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  "body": "\n## Statement\n\nLet $(M, g)$ be a connected complete [[Riemannian manifold|7CBT4SM83Y]] of dimension $3$ that is asymptotically flat in the following sense.\nThere is a compact set $C \\subset M$ and a diffeomorphism from $M \\setminus C$ to $\\R^3$ minus a closed ball.\nIn the coordinates $x = (x^1, x^2, x^3)$ it gives, for some $q \u003e 1/2$,\n$$g_{ij} - \\delta_{ij} = O(\\abs{x}^{-q}), \\quad \\partial_k g_{ij} = O(\\abs{x}^{-q-1}), \\quad \\partial_k \\partial_l g_{ij} = O(\\abs{x}^{-q-2}),$$\nand the [[scalar curvature|ZHX2VYQDZ3]] $R$ of $g$ is integrable on $M$.\nLet the ADM mass of $(M, g)$ be\n$$m = \\frac{1}{16\\pi} \\lim_{r \\to \\infty} \\int_{\\abs{x} = r} \\sum_{i,j=1}^{3} \\bigl(\\partial_i g_{ij} - \\partial_j g_{ii}\\bigr) \\frac{x^j}{r} \\, dA.$$\nIf $R \\ge 0$ on $M$, then $m \\ge 0$.\nMoreover, $m = 0$ if and only if $(M, g)$ is [[isometric|R9YWQS15AA]] to Euclidean space $\\R^3$.\n\n## Remarks\n\nThe theorem was proved by Schoen and Yau in 1979 using minimal surfaces, and by Witten in 1981 using spinors and a Weitzenböck formula for the Dirac operator.\nIn Schoen and Yau's argument, if $m \u003c 0$ one builds a complete stable minimal surface that is asymptotically a plane.\nThe stability inequality, together with $R \\ge 0$ and the Gauss–Bonnet theorem, gives a contradiction.\nBoth methods extend to higher dimensions under extra hypotheses: dimension at most $7$ for minimal hypersurfaces, and spin manifolds for Witten's method.\n\nExample: the Riemannian Schwarzschild metric $g = \\bigl(1 + \\frac{m}{2\\abs{x}}\\bigr)^4 \\delta$ on $\\R^3 \\setminus \\set{0}$, with $m \u003e 0$, has scalar curvature $0$.\nIt has two asymptotically flat ends, as $\\abs{x} \\to \\infty$ and as $\\abs{x} \\to 0$, and each has ADM mass $m$.\nThe theorem extends to manifolds with several such ends: each end has nonnegative mass.\n\nCompleteness is needed: for $m \u003c 0$ the same formula on $\\set{\\abs{x} \u003e -m/2}$ is scalar-flat and asymptotically flat with mass $m \u003c 0$.\nThis metric is not complete, because the conformal factor vanishes at $\\abs{x} = -m/2$, which lies at finite distance.\n\n## Explanations\n\n### L1 human\n\nIn general relativity, a space that has nonnegative energy density everywhere has nonnegative total mass, measured from far away.\nThe only way to have total mass zero is to have no gravity at all: flat space.\n\n### L2 human\n\nThe Riemannian version above is the time-symmetric case of the positive energy theorem for initial data sets of the Einstein equations, where the condition $R \\ge 0$ is the dominant energy condition.\nThe ADM mass is read off from the decay of the metric at infinity and does not depend on the chosen asymptotic coordinates.\nThe theorem is a basic tool in conformal geometry: Schoen used it to finish the solution of the Yamabe problem.\nThe Riemannian Penrose inequality is a sharper version that bounds the mass below by the area of a horizon.\n",
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