The DOI matches its Crossref record; 11 of the 11 references it cites were found in a registry or an index; none of the 2 sources its claims cite is checked for support yet.
Gravitational Collapse and Space-Time Singularities
Open accessAbstract
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.
Overview
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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.
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Claims the paper makes
12 No basis given · 6 Own interpretation · 2 Sources not found
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
§ GRAVITATIONAL COLLAPSE AND SPACE-TIME SINGULARITIES > Abstract
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The impossibility of this follows from the noncompactness of C'.”
§ GRAVITATIONAL COLLAPSE AND SPACE-TIME SINGULARITIES > Abstract
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- 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+.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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'.”
§ GRAVITATIONAL COLLAPSE AND SPACE-TIME SINGULARITIES > Abstract
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- 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+.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Hence B' is compact (closed), being generated by a compact system of finite segments.”
§ GRAVITATIONAL COLLAPSE AND SPACE-TIME SINGULARITIES > Abstract
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- 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+.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“To an outside observer the contraction to ~= 2m appears to take an infinite time.”
§ GRAVITATIONAL COLLAPSE AND SPACE-TIME SINGULARITIES > Abstract
hedgeddirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: unclear — It is unclear why the paper says it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“the existence o a trapped sur ace 1m”
- Also stated in: § Abstract · plain
“that singularities necessarily il develop.metry”
§ GRAVITATIONAL COLLAPSE AND SPACE-TIME SINGULARITIES > Abstract
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- 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.
- Also stated in: § Abstract · plain
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“It will be shown here, in outline, that (i), ~~~, (v) are together inconsistent.”
§ GRAVITATIONAL COLLAPSE AND SPACE-TIME SINGULARITIES > Abstract
boosted- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- 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+.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The empty region surrounding the matter will, in th1s case, Schwarzsc 1 h ld field”
§ GRAVITATIONAL COLLAPSE AND SPACE-TIME SINGULARITIES > Abstract
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: unclear — It is unclear why the paper says it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The existence of a singular1ty can tion such d however, without an assump io inferre, ow r con- as completeness for the manifold under c si eration.”
§ GRAVITATIONAL COLLAPSE AND SPACE-TIME SINGULARITIES > Abstract
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
In the paper
“The recent rotating solution of Kerr' also possesses a physical singularity”
§ GRAVITATIONAL COLLAPSE AND SPACE-TIME SINGULARITIES > Abstract
What the sources say
- Not matchedcites Kerr'
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
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- Not matched
- It may state two separate claims as one. 6
- Probably not a substantial claim. 1
- The quote only partly states this claim. 2
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
§ GRAVITATIONAL COLLAPSE AND SPACE-TIME SINGULARITIES > Abstract
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- 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+.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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'.”
§ GRAVITATIONAL COLLAPSE AND SPACE-TIME SINGULARITIES > Abstract
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Each geodesic thus encounters a caustic.”
§ GRAVITATIONAL COLLAPSE AND SPACE-TIME SINGULARITIES > Abstract
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The quote only partly states this 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+.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
§ GRAVITATIONAL COLLAPSE AND SPACE-TIME SINGULARITIES > Abstract
boosteddirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: unclear — It is unclear why the paper says it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“As measured by local comoving observers”
- Also stated in: § Abstract · plain
“the body passes within its Schwarzschild radius r = 2m.”
§ GRAVITATIONAL COLLAPSE AND SPACE-TIME SINGULARITIES > Abstract
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The quote only partly states this claim.
- Also stated in: § Abstract · plain
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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”
§ GRAVITATIONAL COLLAPSE AND SPACE-TIME SINGULARITIES > Abstract
plainconditional- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- 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.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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.”
§ GRAVITATIONAL COLLAPSE AND SPACE-TIME SINGULARITIES > Abstract
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
§ GRAVITATIONAL COLLAPSE AND SPACE-TIME SINGULARITIES > Abstract
plainconditional- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
In the paper
“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.”
§ GRAVITATIONAL COLLAPSE AND SPACE-TIME SINGULARITIES > Abstract
What the sources say
- Not matchedcites Kerr'
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- Not matched
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.”
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?
The source backs part of the claim, not all of it — or backs it for a different population, condition or comparison.
In the citing paper
“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]
Experimental — accuracy not yet measured
Citations
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Research works printed but not linked: 4
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0 On the topology of untrapped surfaces
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Metadata
- 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)
- 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
- bronze
- doi.org
- 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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