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.
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Notes
It is more convenient for the estimates to use an integration along lines of constant t and \(\theta \) than lines of constant s and \(\theta \). On the light cone, we have \(t=\frac{s}{2}\), so it is why we evaluate the integral at \(t=\frac{s}{2}\)
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Acknowledgements
This paper has benefit from the insight of many people. The author would like to thank in particular Jérémie Szeftel, Qian Wang, Spyros Alexakis, Mihalis Dafermos and Igor Rodnianski for the interesting conversations.
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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 \(g=g_{{\mathfrak {a}}} + \widetilde{g}\)
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
where
Thanks to Klainerman-Sobolev estimates (which are still valid in a region \({\bar{C}}\cap \Sigma _t\)) we have
Thanks to the wave coordinate condition
To improve (A.1) we perform the energy estimate in the background metric, in the region \({\bar{C}}_t\). We obtain
where \({\mathcal {L}}\) is a null vector tangent to \(\partial {\bar{C}}\) and Q is the energy momentum tensor for \(\Box _g\)
Consequently
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
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
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
where we have used (A.3). Consequently the energy estimate yields
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
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
such that the initial data for g given by
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
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
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
We treat the second issue with the help of the weight w:
We now discuss the regularity of \(\partial _t g_{0i}\). The generalized wave coordinate condition can be written
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
contains crossed terms of the form
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
With this choice, \(\partial _t \widetilde{g}_{0i}\) and \(\partial _t \widetilde{g}_{00}\) belong to \(H^N_{\delta +1}\). \(\square \)
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Huneau, C. Stability of Minkowski Space-Time with a Translation Space-Like Killing Field. Ann. PDE 4, 12 (2018). https://doi.org/10.1007/s40818-018-0048-x
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DOI: https://doi.org/10.1007/s40818-018-0048-x