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Cited by 2,977Relativity and Gravitational TheoryCosmology and Gravitation TheoriesBlack Holes and Theoretical Physics

Gravitational Collapse and Space-Time Singularities

Open access
Penrose, Roger·1965·Physical Review Letters
DOI 10.1103/physrevlett.14.57
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Abstract

A similar increase in the low-energy proton in- tensity, in coincidence with the passage of a blast wave, was observed on 12 November 1960 t J. F.

AI snapshot

Written from the paper’s full text.

TL;DR

Penrose argues that once a trapped surface forms during gravitational collapse, spacetime singularities must develop, irrespective of any assumption of spherical or other symmetry. The paper outlines a proof that a set of five plausible physical assumptions is internally inconsistent.

Key findings

  • The five assumptions — (i) M⁺ is a nonsingular manifold with two separate null half-cone systems, (ii) null completeness (every null geodesic extends to arbitrarily large affine parameter), (iii) the Cauchy hypersurface condition (every timelike or null geodesic in M⁺ can be extended into the past until it meets C), (iv) non-negativeness of local energy, and (v) the existence of a trapped surface T² in M⁺ — are together inconsistent.
  • A trapped surface is defined as a closed spacelike 2-surface whose two orthogonal systems of null geodesics converge locally in future directions. In spherical collapse such a surface exists once matter contracts within r = 2m; the Kerr solutions with m > a (angular momentum) also possess trapped surfaces, whereas those with m < a do not.
  • If physical singularities are not permitted, then inside such a collapsing object at least one of the following must hold: (a) negative local energy occurs, (b) Einstein's equations are violated, (c) the spacetime manifold is incomplete, or (d) the concept of spacetime loses its meaning at very high curvatures, possibly because of quantum phenomena.
  • The convergence argument: with p ≤ 0 at T² (trapped), the cross-sectional area A becomes zero at a finite affine distance to the future on each null geodesic, so each geodesic encounters a caustic and B' is compact; approximating B' by a smooth closed spacelike hypersurface B'* and using a degree argument, d(Q) = 1 near both B'* and C, which is impossible because C is noncompact.

Methods

  • Design: a theoretical general-relativity argument, presented as an outline of a proof, deliberately avoiding symmetry assumptions (spherical symmetry precludes discussion of gravitational radiation, which requires at least quadrupole structure).
  • Setup: an initial Cauchy hypersurface C carrying a matter distribution, with Einstein's field equations and suitable equations of state; the only assumption made about the equations of state is the non-negative definiteness of the energy expression.
  • Main analysis: null geodesic convergence and shear for the null geodesic segments generating the boundary B' of the future set of the trapped surface, combined with a topological degree argument on the map from B'* back to C.

Limitations

  • Stated by the paper: the result is given "in outline," with full details of this and other related results to be given elsewhere.
  • Evident from the text: the provided text is a heavily corrupted OCR rendering, and the abstract printed with it concerns an unrelated observation of low-energy proton intensity and a blast wave, so it does not describe this paper's content.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1103/physrevlett.14.57

Cites

11 of 11 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

0 of 2 cited sources checked · 0 supported · 0 flagged

  • Findingreasoning

    After the matter has contracted within r = 2m, a space-like sphere S' can be found in the empty region surrounding the matter.

    “Aft the matter has contracted within r =2m, er e =, = t a space e sp e lik here S' (f =const, 2m &r= cson) can be found in the empty region surrounding the matter.”
    Quote in the paperSubstantial claim
  • Findingreasoning

    The impossibility of the degree map being unity follows from the noncompactness of C'.

    “The impossibility of this follows from the noncompactness of C'.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The manifold M+ is the future time development of an initial Cauchy surface C (the past boundary of the M+ region).
    • The authors state this as an assumption: M+ is a nonsingular C∞ manifold for which the null half-cones form two separate systems (past and future).
    • The authors state this as an assumption: Every null geodesic in M+ can be extended into the future to arbitrarily large affine parameter values (null completeness).
    • The authors state this as an assumption: Every timelike or null geodesic in M+ can be extended into the past until it meets C (Cauchy hypersurface condition).
    • The authors state this as an assumption: At every point of M+, all timelike vectors t satisfy (-R_ab + ... ) t^a t^b ≥ 0 (non-negativeness of local energy).
    • The authors state this as an assumption: There exists a trapped surface T' in M+.
  • Findingreasoning

    B' is null where it is nonsingular, being generated by the null geodesic segments which meet T' orthogonally at a past endpoint and have a future endpoint if this is a singularity (on a caustic or crossing region) of B'.

    “Local considerations show that B' is null where it is nonsingular, being generated by the null geodesic segments which meet T' orthogonally at a past endpoint and have a future endpoint if this is a singularity (on a caustic or crossing region) of B'.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The manifold M+ is the future time development of an initial Cauchy surface C (the past boundary of the M+ region).
    • The authors state this as an assumption: M+ is a nonsingular C∞ manifold for which the null half-cones form two separate systems (past and future).
    • The authors state this as an assumption: Every null geodesic in M+ can be extended into the future to arbitrarily large affine parameter values (null completeness).
    • The authors state this as an assumption: Every timelike or null geodesic in M+ can be extended into the past until it meets C (Cauchy hypersurface condition).
    • The authors state this as an assumption: At every point of M+, all timelike vectors t satisfy (-R_ab + ... ) t^a t^b ≥ 0 (non-negativeness of local energy).
    • The authors state this as an assumption: There exists a trapped surface T' in M+.
  • Findingreasoning

    B' is compact (closed), being generated by a compact system of finite segments.

    “Hence B' is compact (closed), being generated by a compact system of finite segments.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The manifold M+ is the future time development of an initial Cauchy surface C (the past boundary of the M+ region).
    • The authors state this as an assumption: M+ is a nonsingular C∞ manifold for which the null half-cones form two separate systems (past and future).
    • The authors state this as an assumption: Every null geodesic in M+ can be extended into the future to arbitrarily large affine parameter values (null completeness).
    • The authors state this as an assumption: Every timelike or null geodesic in M+ can be extended into the past until it meets C (Cauchy hypersurface condition).
    • The authors state this as an assumption: At every point of M+, all timelike vectors t satisfy (-R_ab + ... ) t^a t^b ≥ 0 (non-negativeness of local energy).
    • The authors state this as an assumption: There exists a trapped surface T' in M+.
  • Premisenone given

    To an outside observer the contraction to r = 2m appears to take an infinite time.

    “To an outside observer the contraction to ~= 2m appears to take an infinite time.”
    Quote in the paperNot a substantial claim
  • May combine two claims. 6
  • Classified as probably not a substantial claim. 1
  • The quote only partly states this claim. 2

  • May combine two claims.

    Findingreasoning

    Since T' is trapped, p ≤ 0 at T', whence A becomes zero at a finite affine distance to the future of T' on each null geodesic.

    “Since T' is trapped, p& 0 at T', whence A becomes zero at a finite affine distance to the future of T' on each null geodesic.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The manifold M+ is the future time development of an initial Cauchy surface C (the past boundary of the M+ region).
    • The authors state this as an assumption: M+ is a nonsingular C∞ manifold for which the null half-cones form two separate systems (past and future).
    • The authors state this as an assumption: Every null geodesic in M+ can be extended into the future to arbitrarily large affine parameter values (null completeness).
    • The authors state this as an assumption: Every timelike or null geodesic in M+ can be extended into the past until it meets C (Cauchy hypersurface condition).
    • The authors state this as an assumption: At every point of M+, all timelike vectors t satisfy (-R_ab + ... ) t^a t^b ≥ 0 (non-negativeness of local energy).
    • The authors state this as an assumption: There exists a trapped surface T' in M+.

  • Classified as probably not a substantial claim.

    Findingnone given

    The sphere S' is an example of a trapped surface T', defined by the property that the two systems of null geodesics which meet T' orthogonally converge locally in future directions at T'.

    “This sphere is an example of what will be called here a trapped surfacedefined T' with the property that the two systems of null eodesics which meet T' orthogonally con- verge locally in future directions at T'.”
    Quote in the paperSubstantial claim

  • The quote only partly states this claim.

    Findingreasoning

    Each geodesic encounters a caustic.

    “Each geodesic thus encounters a caustic.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The manifold M+ is the future time development of an initial Cauchy surface C (the past boundary of the M+ region).
    • The authors state this as an assumption: M+ is a nonsingular C∞ manifold for which the null half-cones form two separate systems (past and future).
    • The authors state this as an assumption: Every null geodesic in M+ can be extended into the future to arbitrarily large affine parameter values (null completeness).
    • The authors state this as an assumption: Every timelike or null geodesic in M+ can be extended into the past until it meets C (Cauchy hypersurface condition).
    • The authors state this as an assumption: At every point of M+, all timelike vectors t satisfy (-R_ab + ... ) t^a t^b ≥ 0 (non-negativeness of local energy).
    • The authors state this as an assumption: There exists a trapped surface T' in M+.

  • May combine two claims.

    Premisenone given

    For a sufficiently great mass, there is no final equilibrium state; when sufficient thermal energy has been radiated away, the body contracts and continues to contract until a physical singularity is encountered at r = 0.

    “For a sufficiently great mass, there is no final equilibrium state.%'hen sufficient thermal energy has been radiated away, the body contracts and continues to contract until a physi- cal singularity is encountered at r = 0.”
    Quote in the paperNot a substantial claim

  • The quote only partly states this claim.

    Premisenone given

    As measured by local comoving observers, the body passes within its Schwarzschild radius r = 2m.

    “As measured by local comoving observers”
    • Also stated in: § Abstract · plain
      “the body passes within its Schwarzschild radius r = 2m.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning

    If actual physical singularities in space-time are not permitted to occur, then inside such a collapsing object at least one of the following holds: negative local energy occurs; Einstein's equations are violated; the space-time manifold is incomplete; or the concept of space-time loses its meaning at very high curvatures possibly because of quantum phenomena.

    “inside such a collapsing ob'ect at least one of the following holds: o ]ec a (a) Negative local energy occurs.stein's equations are v1olated.,c, The s ace- time manifold is incomplete.' (d} The concept”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The only assumption made about the equations of state is the non-negative definiteness of the energy expression T_ab (with or without cosmological terms).
    • The authors state this as an assumption: The matter distribution undergoes gravitational collapse in a way which, at first, qualitatively resembles the spherical case.

  • May combine two claims.

    Premisenone given

    An exterior observer will always see matter outside r = 2m; the collapse through r = 2m to the singularity at r = 0 is invisible to him.

    “Note that an exterior ob- server will always see matter outside r =2m, the collapse through r =2m to the singularity at r =0 being invisible to him.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationnone given

    Trapped surfaces will still exist if the matter region has no sharp boundary or if spherical deviations from the above situation are not too great.

    “ly trapped surfaces will still exist if the matter region has no sharp boundary or if spherica tions from the above situation are not too great.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Premisecitation

    Kerr solutions with m > a (angular momentum) all possess trapped surfaces, whereas those for which m < a do not.

    “Indee, e d th Kerr solutions with m &a angular ) all possess trapped surfac momentum ma .' The whereas os th se for which m ~a do not.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites Kerr'

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

9 of 9

  • The authors state this as an assumption.

    The only assumption made about the equations of state is the non-negative definiteness of the energy expression T_ab (with or without cosmological terms).

    “In fact the only assump- made here about these equations of sta e will be the non-negative definiteness o energy express1on w' (with or without cosmologi- c.”

  • The authors state this as an assumption.

    The matter distribution undergoes gravitational collapse in a way which, at first, qualitatively resembles the spherical case.

    “undergoes gravitational collapse in a way w ic, e s herical- at first, qualitatively resembles the p t ' al case.”

  • The authors state this as an assumption.

    The manifold M+ is the future time development of an initial Cauchy surface C (the past boundary of the M+ region).

    “It will be necessary, here, to sup- 4, 'ch is the future th t the manifold M+, which is development of an initial Cauchy yp time deve opm 4 re ion), sur ace f C~( past boundary of the M+ r g' re.”

  • The authors state this as an assumption.

    M+ is a nonsingular C∞ manifold for which the null half-cones form two separate systems (past and future).

    “(i) M+~i s a nonsingular (+--ifold for which the null half-cones form two separate systems (”

  • The authors state this as an assumption.

    Every null geodesic in M+ can be extended into the future to arbitrarily large affine parameter values (null completeness).

    “(ii) Every nu ge e ' the future to arbitrarily large affine param- into e u ster values (null completeness .”

  • The authors state this as an assumption.

    Every timelike or null geodesic in M+ can be extended into the past until it meets C (Cauchy hypersurface condition).

    “timelike or null geodesic in M+ can e FIG. 1. Spherically symmetrical collapse (one s ace dimension surpressed).The diagram essentially also serves for the discussion o e ym f the as metrical case.into the past until it meets C, (Cauchy hyper- surface condition).”

  • The authors state this as an assumption.

    At every point of M+, all timelike vectors t satisfy (-R_ab + ... ) t^a t^b ≥ 0 (non-negativeness of local energy).

    “(iv) At every point of M+~, all timelike vectors t& satisfy ( -R”
    “non-negativeness of local ener- jLL V gy).”

  • The authors state this as an assumption.

    There exists a trapped surface T' in M+.

    “(v) There exists a trapped surface F' in M+4.”
8 of 8

Claims other papers make about this one

Each is checked against this paper’s own text: does it say what the citing paper says it does?

  • High curvatures occur in collapsing stars and at the beginning of the present universe, and are hidden from us by an event horizon.

    “From the classical singularity theorems [3-6], one would expect such high curvatures to occur in collapsing stars and, in the past, at the beginning of the present espe se e oone o on oet e ne o e e osones of high curvature to be hidden from us by an event horizon [7].”

    From Particle creation by black holes (1975) · cites this paper as [3]

    In the section labeled Abstract (page unavailable), Penrose says an exterior observer cannot see a collapsing body pass within r = 2m to its singularity, supporting the horizon-related point for gravitational collapse. The copy does not address high curvature at the beginning of the present universe or say that such a region is hidden by an event horizon. (LLM, conf=0.84)

    Experimental — accuracy not yet measured

Cites

Flagged first, then by citations

Research works printed but not linked: 4

  1. Ya. B. Zel'dovich, 1964, Dokl. Akad. Nauk SSSR, 155, 1033
  2. I. S. Shklovskiǐ, 1964, Dokl. Akad. Nauk SSSR, 155, 1039
  3. Ya. B. Zel'dovich, 1964, Dokl. Akad. Nauk SSSR, 156, 57
  4. Ya. B. Zel'dovich, 1964, Proceedings of the 1963 Dallas Conference on Gravitational Collapse

Cited by

Sorted by citations

Related

Authors
Penrose, R.
Affiliations
Birkbeck, University of London (GB)
Corresponding author
Penrose, Roger
Venue
Physical Review Letters
Publisher
American Physical Society (APS)
Published
1965-01-18
Volume
14
Issue
3
Pages
57-59
ISSN
0031-9007 (ISSN-L)
DOI
10.1103/physrevlett.14.57
OpenAlex
W2052045615
Semantic Scholar
16e21b5e29346e9413274a9051c44888943c274a
Cited by
2,977
References
11
Citations by year
2026: 270 · 2025: 357 · 2024: 300 · 2023: 277 · 2022: 173 · 2021: 202 · 2020: 148 · 2019: 165 · 2018: 108 · 2017: 93 · 2016: 82 · 2015: 69 · 2014: 55 · 2013: 49 · 2012: 36
FWCI
16.61
Type
journal article
Language
en
Open access
Licence
link.aps.org/licenses/aps-default-license (vor)
Locations
Physical Review Letters (publishedVersion)
Topics
  • Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
  • Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Cosmology and Gravitation Theories
  • Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
Keywords
gravitational collapse, spacetime singularities, blast wave, cosmic ray observation

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