Stability of Minkowski Space-Time with a Translation Space-Like Killing Field

Abstract

In this paper we prove the nonlinear stability of Minkowski space-time with a translation Killing field. In the presence of such a symmetry, the $3+1$ vacuum Einstein equations reduce to the $2+1$ Einstein equations with a scalar field. We work in generalised wave coordinates. In this gauge Einstein’s equations can be written as a system of quasilinear quadratic wave equations. The main difficulty in this paper is due to the decay in $\frac{1}{\sqrt{t}}$ of free solutions to the wave equation in 2 dimensions, which is weaker than in 3 dimensions. This weak decay seems to be an obstruction for proving a stability result in the usual wave coordinates. In this paper we construct a suitable generalized wave gauge in which our system has a “cubic weak null structure”, which allows for the proof of global existence.

Appendices

A Global Existence of Solutions in the Exterior

We denote by ${\bar{C}}$ the complementary of the domain of dependence of $B(0,R+1)$. Let $g_{{\mathfrak {a}}}$ be defined by Theorem 1.3. In generalized wave coordinates $\Box _g x^\alpha = \Box _{g_{{\mathfrak {a}}}} x^{\alpha }$ the system $R_{\mu \nu }=0$ can be written, with the decomposition $g=g_{{\mathfrak {a}}} + \widetilde{g}$

$$\begin{aligned} \Box _g\widetilde{g}_{\mu \nu } = P_{\mu \nu }(g)(\partial \widetilde{g}, \partial \widetilde{g}) + \widetilde{P}_{\mu \nu } (\widetilde{g}, g_{{\mathfrak {a}}}), \end{aligned}$$

where we used the fact that $g_{{\mathfrak {a}}}$ is Ricci flat. We perform a bootsrap argument: let T be such that we have a solution g of this equation on ${\bar{C}}_T$ where ${\bar{C}}_T$ is the restriction of ${\bar{C}}$ to times less than T, and assume that

$$\begin{aligned}&\Vert v^\frac{1}{2}\partial Z^N \widetilde{g}\Vert _{L^2({\bar{C}}\cap \Sigma _t)} \lesssim \varepsilon (1+t)^\rho , \end{aligned}$$

(A.1)

$$\begin{aligned}&\Vert v_1^\frac{1}{2} \partial Z^{N-2} \widetilde{g}\Vert _{L^2({\bar{C}}\cap \Sigma _t)} \lesssim \varepsilon . \end{aligned}$$

(A.2)

where

$$\begin{aligned} v(q)= & {} (1+|q|)^{2+2\delta '},\\ v_1(q)= & {} (1+|q|)^{2+2\delta '-2\sigma }. \end{aligned}$$

Thanks to Klainerman-Sobolev estimates (which are still valid in a region ${\bar{C}}\cap \Sigma _t$) we have

$$\begin{aligned} |\partial Z^{N-2}\widetilde{g}|&\lesssim \frac{\varepsilon }{(1+s)^{\frac{1}{2}-\rho }(1+|q|)^{\frac{3}{2}+\delta '}}, \nonumber \\ |\partial Z^{N-3}\widetilde{g}|&\lesssim \frac{\varepsilon }{(1+s)^{\frac{1}{2}}(1+|q|)^{\frac{3}{2}+\delta '-\sigma }},\nonumber \\ |Z^{N-2}\widetilde{g}|&\lesssim \frac{\varepsilon }{(1+s)^{\frac{1}{2}-\rho }(1+|q|)^{\frac{1}{2}+\delta '}}, \end{aligned}$$

(A.3)

$$\begin{aligned} |\partial Z^{N-3}\widetilde{g}|&\lesssim \frac{\varepsilon }{(1+s)^{\frac{1}{2}}(1+|q|)^{\frac{1}{2}+\delta '-\sigma }}. \end{aligned}$$

(A.4)

Thanks to the wave coordinate condition

$$\begin{aligned} |Z^{N-3}\widetilde{g}_{{\mathcal {T}} {\mathcal {T}}}|&\lesssim \frac{\varepsilon }{(1+s)^{\frac{3}{2}-\rho }(1+|q|)^{-\frac{1}{2}+\delta '}}, \end{aligned}$$

(A.5)

$$\begin{aligned} | Z^{N-4}\widetilde{g}_{{\mathcal {T}} {\mathcal {T}}}|&\lesssim \frac{\varepsilon }{(1+s)^{\frac{3}{2}}(1+|q|)^{-\frac{1}{2}+\delta '-\sigma }}. \end{aligned}$$

(A.6)

To improve (A.1) we perform the energy estimate in the background metric, in the region ${\bar{C}}_t$. We obtain

$$\begin{aligned}&\int _{\Sigma _t} wQ_{TT}+ \int _{\partial {\bar{C}}} wQ_{T{\mathcal {L}}} +C\int _0^t \int w'(q) ({\bar{\partial }} \widetilde{g})^2\\&\quad \lesssim \int _0^t \frac{1}{1+\tau }\int Q_{TT} + \int _0^t \int |\partial _t Z^I \widetilde{g}||\Box _g Z^I \widetilde{g}| \end{aligned}$$

where ${\mathcal {L}}$ is a null vector tangent to $\partial {\bar{C}}$ and Q is the energy momentum tensor for $\Box _g$

$$\begin{aligned} Q_{\alpha \beta }=\partial _\alpha Z^I \widetilde{g} \partial _\beta Z^I \widetilde{g}-\frac{1}{2}g_{\alpha \beta }g^{\mu \nu }\partial _\mu Z^I \widetilde{g} \partial _\nu Z^I \widetilde{g}. \end{aligned}$$

Consequently

$$\begin{aligned} Q_{T{\mathcal {L}}}=T(Z^I \widetilde{g}){\mathcal {L}}(Z^I \widetilde{g})-\frac{1}{2}g_{T{\mathcal {L}}}({\mathcal {L}}(Z^I \widetilde{g}) \underline{{\mathcal {L}}} (Z^I \widetilde{g})+e_\theta (Z^I \phi )^2) \end{aligned}$$

where $\underline{{\mathcal {L}}}$ is null such that $g(\underline{{\mathcal {L}}},{\mathcal {L}})=-2$ and $e_\theta$ is tangent to ${\bar{C}}$ and orthogonal to ${\mathcal {L}}$ and $\underline{{\mathcal {L}}}$. We have

$$\begin{aligned} Q_{T{\mathcal {L}}}&=\frac{1}{2}g_{T{\mathcal {L}}}\underline{{\mathcal {L}}} (Z^I \widetilde{g}){\mathcal {L}}(Z^I \widetilde{g})+\frac{1}{2}g_{T \underline{{\mathcal {L}}}}{\mathcal {L}}(Z^I \widetilde{g}){\mathcal {L}}(Z^I \widetilde{g}) +g_{Te_\theta }{\mathcal {L}}(Z^I \widetilde{g})e_\theta (Z^I \widetilde{g})\\&\qquad -\frac{1}{2}g_{T{\mathcal {L}}}({\mathcal {L}}(Z^I \widetilde{g})\underline{{\mathcal {L}}} (Z^I \widetilde{g})+e_\theta (Z^I \widetilde{g})^2)\\&\quad =\frac{1}{2}g_{T \underline{{\mathcal {L}}}}{\mathcal {L}}(Z^I \widetilde{g}){\mathcal {L}}(Z^I \widetilde{g})-\frac{1}{2}g_{T{\mathcal {L}}}e_\theta (Z^I \phi )^2+g_{Te_\theta }{\mathcal {L}}(Z^I \widetilde{g})e_\theta (Z^I \widetilde{g})\\&\quad \ge (1-C\varepsilon )({\mathcal {L}}(Z^I \widetilde{g})^2+e_\theta (Z^I \widetilde{g})^2)\ge 0. \end{aligned}$$

Since in all our proof, the bootstrap condition (3.25) was not needed in the exterior region, we easily see from Sect. 9.1 that we will be able to improve (A.1).

To improve (A.2) we perform the energy estimate in the flat metric. $\widetilde{Q}$ is now the flat energy-momentum tensor. We now have to be careful with

$$\begin{aligned} \widetilde{Q}_{T{\mathcal {L}}}=&\partial _t Z^I \widetilde{g} {\mathcal {L}}(Z^I \widetilde{g})-\frac{1}{2}m_{T{\mathcal {L}}}\left( \partial _s(Z^I \widetilde{g})\partial _q (Z^I \widetilde{g})+\frac{1}{r^2}(\partial _\theta Z^I \widetilde{g})^2\right) , \end{aligned}$$

which may not be positive. Since ${\mathcal {L}}= (1+O(g-m))\partial _s + O(g_{L L})\partial _q + O(g_{UL})\partial _U,$ we have

$$\begin{aligned} \widetilde{Q}_{T{\mathcal {L}}}&=(1+O(\varepsilon ))\left( (\partial _s Z^I \widetilde{g})^2+\frac{1}{r^2}(\partial _\theta Z^I \widetilde{g})^2\right) +O(g_{ L L})(\partial _q Z^I \widetilde{g})^2\\&\ge (1-\varepsilon )\left( (\partial _s Z^I \widetilde{g})^2+\frac{1}{r^2}(\partial _\theta Z^I \widetilde{g})^2\right) -\frac{\varepsilon ^3}{(1+s)^{\frac{5}{2}-3\rho }} \end{aligned}$$

where we have used (A.3). Consequently the energy estimate yields

$$\begin{aligned}&\int _{\Sigma _t} w(\partial Z^I \widetilde{g})^2+ \int _{\partial {\bar{C}}} w({{\bar{\partial }}} Z^I \widetilde{g})^2 +C\int _0^t \int w'(q) ({\bar{\partial }} \widetilde{g})^2\\&\quad \lesssim \int _0^t \int |\partial _t Z^I \widetilde{g}||\Box _g Z^I \widetilde{g}|+ \int _{0}^t \frac{\varepsilon ^3}{(1+\tau )^{\frac{3}{2}-3\rho }} d\tau . \end{aligned}$$

We then easily see from Sect. 10.3 that we can improve the bootstrap assumption (we can check that the cubic non linearities without null structure in $Q_{\underline{L} \underline{L}}$ are not present).

B Regularity of the Initial Data

To obtain solutions of the constraint equation with an asymptotic behaviour $g=g_{{\mathfrak {b}}}+\widetilde{g}$, we take the exterior solution constructed in the previous section (we denote by $s',q',\theta '$ the coordinates used for this construction), make the change of variable

$$\begin{aligned} s'= & {} (1-\chi (r))s+\chi (r)\left( (1+b(\theta ,s))s-(\partial _\theta b(\theta ,s))^2(1+b(\theta ,s))^{-1}q\right) , \\ q'= & {} (1-\chi (r))q+\chi (r)(1+b(\theta ,s))^{-1}q, \\ \theta '= & {} (1-\chi (r))\theta + \chi (r)\left( \theta - \frac{q}{r}\frac{\partial _\theta b(\theta ,s)}{(1+b(\theta ,s))^2}+ f(\theta ,s)\right) , \end{aligned}$$

and consider the space-like hypersurface, given by $t=0$. We denote by $\Sigma _b$ this hypersurface, and consider ${\bar{g}}=g|_{\Sigma _b}$, and K the second fundamental form of the embedding $\Sigma _b \subset M$. $({\bar{g}},K)$ is a solution to the constraint equations.

Proposition B.1

There exists

$$\begin{aligned} (g_{\alpha \beta })_0,(g_{\alpha \beta })_1 \in H^{N+1}_{\delta }\times H^{N}_{\delta +1} \end{aligned}$$

such that the initial data for g given by

$$\begin{aligned} g=g_{{\mathfrak {b}}}+g_0, \;\partial _t g = \partial _t g_{{\mathfrak {b}}} +g_1, \end{aligned}$$

are such that

• ${\bar{g}}_{ij}=g_{ij}, K_{ij}={\mathcal {L}}_{\beta }g_{ij}$ satisfy the constraint equations (1.4) and (1.5).

• the following generalized wave coordinates condition is satisfied at $t=0$.

  $$\begin{aligned} g^{\lambda \beta }\Gamma ^\alpha _{\lambda \beta }=g_{{\mathfrak {b}}}^{\lambda \beta }(\Gamma _b)^\alpha _{\lambda \beta }+G^\alpha + \widetilde{G}^\alpha , \end{aligned}$$

where $G^\alpha$ is defined by (2.25), (2.26) and (2.27) and $\widetilde{G}$ is the sum of all the crossed term of the form $g_0\frac{\partial _\theta }{r} g_{{\mathfrak {b}}}$ and $g_0\partial _s g_{{\mathfrak {b}}}$ in $g^{\lambda \beta }\Gamma ^\alpha _{\lambda \beta }-g_{{\mathfrak {b}}}^{\lambda \beta }(\Gamma _b)^\alpha _{\lambda \beta }$. Moreover we have the estimate

$$\begin{aligned} \Vert g_0\Vert _{H^{N+1}_{\delta }} +\Vert g_1\Vert _{H^N_{\delta +1}}\lesssim \varepsilon . \end{aligned}$$

Proof

There are two issues to consider for the regularity of $({\bar{g}},K)$.

• We have $t'\sim t-b(\theta ,s)r$, so $|t'|\rightarrow \infty$ as $r\rightarrow \infty$ in $\Sigma _b$. Consequently we have to be careful with the logarithmic growth in $t'$ of the higher energy of $\widetilde{g}$.

• In $\partial ^N_\theta \widetilde{g}$, we have terms of the form $\partial _\theta ^{N+2}b(\theta ,s) \partial _\theta \widetilde{g}$: we have also to be careful with the logarithmic growth in s of $\Vert \partial _\theta ^{N+2}b(\theta ,s) \Vert _{L^2(\mathbb {S}^1)}$.

We treat the first issue. We can estimate $\int _{\Sigma _b} w(q)(\partial Z^N \widetilde{g})^2 rdrd\theta$ by performing the energy estimate on the domain delimited by $\Sigma _0$ and $\Sigma _b$. We denote by $\Omega _b$ this domain. We have

$$\begin{aligned} \int _{\Sigma _b} w(q)(\partial Z^N \widetilde{g})^2\lesssim \int _{\Sigma _0}w(q)(\partial Z^N \widetilde{g})^2 + \int _{\Omega _b} \frac{\varepsilon }{1+s}w(q)(\partial Z^N \widetilde{g})^2. \end{aligned}$$

We note that in the region $\Omega _b\cap \{q>R\}$ we have $|q|>Ct$. Since $w(q)\le v(q)(1+|q|)^{\delta -\delta '}$ we have

$$\begin{aligned} \int _{\Sigma _b} w(q)(\partial Z^N \widetilde{g})^2&\lesssim \int _{\Sigma _0}w(q)(\partial Z^N \widetilde{g})^2 + \int _{\Omega _b} \frac{\varepsilon }{(1+t)^{1+\delta -\delta '}}v(q)(\partial Z^N \widetilde{g})^2\\&\lesssim \int _{\Sigma _0}w(q)(\partial Z^N \widetilde{g})^2 +\varepsilon ^3. \end{aligned}$$

We treat the second issue with the help of the weight w:

$$\begin{aligned} \int w(r)(\partial ^{N+2}b(\theta ,s) \partial \partial _\theta \widetilde{g})^2&\lesssim \int \frac{\varepsilon }{(1+r)^{1+\delta -\delta '}}\Vert \partial _\theta ^{N+2}b(\theta ,r) \Vert ^2_{L^2(\mathbb {S}^1)}\\&\lesssim \int \frac{\varepsilon ^3}{(1+r)^{1+\delta -\delta '-2\rho }}\\&\lesssim \varepsilon ^3 \int w(r)(\partial ^{N+2}\partial _s b(\theta ,s)\partial _\theta \widetilde{g})^2\\&\lesssim \int \frac{\varepsilon }{(1+r)^{\delta -\delta '}}\Vert \partial _s \partial _\theta ^{N+2}b(\theta ,r) \Vert ^2_{L^2(\mathbb {S}^1)}dr\\&\lesssim \varepsilon ^3. \end{aligned}$$

We now discuss the regularity of $\partial _t g_{0i}$. The generalized wave coordinate condition can be written

$$\begin{aligned} g^{\lambda \beta }\Gamma ^\alpha _{\lambda \beta }=(g_{{\mathfrak {b}}})^{\lambda \beta }(\Gamma _b)^\alpha _{\lambda \beta }+G^\alpha + \widetilde{G}^\alpha , \end{aligned}$$

Therefore, if we write it for $\alpha =i$ we obtain a relation for $\partial _tg_{0i}$ and if we write it for $\alpha =0$, we obtain a relation for $\partial _t g_{00}$. However, if we write $g=g_{{\mathfrak {b}}} +\widetilde{g}$, the term

$$\begin{aligned} g^{\lambda \beta }\Gamma ^\alpha _{\lambda \beta }-(g_{{\mathfrak {b}}})^{\lambda \beta }(\Gamma _b)^\alpha _{\lambda \beta } \end{aligned}$$

contains crossed terms of the form

$$\begin{aligned} \widetilde{g} \partial _U g_{{\mathfrak {b}}} \sim \widetilde{g} \frac{\partial ^3_\theta b(\theta )}{r}+s.t., \quad \widetilde{g} \partial _s g_{{\mathfrak {b}}} \sim \widetilde{g}\partial _s^2 \partial _\theta b+s.t. \end{aligned}$$

which do not belong in $H^N_{\delta +1}$ because we are missing a derivative on b. However these terms are removed thanks to the addition of the term $\widetilde{G}$ in the generalized wave coordinate condition. Consequently $\partial _t \widetilde{g}_{00}$ and $\partial _t \widetilde{g}_{0i}$ are given by a sum of terms the form

$$\begin{aligned} K, \;\nabla g_0,\;g_{{\mathfrak {b}}}K,\; g_{{\mathfrak {b}}} \nabla g_0, \; \frac{\chi (r)g_{{\mathfrak {b}}}}{r}g_0. \end{aligned}$$

With this choice, $\partial _t \widetilde{g}_{0i}$ and $\partial _t \widetilde{g}_{00}$ belong to $H^N_{\delta +1}$. $\square$