{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nA spacetime here is a connected, time-oriented, four-dimensional [[Lorentzian manifold|BY4QYYS788]] $(M, g)$ of signature $(-, +, +, +)$.\nIt is timelike (resp. null) geodesically complete if every inextendible timelike (resp. null) [[geodesic|NSYMQMHPDN]] is defined for all values of its affine parameter.\nA closed trapped surface is a compact spacelike surface without boundary in $M$.\nOn it, both families of future-directed null geodesics orthogonal to it must have negative expansion.\nLet $\\operatorname{Ric}$ be the [[Ricci curvature|4GTNM0WHDH]] of $g$.\n\n1. (Hawking) Assume that $(M, g)$ is globally hyperbolic and that $\\operatorname{Ric}(V, V) \\ge 0$ for all timelike vectors $V$.\nLet $\\Sigma$ be a smooth spacelike Cauchy surface in $M$.\nLet $\\theta$ be the expansion on $\\Sigma$ of the past-directed normal geodesics, that is, the mean curvature for the past-directed unit normal.\nAssume that $\\theta \\le C$ on $\\Sigma$ for a constant $C \u003c 0$.\nThen every past-directed timelike curve starting on $\\Sigma$ has length at most $3/\\abs{C}$.\nIn particular, $(M, g)$ is not timelike geodesically complete.\n2. (Penrose) Assume that $(M, g)$ is globally hyperbolic with a noncompact Cauchy surface.\nAssume that $\\operatorname{Ric}(K, K) \\ge 0$ for all null vectors $K$, and that $M$ contains a closed trapped surface.\nThen $(M, g)$ is not null geodesically complete.\n3. (Hawking and Penrose) Assume that $\\operatorname{Ric}(V, V) \\ge 0$ for all timelike and null vectors $V$.\nAssume that $M$ has no closed timelike curves, and that every timelike or null geodesic satisfies the generic condition.\nAssume also that at least one of the conditions (a), (b), (c) below holds.\n(a) $M$ contains a compact achronal set without edge.\n(b) $M$ contains a closed trapped surface.\n(c) There is a point $p$ such that the null geodesics from $p$ all start to reconverge.\nThis means that the expansion of the future-directed null geodesics from $p$ becomes negative along each of them.\nThe same with past-directed geodesics also suffices.\nThen $(M, g)$ contains an incomplete timelike or null geodesic.\n\n## Remarks\n\nThe common core of the proofs is the Raychaudhuri equation.\nBy it, the curvature conditions force the expansion of a congruence of geodesics to become $-\\infty$ in finite affine parameter, which creates conjugate or focal points.\nSuch points prevent geodesics from maximizing length beyond them.\nOn the other hand, global causal assumptions, such as global hyperbolicity, guarantee the existence of maximizing geodesics.\nThe two facts together show that some geodesic cannot be extended.\nThe Schwarzschild black hole contains closed trapped surfaces inside its horizon, and the theorem of Penrose shows that the singularity there persists under perturbations that keep a trapped surface.\nThe theorems prove incompleteness of geodesics; they do not say that the curvature becomes infinite.\n\n## Explanations\n\n### L1 human\n\nThese theorems say that, under reasonable physical conditions, general relativity predicts singularities.\nGravity attracts, so light rays and free-falling particles that start to converge keep converging.\nOnce matter collapses far enough to form a trapped surface, or the whole universe is expanding uniformly, some paths of particles or light must end after a finite time.\nThis is how the theory predicts black holes and a Big Bang.\n\n### L2 human\n\nThe curvature conditions are energy conditions on matter, translated through the Einstein equations into inequalities on the Ricci curvature.\nThe proofs use the comparison theory of geodesics, much like the proof of Myers's theorem in Riemannian geometry, where positive Ricci curvature bounds the length of minimizing geodesics.\nPenrose's theorem dates from 1965, and the joint theorem of Hawking and Penrose from 1970.\nThe third theorem trades global hyperbolicity for the weaker chronology condition, at the cost of the generic condition.\n",
  "commit": "efe85cee8dbc57d85bedbacfb97b99d1b7bee843",
  "front": {
    "about": [
      "BY4QYYS788"
    ],
    "kind": "statement",
    "msc": [
      "83C75",
      "53C50"
    ],
    "name": "Penrose–Hawking singularity theorems",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q1452678",
      "wikipedia": "Penrose–Hawking singularity theorems"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Penrose%E2%80%93Hawking_singularity_theorems"
      }
    ],
    "tag": "WP4YAHJ6V4",
    "uses": [
      "BY4QYYS788",
      "NSYMQMHPDN",
      "4GTNM0WHDH"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/WP4YAHJ6V4/",
  "links": [
    {
      "field": "about",
      "target": "BY4QYYS788"
    },
    {
      "field": "uses",
      "target": "BY4QYYS788"
    },
    {
      "field": "uses",
      "target": "NSYMQMHPDN"
    },
    {
      "field": "uses",
      "target": "4GTNM0WHDH"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/WP4YAHJ6V4.md",
  "msc": [
    "83C75",
    "53C50"
  ],
  "name": "Penrose–Hawking singularity theorems",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/WP4/WP4YAHJ6V4.md",
  "status": null,
  "tag": "WP4YAHJ6V4",
  "uri": "m2://WP4YAHJ6V4",
  "url": "https://math2.tamnd.com/wiki/penrose-hawking-singularity-theorems/",
  "writing": {
    "tag": "WP4YAHJ6V4",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}