned}$$

(3.31c)

Given a symmetric 2-tensor u on $\Omega \subset M$, we begin by calculating the form of

$$\begin{aligned} u_{{\bar{\mu }}{\bar{\nu }};{\bar{\kappa }}} = r^{-s(\mu ,\nu ,\kappa )}\partial _\kappa \bigl (r^{s(\mu ,\nu )}u_{{\bar{\mu }}{\bar{\nu }}}\bigr ) - \Gamma ^{{\bar{\sigma }}}_{{\bar{\kappa }}{\bar{\mu }}}u_{{\bar{\sigma }}{\bar{\nu }}} - \Gamma ^{{\bar{\sigma }}}_{{\bar{\nu }}{\bar{\kappa }}}u_{{\bar{\mu }}{\bar{\sigma }}}. \end{aligned}$$

For $\kappa =0$, note that $r^{-s(\mu ,\nu )}[\partial _0,r^{s(\mu ,\nu )}]\equiv {\tfrac{1}{2}}s(\mu ,\nu ) r^{-1}\bmod r^{-2}\mathcal C^\infty$, which cancels the contribution of the leading order term of (3.31b). Thus, by (3.10),

$$\begin{aligned} u_{{\bar{\mu }}{\bar{\nu }};0}&\in \partial _0 u_{{\bar{\mu }}{\bar{\nu }}} + \bigl (r^{-2}\mathcal C^\infty +\mathcal O_{k-1}^{2+\ell _0,1+\ell _{\!\mathscr {I}}}\bigr )u&\ \subset \ {}&\bigl (\rho _0 x_{\!\mathscr {I}}^2\mathcal C^\infty +\mathcal O_{k-1}^{2+\ell _0,1+\ell _{\!\mathscr {I}}}\bigr )\textrm{Diff}_{\textrm{e,b}}^1(M)u, \end{aligned}$$

(3.32a)

$$\begin{aligned} u_{{\bar{\mu }}{\bar{\nu }};1}&\in \partial _1 u_{{\bar{\mu }}{\bar{\nu }}} + \bigl (r^{-1}\mathcal C^\infty +\mathcal O_{k-1}^{2+\ell _0,1-}\bigr )u&\ \subset \ {}&\bigl (\rho _0\mathcal C^\infty +\mathcal O_{k-1}^{2+\ell _0,1-}\bigr )\textrm{Diff}_{\textrm{e,b}}^1(M)u, \end{aligned}$$

(3.32b)

$$\begin{aligned} u_{{\bar{\mu }}{\bar{\nu }};{\bar{c}}}&\in r^{-1}\partial _c u_{{\bar{\mu }}{\bar{\nu }}} + \bigl (r^{-1}\mathcal C^\infty +\mathcal O_{k-1}^{2+\ell _0,1-})u&\ \subset \ {}&\bigl (\rho _0 x_{\!\mathscr {I}}\mathcal C^\infty +\mathcal O_{k-1}^{2+\ell _0,1-}\bigr )\textrm{Diff}_{\textrm{e,b}}^1(M)u. \end{aligned}$$

(3.32c)

We use this to compute the form of

$$\begin{aligned} \begin{aligned} (\Box _g u)_{{\bar{\mu }}{\bar{\nu }}}&= -r^{-s(\mu ,\nu ,\kappa ,\lambda )}g^{{\bar{\kappa }}{\bar{\lambda }}}\partial _\lambda \bigl (r^{s(\mu ,\nu ,\kappa )}u_{{\bar{\mu }}{\bar{\nu }};{\bar{\kappa }}}\bigr ) \\&\qquad + g^{{\bar{\kappa }}{\bar{\lambda }}}\bigl (\Gamma ^{{\bar{\sigma }}}_{{\bar{\mu }}{\bar{\lambda }}}u_{{\bar{\sigma }}{\bar{\nu }};{\bar{\kappa }}} + \Gamma ^{{\bar{\sigma }}}_{{\bar{\nu }}{\bar{\lambda }}}u_{{\bar{\mu }}{\bar{\sigma }};{\bar{\kappa }}} + \Gamma ^{{\bar{\sigma }}}_{{\bar{\kappa }}{\bar{\lambda }}}u_{{\bar{\mu }}{\bar{\nu }};{\bar{\sigma }}}\bigr ). \end{aligned} \end{aligned}$$

(3.33)

In the second line of (3.33), those terms in which u is covariantly differentiated along $\partial _0,\partial _a$ lie in $(\rho _0^2 x_{\!\mathscr {I}}^3\mathcal C^\infty +\mathcal O_{k-2}^{3+\ell _0,3/2-})\textrm{Diff}_{\textrm{e,b}}^1(M)u$ by (3.31c), (3.32a), and (3.32c) (using that multiplication by $x_{\!\mathscr {I}}$ maps $\mathcal O_{k-2}^{\alpha ,1-}\rightarrow \mathcal O_{k-2}^{\alpha ,3/2-}$). Next, Lemmas 3.23 and 3.25 give $g^{{\bar{\kappa }}{\bar{\lambda }}}\Gamma ^1_{{\bar{\kappa }}{\bar{\lambda }}} \in 2 r^{-1}+\mathcal O_{k-1}^{2+\ell _0,1+\ell _{\!\mathscr {I}}}$; using (3.32b), the terms in the second line of (3.33) involving derivatives of u along $\partial _1$ are thus modulo $(r^{-2}\mathcal C^\infty +\mathcal O_{k-1}^{3+\ell _0,1+\ell _{\!\mathscr {I}}})\textrm{Diff}_{\textrm{e,b}}^1(M)u$ equal to

$$\begin{aligned}&g^{1 0}\Gamma ^{{\bar{\sigma }}}_{{\bar{\mu }} 0}u_{{\bar{\sigma }}{\bar{\nu }};1} + g^{1 0}\Gamma ^{{\bar{\sigma }}}_{{\bar{\nu }} 0}u_{{\bar{\mu }}{\bar{\sigma }};1} + g^{{\bar{\kappa }}{\bar{\lambda }}}\Gamma ^1_{{\bar{\kappa }}{\bar{\lambda }}}u_{{\bar{\mu }}{\bar{\nu }};1} \equiv (-s(\mu ,\nu )+2)r^{-1}\partial _1 u_{{\bar{\mu }}{\bar{\nu }}}. \end{aligned}$$

For the first term on the right in (3.33), all terms with $(\kappa ,\lambda )\ne (0,1),(1,0),(a,b)$ produce terms in $\mathcal O_{k-2}^{3+\ell _0,1+\ell _{\!\mathscr {I}}}\textrm{Diff}_{\textrm{e,b}}^2(M)u$. The remaining terms sum to

with the first line capturing the non-spherical, the second line the spherical terms. Plugging in (3.10) and using $2\ell _{\!\mathscr {I}}<1$ (so $\rho _{\!\mathscr {I}}\rho ^{-3}\mathcal O_{k-2}^{3+\ell _0,3/2-}\rho \subset \mathcal O_{k-2}^{1+\ell _0,\ell _{\!\mathscr {I}}}$), we thus obtain

(3.34)

The coordinate derivatives $\partial _a$ on ${\mathbb {S}}^2$ can be replaced by covariant derivatives , the difference in local coordinates being .

$\bullet {\underline{Modified\, symmetric\, gradient.}}$ Next, consider the second summand in (3.30). We have

$$\begin{aligned} \bigl ((\delta _{g_{\mathfrak {m}},E^\mathcal C}^*-\delta _g^*)\omega \bigr )_{{\bar{\mu }}{\bar{\nu }}}&= \bigl ((\delta _{g_{\mathfrak {m}},E^\mathcal C}^*-\delta _{g_{\mathfrak {m}}}^*)\omega \bigr )_{{\bar{\mu }}{\bar{\nu }}} + \bigl ((\delta _{g_{\mathfrak {m}}}^*-\delta _g^*)\omega \bigr )_{{\bar{\mu }}{\bar{\nu }}} \nonumber \\&=\bigl ((\delta _{g_{\mathfrak {m}},E^\mathcal C}^*-\delta _{g_{\mathfrak {m}}}^*)\omega \bigr )_{{\bar{\mu }}{\bar{\nu }}} + C_{{\bar{\mu }}{\bar{\nu }}}^{{\bar{\kappa }}}\omega _{{\bar{\kappa }}}, \end{aligned}$$

(3.35)

where $C_{{\bar{\mu }}{\bar{\nu }}}^{{\bar{\kappa }}}=\Gamma (g)_{{\bar{\mu }}{\bar{\nu }}}^{{\bar{\kappa }}}-\Gamma (g_{\mathfrak {m}})_{{\bar{\mu }}{\bar{\nu }}}^{{\bar{\kappa }}}$. In the splittings (3.12), we have ${\mathfrak {c}}^\mathcal C={\mathfrak {c}}^\Upsilon =({\tfrac{1}{2}},{\tfrac{1}{2}},0)$, so

$$\begin{aligned} \delta _{g_{\mathfrak {m}},E^\mathcal C}^*-\delta _{g_{\mathfrak {m}}}^* \in \gamma ^\mathcal Cr^{-1} \begin{pmatrix} 1 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad {\tfrac{1}{2}}\\ 0 &{} \quad 1 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad {\tfrac{1}{2}}\\ 1 &{} \quad 1 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 \end{pmatrix} +r^{-2}\mathcal C^\infty (M;{\text {Hom}}(\widetilde{T}^*M,S^2\widetilde{T}^*M)).\nonumber \\ \end{aligned}$$

(3.36)

For the second term in (3.35), we infer from Lemma 3.25 that, modulo $\mathcal O_{k-1}^{2+\ell _0,1+\ell _{\!\mathscr {I}}}$, we have

$$\begin{aligned} C^0_{1 1}\equiv -r^{-1}\partial _1 h_{1 1}, \qquad C^{{\bar{c}}}_{1\bar{b}}=C_{{\bar{b}} 1}^{{\bar{c}}}\equiv {\tfrac{1}{2}}r^{-1}\partial _1 h_{{\bar{b}}}{}^{\bar{c}}, \qquad C^0_{{\bar{a}}{\bar{b}}}\equiv r^{-1}\partial _1 h_{{\bar{a}}{\bar{b}}},\nonumber \\ \end{aligned}$$

(3.37)

while $C^{{\bar{\kappa }}}_{{\bar{\mu }}{\bar{\nu }}}\equiv 0$ for all other $\kappa ,\mu ,\nu$. Using , the operator $\omega \mapsto (C^{{\bar{\kappa }}}_{{\bar{\mu }}{\bar{\nu }}}\omega _{{\bar{\kappa }}})$ is thus

$$\begin{aligned} r^{-1} \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 \\ -\partial _1 h_{1 1} &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad {\tfrac{1}{2}}\partial _1 h_{{\bar{a}}}{}^{{\bar{b}}} \\ 0 &{} \quad 0 &{} \quad 0 \\ \partial _1 h_{{\bar{a}}{\bar{b}}} &{} \quad 0 &{} \quad 0 \end{pmatrix} + \mathcal O_{k-1}^{2+\ell _0,1+\ell _{\!\mathscr {I}}}. \end{aligned}$$

(3.38)

We compute $(\delta _g u)_{{\bar{\mu }}}=-g^{{\bar{\lambda }}{\bar{\kappa }}}u_{{\bar{\mu }}{\bar{\lambda }};{\bar{\kappa }}}$ using (3.32a)–(3.32c) and Lemma 3.23. The terms with $\kappa \ne 1$ contribute $\bigl (\rho _0 x_{\!\mathscr {I}}\mathcal C^\infty +\mathcal O_{k-1}^{2+\ell _0,1-}\bigr )\textrm{Diff}_{\textrm{e,b}}^1(M)u$, as do the terms with $\kappa =1$, $\lambda \ne 0$, so

$$\begin{aligned} \delta _g \in \begin{pmatrix} 2\partial _1 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 2\partial _1 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 2\partial _1 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix} + \bigl (\rho _0 x_{\!\mathscr {I}}\mathcal C^\infty +\mathcal O_{k-1}^{2+\ell _0,1-}\bigr )\textrm{Diff}_{\textrm{e,b}}^1.\nonumber \\ \end{aligned}$$

(3.39)

Lastly, Lemma 3.23 implies

$$\begin{aligned} {\textsf{G}}_g \in \begin{pmatrix} 1 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad {\tfrac{1}{2}}&{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 1 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 1 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 1 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 2 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 1 \end{pmatrix} + r^{-1}\mathcal C^\infty + \mathcal O_k^{1+\ell _0,1-}. \end{aligned}$$

(3.40)

Combining (3.35), (3.36), and (3.38)–(3.40) gives

$$\begin{aligned} \begin{aligned}&\rho _{\!\mathscr {I}}\rho ^{-3}\bigl (2(\delta _{g_{\mathfrak {m}},E^\mathcal C}^*-\delta _g^*)\delta _g{\textsf{G}}_g\bigr )\rho \\&\quad \in 2\begin{pmatrix} 2\gamma ^\mathcal C&{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad \gamma ^\mathcal C&{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ -2\partial _1 h_{1 1} &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad \gamma ^\mathcal C&{} \quad 0 \\ 0 &{} \quad 0 &{} \quad \gamma ^\mathcal C+\partial _1 h_{{\bar{a}}}{}^{{\bar{b}}} &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 2\gamma ^\mathcal C&{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad \gamma ^\mathcal C&{} \quad 0 \\ 2\partial _1 h_{{\bar{a}}{\bar{b}}} &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix}\rho _0^{-1}\partial _1 \\&\quad \qquad + \bigl (x_{\!\mathscr {I}}\mathcal C^\infty +\mathcal O_{k-1}^{1+\ell _0,\ell _{\!\mathscr {I}}}\bigr )\textrm{Diff}_{\textrm{e,b}}^1. \end{aligned} \end{aligned}$$

(3.41)

$\bullet {\underline{Modified\, divergence.}}$ Using Lemma 3.25 with $h=0$, the third summand in (3.30) is

$$\begin{aligned} \delta _{g_{\mathfrak {m}},E^\mathcal C}^*&\in \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 \\ {\tfrac{1}{2}}&{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 1 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad {\tfrac{1}{2}}\\ 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 \end{pmatrix}\partial _1 + \rho _0 x_{\!\mathscr {I}}\textrm{Diff}_{\textrm{e,b}}^1, \end{aligned}$$

(3.42)

$$\begin{aligned} \delta _{g_{\mathfrak {m}},E^\Upsilon }-\delta _{g_{\mathfrak {m}}}&\in \gamma ^\Upsilon r^{-1}\begin{pmatrix} -2 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad -1 &{} 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad -2 &{} \quad 0 &{} \quad -1 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad -2 &{} \quad 0 &{} -2 &{} \quad 0 &{} \quad 0 \end{pmatrix} + r^{-2}\mathcal C^\infty . \end{aligned}$$

(3.43)

Therefore,

$$\begin{aligned}&\rho _{\!\mathscr {I}}\rho ^{-3}\bigl (2\delta _{g_{\mathfrak {m}},E^\mathcal C}^*(\delta _{g_{\mathfrak {m}},E^\Upsilon }-\delta _{g_{\mathfrak {m}}}){\textsf{G}}_{g_{\mathfrak {m}}}\bigr )\rho \nonumber \\&\quad \in 2 \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ -\gamma ^\Upsilon &{} \quad -\gamma ^\Upsilon &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad -2\gamma ^\Upsilon &{} \quad 0 &{} \quad -2\gamma ^\Upsilon &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad -\gamma ^\Upsilon &{} \quad 0 &{} \quad -\gamma ^\Upsilon &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix} (\rho _0\partial _{\rho _0}-\rho _{\!\mathscr {I}}\partial _{\rho _{\!\mathscr {I}}}) \nonumber \\&\quad \qquad + \bigl (x_{\!\mathscr {I}}\mathcal C^\infty +\mathcal O_{k-1}^{1+\ell _0,\ell _{\!\mathscr {I}}}\bigr )\textrm{Diff}_{\textrm{e,b}}^1. \end{aligned}$$

(3.44)

$\bullet {\underline{Term\, involving\, {\mathscr {C}_{g}}}}$. We turn to the fourth summand in (3.30). When calculating $(\mathscr {C}_g u)_{{\bar{\kappa }}}=g_{{\bar{\kappa }}{\bar{\lambda }}}g^{{\bar{\mu }}{\bar{\sigma }}}g^{{\bar{\nu }}{\bar{\tau }}}C^{{\bar{\lambda }}}_{{\bar{\mu }}{\bar{\nu }}} u_{{\bar{\sigma }}{\bar{\tau }}}$, one can replace $g\in g_{\mathfrak {m}}+\mathcal O_k^{1+\ell _0,1-}$ by $g_{\mathfrak {m}}$ at the expense of an error term in $\mathcal O_{k-1}^{3+2\ell _0,2-}$ since $C_{{\bar{\mu }}{\bar{\nu }}}^{{\bar{\lambda }}}\in \mathcal O_{k-1}^{2+\ell _0,1-}$ (cf. (3.37)); furthermore, the components of the tensor C other than those in (3.37) contribute terms in $\mathcal O_{k-1}^{2+\ell _0,1+\ell _{\!\mathscr {I}}}$. Therefore,

$$\begin{aligned} \mathscr {C}_g = \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 2 r^{-1}\partial _1 h_{1 1} &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad -{\tfrac{1}{2}}r^{-1}\partial _1 h^{{\bar{a}}{\bar{b}}} \\ 0 &{} \quad 0 &{} \quad -2 r^{-1}\partial _1 h_{{\bar{a}}}{}^{{\bar{b}}} &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix} + \mathcal O_{k-1}^{2+\ell _0,1+\ell _{\!\mathscr {I}}}. \end{aligned}$$

Together with (3.42), and using again that $\ell _{\!\mathscr {I}}<{\tfrac{1}{2}}$, we thus have

$$\begin{aligned} \begin{aligned} 2\rho _{\!\mathscr {I}}\rho ^{-3}\delta _{g_{\mathfrak {m}},E^\mathcal C}^*\mathscr {C}_g\rho&\in 2\rho _0^{-1}\partial _1 \circ \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 2\partial _1 h_{1 1} &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad -{\tfrac{1}{2}}\partial _1 h^{{\bar{a}}{\bar{b}}} \\ 0 &{} \quad 0 &{} \quad -\partial _1 h_{{\bar{a}}}{}^{{\bar{b}}} &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix} \\&\qquad + \mathcal O_{k-1}^{1+\ell _0,\ell _{\!\mathscr {I}}}\textrm{Diff}_{\textrm{e,b}}^1. \end{aligned}\nonumber \\ \end{aligned}$$

(3.45)

$\bullet {\underline{Term\, involving\, {\mathscr {Y}_{g}}}}$. For the fifth summand in (3.30), note that $\delta _{g_{\mathfrak {m}},E^\mathcal C}^*\in \rho _0\textrm{Diff}_{\textrm{e,b}}^1$ by (3.42). Together with $\Upsilon (g;g_{\mathfrak {m}})^{{\bar{\nu }}}\in \mathcal O_{k-1}^{2+\ell _0,1+\ell _{\!\mathscr {I}}}$ from Lemma 3.28, we get

$$\begin{aligned} -2\rho _{\!\mathscr {I}}\rho ^{-3}\delta _{g_{\mathfrak {m}},E^\mathcal C}^*\mathscr {Y}_g\rho \in \mathcal O_{k-1}^{1+\ell _0,\ell _{\!\mathscr {I}}}\textrm{Diff}_{\textrm{e,b}}^1. \end{aligned}$$

(3.46)

$\bullet {\underline{Curvature\, term.}}$ The final term of (3.30) can be computed using Corollary 3.26. A fortiori, all components of the Riemann and Ricci tensor lie in $r^{-3}\mathcal C^\infty +\mathcal O_{k-2}^{3+\ell _0,1-}$, and hence replacing g by $g_{\mathfrak {m}}$ in the definition of $\mathscr {R}_g$ produces $\mathcal O_{k-2}^{4+\ell _0,2-}$ error terms. One computes

$$\begin{aligned}{} & {} 2\rho _{\!\mathscr {I}}\rho ^{-3}\mathscr {R}_g\rho \in 2\rho _0^{-1} \begin{pmatrix} 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad {\tfrac{1}{2}}\partial _1^2 h^{{\bar{a}}{\bar{b}}} \\ 0 &{} \quad 0 &{} \quad \partial _1^2 h_{{\bar{a}}}{}^{{\bar{b}}} &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 2\partial _1^2 h_{{\bar{a}}{\bar{b}}} &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{pmatrix} \\ {}{} & {} \quad +\rho _0 x_{\!\mathscr {I}}^4\mathcal C^\infty + \mathcal O_{k-2}^{1+\ell _0,\ell _{\!\mathscr {I}}}. \end{aligned}$$

Combining this with (3.34), (3.41), and (3.44)–(3.46), and recalling that $\rho _0^{-1}\partial _1\equiv \rho _0\partial _{\rho _0}-\rho _{\!\mathscr {I}}\partial _{\rho _{\!\mathscr {I}}}\bmod x_{\!\mathscr {I}}\textrm{Diff}_{\textrm{e,b}}^1$, proves the Proposition. $\quad \square$

Definition 3.30

(Forcing terms) For $k\in {\mathbb {N}}_0$ and $\ell _0,\ell _{\!\mathscr {I}}\in {\mathbb {R}}$, $\ell _{\!\mathscr {I}}>0$, we define

$$\begin{aligned} \mathscr {F}^{k,(\ell _0,\ell _{\!\mathscr {I}})}:= & {} \bigl \{ f={\tilde{f}}+f_{1 1}^{(0)}({\mathrm d}x^1)^2 :\tilde{f}\in H_{{\textrm{b}}}^{k,(\ell _0,2\ell _{\!\mathscr {I}})}(\Omega ;S^2\widetilde{T}^*M),\\ {}{} & {} \quad \times f_{1 1}^{(0)}\in H_{{\textrm{b}}}^{k,\ell _0}(\mathscr {I}^+\cap \Omega ) \bigr \}, \end{aligned}$$

with norm $\Vert f\Vert _{\mathscr {F}^{k,(\ell _0,\ell _{\!\mathscr {I}})}}:= \Vert \tilde{f}\Vert _{H_{{\textrm{b}}}^{k,(\ell _0,2\ell _{\!\mathscr {I}})}(\Omega )} + \Vert f_{1 1}^{(0)}\Vert _{H_{{\textrm{b}}}^{k,\ell _0}(\mathscr {I}^+\cap \Omega )}$.

Corollary 3.31

(Nonlinear error term) Let $g=g_{\mathfrak {m}}+r^{-1}h\in \mathscr {G}^{k,(\ell _0,\ell _{\!\mathscr {I}})}$, $k\ge 4$, with h small in ${\tilde{\mathscr {G}}}^{3,(\ell _0,\ell _{\!\mathscr {I}})}$. Then $2 \rho _{\!\mathscr {I}}^{-1}\rho ^3 P_{E^\mathcal C,E^\Upsilon }(g)\in \mathscr {F}^{k-2,(\ell _0,\ell _{\!\mathscr {I}})}$; more precisely,¹⁴

(3.47)

By contrast to [22, Lemma 3.5], it is the leading order term of $h_{1 1}$ that enters in (3.47), rather than a logarithmically divergent term of $h_{1 1}$. The term is captured by the term $(\partial _t\phi _1)^2$ in the equation for $\phi _2$ in (1.8) (with $\phi _1,\phi _2$ being models for , $h_{1 1}$).

Proof of Corollary 3.31

Instead of a direct computation, we integrate up the linearization of $P_{E^\mathcal C,E^\Upsilon }$: the fundamental theorem of calculus gives

$$\begin{aligned} P_{E^\mathcal C,E^\Upsilon }(g_{\mathfrak {m}}+r^{-1}h) = P_{E^\mathcal C,E^\Upsilon }(g_{\mathfrak {m}}) + \int _0^1 P'_{g_{\mathfrak {m}}+r^{-1}s h,E^\mathcal C,E^\Upsilon }(r^{-1}h)\,{\mathrm d}s; \end{aligned}$$

since $P_{E^\mathcal C,E^\Upsilon }(g_{\mathfrak {m}})=0$, we can therefore use Proposition 3.29 to compute

Using Definition 3.20 and (3.29b), the error term of the Proposition contributes

$$\begin{aligned} {\tilde{L}}_{g_{\mathfrak {m}}+r^{-1}s h,E^\mathcal C,E^\Upsilon }(h) \in \mathcal O_{k-2}^{\ell _0,\ell _{\!\mathscr {I}}}. \end{aligned}$$

(3.48)

Regarding the main term (3.29a), the contribution lies, a fortiori, in the space (3.48); and $B_{g_{\mathfrak {m}}+r^{-1}s h,E^\mathcal C,E^\Upsilon }h\in \mathcal O_k^{\ell _0,\ell _{\!\mathscr {I}}}$ since $h_{0 0}\in \mathcal O_k^{\ell _0,\ell _{\!\mathscr {I}}}$. In the first term of (3.29a),

$$\begin{aligned} \rho _{\!\mathscr {I}}\partial _{\rho _{\!\mathscr {I}}}(\rho _0\partial _{\rho _0}-\rho _{\!\mathscr {I}}\partial _{\rho _{\!\mathscr {I}}})h=(\rho _0\partial _{\rho _0}-\rho _{\!\mathscr {I}}\partial _{\rho _{\!\mathscr {I}}})\rho _{\!\mathscr {I}}\partial _{\rho _{\!\mathscr {I}}}h\in \mathcal O_{k-2}^{\ell _0,\ell _{\!\mathscr {I}}} \end{aligned}$$

is an error term as well since $\rho _{\!\mathscr {I}}\partial _{\rho _{\!\mathscr {I}}}$ annihilates the leading order terms of h at $\mathscr {I}^+$; thus,

(3.49)

All coefficients of h in the splitting (3.12) except for $\pi _{1 1}h$ and lie in $\mathcal O_k^{\ell _0,\ell _{\!\mathscr {I}}}$ and thus contribute error terms. The $\pi _{1 1}h$, resp. component only contributes through the (4, 4), resp. (4, 7) entry of $A_{g_{\mathfrak {m}}+r^{-1} s h,E^\mathcal C,E^\Upsilon }$. Therefore, only the 4-th, i.e. $({\mathrm d}x^1)^2$, component of (3.49) does not lie in $\mathcal O_{k-2}^{\ell _0,\ell _{\!\mathscr {I}}}$, and modulo $\mathcal O_{k-2}^{\ell _0,\ell _{\!\mathscr {I}}}$ it equals

$$\begin{aligned}{} & {} 2\rho _0^{-1}\int _0^1\Bigl (-2\gamma ^\Upsilon \partial _1 h_{1 1} - \frac{1}{2}\partial _1(s h^{{\bar{a}}{\bar{b}}})\partial _1 h_{{\bar{a}}{\bar{b}}}\Bigr )\,{\mathrm d}s \\ {}{} & {} \quad = \rho _0^{-1}\Bigl (-4\gamma ^\Upsilon \partial _1 h_{1 1} - \frac{1}{2}\partial _1 h_{\bar{a}{\bar{b}}}\partial _1 h^{{\bar{a}}{\bar{b}}}\Bigr ). \end{aligned}$$

3.5 Tame Energy Estimate

With the modification parameters $E^\mathcal C,E^\Upsilon$ fixed as in Definition 3.27, we shall now drop them from the notation, and thus simply write

$$\begin{aligned} P(g):= P_{E^\mathcal C,E^\Upsilon }(g),\qquad L_g:= L_{g,E^\mathcal C,E^\Upsilon },\qquad A_g:= A_{g,E^\mathcal C,E^\Upsilon },\qquad \text {etc.} \end{aligned}$$

The first key step is an energy estimate for the linearized operator from Definition 3.27 on spaces with fixed weights but arbitrarily high b-regularity; precise decay is obtained in a second step in §3.6.

Proposition 3.32

(Tame energy estimate) Fix $\ell _0,\ell _{\!\mathscr {I}}$ as in Definition , and let $g=g_{\mathfrak {m}}+r^{-1}h\in \mathscr {G}^{k,(\ell _0,\ell _{\!\mathscr {I}})}$, with h small in ${\tilde{\mathscr {G}}}^{8,(\ell _0,\ell _{\!\mathscr {I}})}$. Let $\alpha _0,\alpha _{\!\mathscr {I}}\in {\mathbb {R}}$ with $\alpha _{\!\mathscr {I}}<\min (\alpha _0,0)$, and let $k,m\in {\mathbb {N}}_0$ with $k\ge 8$ and $m\le k-3$. Suppose $f\in H_{{\textrm{b}}}^{m,(\alpha _0,2\alpha _{\!\mathscr {I}})}(\Omega ;S^2\widetilde{T}^*M)$ vanishes near $\Sigma$ (in the notation of Lemma ). Then the unique forward solution u of

$$\begin{aligned} L_g u = f \end{aligned}$$

(3.50)

satisfies $u\in H_{{\textrm{e,b}};{\textrm{b}}}^{(1;m),(\alpha _0,2\alpha _{\!\mathscr {I}})}(\Omega ;S^2\widetilde{T}^*M)$. For $m\ge 3$, we moreover have the tame estimate

$$\begin{aligned} \Vert u\Vert _{H_{{\textrm{e,b}};{\textrm{b}}}^{(1;m),(\alpha _0,2\alpha _{\!\mathscr {I}})}} \le C\Bigl (\Vert f\Vert _{H_{{\textrm{b}}}^{m,(\alpha _0,2\alpha _{\!\mathscr {I}})}} + \Vert h\Vert _{{\tilde{\mathscr {G}}}^{m+3,(\ell _0,\ell _{\!\mathscr {I}})}}\Vert f\Vert _{H_{{\textrm{b}}}^{3,(\alpha _0,2\alpha _{\!\mathscr {I}})}}\Bigr ),\nonumber \\ \end{aligned}$$

(3.51)

where C depends on $m,k,\alpha _0,\alpha _{\!\mathscr {I}},\ell _0,\ell _{\!\mathscr {I}}$, but not on f, h.

We shall give a proof based on elementary (and rather imprecise) considerations.

Lemma 3.33

(Tame product estimate) Write points $x\in {\mathbb {R}}^n={\mathbb {R}}\times {\mathbb {R}}^{n-1}$ as $x=(x_1,x')$. Let $q\le m\in {\mathbb {N}}_0$. For $p\in {\mathbb {N}}_0$, denote by $d_p=\lceil \frac{p+1}{2}\rceil$ the smallest integer $>\frac{p}{2}$. Then there exists a constant $C=C(m,q)$ so that for all $h\in \mathcal C^\infty _{\textrm{c}}({\mathbb {R}}^{n-1})$ and $u\in \mathcal C^\infty _{\textrm{c}}({\mathbb {R}}^n)$,

$$\begin{aligned} \Vert (D^q h)(D^{m-q}u) \Vert _{L^2({\mathbb {R}}^n)}\le & {} C\Bigl (\Vert h\Vert _{H^{d_{n-1}}({\mathbb {R}}^{n-1})}\Vert u\Vert _{H^m({\mathbb {R}}^n)}\\ {}{} & {} \quad + \Vert h\Vert _{H^{m+d_{n-1}}({\mathbb {R}}^{n-1})}\Vert u\Vert _{L^2({\mathbb {R}}^n)}\Bigr ). \end{aligned}$$

Proof

We repeatedly use the following estimate for integers $0\le a<b<c$:

$$\begin{aligned} \Vert D^b u \Vert _{L^2} \le C_{a b c}\Vert D^a u\Vert _{L^2}^{\frac{c-b}{c-a}}\Vert D^c u\Vert _{L^2}^{\frac{b-a}{c-a}}; \end{aligned}$$

(3.52)

this follows by an inductive argument from the base case

$$\begin{aligned} \Vert D u\Vert _{L^2}^2 = \int D(u\overline{D u})\,{\mathrm d}x + \int u\overline{D^2 u}\,{\mathrm d}x = \int u\overline{D^2 u}\,{\mathrm d}x \le \Vert u\Vert _{L^2}\Vert D^2 u\Vert _{L^2}. \end{aligned}$$

We then estimate, using Sobolev embedding $H^{d_{n-1}}({\mathbb {R}}^{n-1})\hookrightarrow L^\infty ({\mathbb {R}}^{n-1})$,

$$\begin{aligned} \Vert (D^q h)(D^{m-q}u) \Vert _{L^2({\mathbb {R}}^n)}^2&\le \Vert D^q h\Vert _{L^\infty ({\mathbb {R}}^{n-1})} \Vert D^{m-q}u\Vert _{L^2({\mathbb {R}}^n)} \\&\lesssim \Vert D^q h\Vert _{H^{d_{n-1}}({\mathbb {R}}^{n-1})} \Vert D^{m-q}u\Vert _{L^2({\mathbb {R}}^n)}. \end{aligned}$$

Since $\Vert D^q h\Vert _{H^{d_{n-1}}}\lesssim \Vert D^{q+d_{n-1}}h\Vert _{L^2}+\Vert D^q h\Vert _{L^2}$, we can further estimate, using (3.52),

$$\begin{aligned} \Vert D^q(D^{d_{n-1}}h)\Vert _{L^2}\Vert D^{m-q}u\Vert _{L^2}&\lesssim \Vert D^{d_{n-1}}h\Vert _{L^2}^{\frac{m-q}{m}}\Vert D^{m+d_{n-1}}h\Vert _{L^2}^{\frac{q}{m}} \Vert u\Vert _{L^2}^{\frac{q}{m}}\Vert D^m u\Vert _{L^2}^{\frac{m-q}{m}} \\&\le \Vert h\Vert _{H^{m+d_{n-1}}} \Vert u\Vert _{L^2} + \Vert h\Vert _{H^{d_{n-1}}}\Vert u\Vert _{H^m}. \end{aligned}$$

We can estimate $\Vert D^q h\Vert _{L^2}\Vert D^{m-q}u\Vert _{L^2}$ by the same right hand side. $\quad \square$

Lemma 3.34

(Commutator identity) Let $\mathcal A$ be an algebra. Let $L,X_1,\ldots ,X_N\in \mathcal A$. Write ${\textrm{ad}}_{X_j}=[-,X_j]$. Then

$$\begin{aligned} {[}L,X_1\cdots X_N] = \sum _{q=1}^N (-1)^{q-1}\sum _{i_1<\cdots <i_q} \bigl ({\textrm{ad}}_{X_{i_1}}\dots {\textrm{ad}}_{X_{i_q}} L\bigr ) \prod _{\genfrac{}{}{0.0pt}{}{j=1}{j\ne i_1,\ldots ,i_q}}^N X_j. \end{aligned}$$

Proof

The case $N=1$ is clear. The inductive step follows from $[L,X_1\cdots X_{N+1}]=[L,X_1\cdots X_N]X_{N+1}+[L,X_{N+1}]X_1\cdots X_N-[{\textrm{ad}}_{X_{N+1}}L,X_1\cdots X_N]$. $\quad \square$

Proof of Proposition 3.32

$\bullet {\underline{Basic\, energy\, estimate.}}$ For fixed $\alpha _0\in {\mathbb {R}}$, we first prove the Proposition for $m=3$ and fixed but large negative $\alpha _{\!\mathscr {I}}<\alpha _0$. We use the vector field multiplier $W=w^2 V$ from (3.22) with, say, $c=1$, the volume density $\underline{\mu }_{\textrm{b}}$ from (3.24), and a pairing calculation analogous to (3.23). Using the $L^2$ inner product on sections of $S^2\widetilde{T}^*_\Omega M\rightarrow \Omega$ relative to $\underline{\mu }_{\textrm{b}}$ and any fixed smooth, positive definite fiber inner product on $S^2\widetilde{T}^*M$, we shall evaluate

$$\begin{aligned}&2{\text {Re}}\langle w L_g u,w V u\rangle = \langle Q u,u\rangle + \text {[boundary terms]},\\&\qquad Q := [L_g,W] - ({\text {div}}_{\underline{\mu }_{\textrm{b}}}W)L_g + (L_g^*-L_g)W, \end{aligned}$$

The first two summands of Q were computed to leading order at $\mathscr {I}^+$ to be equal to $\underline{Q}$ in (3.25); the point now is that for $\alpha _{\!\mathscr {I}}$ sufficiently large and negative, $\underline{Q}$ dominates the principal symbol of the skew-adjoint part $(L_g^*-L_g)W\in w^2 L^\infty (\Omega )\textrm{Diff}_{\textrm{e,b}}^2$ (using Proposition 3.29 and Sobolev embedding) whose bound in this space only depending on $\Vert h\Vert _{{\tilde{\mathscr {G}}}^{5,(\ell _0,\ell _{\!\mathscr {I}})}}$.¹⁵ Following the proof of Proposition 3.17, this gives

$$\begin{aligned} \Vert u\Vert _{H_{\textrm{e,b}}^{1,(\alpha _0,2\alpha _{\!\mathscr {I}})}} \le C_0\Vert L_g u\Vert _{H_{{\textrm{b}}}^{0,(\alpha _0,2\alpha _{\!\mathscr {I}})}}, \end{aligned}$$

(3.53)

with $C_0$ independent of h as long as $\Vert h\Vert _{{\tilde{\mathscr {G}}}^{5,(\ell _0,\ell _{\!\mathscr {I}})}}$ is small. One can commute any number b-derivatives through the equation $L_g u=f$ as in the proof of Proposition 3.17; we give details in a tame setting momentarily. We content ourselves with 3 b-derivatives for now; thus, for a constant $C_3$ only depending on $\Vert h\Vert _{{\tilde{\mathscr {G}}}^{8,(\ell _0,\ell _{\!\mathscr {I}})}}$, we have

$$\begin{aligned} \Vert u\Vert _{H_{{\textrm{e,b}};{\textrm{b}}}^{(1;3),(\alpha _0,2\alpha _{\!\mathscr {I}})}} \le C_3\Vert L_g u\Vert _{H_{{\textrm{b}}}^{3,(\alpha _0,2\alpha _{\!\mathscr {I}})}}. \end{aligned}$$

(3.54)

$\bullet {\underline{Tame\, estimate.}}$ We shall localize to small neighborhoods of $\mathscr {I}^+$ whenever convenient below; proofs of tame estimates away from $\mathscr {I}^+$ follow from simplifications of the arguments below. Recall from the proof of Proposition 3.17 the set of commutators

$$\begin{aligned} \mathcal X= \{ X_0,\ldots ,X_5 \} \subset \textrm{Diff}_{\textrm{b}}^1(M;S^2\widetilde{T}^*M) \end{aligned}$$

given by $X_0=\rho _0\partial _{\rho _0}$, $X_1=\rho _1\partial _{\rho _1}$, spherical vector fields $X_2,X_3,X_4\in \mathcal V({\mathbb {S}}^2)$ (acting by covariant differentiation on spherical 1-forms and symmetric 2-tensors in the splitting (3.12)), and $X_5\equiv 1$. The estimate (3.53) and Lemma 3.34 applied to $V_1\cdots V_l u$ for $l\le m$ and $V_1,\ldots ,V_l\in \mathcal X$ give

$$\begin{aligned}&\Vert u\Vert _{H_{{\textrm{e,b}};{\textrm{b}}}^{(1;m),(\alpha _0,2\alpha _{\!\mathscr {I}})}} \\&\quad \le C_0\Bigl (\Vert L_g u\Vert _{H_{{\textrm{b}}}^{m,(\alpha _0,2\alpha _{\!\mathscr {I}})}} \\&\quad \quad + \sum _{l\le m}\sum _{V_1,\ldots ,V_l\in \mathcal X}\sum _{q=1}^l\sum _{i_1<\cdots <i_q} \Vert ({\textrm{ad}}_{V_{i_1}}\cdots {\textrm{ad}}_{V_{i_q}} L_g) V_1\ldots \widehat{V_{i_1}}\ldots \widehat{V_{i_q}}\\ {}&\quad \quad \ldots V_l u\Vert _{H_{{\textrm{b}}}^{0,(\alpha _0,2\alpha _{\!\mathscr {I}})}}\Bigr ), \end{aligned}$$

which we schematically write as

$$\begin{aligned} \Vert u\Vert _{H_{{\textrm{e,b}};{\textrm{b}}}^{(1;m),(\alpha _0,2\alpha _{\!\mathscr {I}})}} \le C_0\Bigl (\Vert L_g u\Vert _{H_{{\textrm{b}}}^{m,(\alpha _0,2\alpha _{\!\mathscr {I}})}} + \sum _{q\le l\le m}\Vert ({\textrm{ad}}_\mathcal X^q L_g)\mathcal X^{l-q}u\Vert _{H_{{\textrm{b}}}^{0,(\alpha _0,2\alpha _{\!\mathscr {I}})}}\Bigr ).\nonumber \\ \end{aligned}$$

(3.55)

Consider first the contributions from ${\tilde{L}}_g$ to (3.55). We can write ${\tilde{L}}_g$ in (3.29b) as

$$\begin{aligned} {\tilde{L}}_g=\sum (x_{\!\mathscr {I}} a_j+{\tilde{a}}_j)P_j,\qquad a_j\in \mathcal C^\infty ,\quad {\tilde{a}}_j\in H_{{\textrm{b}}}^{k-2,(\ell _0,2\ell _{\!\mathscr {I}})}, \end{aligned}$$

(3.56)

where the operators $P_j\in \textrm{Diff}_{\textrm{e,b}}^2(M;S^2\widetilde{T}^*M)$ span $\textrm{Diff}_{\textrm{e,b}}^2(M;S^2\widetilde{T}^*M)$ over $\mathcal C^\infty (M)$, and so that, for some constant $C=C(k,\ell _0,\ell _{\!\mathscr {I}})$,

$$\begin{aligned} \Vert {\tilde{L}}_g\Vert _{k-2,(\ell _0,2\ell _{\!\mathscr {I}})}:= \max _j\Vert \tilde{a}_j\Vert _{H_{{\textrm{b}}}^{k-2,(\ell _0,2\ell _{\!\mathscr {I}})}}\le C\Vert h\Vert _{{\tilde{\mathscr {G}}}^{k,(\ell _0,\ell _{\!\mathscr {I}})}}; \end{aligned}$$

(3.57)

this uses: (1) the coefficients of ${\tilde{L}}_g$ are rational functions of up to 2 b-derivatives of h; (2) for $k\ge 5$, $H_{{\textrm{b}}}^{k-2}(\Omega )$ is an algebra, with a Moser estimate for the norm of products (which is a consequence of the corresponding result on ${\mathbb {R}}^n$, see e.g. [43, §13, Proposition 3.7], upon passing to coordinates $\log \rho _0$ and $\log \rho _{\!\mathscr {I}}$).

We will use the fact that commutators with elements $X_j\in \mathcal X$ preserve the space $\textrm{Diff}_{\textrm{e,b}}^k(M;S^2\widetilde{T}^*M)$ for all k; this is clear for $j=0,1,5$ (in which case $X_j$ is itself an eb-operator), and for spherical vector fields ($j=2,3,4$) relies on their $\rho _0$-independence, as discussed in (3.19a) and (3.19b). Thus, for $1\le q\le l\le m$, we have (using (3.56))

$$\begin{aligned}&\Vert ({\textrm{ad}}_\mathcal X^q\tilde{L}_g)\mathcal X^{l-q}u\Vert _{H_{{\textrm{b}}}^{0,(\alpha _0,2\alpha _{\!\mathscr {I}})}} \nonumber \\&\quad \le \Vert x_{\!\mathscr {I}}\mathscr {L}_{\textrm{e,b}}^2\mathscr {L}_{\textrm{b}}^{l-q}u\Vert _{H_{{\textrm{b}}}^{0,(\alpha _0,2\alpha _{\!\mathscr {I}})}} + \Vert (\mathscr {L}_{\textrm{b}}^q{\tilde{a}})\mathscr {L}_{\textrm{e,b}}^2 \mathscr {L}_{\textrm{b}}^{l-q}u\Vert _{H_{{\textrm{b}}}^{0,(\alpha _0,2\alpha _{\!\mathscr {I}})}} \nonumber \\&\quad \le C_m\Vert u\Vert _{H_{{\textrm{e,b}};{\textrm{b}}}^{(1;m),(\alpha _0,2\alpha _{\!\mathscr {I}}-1)}} + \Vert (\mathscr {L}_{\textrm{b}}^q\tilde{a})(\mathscr {L}_{\textrm{e,b}}^1\mathscr {L}_{\textrm{b}}^{m-q+1}u)\Vert _{H_{{\textrm{b}}}^{0,(\alpha _0,2\alpha _{\!\mathscr {I}})}}, \end{aligned}$$

(3.58)

where we write $\mathscr {L}_{\textrm{b}}^a$ and $\mathscr {L}_{\textrm{e,b}}^b$ for elements of $\textrm{Diff}_{\textrm{b}}^a$ and $\textrm{Diff}_{\textrm{e,b}}^b$, respectively, whose precise forms do not matter; and in passing to the second line, we used $\textrm{Diff}_{\textrm{e,b}}^1\subset \textrm{Diff}_{\textrm{b}}^1$. The second term of (3.58) is $\le \Vert \mathscr {L}_{\textrm{b}}^q\tilde{a}\Vert _{\rho _0^{\ell _0}x_{\!\mathscr {I}}^{2\ell _{\!\mathscr {I}}}L^\infty }\Vert u\Vert _{H_{{\textrm{e,b}};{\textrm{b}}}^{(1;m-q+1),(\alpha _0-\ell _0,2(\alpha _{\!\mathscr {I}}-\ell _{\!\mathscr {I}}))}}$; due to the weaker weight at $\mathscr {I}^+$, we can conclude that upon working in a sufficiently small neighborhood of $\mathscr {I}^+$, this is bounded by a small constant times $\Vert u\Vert _{H_{{\textrm{e,b}};{\textrm{b}}}^{(1;m),(\alpha _0,2\alpha _{\!\mathscr {I}})}}$ and can thus be absorbed into the left hand side of (3.55); similarly for the first term. To get a tame estimate, we use [43, §13, Proposition 3.6] and Sobolev embedding (3.3) to bound the second term in (3.58) by

$$\begin{aligned}&C_m\Bigl (\Vert \mathscr {L}_{\textrm{b}}^1\tilde{a}\Vert _{\rho _0^{\ell _0}x_{\!\mathscr {I}}^{2\ell _{\!\mathscr {I}}}L^\infty } \Vert \mathscr {L}_{\textrm{e,b}}^1 u\Vert _{H_{{\textrm{b}}}^{m,(\alpha _0-\ell _0,2(\alpha _{\!\mathscr {I}}-\ell _{\!\mathscr {I}}))}} + \Vert \mathscr {L}_{\textrm{b}}^1\tilde{a}\Vert _{H_{{\textrm{b}}}^{m,(\ell _0,2\ell _{\!\mathscr {I}})}}\Vert \mathscr {L}_{\textrm{e,b}}^1 u\Vert _{\rho _0^{\alpha _0}x_{\!\mathscr {I}}^{2\alpha _{\!\mathscr {I}}}L^\infty }\Bigr ) \\&\quad \le C_m'\Bigl (\Vert \tilde{a}\Vert _{H_{{\textrm{b}}}^{4,(\ell _0,2\ell _{\!\mathscr {I}})}}\Vert u\Vert _{H_{{\textrm{e,b}};{\textrm{b}}}^{(1;m),(\alpha _0,2(\alpha _{\!\mathscr {I}}-\ell _{\!\mathscr {I}}))}} + \Vert \tilde{a}\Vert _{H_{{\textrm{b}}}^{m+1,(\ell _0,2\ell _{\!\mathscr {I}})}}\Vert u\Vert _{H_{{\textrm{e,b}};{\textrm{b}}}^{(1;3),(\alpha _0,2\alpha _{\!\mathscr {I}})}}\Bigr ) \\&\quad \le C_m''\Bigl (\Vert h\Vert _{{\tilde{\mathscr {G}}}^{6,(\ell _0,\ell _{\!\mathscr {I}})}}\Vert u\Vert _{H_{{\textrm{e,b}};{\textrm{b}}}^{(1;m),(\alpha _0,2(\alpha _{\!\mathscr {I}}-\ell _{\!\mathscr {I}}))}} + \Vert h\Vert _{{\tilde{\mathscr {G}}}^{m+3,(\ell _0,\ell _{\!\mathscr {I}})}}\Vert L_g u\Vert _{H_{{\textrm{b}}}^{3,(\alpha _0,2\alpha _{\!\mathscr {I}})}}\Bigr ), \end{aligned}$$

where in passing to the final line we used the estimates (3.57) and (3.54). The first term only involves a fixed low regularity norm of h, and upon localizing to a sufficiently small (only depending on m) neighborhood of $\mathscr {I}^+$ can be absorbed into the left hand side of (3.55). The second term already fits into the estimate (3.51).

Next, we decompose the main term $L_g^0$ of $L_g$ in (3.29a) into $L_{g_{\mathfrak {m}}}^0+(L_g^0-L_{g_{\mathfrak {m}}}^0)$, with the first term capturing the smooth terms and the second term capturing the terms involving h. Using $[L_{g_{\mathfrak {m}}}^0,X_i]\in x_{\!\mathscr {I}}\textrm{Diff}_{{\textrm{e,b}};{\textrm{b}}}^{1,1}$ as in Lemma 3.16, the contribution of $L_{g_{\mathfrak {m}}}^0$ to the second summand on the right in (3.55) can be estimated by $C_m\Vert x_{\!\mathscr {I}} u\Vert _{H_{{\textrm{e,b}};{\textrm{b}}}^{(1;m),(\alpha _0,2\alpha _{\!\mathscr {I}})}}=C_m\Vert u\Vert _{H_{{\textrm{e,b}};{\textrm{b}}}^{(1;m),(\alpha _0,2\alpha _{\!\mathscr {I}}-1)}}$, which can again be absorbed into the left hand side of (3.55) for small $x_{\!\mathscr {I}}$.

Turning to the term $A_h\mathscr {L}_{\textrm{e,b}}^1$ of $L_g^0-L_{g_{\mathfrak {m}}}^0$, where $A_h:=A_g-A_{g_{\mathfrak {m}}}$, we decompose $A_h=A_h^0+{\tilde{A}}_h$ into the $\rho _{\!\mathscr {I}}$-independent leading order term $A_h^0\in H_{{\textrm{b}}}^{k-1,1+\ell _0}(\mathscr {I}^+\cap \Omega )$ plus a remainder term ${\tilde{A}}_h\in H_{{\textrm{b}}}^{k-1,(1+\ell _0,2\ell _{\!\mathscr {I}})}(\Omega )$. The contribution from ${\tilde{A}}_h$ to the second term on the right in (3.55) can be treated like the contribution from ${\tilde{L}}_g$. For the contribution from $A_h^0$, which is linear in in the notation of Definition 3.20, we apply Lemma 3.33 in logarithmic coordinates $\log \rho _0,\log \rho _{\!\mathscr {I}}$ (with $\log \rho _{\!\mathscr {I}}$ playing the role of $x_1$ in the lemma) with $n=4$, and h, m there replaced by where $l\le m$, so

We can then use (3.53) to bound the second term on the right. The contribution from $B_g$ to the right hand side of (3.55) is analyzed similarly. This finishes the proof of (3.51).

$\bullet {\underline{Estimate\, with\, sharp\, weights.}}$ $A_g$ is lower triangular in the bundle splitting , with scalar diagonal entries that are independent of h; see (3.59a) below for the explicit expression.¹⁶ We may thus choose a positive definite fiber inner product on $S^2\widetilde{T}^*M$ with respect to which the skew-adjoint part of $A_g$ is as small as we like along $\mathscr {I}^+$ in $\rho _0^{1+\ell _0}L^\infty (\mathscr {I}^+\cap \Omega )$ (using only that $\Vert h\Vert _{{\tilde{\mathscr {G}}}^{3,(\ell _0,\ell _{\!\mathscr {I}})}}\lesssim 1$). The calculation (3.25) thus shows that the condition $\alpha _{\!\mathscr {I}}<\min (\alpha _0,0)$ suffices to obtain the estimate (3.51). $\square$

3.6 Recovery of Decay; Proof of Nonlinear Stability

In the splitting , the endomorphisms $A_g$ and $B_g$ from Proposition 3.29 are

(3.59a)

(3.59b)

Theorem 3.35

(Tame estimate with sharp decay) Fix $\ell _0,\ell _{\!\mathscr {I}}$ as in Definition . Let $k,m\in {\mathbb {N}}_0$ with $k\ge m+11$. Let $g=g_{\mathfrak {m}}+r^{-1}h\in \mathscr {G}^{k,(\ell _0,\ell _{\!\mathscr {I}})}$, with h small in ${\tilde{\mathscr {G}}}^{8,(\ell _0,\ell _{\!\mathscr {I}})}$. Consider $f\in \mathscr {F}^{m+8,(\ell _0,\ell _{\!\mathscr {I}})}$ (see Definition ) which vanishes near $\Sigma$. Then the unique forward solution u of $L_g u=f$ satisfies $u\in {\tilde{\mathscr {G}}}^{m,(\ell _0,\ell _{\!\mathscr {I}})}$ and a tame estimate

$$\begin{aligned} \Vert u\Vert _{{\tilde{\mathscr {G}}}^{m,(\ell _0,\ell _{\!\mathscr {I}})}} \le C\Bigl (\Vert f\Vert _{\mathscr {F}^{m+8,(\ell _0,\ell _{\!\mathscr {I}})}} + \Vert h\Vert _{{\tilde{\mathscr {G}}}^{m+11,(\ell _0,\ell _{\!\mathscr {I}})}}\Vert f\Vert _{\mathscr {F}^{3,(\ell _0,\ell _{\!\mathscr {I}})}}\Bigr ).\qquad \end{aligned}$$

(3.60)

Proof

For $\alpha _0=\ell _0$ and $\alpha _{\!\mathscr {I}}\in (-\ell _{\!\mathscr {I}},0)$, we can apply Proposition 3.32 to obtain $u\in H_{{\textrm{e,b}};{\textrm{b}}}^{(1;m+8),(\ell _0,2\alpha _{\!\mathscr {I}})}(\Omega )$ satisfying the estimate (3.51) with $m+8$ in place of m. Write

$$\begin{aligned}{} & {} L_g=-2(\rho _{\!\mathscr {I}}\partial _{\rho _{\!\mathscr {I}}}-A_g)(\rho _0\partial _{\rho _0}-\rho _{\!\mathscr {I}}\partial _{\rho _{\!\mathscr {I}}})+2 B_g+L_g^\flat ,\qquad \\{} & {} L_g^\flat \in (x_{\!\mathscr {I}}\mathcal C^\infty (\Omega )+H_{{\textrm{b}}}^{k-2,(\ell _0,2\ell _{\!\mathscr {I}})}(\Omega ))\textrm{Diff}_{\textrm{b}}^2, \end{aligned}$$

where spherical derivatives are error terms since we work in the b-setting now. We now use

$$\begin{aligned} 2(\rho _{\!\mathscr {I}}\partial _{\rho _{\!\mathscr {I}}}-A_g)(\rho _0\partial _{\rho _0}-\rho _{\!\mathscr {I}}\partial _{\rho _{\!\mathscr {I}}})u = f + 2 B_g u + L_g^\flat u \end{aligned}$$

(3.61)

repeatedly, together with the spectral information on $A_g$ given in (3.59a), to prove sharp decay for the various components of h at $\mathscr {I}^+$.

$\bullet {\underline{First\, improvement.}}$ Applying $\pi ^{\mathcal C\Upsilon }$ to (3.61), we get

$$\begin{aligned} (\rho _{\!\mathscr {I}}\partial _{\rho _{\!\mathscr {I}}}-A^{\mathcal C\Upsilon })(\rho _0\partial _{\rho _0}-\rho _{\!\mathscr {I}}\partial _{\rho _{\!\mathscr {I}}})(\pi ^{\mathcal C\Upsilon } u)&\in H_{{\textrm{b}}}^{m+8,(\ell _0,2\ell _{\!\mathscr {I}})} + H_{{\textrm{b}}}^{m+6,(\ell _0,2(\alpha _{\!\mathscr {I}}+\ell _{\!\mathscr {I}}))} \\&\subset H_{{\textrm{b}}}^{m+6,(\ell _0,2(\alpha _{\!\mathscr {I}}+\ell _{\!\mathscr {I}}))}. \end{aligned}$$

Definition 3.20 ensures that all eigenvalues of $A^{\mathcal C\Upsilon }$ are $>\ell _{\!\mathscr {I}}$. Thus, we get improved decay $(\rho _0\partial _{\rho _0}-\rho _{\!\mathscr {I}}\partial _{\rho _{\!\mathscr {I}}})(\pi ^{\mathcal C\Upsilon }u)\in H_{{\textrm{b}}}^{m+6,(\ell _0,2(\alpha _{\!\mathscr {I}}+\ell _{\!\mathscr {I}}))}$ at the cost of 2 b-derivatives. Integrating this from $\Sigma$ (see [22, Lemma 7.7(1)]) and using that $\alpha _{\!\mathscr {I}}+\ell _{\!\mathscr {I}}<\ell _{\!\mathscr {I}}<\ell _0$ gives

$$\begin{aligned} \pi ^{\mathcal C\Upsilon }u\in H_{{\textrm{b}}}^{m+6,(\ell _0,2(\alpha _{\!\mathscr {I}}+\ell _{\!\mathscr {I}}))}. \end{aligned}$$

(3.62)

Applying to (3.61) and using (3.62) to estimate the contributions from and , we obtain

Integrating $\rho _0\partial _{\rho _0}-\rho _{\!\mathscr {I}}\partial _{\rho _{\!\mathscr {I}}}$ gives and therefore

(3.63)

Lastly, we apply $\pi _{1 1}$ to (3.61) and use (3.62)–(3.63), and note that is coupled to $\pi _{1 1}u$ via to obtain

$$\begin{aligned} (\rho _{\!\mathscr {I}}\partial _{\rho _{\!\mathscr {I}}}-A_{1 1})(\rho _0\partial _{\rho _0}-\rho _{\!\mathscr {I}}\partial _{\rho _{\!\mathscr {I}}})(\pi _{1 1}u)\in & {} H_{{\textrm{b}}}^{m+4,1+2\ell _0}(\mathscr {I}^+\cap \Omega ) \\ {}{} & {} \quad + H_{{\textrm{b}}}^{m+4,(\ell _0,2(\alpha _{\!\mathscr {I}}+\ell _{\!\mathscr {I}}))}(\Omega ). \end{aligned}$$

Since $A_{1 1}=-2\gamma ^\Upsilon>\ell _{\!\mathscr {I}}>\alpha _{\!\mathscr {I}}+\ell _{\!\mathscr {I}}$, integration of this implies

$$\begin{aligned} \pi _{1 1}u= & {} u_{1 1}^{(0)} + {\tilde{u}}_{1 1},\qquad u_{1 1}^{(0)} \in H_{{\textrm{b}}}^{m+4,1+2\ell _0}(\mathscr {I}^+\cap \Omega ),\quad \nonumber \\ {\tilde{u}}_{1 1}\in & {} H_{{\textrm{b}}}^{m+4,(\ell _0,2(\alpha _{\!\mathscr {I}}+\ell _{\!\mathscr {I}}))}(\Omega ). \end{aligned}$$

(3.64)

$\bullet {\underline{Second\, improvement.}}$ We again apply $\pi ^{\mathcal C\Upsilon }$ to (3.61); exploiting the sharper (as far as decay is concerned) information (3.62)–(3.64), we now get

$$\begin{aligned} (\rho _{\!\mathscr {I}}\partial _{\rho _{\!\mathscr {I}}}-A^{\mathcal C\Upsilon })(\rho _0\partial _{\rho _0}-\rho _{\!\mathscr {I}}\partial _{\rho _{\!\mathscr {I}}})(\pi ^{\mathcal C\Upsilon }u) \in H_{{\textrm{b}}}^{m+8,(\ell _0,2\ell _{\!\mathscr {I}})} + H_{{\textrm{b}}}^{m+2,(\ell _0,2\ell _{\!\mathscr {I}})}, \end{aligned}$$

with the second term coming from the second order operator $L_g^\flat$ acting on , $\pi _{1 1}u$. Integrating this gives $\pi ^{\mathcal C\Upsilon }u\in H_{{\textrm{b}}}^{m+2,(\ell _0,2\ell _{\!\mathscr {I}})}$. For , this improved information gives

which implies that in (3.63). This in turn gives

$$\begin{aligned} (\rho _{\!\mathscr {I}}\partial _{\rho _{\!\mathscr {I}}}-A_{1 1})(\rho _0\partial _{\rho _0}-\rho _{\!\mathscr {I}}\partial _{\rho _{\!\mathscr {I}}})(\pi _{1 1}u)\in & {} H_{{\textrm{b}}}^{m+4,1+2\ell _0}(\mathscr {I}^+\cap \Omega ) \\ {}{} & {} \quad + H_{{\textrm{b}}}^{m,(\ell _0,2\ell _{\!\mathscr {I}})}(\Omega ), \end{aligned}$$

and hence ${\tilde{u}}_{1 1}\in H_{{\textrm{b}}}^{m,(\ell _0,2\ell _{\!\mathscr {I}})}(\Omega )$ in (3.64). This demonstrates that $u\in \mathscr {G}^{m,(\ell _0,\ell _{\!\mathscr {I}})}$. The tame estimate (3.60) follows from that in Proposition 3.32 together with tame estimates for products, as already exploited in the proof of Proposition 3.32. $\square$

Corollary 3.36

(Nonlinear stability near the far end) Let $\Omega$ and $\Sigma$ be as in Definition  and Lemma , and consider the quasilinear wave operator P(g) from Definition . Let $\ell _0>0$, and let $\ell _{\!\mathscr {I}}\in (0,\min (\ell _0,{\tfrac{1}{2}}))$. Suppose $h_0,h_1\in H_{{\textrm{b}}}^{\infty ,\ell _0}(\Sigma ;S^2\widetilde{T}^*M)$; putting $\Vert (h_0,h_1)\Vert _m:=\Vert h_0\Vert _{H_{{\textrm{b}}}^{m+1,\ell _0}}+\Vert h_1\Vert _{H_{{\textrm{b}}}^{m,\ell _0}}$, assume that $(h_0,h_1)$ is small in the sense that $\Vert (h_0,h_1)\Vert _{22}<C$ where $C=C(\Vert (h_0,h_1)\Vert _{2433})$, with $C=C(q)$ positive and continuous in $q\in [0,\infty )$.¹⁷ Then the initial value problem

$$\begin{aligned} P(g_{\mathfrak {m}}+r^{-1}h) = 0,\qquad (h,\mathcal L_{x_{\!\mathscr {I}}}h)|_{\Sigma } = (h_0,h_1) \end{aligned}$$

(3.65)

has a unique solution $h\in {\tilde{\mathscr {G}}}^{\infty ,(\ell _0,\ell _{\!\mathscr {I}})}$.¹⁸

In particular, if the induced metric and second fundamental form of $g:=g_{\mathfrak {m}}+r^{-1}h\in \mathscr {G}^{\infty ,(\ell _0,\ell _{\!\mathscr {I}})}$ at $\Sigma$ satisfy the constraint equations, and $\Upsilon _{E^\Upsilon }(g;g_{\mathfrak {m}})=0$ at $\Sigma$, then g solves the Einstein vacuum equations $\textrm{Ric}(g)=0$ in the gauge $\Upsilon _{E^\Upsilon }(g;g_{\mathfrak {m}})=0$.

Remark 3.37

(Initial data) Given geometric initial data (i.e. a Riemannian metric and second fundamental form) on $\Sigma$ satisfying the constraint equations, it is easy to construct $h_0,h_1$ so that $g=g_{\mathfrak {m}}+r^{-1}h$, with h having initial data $h_0,h_1$ at $\Sigma$, attains these data at $\Sigma$ and satisfies $\Upsilon _{E^\Upsilon }(g;g_{\mathfrak {m}})=0$ at $\Sigma$, see e.g. [22, Lemma 6.2] (for a slightly different choice of gauge).

Proof of Corollary 3.36

While so far we have only discussed forcing problems, our energy estimate based arguments apply to initial value problems as well. Alternatively, one can piece together a short time solution $h_\textrm{in}$ on $x_{\!\mathscr {I}}^{-1}([{\tfrac{1}{2}}c,c])$, say, with the forward solution of

$$\begin{aligned} P_{\textrm{fw}}(h):= \rho _{\!\mathscr {I}}\rho ^{-3}P\bigl (g_{\mathfrak {m}}+r^{-1}(\chi h_{\textrm{in}}+h)\bigr )=0, \end{aligned}$$

where $\chi \in \mathcal C^\infty _{\textrm{c}}(({\tfrac{1}{2}}c,c])$ is 1 on $[\frac{3}{4} c,c]$. Since $P_{\textrm{fw}}(h)\in H_{{\textrm{b}}}^{\infty ,(\ell _0,\infty )}(\Omega ;S^2\widetilde{T}^*M)$ is small in $H_{{\textrm{b}}}^{22,(\ell _0,1)}(\Omega )$ and has support in $x_{\!\mathscr {I}}^{-1}([{\tfrac{1}{2}}c,\tfrac{3}{4} c])$, Nash–Moser iteration can be applied to the nonlinear map

$$\begin{aligned} {\tilde{\mathscr {G}}}^{\infty ,(\ell _0,\ell _{\!\mathscr {I}})}\ni h\mapsto P_\textrm{fw}(h)\in \mathscr {F}^{\infty ,(\ell _0,\ell _{\!\mathscr {I}})} \end{aligned}$$

in view of Corollary 3.31 and Theorem 3.35, upon restricting to inputs h vanishing on $x_{\!\mathscr {I}}^{-1}([{\tfrac{1}{2}}c,c])$. Indeed, applying the main theorem of [42] with loss of derivatives parameter $d=11$ (cf. (3.60)) produces the solution of (3.65); here, $2433=16 d^2+43 d+24$.

The second part is standard: given a solution $g=g_{\mathfrak {m}}+r^{-1}h$ of (3.65) satisfying the constraint equations and the gauge condition initially, one first concludes that also $\mathcal L_{\partial _{x_{\!\mathscr {I}}}}\Upsilon _{E^\Upsilon }(g;g_{\mathfrak {m}})=0$ at $\Sigma$. The second Bianchi identity implies the homogeneous wave-type equation $2\delta _g{\textsf{G}}_g\delta _{g,E^\mathcal C}^*(\Upsilon _{E^\Upsilon }(g;g_{\mathfrak {m}}))=0$ which gives $\Upsilon _{E^\Upsilon }(g;g_{\mathfrak {m}})\equiv 0$ and therefore, by definition of P(g), also $\textrm{Ric}(g)=0$. $\quad \square$

Remark 3.38

(Gravitational radiation and Bondi mass) Given a Ricci-flat metric $g=g_{\mathfrak {m}}+r^{-1}h\in \mathscr {G}^{\infty ,(\ell _0,\ell _{\!\mathscr {I}})}$ in the gauge $\Upsilon _{E^\Upsilon }(g;g_{\mathfrak {m}})=0$, one can (with some effort) adapt the arguments in [22, §8] to identify the Bondi mass at retarded time $u:=-\rho _0^{-1}=t-r_*$ as

using the notation of Definition 3.20. By (3.47), $M_{\textrm{B}}(u)$ satisfies the mass loss formula

Remark 3.39

(Polyhomogeneity of the metric) The methods of [22, §7] apply, mutatis mutandis, to demonstrate the polyhomogeneity of the spacetime metric $g=g_{\mathfrak {m}}+r^{-1}h$ on M provided the initial data $h_0,h_1$ are polyhomogeneous. Since the metric perturbation h here has stronger decay at $\mathscr {I}^+$ compared to the reference, the index sets will be smaller than in [22, Theorem 7.1].

Appendix A. Constraint Damping and Gauge Change for the Maxwell Equations

As a simpler analogue to the Einstein vacuum equations, we consider the Maxwell equations for a 1-form (gauge potential) A on the domain of dependence $t<r-1$ of the complement of the ball of radius 1 inside $t^{-1}(0)$ inside Minkowski space $({\mathbb {R}}^4,g)$, ,

$$\begin{aligned} \delta _g{\mathrm d}A = 0. \end{aligned}$$

(A.1)

A standard way to break the gauge invariance $A\rightarrow A+{\mathrm d}\chi$ is the imposition of the Lorenz gauge $\delta _g A=0$. The most simple-minded gauge-fixed Maxwell equations are then $\delta _g{\mathrm d}A+{\mathrm d}\delta _g A=0$; this is the tensor wave equation on 1-forms on $({\mathbb {R}}^4,g)$. Constraint damping and a gauge change amount to modifications in the second, gauge breaking, term: letting

$$\begin{aligned} {\mathrm d}_{E^\mathcal C}:= {\mathrm d}+ 2\gamma ^\mathcal C{\mathfrak {c}}^\mathcal C,\qquad \delta _{g,E^\Upsilon }:= \delta _g + 2\gamma ^\Upsilon \iota _{{\mathfrak {c}}^\Upsilon } \end{aligned}$$

for $E^\bullet =({\mathfrak {c}}^\bullet ,\gamma ^\bullet )$, $\bullet =\mathcal C,\Upsilon$, we consider

$$\begin{aligned} P'_{E^\mathcal C,E^\Upsilon }A:= \delta _g{\mathrm d}A + {\mathrm d}_{E^\mathcal C}\delta _{g,E^\Upsilon }A = 0. \end{aligned}$$

(A.2)

On Minkowski space, we concretely take ${\mathfrak {c}}^\mathcal C={\mathfrak {c}}^\Upsilon =r^{-1}\,{\mathrm d}t$, $\gamma ^\mathcal C>0$, $\gamma ^\Upsilon <0$. Writing 1-forms in the splitting (3.12), we compute on functions, resp. 1-forms,

In the notation of §3.1 and Lemma 3.6, recall that decay at $\mathscr {I}^+$, whereas $\partial _1=\rho _0(\rho _0\partial _{\rho _0}-\rho _{\!\mathscr {I}}\partial _{\rho _{\!\mathscr {I}}})$ does not; the analogue of Proposition 3.29 then reads

Thus, if solves $L\omega =0$ with sufficiently decaying initial data, then $\omega _0$ (the Maxwell analogue of $\pi ^\mathcal Ch$ in Definition 3.20 or $\phi _3$ in the model (1.8)) and $\omega _1$ (the Maxwell analogue of $\pi ^\Upsilon h$ or $\phi _4$) decay at $\mathscr {I}^+$, while has a leading order term at $\mathscr {I}^+$. For initial data satisfying the Maxwell constraint equations and the gauge condition $\delta _{g,E^\Upsilon }\omega =0$ initially, $A:=r^{-1}\omega$ solves (A.1) and globally satisfies the gauge condition (using an argument in which the Bianchi identity $\delta _g{\textsf{G}}_g\textrm{Ric}(g)\equiv 0$ is replaced by $\delta _g\delta _g\equiv 0$)

$$\begin{aligned} \delta _{g,E^\Upsilon }A = 0. \end{aligned}$$

(A.3)

Now to leading order at $\mathscr {I}^+$, this gauge condition reads $\partial _1\omega _0=0$ (independently of $\gamma ^\Upsilon$) and thus, by itself, only recovers the improved decay of $\omega _0$ at $\mathscr {I}^+$. The improvement coming from the gauge change encoded by $E^\Upsilon$ only arises once one considers (A.3) together with the Maxwell equations (A.1); on an algebraic level, the gauge condition (A.3) allows one to exchange occurrences of $\partial _1\omega _0$ in (A.1) (in particular in second order terms $\partial _1^2\omega _0$, which do appear for (A.1) but not for gauge-fixed equations such as (A.2)) by lower order terms in the sense of decay, and one particular such combination is (A.2) which implies the desired improved decay of $\omega _1$ due to the structure of $S_{E^\mathcal C,E^\Upsilon }$.

We give a more conceptual (but more abstract) reason for the fact that the gauge change improves decay for a component ($\omega _1$) other than $\omega _0$ (which is affected by constraint damping and the accompanying improved decay of the gauge condition), based on duality considerations. Namely, since ${\mathrm d}_{E^\mathcal C}^*=\delta _{g,E^\mathcal C}$, constraint damping with strength $\gamma ^\mathcal C>0$, resp. a gauge change with strength $\gamma ^\Upsilon <0$, is dual to a gauge change with strength $\gamma ^\mathcal C>0$, resp. constraint damping with strength $\gamma ^\Upsilon <0$ (note the ‘wrong’ signs), in the sense that $E^\Upsilon$ and $E^\mathcal C$ get interchanged when passing from $P'_{E^\mathcal C,E^\Upsilon }$ in (A.2) to its adjoint $(P'_{E^\mathcal C,E^\Upsilon })^*=P'_{E^\Upsilon ,E^\mathcal C}$. Taking for simplicity $E^\mathcal C=0$, ‘negative’ constraint damping (encoded by $E^\Upsilon$ for the adjoint operator) allows one to solve the adjoint (thus, backwards) forcing problem $(P'_{0,E^\Upsilon })^*{\tilde{u}}=\tilde{f}$ for ${\tilde{u}},{\tilde{f}}$ having additional terms with more growth at $\mathscr {I}^+$ than without ‘inverse’ constraint damping—concretely, ${\tilde{f}}$ may have growing contributions which are sections of the bundle $\langle {\mathrm d}x^0+{\mathrm d}x^1\rangle$ spanned by the nonzero eigenvector of $S_{E^\Upsilon ,0}$, or equivalently $\tilde{f}$ lies in a space of more growing 1-forms so that $\pi {\tilde{f}}$ has standard bounds, with $\pi$ any bundle projection with $\ker \pi =\langle {\mathrm d}x^0+{\mathrm d}x^1\rangle$. Dually, this means that one can solve the forward problem for $P_{0,E^\Upsilon }\omega =f$ on function spaces encoding extra decay in certain components—concretely, for suitably decaying f, the solution $\omega$ is the sum of a 1-form with improved decay and a 1-form valued in ${\text {ran}}\pi ^*=(\ker \pi )^\perp =\langle {\mathrm d}x^0-{\mathrm d}x^1\rangle \oplus r T^*{\mathbb {S}}^2$ with standard $r^{-1}$ decay at $\mathscr {I}^+$. Since the annihilator of ${\text {ran}}\pi ^*$ is $\partial _0+\partial _1$, this means that $\omega _0+\omega _1$ has improved decay. Combined with constraint damping (which gives the improved decay of $\omega _0$ as discussed after (A.3)), this finally provides improved decay of $\omega _1$. Analogous remarks apply to the (linearized) gauge-fixed Einstein equations; see Remark 2.3.