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  "body": "\n## Statement\n\nConsider the Einstein vacuum equations $\\operatorname{Ric}(g) = 0$ for a [[Lorentzian metric|BY4QYYS788]] $g$ on a $4$-dimensional manifold, the vacuum case of the [[Einstein field equations|3MCZ97SW6V]].\nLet $(\\Sigma, \\bar g, k)$ be an initial data set for these equations, with $\\Sigma$ diffeomorphic to $\\R^3$.\nHere $\\bar g$ is a Riemannian metric on $\\Sigma$, $k$ is a symmetric $2$-tensor on $\\Sigma$, and the constraint equations hold.\nAssume that the data are strongly asymptotically flat and maximal ($\\operatorname{tr}_{\\bar g} k = 0$).\nAssume also that they are close enough to the data $(\\R^3, \\delta, 0)$ of Minkowski space in a suitable weighted Sobolev norm.\nThen the maximal globally hyperbolic development of the data is a geodesically complete solution of the vacuum equations.\nMoreover, it approaches Minkowski space in every direction as one goes to infinity, with precise decay rates for the curvature.\n\n## Remarks\n\nThe result was proved by Demetrios Christodoulou and Sergiu Klainerman, and published as a monograph in 1993.\nIn a suitable gauge the vacuum equations form a system of nonlinear wave equations, so the theorem is a global existence result for small data of a [[hyperbolic|92C3S03AQZ]] system.\nThe nonlinearity is critical in three space dimensions.\nFor general quadratic nonlinearities small data can blow up in finite time, so global existence uses a special structure of the Einstein equations.\nThe proof uses a foliation of spacetime by maximal hypersurfaces and by null cones, and energy estimates for the Bel–Robinson tensor contracted with approximate symmetries of the solution.\nLindblad and Rodnianski later gave a different proof in harmonic (wave) gauge, under somewhat different assumptions on the data.\n\n## Explanations\n\n### L1 human\n\nIn general relativity, empty flat spacetime is the simplest solution of Einstein's equations.\nThe theorem says that this solution is stable: if one disturbs it a little, for instance by weak gravitational waves, the disturbance spreads out and dies away.\nThe spacetime does not collapse into a black hole or develop other singularities.\nFar away and late in time it looks again like flat spacetime.\n\n### L2 human\n\nProving stability means proving global existence and decay for a quasilinear hyperbolic system with small data.\nLinear waves in three space dimensions decay like $t^{-1}$.\nThis is just barely too slow to control a general quadratic nonlinearity, as examples of John show for some nonlinear wave equations.\nIn harmonic gauge the Einstein equations do not satisfy the classical null condition of Klainerman, but their nonlinear terms still have enough structure to prevent blow-up.\nThe techniques of the proof, the vector field method and geometric energy estimates, became standard tools for nonlinear wave equations.\nThey are also used in later stability problems, such as the stability of black holes.\n",
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