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  "body": "\n## Example\n\nLet $\\eta = \\operatorname{diag}(1, -1, -1, -1)$.\nGive $\\R^4$ with coordinates $(t, x, y, z)$ the quadratic form\n$$Q(t, x, y, z) = t^2 - x^2 - y^2 - z^2,$$\nwhich makes it Minkowski space (in units where the speed of light is $1$).\nThe *Lorentz group* is the group of Lorentz transformations, the linear maps of $\\R^4$ preserving $Q$:\n$$O(1, 3) = \\set{\\Lambda \\in GL_4(\\R) : \\Lambda^T \\eta \\Lambda = \\eta}.$$\nIt is the [[indefinite orthogonal group|M2900DMEW6]] of signature $(1, 3)$, a [[Lie group|7JBRGCVJRB]] of dimension $6$.\n\nCheck with a boost along the $x$-axis with rapidity $\\phi \\in \\R$:\n$$\\Lambda = \\begin{pmatrix} \\cosh\\phi \u0026 \\sinh\\phi \u0026 0 \u0026 0 \\\\ \\sinh\\phi \u0026 \\cosh\\phi \u0026 0 \u0026 0 \\\\ 0 \u0026 0 \u0026 1 \u0026 0 \\\\ 0 \u0026 0 \u0026 0 \u0026 1 \\end{pmatrix}.$$\nIt sends $(t, x, y, z)$ to $(t \\cosh\\phi + x \\sinh\\phi, \\ t \\sinh\\phi + x \\cosh\\phi, \\ y, \\ z)$, and\n$$(t \\cosh\\phi + x \\sinh\\phi)^2 - (t \\sinh\\phi + x \\cosh\\phi)^2 = (t^2 - x^2)(\\cosh^2\\phi - \\sinh^2\\phi) = t^2 - x^2,$$\nso $Q$ is preserved.\n\n## Remarks\n\nThe rotations of space, $\\Lambda = \\operatorname{diag}(1, R)$ with $R \\in$ [[SO(3)|0CGYNDCWRR]], form a compact subgroup, and the boosts make the group noncompact.\n$O(1, 3)$ has four connected components, distinguished by the sign of $\\det \\Lambda$ and the sign of the entry $\\Lambda_{00}$.\nThe identity component $SO^+(1, 3)$ is the *proper orthochronous* Lorentz group: it keeps the orientation of space-time and the direction of time.\nThe other components are reached from it by space inversion, time reversal, or both.\n$SO^+(1, 3)$ is isomorphic to $PSL(2, \\C)$, and [[SL(2,C)|J58ED1GV6N]] is its universal cover, a double cover.\n\n## Explanations\n\n### L1 human\n\nThe Lorentz group collects the changes of reference frame of special relativity that fix the origin of space-time.\nThese are the rotations of space and the boosts, which pass to an observer moving at constant velocity.\nBecause they preserve $t^2 - x^2 - y^2 - z^2$, they map the light cone $t^2 = x^2 + y^2 + z^2$ to itself, so every observer measures the same speed of light.\nA boost mixes time with one space coordinate in the way a rotation mixes two space coordinates, but with hyperbolic functions in place of sines and cosines.\n",
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