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  "body": "\n## Example\n\nFix a mass parameter $m \u003e 0$, in units where the gravitational constant and the speed of light are $1$.\nOn $M = \\R \\times (2m, \\infty) \\times S^2$ with coordinates $(t, r, \\theta, \\phi)$, the *Schwarzschild metric* is\n$$g = -\\left(1 - \\frac{2m}{r}\\right) dt^2 + \\left(1 - \\frac{2m}{r}\\right)^{-1} dr^2 + r^2 \\left(d\\theta^2 + \\sin^2\\theta \\, d\\phi^2\\right).$$\nCheck that $(M, g)$ is a [[Lorentzian manifold|BY4QYYS788]]: for $r \u003e 2m$ the number $1 - 2m/r$ is positive.\nSo the coefficient of $dt^2$ is negative and the coefficient of $dr^2$ is positive.\nThe last term is $r^2$ times the round metric of $S^2$, which is positive definite.\nHence $g$ has signature $(-, +, +, +)$ at every point.\nA direct computation of the Christoffel symbols shows that the [[Ricci curvature|4GTNM0WHDH]] of $g$ is zero, so $g$ solves the vacuum Einstein equations.\nIt is static and spherically symmetric, and $g$ tends to the Minkowski metric as $r \\to \\infty$.\n\nThe slice $t = 0$ carries the Riemannian metric $(1 - 2m/r)^{-1} dr^2 + r^2 (d\\theta^2 + \\sin^2\\theta\\, d\\phi^2)$, the spatial Schwarzschild metric.\nIt has zero scalar curvature, and in the coordinate $r$ extended to $r \\ge 2m$ its boundary sphere $r = 2m$ is a minimal surface of area $16\\pi m^2$.\n\n## Explanations\n\n### L1 human\n\nThe Schwarzschild metric describes space and time outside a single round, non-rotating, uncharged mass, such as a star or a black hole.\nFar away it looks like flat spacetime, and near the mass time runs slower and space is stretched.\nThe sphere $r = 2m$ is the event horizon of the black hole: the coordinates above break down there, although the spacetime itself does not.\nIt was the first exact solution of Einstein's equations, found by Karl Schwarzschild in 1916.\n",
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