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The global stability of Minkowski space-time in harmonic gauge. (English) Zbl 1192.53066
This paper considers the question of stability of Minkowski space-time for the system of the Einstein-scalar field equations
\[ R_{\mu\nu}-\frac12g_{\mu\nu}R=T_{\mu\nu}.\tag{\(*\)} \] The equations connect the gravitational tensor \(G_{\mu\nu}=R_{\mu\nu}-\frac12g_{\mu\nu}\) given in terms of the Ricci tensor \(R_{\mu\nu}\) and the scalar curvature \(R=g^{\mu\nu}R_{\mu\nu}\) of an unknown Lorentzian metric \(g_{\mu\nu}\) and the energy-momentum tensor \(T_{\mu\nu}\) of a matter field \(\psi\) given by \(T_{\mu\nu}=\partial_\mu\psi \partial_\nu\psi-\frac12g_{\mu\nu}\left(g^{\alpha\beta}\partial_\alpha\psi \partial_\beta\psi\right)\). In the problem of stability of Minkowski space, using the Cauchy formulation of the Einstein equations for a \(3\)-dimensional manifold \(\Sigma_0\) with a Riemannian metric \(g_ 0\), a symmetric \(2\)-tensor \(k_ 0\), and the initial data \((\psi_0,\psi_ 1)\) for the scalar field, it is needed to find a \(4\)-dimensional manifold \(M\) with a Lorentzian metric \(g\) and a scalar field \(\psi\) satisfying the Einstein equations \((*)\), an embedding \(\Sigma_ 0\subset M\) such that \(g_ 0\) is the restriction of \(g\) to \(\Sigma_ 0\), \(k_ 0\) is the second fundamental form of \(\Sigma_ 0\), and the restriction of \(\psi\) to \(\Sigma_ 0\) gives rise to the data \((\psi_0,\psi_ 1)\).
The authors present a new proof of the global stability of Minkowski space for the Einstein-vacuum equations established in the work [Princeton Mathematical Series. 41. Princeton, NJ: Princeton University Press (1993; Zbl 0827.53055)] of D. Christodoulou and S. Klainerman for strongly asymptotic initial data. The new approach, relying on the classical harmonic gauge, shows that the Einstein-vacuum and the Einstein-scalar field equations with asymptotically flat initial data satisfying a global smallness condition produce global causally geodesically complete solutions asymptotically convergent to the Minkowski space-time.

MSC:
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
35Q99 Partial differential equations of mathematical physics and other areas of application
83C99 General relativity
Citations:
Zbl 0827.53055
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