Global Stability of Minkowski Space for the Einstein–Maxwell–Klein–Gordon System in Generalized Wave Coordinates

Abstract

We prove global existence for Einstein’s equations with a charged scalar field for initial conditions sufficiently close to the Minkowski spacetime without matter. The proof relies on generalized wave coordinates adapted to the outgoing Schwarzschild light cones and the estimates for the massless Maxwell–Klein–Gordon system, on the background of metrics asymptotically approaching Schwarzschild at null infinity in such coordinates, by Kauffman (Global stability for charged scalar fields in an asymptotically flat metric in harmonic gauge, preprint, 2018). The generalized wave coordinates are obtained from a change of variables, introduced in Lindblad (Commun Math Phys 353(1):135–184, 2017), to asymptotically Schwarzschild coordinates at null infinity. The main technical advances are that the change of coordinates makes critical components of the metric decay faster, making the quasilinear wave operator closer to the flat wave operator, and that commuting with modified Lie derivatives preserves the geometric null structure, improving the error terms. This improved decay of the metric is essential for proving the estimates in Kauffman (2018) and will likely be useful in other contexts as well.

Appendices

Appendix A. The Ricci Curvature in Terms of Generalized Wave Coordinates

Here we derive the expression for Einstein’s equations for the metric g in terms of a quantity that is assumed to be under control by a generalized wave coordinate condition. In this paper, we will only use these equations in the case this quantity vanishes. However, in Appendix 16.1.5 we show that the equations we use can alternatively be derived as expressing the metric in generalized wave coordinates.

We consider the generalized harmonic coordinate condition

$$\begin{aligned} \Gamma ^\alpha = \Gamma ^{\alpha }_{\beta \gamma }g^{\beta \gamma }, \end{aligned}$$

(A.1a)

where $\Gamma$ is a known vector. Using this notation, the harmonic coordinate condition is equivalent to the condition $\Gamma ^\alpha = 0$. We can rewrite (A.1a) as

$$\begin{aligned} g^{\alpha \beta }\partial _\alpha g_{\beta \gamma } = \Gamma _\gamma + \Gamma ^{\delta }_{\gamma \delta }. \end{aligned}$$

(A.1b)

The Ricci curvature tensor

$$\begin{aligned} R_{\alpha \beta } = \partial _\gamma \Gamma ^{\gamma }_{\alpha \beta } - \partial _\beta \Gamma ^{\gamma }_{\alpha \gamma } + \Gamma ^\gamma _{\gamma \rho }\Gamma ^{\rho }_{\alpha \beta } - \Gamma ^\gamma _{\alpha \rho }\Gamma ^{\rho }_{\beta \gamma }, \end{aligned}$$

satisfies the following identity

Lemma A.1

$$\begin{aligned} R_{\alpha \beta }= & {} -\frac{1}{2} {\square }^{\,g} g_{\alpha \beta } +\frac{1}{2} \Gamma ^\delta \partial _\delta g_{\alpha \beta } + \frac{1}{2}(\partial _\alpha \Gamma _\beta + \partial _\beta \Gamma _\alpha ) -\!\frac{1}{2} \Gamma _\alpha \Gamma _\beta \\{} & {} + \frac{1}{2}\Gamma ^\gamma _{\alpha \gamma }\Gamma ^{\rho }_{\beta \rho }\! - \frac{1}{4} g^{\gamma \delta }g^{\rho \sigma }\partial _\alpha g_{\delta \rho }\partial _\beta g_{\sigma \gamma }\!+ Q_{\alpha \beta }, \end{aligned}$$

where $Q_{\alpha \beta }$ is a linear combination of classical null forms.

Remark A.2

Note that the second term on the right is exactly the difference between the geometric wave operator and the reduced wave operator. Having this term together with the other semilinear terms provides an additional cancelation, c.f. [16].

Proof

For the proof, we expand each of the four terms in (A). First,

$$\begin{aligned} \partial _\gamma \Gamma ^{\gamma }_{\alpha \beta }&= \frac{1}{2}(\partial _\gamma g^{\gamma \delta })(\partial _\alpha g_{\beta \delta } + \partial _\beta g_{\alpha \delta } - \partial _\delta g_{\alpha \beta }) \\&\quad + \frac{1}{2} g^{\gamma \delta }(\partial _\alpha \partial _\gamma g_{\beta \delta } + \partial _\beta \partial _\gamma g_{\alpha \delta } - \partial _\gamma \partial _\delta g_{\alpha \beta }) \\&= \frac{1}{2}(\partial _\gamma g^{\gamma \delta })(\partial _\alpha g_{\beta \delta } + \partial _\beta g_{\alpha \delta } - \partial _\delta g_{\alpha \beta }) \\&\quad + \frac{1}{2} (\partial _\alpha (g^{\gamma \delta }\partial _\gamma g_{\beta \delta }) + \partial _\beta (g^{\gamma \delta }\partial _\gamma g_{\alpha \delta }) - g^{\gamma \delta }\partial _\gamma \partial _\delta g_{\alpha \beta }) \\&\quad - \frac{1}{2}\partial _\alpha g^{\gamma \delta }\partial _\gamma g_{\beta \delta } - \frac{1}{2}\partial _\beta g^{\gamma \delta }\partial _\gamma g_{\alpha \delta }. \end{aligned}$$

It follows from (A.1b) that

$$\begin{aligned} -\frac{1}{2}\partial _\gamma g^{\gamma \delta } = \frac{1}{2} \Gamma ^\delta + \frac{1}{4} g^{\rho \sigma }g^{\delta \gamma }\partial _\gamma g_{\rho \sigma }, \end{aligned}$$

and therefore,

$$\begin{aligned} -\frac{1}{2}\partial _\gamma g^{\gamma \delta }\partial _\delta g_{\alpha \beta } = \frac{1}{2} \Gamma ^\delta \partial _\delta g_{\alpha \beta } + \frac{1}{4} g^{\rho \sigma }g^{\delta \gamma }\partial _\gamma g_{\rho \sigma }\partial _\delta g_{\alpha \beta }. \end{aligned}$$

If we define

$$\begin{aligned} Q^1_{\alpha \beta }= & {} \frac{1}{2}(\partial _\gamma g^{\gamma \delta }\partial _\alpha g_{\beta \gamma } - \partial _\alpha g^{\gamma \delta }\partial _\gamma g_{\beta \gamma }) +\frac{1}{2}( \partial _\gamma g^{\gamma \delta }\partial _\beta g_{\alpha \delta } - \partial _\beta g^{\gamma \delta }\partial _\gamma g_{\beta \delta })\\{} & {} + \frac{1}{4} g^{\rho \sigma }g^{\delta \gamma }\partial _\gamma g_{\rho \sigma }\partial _\delta g_{\alpha \beta }, \end{aligned}$$

and apply (A.1b) to rewrite

$$\begin{aligned} \frac{1}{2}\partial _\alpha (g^{\gamma \delta }\partial _\gamma g_{\beta \delta })&= \frac{1}{2}\partial _\alpha \Gamma _\beta + \frac{1}{4} \partial _\alpha g^{\gamma \delta }\partial _\beta g_{\gamma \delta } + \frac{1}{4}g^{\gamma \delta }\partial _\alpha \partial _\beta g_{\gamma \delta }, \\ \frac{1}{2}\partial _\beta (g^{\gamma \delta }\partial _\gamma g_{\alpha \delta })&= \frac{1}{2}\partial _\beta \Gamma _\alpha + \frac{1}{4} \partial _\beta g^{\gamma \delta }\partial _\alpha g_{\gamma \delta } + \frac{1}{4}g^{\gamma \delta }\partial _\alpha \partial _\beta g_{\gamma \delta }, \end{aligned}$$

we have the following identity:

$$\begin{aligned} \partial _\gamma \Gamma ^\gamma _{\alpha \beta }= & {} \!\frac{1}{2} \Gamma ^\delta \partial _\delta g_{\alpha \beta } \! + \! \frac{1}{2}(\partial _\alpha \Gamma _\beta + \partial _\beta \Gamma _\alpha ) \! + \!\frac{1}{4}(\partial _\alpha g^{\gamma \delta } \partial _\beta g_{\gamma \delta } \! + \!\partial _\beta g^{\gamma \delta }\partial _\alpha g_{\gamma \delta }) \!\\{} & {} + \!\frac{1}{2} g^{\gamma \delta }\partial _\alpha \partial _\beta g_{\gamma \delta } \!- \!\frac{1}{2} {\widetilde{\square }}_g g_{\alpha \beta } \! + \! Q^1_{\alpha \beta }. \end{aligned}$$

Now we recall the identity

$$\begin{aligned} \Gamma _{\alpha \gamma }^\gamma = \frac{1}{2} g^{\gamma \delta }\partial _\alpha g_{\gamma \delta }. \end{aligned}$$

It follows that

$$\begin{aligned} -\partial _\beta \Gamma ^{\gamma }_{\alpha \gamma } = -\frac{1}{2}\partial _\beta g^{\gamma \delta } \partial _\alpha g_{\gamma \delta } - \frac{1}{2} g^{\gamma \delta }\partial _\alpha \partial _\beta g_{\gamma \delta }. \end{aligned}$$

We can define

$$\begin{aligned} Q^2_{\alpha \beta }= & {} \frac{1}{4}(\partial _\alpha g^{\gamma \delta }\partial _\beta g_{\gamma \delta } - \partial _\beta g^{\gamma \delta }\partial _\alpha g_{\gamma \delta }) \end{aligned}$$

in order to rewrite

$$\begin{aligned} -\partial _\beta \Gamma ^{\gamma }_{\alpha \gamma } = -\frac{1}{4}(\partial _\alpha g^{\gamma \delta }\partial _\beta g_{\gamma \delta } + \partial _\beta g^{\gamma \delta }\partial _\alpha g_{\gamma \delta }) - \frac{1}{2} g^{\gamma \delta }\partial _\alpha \partial _\beta g_{\gamma \delta } + Q^2_{\alpha \beta }. \end{aligned}$$

Next, we expand

$$\begin{aligned} \Gamma ^\gamma _{\gamma \rho }\Gamma ^{\rho }_{\alpha \beta }&= \frac{1}{4} g^{\rho \sigma }g^{\gamma \delta }\partial _\rho g_{\gamma \delta }\left( \partial _\alpha g_{\beta \sigma } + \partial _\beta g_{\alpha \sigma } - \partial _\sigma g_{\alpha \beta }\right) . \end{aligned}$$

We define

$$\begin{aligned} Q^3_{\alpha \beta }= & {} \frac{1}{4}g^{\rho \sigma }g^{\gamma \delta }\left( (\partial _\rho g_{\gamma \delta } \partial _\alpha g_{\beta \sigma } - \partial _\alpha g_{\gamma \delta }\partial _\rho g_{\beta \sigma })\right. \\{} & {} \left. + (\partial _\rho g_{\gamma \delta }\partial _\beta g_{\alpha \sigma } - \partial _\beta g_{\gamma \delta }\partial _\rho g_{\alpha \sigma }) - \partial _\rho g_{\gamma \delta } \partial _\sigma g_{\alpha \beta }\right) \end{aligned}$$

and rewrite

$$\begin{aligned} \frac{1}{4} g^{\gamma \delta }\partial _\alpha g_{\gamma \delta }g^{\rho \sigma }\partial _\rho g_{\beta \sigma }&= \frac{1}{4} g^{\gamma \delta }\partial _\alpha g_{\gamma \delta }\Big (\Gamma _\beta + \frac{1}{2} g^{\rho \sigma }\partial _\beta g_{\rho \sigma }\Big )\\ \frac{1}{4} g^{\gamma \delta }\partial _\beta g_{\gamma \delta }g^{\rho \sigma }\partial _\rho g_{\alpha \sigma }&=\frac{1}{4} g^{\gamma \delta }\partial _\beta g_{\gamma \delta }\Big (\Gamma _\alpha + \frac{1}{2} g^{\rho \sigma }\partial _\alpha g_{\rho \sigma }\Big ). \end{aligned}$$

We therefore have

$$\begin{aligned} \Gamma ^\gamma _{\gamma \rho }\Gamma ^{\rho }_{\alpha \beta } = Q^3_{\alpha \beta } + \frac{1}{2}(\Gamma ^{\gamma }_{\alpha \gamma }\Gamma _\beta + \Gamma ^{\gamma }_{\beta \gamma }\Gamma _\alpha ) + \Gamma ^\gamma _{\alpha \gamma }\Gamma ^{\rho }_{\beta \rho }. \end{aligned}$$

Now we take the final term,

$$\begin{aligned} -\Gamma ^\gamma _{\alpha \rho }\Gamma ^\rho _{\beta \gamma } = -\frac{1}{4} g^{\gamma \delta }g^{\rho \sigma }(\partial _\alpha g_{\delta \rho } + \partial _\rho g_{\alpha \delta } - \partial _\delta g_{\alpha \rho })(\partial _\beta g_{\sigma \gamma } + \partial _\gamma g_{\beta \sigma }-\partial _\sigma g_{\beta \gamma }). \end{aligned}$$

We expand it and deal with it term by term. We first take

$$\begin{aligned} -\frac{1}{4} g^{\gamma \delta }g^{\rho \sigma }\partial _\alpha g_{\delta \rho }(\partial _\gamma g_{\beta \sigma } - \partial _\sigma g_{\beta \gamma })= & {} -\frac{1}{4}g^{\gamma \delta }g^{\rho \sigma }((\partial _\alpha g_{\delta \rho }\partial _\gamma g_{\beta \sigma } - \partial _\gamma g_{\delta \rho }\partial _\alpha g_{\beta \sigma }) \\{} & {} -(\partial _\alpha g_{\delta \rho }\partial _\sigma g_{\beta \gamma } - \partial _\sigma g_{\delta \rho }\partial _\alpha g_{\beta \gamma })) \quad \\{} & {} -\frac{1}{4} g^{\gamma \delta }g^{\rho \sigma }(\partial _\gamma g_{\delta \rho }\partial _{\alpha }g_{\beta \sigma } - \partial _\sigma g_{\delta \rho }\partial _\alpha g_{\beta \gamma }), \\= & {} -\frac{1}{4}g^{\gamma \delta }g^{\rho \sigma }((\partial _\alpha g_{\delta \rho }\partial _\gamma g_{\beta \sigma } - \partial _\gamma g_{\delta \rho }\partial _\alpha g_{\beta \sigma })\\{} & {} -(\partial _\alpha g_{\delta \rho }\partial _\sigma g_{\beta \gamma } - \partial _\sigma g_{\delta \rho }\partial _\alpha g_{\beta \gamma })). \end{aligned}$$

The two terms that cancel out are identical, which follows straightforwardly from renaming contracted indices and noting symmetry of g. Similarly,

$$\begin{aligned} -\frac{1}{4} g^{\gamma \delta }g^{\rho \sigma }\partial _\beta g_{\sigma \gamma }(\partial _\rho g_{\alpha \delta }\! - \partial _\delta g_{\alpha \rho })= & {} -\frac{1}{4}g^{\gamma \delta }g^{\rho \sigma }((\partial _\beta g_{\sigma \gamma }\partial _\rho g_{\alpha \gamma } \!- \partial _\rho g_{\sigma \gamma }\partial _\beta g_{\alpha \gamma }) \\{} & {} - (\partial _\beta g_{\sigma \gamma }\partial _\delta g_{\alpha \rho }\! - \partial _\delta g_{\sigma \gamma }\partial _\beta g_{\alpha \rho })). \end{aligned}$$

Next, we take

$$\begin{aligned} -\frac{1}{4} g^{\gamma \delta }g^{\rho \sigma }(\partial _\rho g_{\alpha \delta }\partial _\gamma g_{\beta \sigma } + \partial _\delta g_{\alpha \rho }\partial _\sigma g_{\beta \gamma }). \end{aligned}$$

By renaming indices, we have

$$\begin{aligned} -\frac{1}{4} g^{\gamma \delta }g^{\rho \sigma }(\partial _\rho g_{\alpha \delta }\partial _\gamma g_{\beta \sigma } + \partial _\delta g_{\alpha \rho }\partial _\sigma g_{\beta \gamma })= & {} -\frac{1}{2} g^{\gamma \delta }g^{\rho \sigma }(\partial _\rho g_{\alpha \delta }\partial _\gamma g_{\beta \sigma }) \\= & {} -\frac{1}{2} g^{\gamma \delta }g^{\rho \sigma } ((\partial _\rho g_{\alpha \delta }\partial _\gamma g_{\beta \sigma } - \partial _\gamma g_{\alpha \delta } \partial _\rho g_{\beta \sigma })\\{} & {} + \partial _\gamma g_{\alpha \delta }\partial _\rho g_{\beta \sigma }) \\= & {} -\frac{1}{2} g^{\gamma \delta }g^{\rho \sigma }(\partial _\rho g_{\alpha \delta }\partial _\gamma g_{\beta \sigma } - \partial _\gamma g_{\alpha \delta } \partial _\rho g_{\beta \sigma })\\{} & {} -\frac{1}{2}(\Gamma _\alpha + \Gamma ^{\gamma }_{\alpha \gamma })(\Gamma _\beta + \Gamma ^\delta _{\beta \delta }). \end{aligned}$$

Defining

$$\begin{aligned} Q^4_{\alpha \beta }&= -\frac{1}{4}g^{\gamma \delta }g^{\rho \sigma }((\partial _\alpha g_{\delta \rho }\partial _\gamma g_{\beta \sigma } - \partial _\gamma g_{\delta \rho }\partial _\alpha g_{\beta \sigma })-(\partial _\alpha g_{\delta \rho }\partial _\sigma g_{\beta \gamma } - \partial _\sigma g_{\delta \rho }\partial _\alpha g_{\beta \gamma })) \\&\quad -\frac{1}{4}g^{\gamma \delta }g^{\rho \sigma }((\partial _\beta g_{\sigma \gamma }\partial _\rho g_{\alpha \gamma } - \partial _\rho g_{\sigma \gamma }\partial _\beta g_{\alpha \gamma }) - (\partial _\beta g_{\sigma \gamma }\partial _\delta g_{\alpha \rho } - \partial _\delta g_{\sigma \gamma }\partial _\beta g_{\alpha \rho })) \\&\quad + \frac{1}{4} g^{\gamma \delta }g^{\rho \sigma }\partial _\rho g_{\alpha \delta }\partial _\sigma g_{\beta \gamma } + \frac{1}{4} g^{\gamma \delta }g^{\rho \sigma }\partial _\delta g_{\alpha \rho }\partial _\gamma g_{\beta \sigma }\\&\quad -\frac{1}{2} g^{\gamma \delta }g^{\rho \sigma }(\partial _\rho g_{\alpha \delta }\partial _\gamma g_{\beta \sigma } - \partial _\gamma g_{\alpha \delta } \partial _\rho g_{\beta \sigma }), \end{aligned}$$

we can expand

$$\begin{aligned} -\Gamma ^\gamma _{\alpha \rho }\Gamma ^\rho _{\beta \gamma } = -\frac{1}{4} g^{\gamma \delta }g^{\rho \sigma }\partial _\alpha g_{\delta \rho }\partial _\beta g_{\sigma \gamma } - \frac{1}{2} (\Gamma _\alpha + \Gamma ^{\gamma }_{\alpha \gamma })(\Gamma _\beta + \Gamma ^\delta _{\beta \delta })+ Q^4_{\alpha \beta }. \end{aligned}$$

We can combine everything to get

$$\begin{aligned} R_{\alpha \beta }&= -\frac{1}{2} {\square }^{\,g} g_{\alpha \beta } +\frac{1}{2} \Gamma ^\delta \partial _\delta g_{\alpha \beta }\\&\quad + \frac{1}{2}(\partial _\alpha \Gamma _\beta + \partial _\beta \Gamma _\alpha ) -\frac{1}{2} \Gamma _\alpha \Gamma _\beta \\&\quad + \frac{1}{2}\Gamma ^\gamma _{\alpha \gamma }\Gamma ^{\rho }_{\beta \rho } - \frac{1}{4} g^{\gamma \delta }g^{\rho \sigma }\partial _\alpha g_{\delta \rho }\partial _\beta g_{\sigma \gamma }+ \\&\quad + Q^1_{\alpha \beta } + Q^2_{\alpha \beta } + Q^3_{\alpha \beta } + Q^4_{\alpha \beta }. \end{aligned}$$

$\square$

1.1 A.0.6. The Expression for Ricci Curvature in Terms of the Wave Operator

For a general metric, g twice the Ricci curvature can by Lemma A.1 be written

$$\begin{aligned} 2 R_{\mu \nu }= & {} -\square ^{\,g} g_{\mu \nu }+\partial _\mu \Gamma _{\!\nu }+\partial _\nu \Gamma _{\!\mu } +F_{\!\mu \nu }(g)[\partial g,\partial g]\nonumber \\{} & {} -\Gamma _{\!\mu } \Gamma _{\!\nu }+\Gamma ^\delta \partial _\delta g_{\mu \nu }, \quad \text {where} \quad \Gamma _{\!\mu }\!=g_{\mu \delta }g^{\alpha \beta }\Gamma _{\!\alpha \beta }^\delta , \end{aligned}$$

(A.3)

and $F_{\mu \nu }$ is as in (1.6). The Einstein vacuum equations in harmonic coordinates are $R_{\mu \nu }\!=0$ and $\Gamma _{\!\mu }\!=0$. The Minkowski metric and the Schwarzschild expressed in harmonic coordinates satisfy these. Since $m_0$ in (4.5) is the leading term in the expansion of the Schwarzschild metric, it is therefore not surprising that it approximately satisfies these. By a similar calculation as above using (4.2), we have

$$\begin{aligned} \Gamma ^{0\mu }= & {} m_0^{\alpha \beta } \Gamma ^{0\mu }_{\alpha \beta } =\partial _\alpha m_0^{\alpha \beta } -\tfrac{1}{2}m_0^{\alpha \beta } m^{-1}_{0\gamma \mu }\,\partial _\alpha m_0^{\gamma \mu } \\= & {} \partial _\alpha H_0^{\alpha \mu } -\tfrac{1}{2}m^{\gamma \mu } m_{\alpha \beta }\,\partial _\gamma H_0^{\alpha \beta } +W^\mu (H_0)[H_0,\partial H_0]\\= & {} {\chi }^{\,\prime \!}\big (\tfrac{r}{t+1}\big )M\delta ^{\mu 0} r^{-2} +\chi ^{\mu }\big (\tfrac{r}{1+t},\omega ,\tfrac{M}{r}\big )M^2r^{-3}. \end{aligned}$$

Also using (4.7), it follows that

$$\begin{aligned} 2 R_{\mu \nu }^0= & {} -\square ^{m_0} m^0_{\mu \nu }+\partial _\mu \Gamma ^0_\nu +\partial _\nu \Gamma ^0_\mu +\Gamma ^{0\delta }\partial _\delta m^0_{\mu \nu } +F_{\mu \nu }(m_0)[\partial m^0,\partial m^0]-\Gamma ^0_\mu \Gamma ^0_\nu \\= & {} \chi ^\prime _{2,\,\mu \nu }\big (\tfrac{r}{1+t},\omega ,\tfrac{M}{r}\big )M r^{-3} +\chi _{2,\,\mu \nu }\big (\tfrac{r}{1+t},\omega ,\tfrac{M}{r}\big )M^2 r^{-4}. \end{aligned}$$

Appendix B. Einstein’s Equations in Generalized Wave Coordinates

There is a different way of deriving the reduced Einstein’s equations in the new coordinates. Instead of using covariant derivatives with respect to the new coordinates, one can consider the new coordinates as generalized wave coordinates and use the procedure for deriving Einstein’s equations in generalized wave coordinates. Although we decided not to do it that way, here we include the calculation since it is of interest that it can be interpreted this way and it could be used elsewhere.

1.1 B.0.7. The Reduced Einstein’s Equations in Generalized Wave Coordinates

By (A.3), Einstein’s equations ${\widetilde{R}}_{ab}=0$ in generalized wave coordinates are

$$\begin{aligned}{} & {} {\widetilde{\square }}^{\,{\widetilde{g}}}\, {\widetilde{g}}_{ab} -{\widetilde{\partial }}_a{\mathcal {W}}_{\!b}({{\widetilde{g}}})-{\widetilde{\partial }}_b {\mathcal {W}}_{\!a}({{\widetilde{g}}})-{\mathcal {W}}^{c}({{\widetilde{g}}}) {\widetilde{\partial }}_c{\widetilde{g}}_{ab}\nonumber \\{} & {} \quad ={\widetilde{F}}_{\!ab} ({{\widetilde{g}}}) [{\widetilde{\partial }}{\widetilde{g}}, {\widetilde{\partial }}{\widetilde{g}}]-{\mathcal {W}}_{\!a}({{\widetilde{g}}}){\mathcal {W}}_{\!b}({{\widetilde{g}}})+{\widetilde{T}}_{ab}, \end{aligned}$$

(B.1)

where ${\widetilde{\square }}^{\,{{\widetilde{g}}}\!}\!= {\widetilde{g}}^{ab}{\widetilde{\partial }}_a{\widetilde{\partial }}_b$, ${\mathcal {W}}_{\!c\!}\!={{\widetilde{g}}}_{\!cd}{\mathcal {W}}^{d}$ and ${\mathcal {W}}^c$ are some given functions of the coordinate ${\widetilde{x}}$ and the metric ${\widetilde{g}}$ not depending on its derivative. One can show that if ${\widetilde{g}}$ satisfy the reduced equations (B.1) and ${\widetilde{\Gamma }}^{c}_{ab}$ are its Christoffel symbols, then the generalized wave coordinate condition

$$\begin{aligned} {\widetilde{g}}^{ab}{\widetilde{\Gamma }}^{c}_{ab} ={\mathcal {W}}^c({{\widetilde{g}}}) \end{aligned}$$

holds if it holds initially. In particular if g is the metric expressed in wave coordinates x, and we choose new coordinates ${\widetilde{x}}={\widetilde{x}}(x)$ which are given fixed functions of the original coordinates then by (5.5)–(5.4)

$$\begin{aligned} {\mathcal {W}}^c({{\widetilde{g}}}) ={{\widetilde{g}}}^{ ab}{\widehat{\Gamma }}^{c}_{ab} ={\mathcal {W}}^c({{\widetilde{g}}})[{\widehat{\Gamma }}], \end{aligned}$$

(B.2)

where ${\widehat{\Gamma }}$ given by (5.4) are the Christoffel symbols of the fixed metric ${\widetilde{m}}$ and the last equality indicates that it is linear in ${\widehat{\Gamma }}$ which by itself is $O(M\langle t+r\rangle ^{-2}\ln {\langle t+r\rangle })$. By (B.1) and (B.2),

$$\begin{aligned}{} & {} {\widetilde{\square }}^{\,{\widetilde{g}}}\, {\widetilde{g}}_{ab} -{\widetilde{\partial }}_a{\mathcal {W}}_b({{\widetilde{g}}})[{\widehat{\Gamma }}] -{\widetilde{\partial }}_b{\mathcal {W}}_a({{\widetilde{g}}})[{\widehat{\Gamma }}] -{\mathcal {W}}^c({{\widetilde{g}}})[{\widehat{\Gamma }}]\, {\widetilde{\partial }}_c{\widetilde{g}}_{ab} \\{} & {} \quad ={\widetilde{F}}_{ab} ({{\widetilde{g}}}) [{\widetilde{\partial }}{\widetilde{g}},{\widetilde{\partial }}{\widetilde{g}}] -{\mathcal {W}}_{ab}({{\widetilde{g}}})[{\widehat{\Gamma }},{\widehat{\Gamma }}] +{\widetilde{T}}_{ab}, \end{aligned}$$

where

$$\begin{aligned} {\mathcal {W}}_c({{\widetilde{g}}})[{\widehat{\Gamma }}] ={\widetilde{g}}_{cd}{\mathcal {W}}^d({{\widetilde{g}}})[{\widehat{\Gamma }}], \quad \text {and}\quad {\mathcal {W}}_{ab}({{\widetilde{g}}})[{\widehat{\Gamma }},{\widehat{\Gamma }}] ={\mathcal {W}}_a({{\widetilde{g}}})[{\widehat{\Gamma }}]\, {\mathcal {W}}_b({{\widetilde{g}}})[{\widehat{\Gamma }}] \end{aligned}$$

Now let $h_{\alpha \beta }=g_{\alpha \beta }-m_{\alpha \beta }$ and $H^{\alpha \beta }=g^{\alpha \beta }-m^{\alpha \beta }=-h_{\alpha \beta }+O(h^2)$

$$\begin{aligned} {\widetilde{h}}_{ab}={\widetilde{g}}_{ab}-{\widetilde{m}}_{ab}= A^{\,\,\alpha }_{ a} A^{\,\,\beta }_{ b} h_{\alpha \beta } \quad \text {and}\quad {\widetilde{H}}^{ab}={\widetilde{g}}^{ab}-{\widetilde{m}}^{ab}= A_{\,\,\alpha }^{ a} A_{\,\,\beta }^{ b} H^{\alpha \beta }. \end{aligned}$$

Since in particular the Minkowski metric is a solution of the vacuum equations, we have by (B.1) and (B.2) with ${\widetilde{m}}$ in place of ${{\widetilde{g}}}$

$$\begin{aligned}{} & {} {\widetilde{\square }}^{\,{\widetilde{m}}}\, {\widetilde{m}}_{ab} -{\widetilde{\partial }}_a{\mathcal {W}}_b({\widetilde{m}})[{\widehat{\Gamma }}] -{\widetilde{\partial }}_b{\mathcal {W}}_a({\widetilde{m}})[{\widehat{\Gamma }}] -{\mathcal {W}}^c({\widetilde{m}})[{\widehat{\Gamma }}]\, {\widetilde{\partial }}_c{\widetilde{m}}_{ab}\\{} & {} \quad ={\widetilde{F}}_{ab} ({\widetilde{m}}) [{\widetilde{\partial }}{\widetilde{m}}, {\widetilde{\partial }}{\widetilde{m}}] -{\mathcal {W}}_{ab}({\widetilde{m}})[{\widehat{\Gamma }},{\widehat{\Gamma }}]. \end{aligned}$$

Subtracting the two equations, we get an equation for the difference

$$\begin{aligned}{} & {} \!\!\!\!\!\!{\widetilde{\square }}^{\,{\widetilde{g}}} {\widetilde{h}}_{ab}-{\widetilde{H}}^{cd}{\widetilde{\partial }}_c{\widetilde{\partial }}_d {\widetilde{m}}_{ab} -{\widetilde{\partial }}_a{\widetilde{W}}_{\!b}({\widetilde{h}},{\widetilde{m}}) [{\widetilde{h}},{\widehat{\Gamma }}] -{\widetilde{\partial }}_b{\widetilde{W}}_{\!a}({\widetilde{h}},{\widetilde{m}}) [{\widetilde{h}},{\widehat{\Gamma }}]\\{} & {} \qquad -{\mathcal {W}}^{c\!}({{\widetilde{g}}}\!)[{\widehat{\Gamma }}]\, {\widetilde{\partial }}_c{\widetilde{h}}_{ab} -{\widetilde{W}}_{2}^c[{\widetilde{H}},{\widehat{\Gamma }}]{\widetilde{\partial }}_c{\widetilde{m}}_{ab}\\{} & {} \quad ={\widetilde{F}}_{ab} ({{\widetilde{g}}}) [{\widetilde{\partial }}{\widetilde{g}},{\widetilde{\partial }}{\widetilde{g}}] -{\widetilde{F}}_{ab} ({\widetilde{m}}) [{\widetilde{\partial }}{\widetilde{m}}, {\widetilde{\partial }}{\widetilde{m}}] -{\widetilde{W}}_{ab}({\widetilde{h}},{\widetilde{m}}) [{\widetilde{h}},{\widehat{\Gamma }},{\widehat{\Gamma }}] +{\widetilde{T}}_{ab}, \end{aligned}$$

where

$$\begin{aligned} {\widetilde{W}}_c({\widetilde{h}},{\widetilde{m}})[{\widetilde{h}},{\widehat{\Gamma }}]&={\mathcal {W}}_c({{\widetilde{g}}})[{\widehat{\Gamma }}] -{\mathcal {W}}_c({\widetilde{m}})[{\widehat{\Gamma }}],\\ {\widetilde{W}}_{ab}({\widetilde{h}},{\widetilde{m}}) [{\widetilde{h}},{\widehat{\Gamma }},{\widehat{\Gamma }}]&={\mathcal {W}}_{ab}({{\widetilde{g}}})[{\widehat{\Gamma }},{\widehat{\Gamma }}] -{\mathcal {W}}_{ab}({\widetilde{m}})[{\widehat{\Gamma }},{\widehat{\Gamma }}]\\ {\mathcal {W}}^c({{\widetilde{g}}})[{\widehat{\Gamma }}]\,{\widetilde{\partial }}_c{\widetilde{h}}_{ab} +{\widetilde{W}}_{2}^c[{\widetilde{H}},{\widehat{\Gamma }}]{\widetilde{\partial }}_c{\widetilde{m}}_{ab}&={\mathcal {W}}^c({{\widetilde{g}}})[{\widehat{\Gamma }}]\, {\widetilde{\partial }}_c{\widetilde{g}}_{ab} -{\mathcal {W}}^c({\widetilde{m}})[{\widehat{\Gamma }}]\, {\widetilde{\partial }}_c{\widetilde{m}}_{ab}. \end{aligned}$$

Here $W(\cdots )[u_1,\cdots ,u_k]$ stands for functions that are separately linear in each of the arguments $u_1,\dots ,u_k$.

1.2 B.0.8. The Generalized Wave Coordinate Condition

We have

$$\begin{aligned} {\widetilde{\partial }}_a {{\widetilde{g}}}^{ac} -\tfrac{1}{2}{{\widetilde{g}}}^{ac} {{\widetilde{g}}}_{bd}\,{\widetilde{\partial }}_a {{\widetilde{g}}}^{bd} ={\mathcal {W}}^c({{\widetilde{g}}})[{\widehat{\Gamma }}] \end{aligned}$$

and in particular

$$\begin{aligned} {\widetilde{\partial }}_a {\widetilde{m}}^{ac} -\tfrac{1}{2}{\widetilde{m}}^{ac} {\widetilde{m}}_{bd}\, {\widetilde{\partial }}_a {\widetilde{m}}^{bd} ={\mathcal {W}}^c({\widetilde{m}})[{\widehat{\Gamma }}]. \end{aligned}$$

It follows that ${\widetilde{H}}^{ab}={{\widetilde{g}}}^{ab}-{\widetilde{m}}^{ab}$ satisfy

$$\begin{aligned} {\widetilde{\partial }}_a \big ({\widetilde{H}}^{ac} -\tfrac{1}{2}{\widetilde{m}}^{ac} {\widetilde{m}}_{bd}\, {\widetilde{H}}^{bd}\big ) ={\widetilde{W}}_1^c({\widetilde{m}},{\widetilde{H}})[{\widetilde{H}},{\widetilde{\partial }}{\widetilde{m}}] +{\widetilde{W}}_2^c({\widetilde{m}},{\widetilde{H}})[{\widetilde{H}},{\widetilde{\partial }}{\widetilde{H}}]. \end{aligned}$$

Appendix C. Additional Interior Asymptotics

1.1 C.0.9. Subtracting off a Better Approximation from the Homogeneous Wave Equation Picking up the Mass in the Exterior

We can get an improved approximation to the solution h of the homogeneous wave equation in the interior of the light cone by instead of having a cutoff which is a function of r/t have a cutoff which is a function of $r^*-t$. This, however, only works in the wave equation but not in the solution of the wave coordinate condition H. This is not needed for the energy estimate and existence but only for more precise asymptotics in the interior, see [35] for the detailed proof. Here we just want to show how the error terms are under control because of the wave coordinate condition. One can use this and the other results in this paper to give a more direct proof of the asymptotics in [35].

Let $\square ^*={\widehat{m}}^{ab}{\widetilde{\partial }}_a{\widetilde{\partial }}_b$ and

$$\begin{aligned} {\widetilde{h}}^{1\, e}_{ab}={\widetilde{h}}_{ab}-{\widetilde{h}}^{0\, e}_{ab},\qquad \text {where}\quad {\widetilde{h}}^{0\, e}_{ab}=\frac{M}{r^*} {\widetilde{\chi }}(r^*\!\!-t) \delta _{ab} \end{aligned}$$

We have

$$\begin{aligned} \square ^* {\widetilde{h}}^{0\, e}_{cd}=0 \end{aligned}$$

Moreover, with $L^*_a={\widetilde{\partial }}_\alpha (r^{\!*}\!-t)$, $\omega _0=0$, $\omega _i=x_i/|x|$, for $i\ge 1$ and and

Hence, with $R=(0,\omega )$