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
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
Cites
Statistics
Verification
Claims the paper makes
0 of 2 cited sources checked · 0 supported · 0 flagged
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
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesThe 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
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- 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
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- 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
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- 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
Strength:hedged· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear? Based: unclearA 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
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- 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
Load-bearingStrength:boosted· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- 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
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes? Based: unclearA 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
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIn 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.
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- 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
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt 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
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesProbably 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
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesThe 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
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes? Based: unclearIt 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
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesThe 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
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt 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
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt 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
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt 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.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt 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
Cites
Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics
Kerr, R. P. (1963) · Physical Review Letters · doi:10.1103/physrevlett.11.237 · Cited by 3,563
An Approach to Gravitational Radiation by a Method of Spin Coefficients
Newman, E., & Penrose, R. (1962) · Journal of Mathematical Physics · doi:10.1063/1.1724257 · Cited by 2,702
Self-citationOn Continued Gravitational Contraction
Oppenheimer, J. R., & Snyder, H. S. (1939) · Physical Review · doi:10.1103/physrev.56.455 · Cited by 2,054
Relativity, groups and topology
L.R. (1965) · Nuclear Physics · doi:10.1016/0029-5582(65)90020-9 · Cited by 1,104
Investigations in relativistic cosmology
LIFSHITZ, E. M., & Khalatnikov, I. M. (1963) · Advances In Physics · doi:10.1080/00018736300101283 · Cited by 915
Past-Future Asymmetry of the Gravitational Field of a Point Particle
Finkelstein, D. (1958) · Physical Review · doi:10.1103/physrev.110.965 · Cited by 542
On the Nature of Strong Radio Sources
Hoyle, F., & Fowler, W. A. (1962) · Monthly Notices of the Royal Astronomical Society · doi:10.1093/mnras/125.2.169 · Cited by 228
HOYLE, F., Fowler, W. A., Burbidge, G. R., & Burbidge, E. M. (1964) · The Astrophysical Journal · doi:10.1086/147825 · Cited by 185
Massive Stars, Relativistic Polytropes, and Gravitational Radiation
Fowler, W. A. (1964) · Reviews of Modern Physics · doi:10.1103/revmodphys.36.545 · Cited by 92
On the avoidance of singularities in C -field cosmology
HOYLE, F., & Narlikar, J. V. (1964) · Proceedings of the Royal Society of London A Mathematical and Physical Sciences · doi:10.1098/rspa.1964.0076 · Cited by 80
Flagged first, then by citations
Research works printed but not linked: 4
- Ya. B. Zel'dovich, 1964, Dokl. Akad. Nauk SSSR, 155, 1033
- I. S. Shklovskiǐ, 1964, Dokl. Akad. Nauk SSSR, 155, 1039
- Ya. B. Zel'dovich, 1964, Dokl. Akad. Nauk SSSR, 156, 57
- Ya. B. Zel'dovich, 1964, Proceedings of the 1963 Dallas Conference on Gravitational Collapse
Cited by
Sorted by citations
Items 301–400 of 2,958
From geometry to numerics: interdisciplinary aspects in mathematical and numerical relativity
Jaramillo, J. L., Kroon, J. A. V., & Gourgoulhon, É. (2008) · Classical and Quantum Gravity · doi:10.1088/0264-9381/25/9/093001 · Cited by 53
Hyperbolic Geometric Flow (I): Short-time Existence and Nonlinear Stability
Dai, W.-R., Kong, D.-X., & Liu, K. (2010) · Pure and Applied Mathematics Quarterly · doi:10.4310/pamq.2010.v6.n2.a3 · Cited by 52
Chen, B. (2007) · Journal of Mathematical Analysis and Applications · doi:10.1016/j.jmaa.2007.09.021 · Cited by 52
Generalisation for regular black holes on general relativity to f(R) gravity
Rodrigues, M. E., Fabris, J. C., Ednaldo, L. B., & Marques, G. T. (2016) · The European Physical Journal C · doi:10.1140/epjc/s10052-016-4085-x · Cited by 52
Manchak, J. B. (2020) · Cambridge University Press eBooks · doi:10.1017/9781108876070 · Cited by 51
Quasinormal modes and gray-body factors of regular black holes in asymptotically safe gravity
Stashko, O. S. (2024) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.110.084016 · Cited by 51
Hoop conjecture for black-hole horizon formation
Flanagan, É. É. (1991) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.44.2409 · Cited by 50
Wave character of metrics and hyperbolic geometric flow
Kong, D.-X., & Liu, K. (2007) · Journal of Mathematical Physics · doi:10.1063/1.2795839 · Cited by 51
General Relativity and Einstein's Equations
Choquet–Bruhat, Y. (2008) · Oxford University Press eBooks · doi:10.1093/acprof:oso/9780199230723.003.0003 · Cited by 51
On the structure and applications of the Bondi–Metzner–Sachs group
Alessio, F., & Esposito, G. V. M. (2017) · International Journal of Geometric Methods in Modern Physics · doi:10.1142/s0219887818300027 · Cited by 51
Marginally trapped surfaces in Lorentzian space forms with positive relative nullity
Chen, B., & der Veken, J. V. (2006) · Classical and Quantum Gravity · doi:10.1088/0264-9381/24/3/003 · Cited by 51
GRAVITATIONAL PERFECT FLUID COLLAPSE WITH COSMOLOGICAL CONSTANT
Sharif, M., & Ahmad, Z. (2007) · Modern Physics Letters A · doi:10.1142/s0217732307021834 · Cited by 51
Nonpolynomial Lagrangian approach to regular black holes
Colléaux, A., Chinaglia, S., & Zerbini, S. (2018) · International Journal of Modern Physics D · doi:10.1142/s0218271818300021 · Cited by 60
Constraints on singularity resolution by nonlinear electrodynamics
Bokulić, A., Smolić, I., & Jurić, T. (2022) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.106.064020 · Cited by 50
Övgün, A., Pantig, R. C., & Rincón, Á. (2023) · The European Physical Journal Plus · doi:10.1140/epjp/s13360-023-03793-w · Cited by 50
Heidari, N., Filho, A. A. A., Pantig, R. C., & Övgün, A. (2025) · Physics of the Dark Universe · doi:10.1016/j.dark.2025.101815 · Cited by 58
Weak cosmic censorship, superradiance, and quantum particle creation
Semiz, İ., & Düztaş, K. (2015) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.92.104021 · Cited by 50
Trapped surfaces in vacuum spacetimes
Beig, R., & Murchadha, N. Ó. (1994) · Classical and Quantum Gravity · doi:10.1088/0264-9381/11/2/013 · Cited by 50
BEHAVIOR OF THE SCHWARZSCHILD SINGULARITY IN SUPERIMPOSED GRAVITATIONAL FIELDS
Mysak, L. A., & Szekeres, G. (1966) · Canadian Journal of Physics · doi:10.1139/p66-051 · Cited by 50
Relativistic SU(6) wave functions as the basis of modern approaches to hadronic wave functions
Hussain, F., Körner, J. G., & Thompson, G. (1991) · Annals of Physics · doi:10.1016/0003-4916(91)90004-r · Cited by 50
Holonomy corrected Schwarzschild black hole lensing
Soares, A. R., Pereira, C. F. S., Vitória, R. L. L., & Rocha, E. M. (2023) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.108.124024 · Cited by 56
Acceleration of particles by Janis-Newman-Winicour singularities
Patil, M., & Joshi, P. S. (2012) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.85.104014 · Cited by 49
AN INTRODUCTION TO QUANTUM GRAVITY
DeWitt, B. S., & Esposito, G. V. M. (2008) · International Journal of Geometric Methods in Modern Physics · doi:10.1142/s0219887808002679 · Cited by 59
Quasinormal modes and their overtones at the common horizon in a binary black hole merger
Mourier, P., Forteza, X. J., Pook-Kolb, D., Krishnan, B., & Schnetter, E. (2021) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.103.044054 · Cited by 49
Black hole singularity resolution via the modified Raychaudhuri equation in loop quantum gravity
Blanchette, K., Das, S., Hergott, S., & Rastgoo, S. (2021) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.103.084038 · Cited by 53
No Cauchy horizon theorem for nonlinear electrodynamics black holes with charged scalar hairs
An, Y.-S., Li, L., & Yang, F.-G. (2021) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.104.024040 · Cited by 49
Aguirre, A. N., & Johnson, M. C. (2006) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.73.123529 · Cited by 52
83 years of general relativity and cosmology: progress and problems
Ellis, G. (1999) · Classical and Quantum Gravity · doi:10.1088/0264-9381/16/12a/303 · Cited by 46
The stochastic gravitational-wave background in the absence of horizons
Barausse, E., Brito, R., Cardoso, V., Dvorkin, I., & Pani, P. (2018) · MPG.PuRe (Max Planck Society) · Cited by 48
Gravitational collapse of a charged fluid sphere
Mashhoon, B., & Partovi, M. H. (1979) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.20.2455 · Cited by 48
Stachel, J. J. (2004) · Institute of Physics eBooks · doi:10.1887/0750303530/b213v1c4 · Cited by 48
Nonsingularity of the exact two-dimensional string black hole
Perry, M. J., & Teo, E. (1993) · Physical Review Letters · doi:10.1103/physrevlett.70.2669 · Cited by 51
Black Holes and Gravitational Theory
Penrose, R. (1972) · Nature · doi:10.1038/236377a0 · Cited by 47
Self-citationBouncing cosmology in modified gravity with higher-order curvature terms
Singh, J. K., Balhara, H., Bamba, K., & Jena, J. (2023) · Journal of High Energy Physics · doi:10.1007/jhep03(2023)191 · Cited by 51
Fundamental physics opportunities with the next-generation Event Horizon Telescope
Ayzenberg, D., Blackburn, L., Brito, R., Britzen, S., Broderick, A. E., Carballo-Rubio, R., Cardoso, V., Chael, A., Chatterjee, K., Chen, Y., et al. (2025) · Living Reviews in Relativity · doi:10.1007/s41114-025-00057-0 · Cited by 109
Fundamental Physics and General Relativity with the LARES and LAGEOS satellites
Ciufolini, I., Paolozzi, A., Koenig, R., PAVLIS, E. C., Ries, J. C., Matzner, R. A., Gurzadyan, V., Penrose, R., Sindoni, G., & Paris, C. (2013) · Nuclear Physics B - Proceedings Supplements · doi:10.1016/j.nuclphysbps.2013.09.005 · Cited by 47
Self-citationSpace–time singularities and cosmic censorship conjecture: A Review with some thoughts
Ong, Y. C. (2020) · International Journal of Modern Physics A · doi:10.1142/s0217751x20300070 · Cited by 50
Scalar perturbations and stability of a loop quantum corrected Kruskal black hole
Daghigh, R. G., Green, M. D., & Kunstatter, G. (2021) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.103.084031 · Cited by 52
WGC and WCCC of black holes with quintessence and cloud strings in RPS space
Alipour, M. R., Sadeghi, J., & Shokri, M. (2023) · Nuclear Physics B · doi:10.1016/j.nuclphysb.2023.116184 · Cited by 43
Primordial regular black holes as all the dark matter. I. Time-radial-symmetric metrics
Calzà, M., Pedrotti, D., & Vagnozzi, S. (2025) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.111.024009 · Cited by 49
Matter conditions for regular black holes in f(T) gravity
Aftergood, J., & DeBenedictis, A. (2014) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.90.124006 · Cited by 46
Rodnianski, I., & Speck, J. (2017) · Annals of Mathematics · doi:10.4007/annals.2018.187.1.2 · Cited by 46
The shadow of M87∗ black hole within rational nonlinear electrodynamics
Kruglov, S. I. (2020) · Modern Physics Letters A · doi:10.1142/s0217732320502910 · Cited by 46
Spacetime near isolated and dynamical trapping horizons
Booth, I. (2013) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.87.024008 · Cited by 50
Bogojević, A., & Stojković, D. (2000) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.61.084011 · Cited by 46
Black holes in three-dimensional dilaton gravity theories
Sá, P. M., Kleber, A., & Lemos, J. P. S. (1996) · Classical and Quantum Gravity · doi:10.1088/0264-9381/13/1/011 · Cited by 50
Singularities from colliding plane gravitational waves
Tipler, F. J. (1980) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.22.2929 · Cited by 46
Exploring New Physics Frontiers Through Numerical RelativityPreprint
Cardoso, V., Gualtieri, L., Herdeiro, C., & Sperhake, U. (2015) · Cited by 55
Topology of nonlinearly charged black hole chemistry via massive gravity
Zhang, M., Chen, H., HASSANABADI, H., Long, Z., & Yang, H. (2023) · The European Physical Journal C · doi:10.1140/epjc/s10052-023-11933-w · Cited by 51
Particle motion, shadows and thermodynamics of regular black hole in pure gravity
Ditta, A., Bouzenada, A., Mustafa, G., Alanazi, Y. M., & Mushtaq, F. (2024) · Physics of the Dark Universe · doi:10.1016/j.dark.2024.101573 · Cited by 46
Mustafa, G., Maurya, S. K., Ditta, A., Ray, S., & Atamurotov, F. (2024) · The European Physical Journal C · doi:10.1140/epjc/s10052-024-13055-3 · Cited by 46
Regular black holes from thin-shell collapse
Bueno, P., Cano, P. A., Hennigar, R. A., & Murcia, Á. (2025) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.111.104009 · Cited by 45
Black hole spectroscopy: from theory to experiment
Berti, E., Cardoso, V., Carullo, G., Abedi, J., Afshordi, N., Albanesi, S., Baibhav, V., Bhagwat, S., Blázquez-Salcedo, J. L., Bonga, B., et al. (2026) · Classical and Quantum Gravity · doi:10.1088/1361-6382/ae59e2 · Cited by 248
On singular semi-Riemannian manifolds
Stoica, O. C. (2014) · International Journal of Geometric Methods in Modern Physics · doi:10.1142/s0219887814500418 · Cited by 45
Gravitational collapse in Rastall gravity
Ziaie, A. H., Moradpour, H., & Ghaffari, S. (2019) · Physics Letters B · doi:10.1016/j.physletb.2019.04.055 · Cited by 45
Shadow cast by a rotating black hole with anisotropic matter
Lee, B.-H., Lee, W., & Myung, Y. S. (2021) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.103.064026 · Cited by 45
Plate with a hole obeys the averaged null energy condition
Graham, N., & Olum, K. D. (2005) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.72.025013 · Cited by 45
Astrophysical evidence for the existence of black holes
Celotti, A., Miller, J. C., & Sciama, D. W. (2000) · CERN Document Server (European Organization for Nuclear Research) · Cited by 45
Horizon energy and angular momentum from a Hamiltonian perspective
Booth, I., & Fairhurst, S. (2005) · Classical and Quantum Gravity · doi:10.1088/0264-9381/22/21/006 · Cited by 47
Two-dimensional quantum-corrected eternal black hole
Solodukhin, S. N. (1996) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.53.824 · Cited by 46
Chinea, F. J., Fernández-Jambrina, L., & Senovilla, J. M. M. (1992) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.45.481 · Cited by 45
Shadow and deflection angle of asymptotic, magnetically-charged, non-singular black hole
Kumaran, Y., & Övgün, A. (2023) · The European Physical Journal C · doi:10.1140/epjc/s10052-023-12001-z · Cited by 44
Quasinormal modes of quantum-corrected black holes
Gong, H., Li, S., Zhang, D., Fu, G., & Wu, J.-P. (2024) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.110.044040 · Cited by 63
New solutions of charged regular black holes and their stability
Uchikata, N., Yoshida, S., & Futamase, T. (2012) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.86.084025 · Cited by 44
A reformulation of the hoop conjecture
Senovilla, J. M. M. (2007) · Europhysics Letters (EPL) · doi:10.1209/0295-5075/81/20004 · Cited by 44
Tensile Strength and the Mining of Black Holes
Brown, A. R. (2013) · Physical Review Letters · doi:10.1103/physrevlett.111.211301 · Cited by 49
Cosmic Censorship for Gowdy Spacetimes
Ringström, H. (2010) · Living Reviews in Relativity · doi:10.12942/lrr-2010-2 · Cited by 44
Cavalletti, F., & Mondino, A. (2024) · Cambridge Journal of Mathematics · doi:10.4310/cjm.2024.v12.n2.a3 · Cited by 44
Accurate mapping of spherically symmetric black holes in a parametrized framework
Kocherlakota, P., & Rezzolla, L. (2020) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.102.064058 · Cited by 45
Coley, A. A., & Ellis, G. (2019) · Classical and Quantum Gravity · doi:10.1088/1361-6382/ab49b6 · Cited by 42
Mathematical Foundations of the Theory of Relativistic Stellar and Black Hole Configurations
Carter, B. (1987) · NATO ASI series. Series B : Physics · doi:10.1007/978-1-4613-1897-2_2 · Cited by 44
THE NATURE OF SPACETIME SINGULARITIES
Rendall, A. D. (2005) · WORLD SCIENTIFIC eBooks · doi:10.1142/9789812700988_0003 · Cited by 43
Eichhorn, A., Held, A., & Johannsen, P.-V. (2023) · Journal of Cosmology and Astroparticle Physics · doi:10.1088/1475-7516/2023/01/043 · Cited by 54
Quasinormal modes of a holonomy corrected Schwarzschild black hole
Moreira, Z. S., Lima, H. C. D., Crispino, L. C. B., & Herdeiro, C. (2023) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.107.104016 · Cited by 51
Topological classes of thermodynamics of black holes in perfect fluid dark matter background
Rizwan, M. A., & Jusufi, K. (2023) · The European Physical Journal C · doi:10.1140/epjc/s10052-023-12126-1 · Cited by 43
Optical appearance of black holes surrounded by a dark matter halo
Macedo, C. F. B., Rosa, J. L., & Rubiera-García, D. (2024) · Journal of Cosmology and Astroparticle Physics · doi:10.1088/1475-7516/2024/07/046 · Cited by 43
On black bounce space-times in non-linear electrodynamics
Alencar, G., Bronnikov, K. A., Rodrigues, M. E., Gómez, D. S.-C., & Silva, M. V. D. S. (2024) · The European Physical Journal C · doi:10.1140/epjc/s10052-024-13119-4 · Cited by 43
Ellipsoidal collapse and primordial black hole formation
Kühnel, F., & Sandstad, M. (2016) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.94.063514 · Cited by 43
WETTERICH, C. (2014) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.90.043520 · Cited by 43
No Page curves for the de Sitter horizon
Kames-King, J., Verheijden, E., & Verlinde, E. (2022) · Journal of High Energy Physics · doi:10.1007/jhep03(2022)040 · Cited by 53
Complete classification of parallel surfaces in 4-dimensional Lorentzian space forms
Chen, B., & der Veken, J. V. (2009) · Tohoku Mathematical Journal · doi:10.2748/tmj/1238764545 · Cited by 43
Quantum evolution of black hole initial data sets: Foundations
Alesci, E., Bahrami, S., & Pranzetti, D. (2018) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.98.046014 · Cited by 43
Quasiperiodic oscillations for spherically symmetric regular black holes
Boshkayev, K., Idrissov, A., Luongo, O., & Muccino, M. (2023) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.108.044063 · Cited by 42
Weak cosmic censorship in Kerr-Sen black hole under charged scalar field
Gwak, B. (2020) · Journal of Cosmology and Astroparticle Physics · doi:10.1088/1475-7516/2020/03/058 · Cited by 42
Stability of magnetic black holes in general nonlinear electrodynamics
Nomura, K., Yoshida, D., & Soda, J. (2020) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.101.124026 · Cited by 42
On the Inner Horizon Instability of Non-Singular Black Holes
Filippo, F. D., Carballo-Rubio, R., Liberati, S., Pacilio, C., & Visser, M. (2022) · Universe · doi:10.3390/universe8040204 · Cited by 41
A connection between regular black holes and horizonless ultracompact stars
Carballo-Rubio, R., Filippo, F. D., Liberati, S., & Visser, M. (2023) · Journal of High Energy Physics · doi:10.1007/jhep08(2023)046 · Cited by 41
Topological classes of black holes in de-Sitter spacetime
Du, Y., & Zhang, X. (2023) · The European Physical Journal C · doi:10.1140/epjc/s10052-023-12114-5 · Cited by 55
Shadow and greybody bounding of a regular scale-dependent black hole solution
Övgün, A., Pantig, R. C., & Rincón, Á. (2024) · Annals of Physics · doi:10.1016/j.aop.2024.169625 · Cited by 41
Experimental studies of black holes: status and future prospects
Genzel, R., Eisenhauer, F., & Gillessen, S. (2024) · The Astronomy and Astrophysics Review · doi:10.1007/s00159-024-00154-z · Cited by 48
Cosmological perturbations and stability of nonsingular cosmologies with limiting curvature
Yoshida, D., Quintin, J., Yamaguchi, M., & Brandenberger, R. (2017) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.96.043502 · Cited by 41
Charged black hole in gravity's rainbow: Violation of weak cosmic censorship
Gim, Y., & Gwak, B. (2019) · Physics Letters B · doi:10.1016/j.physletb.2019.05.039 · Cited by 41
On the assumptions leading to the information loss paradox
Buoninfante, L., Filippo, F. D., & Shinji Mukohyama, A. (2021) · Journal of High Energy Physics · doi:10.1007/jhep10(2021)081 · Cited by 44
Shadows and precession of orbits in rotating Janis–Newman–Winicour spacetime
Solanki, D. N., Bambhaniya, P., Dey, D., Joshi, P. S., & Pathak, K. (2022) · The European Physical Journal C · doi:10.1140/epjc/s10052-022-10045-1 · Cited by 41
Black brane entropy and hydrodynamics: The boost-invariant case
Booth, I., Heller, M. P., & Spaliński, M. (2009) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.80.126013 · Cited by 41
Hawking’s singularity theorem for C 1,1 -metrics
Kunzinger, M., Steinbauer, R., Stojkovic, М., & Vickers, J. A. (2015) · Classical and Quantum Gravity · doi:10.1088/0264-9381/32/7/075012 · Cited by 41
Curvature estimates for stable marginally trapped surfaces
Andersson, L., & Metzger, J. (2010) · Journal of Differential Geometry · doi:10.4310/jdg/1274707313 · Cited by 41
A CP5 calculus for space-time fields
Hughston, L. P. (1983) · Physics Reports · doi:10.1016/0370-1573(83)90003-0 · Cited by 41
Gravitational Radiation from a Naked Singularity: Odd-Parity Perturbation
Iguchi, H., Harada, T., & Nakao, K.-I. (1999) · Progress of Theoretical Physics · doi:10.1143/ptp.101.1235 · Cited by 41
Bouncing unitary cosmology I. Mini-superspace general solution
Gryb, S., & Thébault, K. P. Y. (2018) · Classical and Quantum Gravity · doi:10.1088/1361-6382/aaf823 · Cited by 43
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
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
A language model drew the claims from the paper on 2026-10-30, under version 3 of the claim definition (model deepseek-v4.1-flash, prompt extract-v3).
A second model labelled each claim on 2026-10-30 (model jev-1.13-20260917, definition revision gold-guideline-v0.2).
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
