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The influence of Penrose’s singularity theorem in general relativity

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Abstract

Penrose’s crucial contributions to General Relativity, symbolized by his 1965 singularity theorem, received (half of) the 2020 Nobel prize in Physics. A renewed interest in the ideas and implications behind that theorem, its later developments, and other Penrose’s ideas improving our understanding of the gravitational field thereby emerged. In this paper I highlight some of the advancements motivated by the theorem that were developed over the years. I also identify some common misconceptions about the theorem’s implications. A modern perspective on the concept of closed trapped submanifolds, based on the mean curvature vector, is advocated.

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Notes

  1. Vacuum or electromagnetic plane waves’ Ricci tensor takes the form \(R_{\mu \nu }= -(A(u)+C(u)) k_\mu k_\nu \) where \(k^\mu \) is the parallel vector field of the spacetime

  2. The original definition is still used in most of the physics literature, and in a large part of the mathematical one. The reasons behind this are obscure to me. The characterization with the mean curvature vector is clearly neater and provides more information, apart from unifying the concept for arbitrary dimensional manifolds. A (future) trapped submanifold has all possible (future) expansions negative, not only the null expansions. Moreover, the computation of null expansions and the choice of null directions complicates the explicit calculations. Computing the mean curvature vector is far simpler! [136]

  3. Here, what I mean by the ‘boundary of the future’ of \(\zeta \) is \(E^+(\zeta )\) [72, 73, 121, 135], defined as the set of points that can be reached from \(\zeta \) causally, but not through a timelike curve. If \(E^+(\zeta )\) is compact, then \(\zeta \) is a trapped set. Not to be confused with a trapped submanifold. Trapped submanifolds become trapped sets precisely under the appropriate curvature condition –if spacetime is null complete–, as explained in the text.

  4. If one prefers the traditional notion with negative expansions, they are also strict inequalities

References

  1. Aazami, A.B., Javaloyes, M.A.: Penrose’s singularity theorem in a Finsler spacetime Class. Quantum Gravit. 33, 025003 (2016)

  2. Andersson, L., Blue, P., Wyatt, Z., Yau, S.-T.: Global stability of spacetimes with supersymmetric compactifications (2020) arXiv:2006.00824

  3. Andersson, L., Mars, M., Simon, W.: Local existence of dynamical and trapping horizons. Phys. Rev. Lett. 95, 111102 (2005)

    Article  ADS  Google Scholar 

  4. Andersson, L., Mars, M., Simon, W.: Stability of marginally outer trapped surfaces and existence of marginally outer trapped tubes. Adv. Theor. Math. Phys. 12, 853 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Andersson, L., Mars, M., Metzger, J., Simon, W.: The time evolution of marginally trapped surfaces. Class. Quantum Gravit. 26, 085018 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Ashtekar, A., Krishnan, B.: Isolated and dynamical horizons and their applications. Living Rev. Relativ. 7, 10 (2004)

    Article  ADS  MATH  Google Scholar 

  7. Ashtekar, A., Galloway, G.J.: Some uniqueness results for dynamical horizons. Adv. Theor. Math. Phys. 9, 1 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Barceló, C., Liberati, S., Visser, M.: Analogue gravity. Living Rev. Relativ. 14, 3 (2011)

    Article  ADS  MATH  Google Scholar 

  9. Baumgarte, T.W., Shapiro, S.L.: Numerical Relativity: Solving Einstein’s Equations on the Computer. Cambridge University Press, Cambridge (2010)

  10. Beem, J.K., Ehrlich, P.E., Easley, K.L.: Global Lorentzian Geomertry, Pure and Applied Mathematics, vol. 202. Marcel Dekker, New York (1996)

    Google Scholar 

  11. Bengtsson, I.: Some examples of trapped surfaces, Chapter 9 in [75] (2013)

  12. Bengtsson, I., Senovilla, J.M.M.: Region with trapped surfaces in spherical symmetry, its core, and their boundaries. Phys. Rev. D 83, 044012 (2011)

    Article  ADS  Google Scholar 

  13. Bernal, A.N., Sánchez, M.: On smooth Cauchy hypersurfaces and Geroch’s splitting theorem. Commun. Math. Phys. 243, 461–70 (2003)

  14. Bernal, A.N., Sánchez, M.: Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes. Commun. Math. Phys. 257, 43–50 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Bernal, A.N., Sánchez, M.: Globally hyperbolic spacetimes can be defined as ‘causal’ instead of ‘strongly causal’. Class. Quantum Gravit. 24, 745–49 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Bojowald, M.: Singularities and quantum gravity (lecture course at the XIIth Brazilian School on Cosmology and Gravitation 2006). AIP Conf. Proc. 910, 294–333 (2007)

    Article  ADS  Google Scholar 

  17. Booth, I.: Black hole boundaries. Can. J. Phys. 83, 1073–1099 (2005)

    Article  ADS  Google Scholar 

  18. Borde, A., Vilenkin, A.: Singularities in inflationary cosmology: a review. Int. J. Mod. Phys. D 5, 813 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  19. Branding, V., Fajman, D., Kröncke, K.: Stable cosmological Kaluza–Klein spacetimes. Commun. Math. Phys. 368, 1087–120 (2019)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Carroll, S.M., Geddes, J., Hoffman, M.B., Wald, R.M.: Classical stabilization of homogeneous extra dimensions. Phys. Rev. D 66, 024036 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  21. Carter, B.: Causal structure in space-time. Gen. Relativ. Gravit. 1, 349 (1971)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Chinea, F.J., Fernández-Jambrina, L., Senovilla, J.M.M.: Singularity-free space-time. Phys. Rev. D 45, 481 (1992)

    Article  ADS  MathSciNet  Google Scholar 

  23. Choquet-Bruhat, Y., Geroch, R.: Global aspects of the Cauchy problem in general relativity. Commun. Math. Phys. 14, 329–35 (1969)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. Christodoulou, D.: The Formation of Black Holes in General Relativity, EMS Monographs in Mathematics. European Mathematics, Soc (Zürich) (2009).. (arXiv:0805.3880)

    MATH  Google Scholar 

  25. Chruściel, P.T., Li, Y., Weinstein, G.: Mass and angular-momentum inequalities for axi-symmetric initial data sets: II. Angular-Momentum Ann. Phys. 323, 2591–2613 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. Cipriani, N., Senovilla, J.M.M.: Singularity theorems for warped products and the stability of spatial extra dimensions. JHEP 04, 175 (2019)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. Clarke, C.J.S.: The Analysis of Space-Time Singularities, (Cambridge Lectures Notes in Physics 1, Cambridge) (1993)

  28. Clarke, C.J.S., Schmidt, B.G.: Singularities: the state of the art. Gen. Relativ. Gravit. 8, 129 (1977)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. Claudel, C.-M.: Black holes and closed trapped surfaces: a revision of a classic theorem. arXiv:gr-qc/0005031 (2000)

  30. Dafermos, M.: Spherically symmetric spacetimes with a trapped surface. Class. Quantum Gravit. 22, 2221–2232 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. Dafermos, M., Luk, J.: The interior of dynamical vacuum black holes I: The \(C^0\)-stability of the Kerr Cauchy horizon. arXiv: 1710.01722 (2017)

  32. Dain, S.: Angular-momentum mass inequality for axisymmetric black holes. Phys. Rev. Lett. 96, 101101 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. Dain, S.: Proof of the angular momentum-mass inequality for axisymmetric black holes. J. Differ. Geom. 79, 33–67 (2008)

    MathSciNet  MATH  Google Scholar 

  34. Dain, S.: Geometric inequalities for axially symmetric black holes. Class. Quantum Gravit. 29, 073001 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. Dain, S., Jaramillo, J.L., Reiris, M.: Area-charge inequality for black holes. Class Quantum Gravit. 29, 035013 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  36. Dain, S., Reiris, M.: Area-angular-momentum inequality for axisymmetric black holes. Phys. Rev. Lett. 107, 051101 (2011)

    Article  ADS  Google Scholar 

  37. Eichmair, M.: The Plateau problem for marginally trapped surfaces. J. Differ. Geom. 83, 551–584 (2009)

    MathSciNet  MATH  Google Scholar 

  38. Einstein, A.: Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie, Sitzungsber. Preuss. Akad. Wiss. phys.-math Klasse VI 142–152 (1917)

  39. Exton, A.R., Newman, E.T., Penrose, R.: Conserved quantities in the Einstein–Maxwell theory. J. Math. Phys. 10, 1566–70 (1969)

    Article  ADS  MathSciNet  Google Scholar 

  40. Fayos, F., Martín-Prats, M.M., Senovilla, J.M.M.: On the extension of Vaidya and Vaidya–Reissner–Nordström spacetimes. Class. Quantum Gravit. 12, 2565–2576 (1995)

    Article  ADS  MATH  Google Scholar 

  41. Fayos, F., Senovilla, J.M.M., Torres, R.: General matching of two spherically symmetric spacetimes. Phys. Rev. D 54, 4862 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  42. Flores, J.L.: The causal boundary of spacetimes revisited. Commun. Math. Phys. 276, 611–643 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  43. Flores, J.L., Harris, S.G.: Topology of the causal boundary for standard static spacetimes. Class. Quantum Gravit. 24, 1211–1260 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  44. Flores, J.L., Herrera, J., Sánchez, M.: Isocausal spacetimes may have different causal boundaries. Class. Quantum Gravit. 28, 175016 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  45. Flores, J.L., Herrera, J., Sánchez, M.: On the final definition of the causal boundary and its relation with the conformal boundary. Adv. Theor. Math. Phys. 15, 991–1057 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  46. Flores, J.L., Herrera, J., Sánchez, M.: Gromov, Cauchy and causal boundaries for Riemannian, Finslerian and Lorentzian manifolds. Mem. Am. Math. Soc. 226(1064) (2013)

  47. Flores, J.L., Sánchez, M.: Causality and conjugate points in general plane waves. Class. Quantum Gravit. 20, 2275–91 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  48. Fourès-Bruhat, Y.: Théorèm d’existence pour certain systèmes d’équations aux derivées partielles non linéaires. Acta Math. 88, 141–225 (1952)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  49. Frauendiener, J.: Conformal infinity. Living Rev. Relativ. 7, 1 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  50. Gabach Clement, M.E., Jaramillo, J.L., Reiris, M.: Proof of the area-angular momentum-charge inequality for axisymmetric black holes. Class. Quantum Gravit. 30, 065017 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  51. Galloway, G.J., Senovilla, J.M.M.: Singularity theorems based on trapped submanifolds of arbitrary co-dimension. Class Quantum Gravit. 27, 152002 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  52. García-Parrado, A., Sánchez, M.: Further properties of causal relationship: causal structure stability, new criteria for isocausality and counterexamples. Class. Quantum Gravit. 22, 4589–619 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  53. García-Parrado, A., Senovilla, J.M.M.: Causal relationship: a new tool for the causal characterization of Lorentzian manifolds. Class. Quantum Gravit. 20, 625–64 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  54. García-Parrado, A., Senovilla, J.M.M.: Causal structures and causal boundaries. Class. Quantum Gravit. 22, R1–R84 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  55. Garfinkle, D., Harris, S.G.: Ricci fall-off in static and stationary, globally hyperbolic, non-singular spacetimes. Class. Quantum Gravit. 14, 139 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  56. Geroch, R.P.: Singularities in closed universes. Phys. Rev. Lett. 17, 445 (1966)

    Article  ADS  MATH  Google Scholar 

  57. Geroch, R.P.: What is a singularity in general relativity? Ann. Phys. (N.Y.) 48, 526 (1968)

    Article  ADS  MATH  Google Scholar 

  58. Geroch, R.P.: Domain of dependence. J. Math. Phys. 11, 437 (1970)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  59. Geroch, R.P., Kronheimer, E.H., Penrose, R.: Ideal points in space-time. Proc. R. Soc. Lond. A 327, 545 (1972)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  60. Gibbons, G.W.: The isoperimetric and Bogomolny inequalities for black holes, In: Global Riemannian Geometry edited by Willmore T. J. and Hitchin N. J. (Ellis Harwood, Chichester), p. 194 (1984)

  61. Gibbons, G.W.: Collapsing shells and the isoperimetric inequality. CQG 14, 2905 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  62. Graf, M.: Singularity theorems for \(C^1\)-Lorentzian metrics. Commun. Math. Phys. 378, 1417–1450 (2020)

    Article  ADS  MATH  Google Scholar 

  63. Graf, M., Grant, J.D.E., Kunzinger, M., Steinbauer, R.: The Hawking-Penrose singularity theorem for \(C^{1,1}\)-Lorentzian metrics. Commun. Math. Phys. 360, 1009–1042 (2018)

    Article  ADS  MATH  Google Scholar 

  64. Griffiths, J.B.: Colliding Plane Waves in General Relativity. Oxford University Press, Oxford (1991)

    MATH  Google Scholar 

  65. Griffiths, J.B., Podolský, J.: Exact Spacetimes in Einstein’s General Relativity. Cambridge University Press (2009)

  66. Hawking, S.W.: Occurrence of singularities in open universes. Phys. Rev. Lett. 15, 689 (1965)

    Article  ADS  MathSciNet  Google Scholar 

  67. Hawking, S.W.: Singularities in the Universe. Phys. Rev. Lett. 17, 443 (1966)

    Article  ADS  Google Scholar 

  68. Hawking, S.W.: The occurrence of singularities in Cosmology. I. Proc. R. Soc. Lond. A 294, 511 (1966)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  69. Hawking, S.W.: The occurrence of singularities in Cosmology. II. Proc. R. Soc. Lond. A 295, 490 (1966)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  70. Hawking, S.W.: The occurrence of singularities in cosmology. III. Causality and singularities. Proc. R. Soc. Lond. A 300, 187 (1967)

  71. Hawking, S.W.: Black holes in general relativity. Commun. Math. Phys. 25, 152 (1972)

    Article  ADS  MathSciNet  Google Scholar 

  72. Hawking, S.W., Ellis, G.F.R.: The Large Scale Structure of Space-Time. Cambridge University Press, Cambridge (1973)

    Book  MATH  Google Scholar 

  73. Hawking, S.W., Penrose, R.: The singularities of gravitational collapse and cosmology. Proc. R. Soc. Lond. A 314, 529 (1970)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  74. Hayward, S.A.: General laws of black-hole dynamics. Phys. Rev. D 49, 6467 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  75. Hayward, S.A. (ed.): Black Holes: New Horizons. World Scientific, Singapore (2013)

  76. Israel, W.: Gravitational collapse of a radiating star. Phys. Lett. A 24, 184–6 (1967)

    Article  ADS  Google Scholar 

  77. Jaramillo, J.L.: An Introduction to Local Black Hole Horizons in the 3+1 Approach to General Relativity. Chapter 1 in [75] (2013)

  78. Jaramillo, J.L., Reiris, M., Dain, S.: Black hole area-angular momentum inequality in non-vacuum spacetimes. Phys. Rev. D 84, 121503(R) (2011)

    Article  ADS  Google Scholar 

  79. Jaramillo, J.L., Valiente Kroon, J.A., Gourgoulhon, E.: From geometry to numerics: interdisciplinary aspects in mathematical and numerical relativity. Class. Quantum Gravit. 25, 093001 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  80. Kánnár, J., Rácz, I.: On the strength of space-time singularities. J. Math. Phys. 33, 2842–2848 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  81. Khan, K., Penrose, R.: Scattering of two impulsive gravitational plane waves. Nature 229, 185–186 (1971)

    Article  ADS  Google Scholar 

  82. Klainerman, S., Rodnianski, I.: On the formation of trapped surfaces. Acta Math. 208, 211–333 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  83. Komar, A.: Necessity of singularities in the solution of the field equations of general relativity. Phys. Rev. 104, 544 (1956)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  84. Kriele, M.: Spacetime. Springer, Berlin (1999)

    MATH  Google Scholar 

  85. Kronheimer, E.H., Penrose, R.: On the structure of causal spaces. Proc. Camb. Philos. Soc. 63, 481 (1967)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  86. Kunzinger, M., Ohanyan, A., Schinnerl, B., Steinbauer, R.: The Hawking–Penrose singularity theorem for \(C^1\)-Lorentzian metrics. arXiv:2110.09176 (2022)

  87. Kunzinger, M., Steinbauer, R., Stojković, M., Vickers, J.A.: A regularisation approach to causality theory for \(C^{1,1}\) Lorentzian metrics. Gen. Relativ. Gravit. 46, 1738 (2014)

    Article  ADS  MATH  Google Scholar 

  88. Kunzinger, M., Steinbauer, R., Stojković, M., Vickers, J.A.: Hawking’s singularity theorem for \(C^{1,1}\)-metrics. Class. Quantum Gravit. 32, 075012 (2015)

    Article  ADS  MATH  Google Scholar 

  89. Kunzinger, M., Steinbauer, R., Vickers, J.A.: The Penrose singularity theorem in regularity \(C^{1,1}\). Class. Quantum Gravit. 32, 155010 (2015)

    Article  ADS  MATH  Google Scholar 

  90. Landsman, K.: Penrose’s 1965 singularity theorem: From geodesic incompleteness to cosmic censorship. Gen. Relativ. Gravit. 54, 115 (2022)

  91. Leray, J.: Hyperbolic Differential Equations, Institute for Advanced Study, (Princeton Preprint) (1952)

  92. Lesourd, M.: A new singularity theorem for black holes which allows chronology violation in the interior. Class. Quantum Gravit. 35, 245003 (2018)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  93. Maeda, H.: Quest for realistic non-singular black-hole geometries: regular-center type. arXiv:2107.04791 (2021)

  94. Malec, E.: Isoperimetric inequalities in the physics of black holes. Acta Phys. Pol. B 22, 829–858 (1991)

    ADS  MathSciNet  Google Scholar 

  95. Mars, M.: Present status of the Penrose inequality. Class. Quantum Gravit. 26, 193001 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  96. Mars, M., Martín-Prats, M.M., Senovilla, J.M.M.: Models of regular Schwarzschild black holes satisfying weak energy conditions. Class. Quantum Gravit. 13, L51–L58 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  97. Mars, M., Senovilla, J.M.M.: Geometry of general hypersurfaces in spacetime: junction conditions. Class. Quantum Gravit. 10, 1865–1897 (1993)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  98. Mars, M., Senovilla, J.M.M.: Trapped surfaces and symmetries. Class. Quantum Gravit. 20, L293 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  99. Minguzzi, E.: Convex neighborhoods for Lipschitz connections and sprays. Monatsh Math. 177, 569–625 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  100. Minguzzi, E.: The boundary of the chronology violating set. Class. Quantum Gravit. 33, 225004 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  101. Minguzzi, E.: Lorentzian causality theory. Living reviews in relativity vol. 22:3, 1–202. Springer (2019)

  102. Minguzzi, E.: A gravitational collapse singularity theorem consistent with black hole evaporation. Lett. Math. Phys. 110, 2383–2396 (2020)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  103. Minguzzi, E., Sánchez, M.: The causal hierarchy of spacetimes, in Recent developments in pseudo-Riemannian geometry, H. Baum and D. Alekseevsky (eds.), ESI Lect. Math. Phys. (Eur. Math. Soc. Publ. House, Zurich), pp. 299–358 (2008)

  104. Misner, C.W., Thorne, K.S., Wheeler, J.A.: Gravitation. W.H. Freeman and Co., New York (1973)

    Google Scholar 

  105. Newman, E.T., Penrose, R.: An approach to gravitational radiation by a method of spin coefficients J. Math. Phys.3 566–78; erratum4 998 (1962)

  106. Newman, E.T., Penrose, R.: 10 exact gravitationally conserved quantities. Phys. Rev. Lett. 15, 231–33 (1965)

    Article  ADS  MathSciNet  Google Scholar 

  107. Newman, E.T., Penrose, R.: New conservation laws for zero rest-mass fields in asymptotically flat space-time. Proc. R. Soc. Lond. A 305, 175–204 (1968)

    Article  ADS  Google Scholar 

  108. Newman, R.P.A.C.: Persistent curvature and cosmic censorship. Gen. Relativ. Gravit. 16, 1177 (1984)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  109. Newman, R.P.A.C.: Topology and stability of marginal 2-surfaces. Class. Quantum Gravit. 4, 277 (1987)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  110. O’Neill, B.: Semi-Riemannian Geometry: With Applications to Relativity. Academic Press, Cambridge (1983)

    MATH  Google Scholar 

  111. Oppenheimer, J.R., Snyder, H.: On continued gravitational contraction. Phys. Rev. 56, 455–459 (1939)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  112. Oppenheimer, J.R., Volkoff, G.M.: On massive neutron cores. Phys. Rev. 55, 374 (1939)

    Article  ADS  MATH  Google Scholar 

  113. Penrose, R.: A spinor approach to general relativity. Ann. Pays. (N.Y.) 10, 171–201 (1960)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  114. Penrose, R.: Asymptotic properties of fields and space-times. Phys. Rev. Lett. 10, 66–68 (1963)

    Article  ADS  MathSciNet  Google Scholar 

  115. Penrose, R.: Conformal treatment of infinity, In: Relativity Groups and Topology eds. C M de Witt and B de Witt (New York: Gordon and Breach) pp. 566–84 (1964)

  116. Penrose, R.: Gravitational collapse and space-time singularities. Phys. Rev. Lett. 14, 57 (1965)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  117. Penrose, R.: A remarkable property of plane waves in General Relativity. Rev. Mod. Phys. 37, 215 (1965)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  118. Penrose, R.: Zero rest-mass fields including gravitation: asymptotic behaviour. Proc. R. Soc. Lond. Ser. A 284, 159–203 (1965)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  119. Penrose, R.: Structure of space-time. In: Batelle Rencontres, C. M. de Witt and J. A. Wheeler, eds. (Benjamin, New York) (1968)

  120. Penrose, R.: Gravitational collapse: the role of general relativity. Riv. Nuovo Cimento 1, 252 (1969)

    Google Scholar 

  121. Penrose, R.: Techniques of Differential Topology in Relativity, Regional Conference Series in Applied Math. 7 (SIAM, Philadelphia) (1972)

  122. Penrose, R.: Naked singularities. Ann. N. Y. Acad. Sci. 224, 125 (1973)

    Article  ADS  MATH  Google Scholar 

  123. Penrose, R.: Singularities and time asymmetry, In: General Relativity: an Einstein Centenary Survey, S. W. Hawking and W. Israel, eds. (Cambridge University Press, Cambridge) (1979)

  124. Penrose, R.: The question of cosmic censorship. J. Astrophys. Astron. 20, 233–248 (1999)

    Article  ADS  Google Scholar 

  125. Penrose, R.: On the instability of extra space dimensions, in the future of the theoretical physics and cosmology, 185–201. Cambridge University Press, Cambridge (2003)

    Google Scholar 

  126. Penrose, R., Floyd, R.M.: Extraction of rotational energy from a black hole. Nat. Phys. Sci. 229, 177 (1971)

    Article  ADS  Google Scholar 

  127. Penrose, R., Rindler, W.: Spinors and Spacetime, vol. 1. Cambridge University Press, Cambridge (1984)

    Book  MATH  Google Scholar 

  128. Penrose, R., Rindler, W.: Spinors and Spacetime, vol. 2. Cambridge University Press, Cambridge (1986)

    Book  MATH  Google Scholar 

  129. Poisson, E., Israel, W.: Inner-horizon instability and mass inflation in black holes. Phys. Rev. Lett. 63, 1663 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  130. Raychaudhuri, A.K.: Relativistic cosmology I. Phys. Rev. 98, 1123 (1955)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  131. Raychaudhuri, A.K.: Singular state in relativistic cosmology. Phys. Rev. 106, 172 (1957)

    Article  ADS  Google Scholar 

  132. Reiterer, M., Trubowitz, E.: Strongly focused gravitational waves. Commun. Math. Phys. 307, 275 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  133. Ringström, H.: Origin and development of the Cauchy problem in general relativity. Class. Quantum Gravit. 32, 124003 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  134. Senovilla, J.M.M.: New class of inhomogeneous cosmological perfect-fluid solutions without big-bang singularity. Phys. Rev. Lett. 64, 2219 (1990)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  135. Senovilla, J.M.M.: Singularity theorems and their consequences. Gen. Relativ. Gravit. 30, 701–48 (1998)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  136. Senovilla, J.M.M.: Trapped surfaces, horizons and exact solutions in higher dimensions. Class. Quantum Gravit. 19, L113 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  137. Senovilla, J.M.M.: On the existence of horizons in spacetimes with vanishing curvature invariants. J. High Energy Phys. 11, 046 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  138. Senovilla, J.M.M.: Classification of spacelike surfaces in spacetime. Class. Quantum Gravit. 24, 3091–3124 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  139. Senovilla, J.M.M.: A singularity theorem based on spatial averages. Pramana 69, 31–47 (2007)

    Article  ADS  Google Scholar 

  140. Senovilla, J.M.M.: A new type of singularity theorem. In: Proceedings of 30th Spanish Relativity Meeting ERE2007, A. Oscoz, E. Mediavilla and M. Serra-Ricart eds., EAS Publ.Ser. 30 (2008). arXiv:0712.1428

  141. Senovilla, J.M.M.: Trapped surfaces. Int. J. Mod. Phys. D 20, 2139–2168 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  142. Senovilla, J.M.M.: Singularity theorems in general relativity: achievements and open questions, Chapter 15 of Einstein and the Changing Worldviews of Physics (eds. C Lehner, J Renn, M Schemmel), Einstein Studies 12, (Birkhäuser) (2012)

  143. Senovilla, J.M.M.: A critical appraisal of the singularity theorems. Philos. Trans. R. Soc. A 380, 20210174 (2022). https://doi.org/10.1098/rsta.2021.0174

    Article  ADS  MathSciNet  Google Scholar 

  144. Senovilla, J.M.M., Garfinkle, D.: The 1965 Penrose singularity theorem. Class. Quantum Gravit. 32, 124008 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  145. Simon, W.: Bounds on area and charge for marginally trapped surfaces with cosmological constant. Class. Quantum Gravit. 29, 062001 (2012)

    Article  ADS  MATH  Google Scholar 

  146. Steinbauer, R.: The singularity theorems of General Relativity and their low regularity extensions. Jahresber. Dtsch. Math. Ver. (2022)

  147. Stephani, H., Kramer, D., MacCallum, M.A.H., Hoenselaers, C., Herlt, E.: Exact Solutions to Einstein’s Field Equations, 2nd edn. Cambridge University Press, Cambridge (2003)

  148. Szabados, L.B.: On singularity theorems and curvature growth. J. Math. Phys. 28, 142 (1987)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  149. Thornburg, J.: Event and apparent horizon finders for 3+1 numerical relativity. Living Rev. Relativ. 10, 3 (2007)

    Article  ADS  MATH  Google Scholar 

  150. Thorpe, J.A.: Curvature invariants and spacetime singularities. J. Math. Phys. 18, 960 (1977)

    Article  ADS  Google Scholar 

  151. Tipler, F.J.: On the nature of singularities in general relativity. Phys. Rev. D 15, 942 (1977)

    Article  ADS  MathSciNet  Google Scholar 

  152. Tipler, F.J.: Singularities from colliding plane gravitational waves. Phys. Rev. D 22, 2929 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  153. Tipler, F.J., Clarke, C.J.S., Ellis, G.F.R.: Singularities and Horizons—a review article. In: General Relativity and Gravitation: One Hundred Years After the Birth of Albert Einstein, A. Held, ed. (Plenum Press, New York) (1980)

  154. Vilenkin, A., Wall, A.C.: Cosmological singularity theorems and black holes. Phys. Rev. D 89, 064035 (2014)

    Article  ADS  Google Scholar 

  155. Wald, R.M.: General Relativity. The University of Chicago Press, Chicago (1984)

    Book  MATH  Google Scholar 

  156. Wald, R.M.: Gravitational collapse and cosmic censorship, In: Black Holes, Gravitational Radiation and the Universe, edited by B.R. Iyer and B. Bhawal (Springer, Berlin) arXiv: gr-qc/9710068 (1998)

  157. Wall, A.C.: The generalized second law implies a quantum singularity theorem. Class. Quantum Gravit. 30, 165003 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  158. Witten, E.: Instability of the Kaluza–Klein vacuum. Nucl. Phys. B 195, 481 (1982)

    Article  ADS  MATH  Google Scholar 

  159. Wyatt, Z.: The weak null condition and Kaluza–Klein spacetimes. J. Hyperbolic Differ. Equ. 15, 219–58 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  160. Yurtsever, U.: Singularities in the collisions of almost-plane gravitational waves. Phys. Rev. D 38, 1731 (1988)

    Article  ADS  MathSciNet  Google Scholar 

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Acknowledgements

This article is freely based on my contribution The 1965 singularity theorem and its legacy to the conference “Singularity theorems, causality, and all that: A tribute to Roger Penrose” (SCRI21), June 14–18, 2021. I am grateful to the organizers for giving me the opportunity to deliver the opening talk.

Funding

Supported by the Basque Government grant number IT1628-22, and by Grants FIS2017-85076-P and PID2021-123226NB-I00 funded by the Spanish MCIN/AEI/10.13039/501100011033 together with “ERDF A way of making Europe”.

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  1. Departamento de Física, University of the Basque Country UPV/EHU, Apartado 644, 48080, Bilbao, Spain

    José M. M. Senovilla

  2. EHU Quantum Center, University of the Basque Country UPV/EHU, Leioa, Spain

    José M. M. Senovilla

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Senovilla, J.M.M. The influence of Penrose’s singularity theorem in general relativity. Gen Relativ Gravit 54, 151 (2022). https://doi.org/10.1007/s10714-022-03038-8

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