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  "body": "\n## Definition\n\nLet $(M, g)$ be a [[pseudo-Riemannian manifold|GH6CXWFNDJ]] of dimension $n \\ge 2$.\nIt is a *Lorentzian manifold* if, at each point $p \\in M$, the form $g_p$ has [[signature|07EZA80NEH]] $(n - 1, 1)$.\nThat is, in a suitable basis of $T_pM$ the matrix of $g_p$ is $\\operatorname{diag}(-1, 1, \\ldots, 1)$.\n\n## Remarks\n\nExample: Minkowski space is $\\R^n$ with coordinates $(t, x_1, \\ldots, x_{n-1})$ and $g = -dt^2 + dx_1^2 + \\cdots + dx_{n-1}^2$.\nIts matrix is $\\operatorname{diag}(-1, 1, \\ldots, 1)$ at every point, with one negative and $n - 1$ positive entries.\n\nNon-example: $\\R^4$ with $g = -dt^2 - dx^2 + dy^2 + dz^2$ is pseudo-Riemannian, but its signature has two negative entries, so it is not Lorentzian.\n\nMany physics texts use the opposite convention, signature $(1, n - 1)$, that is $\\operatorname{diag}(1, -1, \\ldots, -1)$; the two conventions differ by replacing $g$ with $-g$.\nA nonzero tangent vector $v$ is called timelike if $g(v, v) \u003c 0$, null if $g(v, v) = 0$, and spacelike if $g(v, v) \u003e 0$ (in the convention used here).\n",
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