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  "body": "\n## Definition\n\nLet $M$ be a smooth oriented $4$-manifold, and let $g$ be a [[Lorentzian metric|BY4QYYS788]] on $M$ of signature $(-,+,+,+)$.\nLet $R$ be the [[scalar curvature|ZHX2VYQDZ3]] of $g$, and let $\\sqrt{-\\det g}\\; d^4x$ be its volume form in local coordinates $x^0, \\dots, x^3$.\nLet $G$ be Newton's gravitational constant, $c$ the speed of light, and $\\kappa = 8\\pi G / c^4$.\nFor a compact region $K \\subseteq M$, the *Einstein–Hilbert action* of $g$ on $K$ is\n$$S_{EH}[g] = \\frac{1}{2\\kappa} \\int_K R \\,\\sqrt{-\\det g}\\; d^4x.$$\n\n## Remarks\n\nLet $g_t$ be a variation of the metric supported in the interior of $K$.\nWrite $\\delta g^{\\mu\\nu} = \\frac{d}{dt}\\big|_{t=0} g_t^{\\mu\\nu}$.\nThen\n$$\\frac{d}{dt}\\Big|_{t=0} S_{EH}[g_t] = \\frac{1}{2\\kappa} \\int_K \\Big(R_{\\mu\\nu} - \\tfrac12 R\\, g_{\\mu\\nu}\\Big)\\, \\delta g^{\\mu\\nu} \\sqrt{-\\det g}\\; d^4x,$$\nwhere $R_{\\mu\\nu}$ is the Ricci tensor; the term with the variation of $R_{\\mu\\nu}$ is a divergence and drops out.\nSo $g$ is a critical point of $S_{EH}$ for all such variations if and only if $R_{\\mu\\nu} - \\tfrac12 R g_{\\mu\\nu} = 0$, the vacuum [[Einstein field equations|3MCZ97SW6V]].\nExample: Minkowski space $\\R^4$ with $g = -dt^2 + dx^2 + dy^2 + dz^2$ is flat, so its Ricci tensor vanishes, and it is a critical point.\nNon-example: the Euclidean metric $dt^2 + dx^2 + dy^2 + dz^2$ is not Lorentzian, so the definition above does not apply to it; the analogous Riemannian functional $\\int R \\sqrt{\\det g}\\, d^4x$ is a different object.\nWith a cosmological constant $\\Lambda$ one replaces $R$ by $R - 2\\Lambda$.\nAdding a matter action $S_M$ and setting $T_{\\mu\\nu} = -\\frac{2}{\\sqrt{-\\det g}} \\frac{\\delta S_M}{\\delta g^{\\mu\\nu}}$ gives $R_{\\mu\\nu} - \\tfrac12 R g_{\\mu\\nu} = \\kappa T_{\\mu\\nu}$.\nOn a region with boundary, one adds the Gibbons–Hawking–York boundary term to obtain a well-posed variational problem with the metric fixed on the boundary.\n",
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