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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
Zbl 0827.53055
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