Weak cosmic censorship, trapped surfaces, and naked singularities for the Einstein vacuum equations

Yakov Shlapentokh-Rothman

doi:10.5802/crmeca.284

Article de synthèse

Weak cosmic censorship, trapped surfaces, and naked singularities for the Einstein vacuum equations
[Censure cosmique faible, surfaces piégées et singularités nues pour les équations d’Einstein dans le vide]

Yakov Shlapentokh-Rothman ^1,²

¹ Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, ON, Canada
² Department of Mathematical and Computational Sciences, University of Toronto Mississauga, 3359 Mississauga Road, Mississauga, ON, Canada

Comptes Rendus. Mécanique, Volume 353 (2025), pp. 379-410.

Cet article fait partie du numéro thématique Avancées récentes dans le domaine de la relativité générale : numéro en l'honneur d'Yvonne Choquet-Bruhat coordonné par Jérémie Szeftel.

Résumé
Abstract

La conjecture de censure cosmique faible postule que, de manière générale, toutes les singularités en relativité générale provenant de données initiales régulières asymptotiquement plates devraient avoir un infini nul futur complet. Bien que cette conjecture reste largement ouverte, elle a inspiré de nombreux travaux mathématiques portant sur des sujets tels que la formation de surfaces piégées et la construction de singularités nues. Dans cet article, nous passerons en revue certains de ces travaux et tenterons de mettre en évidence leurs interconnexions. interconnectedness.

The weak cosmic censorship conjecture posits that, generically, all singularities in General Relativity arising from regular asymptotically flat initial data should have a complete future null infinity. While this conjecture remains wide open, it has inspired many mathematical works concerning topics such as trapped surface formation and the construction of naked singularities. In this article we will review some of these works and attempt to emphasize their interconnectedness.

Reçu le : 2024-10-11
Révisé le : 2025-01-13
Accepté le : 2025-01-13
Publié le : 2025-02-06

DOI : 10.5802/crmeca.284

Keywords: Weak cosmic censorship, Trapped surfaces, Naked singularities
Mots-clés : Censure cosmique faible, Surfaces piégées, Singularités nues

Affiliations des auteurs :

Yakov Shlapentokh-Rothman ^{1, 2}

¹ Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, ON, Canada
² Department of Mathematical and Computational Sciences, University of Toronto Mississauga, 3359 Mississauga Road, Mississauga, ON, Canada

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Weak cosmic censorship, trapped surfaces, and naked singularities for the {Einstein} vacuum equations},
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Yakov Shlapentokh-Rothman. Weak cosmic censorship, trapped surfaces, and naked singularities for the Einstein vacuum equations. Comptes Rendus. Mécanique, Volume 353 (2025), pp. 379-410. doi : 10.5802/crmeca.284. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.284/

Bibliographie
Cité par

[1] Y. Fourès-Bruhat Théorème d’existence pour certains systèmes d’équations aux dérivées partielles non linéaires, Acta Math., Volume 88 (1952), pp. 141-225 | DOI | Zbl

[2] Y. Choquet-Bruhat; R. P. Geroch Global aspects of the Cauchy problem in general relativity, Commun. Math. Phys., Volume 14 (1969), pp. 329-335 | DOI | Zbl

[3] R. M. Wald General Relativity, The University of Chicago Press, Chicago, 1984 | DOI

[4] K. Schwarzschild Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie, Sitzungsber. K. Preuss. Akad. Wiss., Volume 1 (1916), pp. 189-196 | Zbl

[5] M. Dafermos; I. Rodnianski Lectures on black holes and linear waves, Evolution Equations (Clay Mathematics Proceedings), Volume 17, American Mathematical Society, Providence, RI, 2013, pp. 97-205 | Zbl

[6] J. Sbierski The C $_{0}$ -inextendibility of the Schwarzschild spacetime and the spacelike diameter in Lorentzian geometry, J. Differ. Geom., Volume 108 (2018) no. 2, pp. 319-378 | DOI | Zbl

[7] J. Sbierski On the proof of the C $^{0}$ -inextendibility of the Schwarzschild spacetime, J. Phys. Conf. Ser., Volume 968 (2018), 012012 | DOI

[8] R. Penrose Gravitational collapse and space-time singularities, Phys. Rev. Lett., Volume 14 (1965), pp. 57-59 | DOI | Zbl

[9] D. Christodoulou; S. Klainerman The Global Nonlinear Stability of the Minkowski Space, Princeton Mathematical Series, 41, Princeton University Press, Princeton, NJ, 1993, x+514 pages

[10] S. Klainerman; F. Nicolò The Evolution Problem in General Relativity, Progress in Mathematical Physics, 25, Birkhäuser Boston Inc., Boston, MA, 2003, xiv+385 pages | DOI

[11] D. Christodoulou On the global initial value problem and the issue of singularities, Class. Quantum Gravity, Volume 16 (1999), p. A23-A35 | DOI | Zbl

[12] D. Shen Global stability of Minkowski spacetime with minimal decay, preprint, 2023 | arXiv

[13] R. Penrose Gravitational collapse: the role of general relativity, Riv. Nuovo Cim., Volume 1 (1969), pp. 252-276

[14] D. Christodoulou Examples of naked singularity formation in the gravitational collapse of a scalar field, Ann. Math. (2), Volume 140 (1994) no. 3, pp. 607-653 | DOI | Zbl

[15] M. Dafermos; J. Luk The interior of dynamical vacuum black holes I: the C $^{0}$ -stability of the Kerr Cauchy horizon, Ann. Math. (2017) (to appear)

[16] D. Christodoulou The formation of black holes and singularities in spherically symmetric gravitational collapse, Commun. Pure Appl. Math., Volume 44 (1991) no. 3, pp. 339-373 | DOI | Zbl

[17] D. Christodoulou Bounded variation solutions of the spherically symmetric Einstein-scalar field equations, Commun. Pure Appl. Math., Volume 46 (1993) no. 8, pp. 1131-1220 | DOI | Zbl

[18] D. Christodoulou The instability of naked singularities in the gravitational collapse of a scalar field, Ann. Math., Volume 149 (1999), pp. 183-217 | DOI | Zbl

[19] J. Singh High regularity waves on self-similar naked singularity interiors: decay and the role of blue-shift, preprint, 2024 | arXiv

[20] M. Dafermos Spherically symmetric spacetimes with a trapped surface, Class. Quantum Gravity, Volume 22 (2005) no. 11, pp. 2221-2232 | DOI | Zbl

[21] J. Singh A construction of approximately self-similar naked singularities for the spherically symmetric Einstein-scalar field system, Ann. Henri Poincaré (2024) | DOI

[22] D. Christodoulou Violation of cosmic censorship in the gravitational collapse of a dust cloud, Commun. Math. Phys., Volume 93 (1984) no. 2, pp. 171-195 | DOI

[23] Y. Guo; M. Hadzic; J. Jang Naked singularities in the Einstein–Euler system, Ann. PDE, Volume 9 (2023) no. 1, 4 | DOI | Zbl

[24] P. Bizoń; A. Wasserman On the existence of self-similar spherically symmetric wave maps coupled to gravity, Class. Quantum Gravity, Volume 19 (2002) no. 12, pp. 3309-3321 | DOI | Zbl

[25] M. Choptuik Universality and scaling in gravitational collapse of a massive scalar field, Phys. Rev. Lett., Volume 70 (1993), pp. 9-12 | DOI

[26] C. Gundlach; J. Martin-Garcia Critical phenomena in gravitational collapse, Living Rev. Relativ., Volume 10 (2007), 5 | DOI | Zbl

[27] J. Liu; J. Li A robust proof of the instability of naked singularities of a scalar field in spherical symmetry, Commun. Math. Phys., Volume 363 (2018) no. 2, pp. 561-578 | DOI | Zbl

[28] J. Li; J. Liu Instability of spherical naked singularities of a scalar field under gravitational perturbations, J. Differ. Geom., Volume 120 (2022) no. 1, pp. 97-197 | DOI | Zbl

[29] X. An Naked singularity censoring with aniostropic apparent horizon, Ann. Math. (2024) (to appear)

[30] V. Cardoso; J. A. L. Costa; K. Destounis; P. Hintz; A. Jansen Quasinormal modes and strong cosmic censorship, Phys. Rev. Lett., Volume 120 (2018), 031103 | DOI

[31] M. Dafermos; Y. Shlapentokh-Rothman Rough initial data and the strength of the blue-shift instability on cosmological black holes with $Λ$ ensuremath> 0, Class. Quantum Gravity, Volume 35 (2018) no. 19, 195010 | DOI | Zbl

[32] O. J. C. Dias; H. S. Reall; J. E. Santos Strong cosmic censorship: taking the rough with the smooth, J. High Energy Phys. (2018) no. 10, 1 | DOI | Zbl

[33] D. Christodoulou The Formation of Black Holes in General Relativity, EMS Monographs in Mathematics, European Mathematical Society (EMS), Zürich, 2009, x+589 pages | DOI

[34] X. Chen; S. Klainerman Formation of trapped surfaces in geodesic foliation, preprint, 2024 | arXiv

[35] M. Dafermos The formation of black holes in general relativity [after D. Christodoulou], Sémin. Bourbaki, Volume 2011/2012 (2013) no. 352, pp. 243-313 (Exposés 1043–1058, p. Exp. No. 1051, viii) | Zbl

[36] J. Li; P. Yu Construction of Cauchy data of vacuum Einstein field equations evolving to black holes, Ann. Math. (2), Volume 181 (2015) no. 2, pp. 699-768 | DOI | Zbl

[37] J. Corvino; R. M. Schoen On the asymptotics for the vacuum Einstein constraint equations, J. Differ. Geom., Volume 73 (2006) no. 2, pp. 185-217 | DOI | Zbl

[38] C. Kehle; R. Unger Event horizon gluing and black hole formation in vacuum: the very slowly rotating case, Adv. Math., Volume 452 (2024), 109816 | DOI | Zbl

[39] S. Klainerman; I. Rodnianski On the formation of trapped surfaces, Acta Math., Volume 208 (2012) no. 2, pp. 211-333 | DOI | Zbl

[40] S. Klainerman; I. Rodnianski On emerging scarred surfaces for the Einstein vacuum equations, Discrete Contin. Dyn. Syst., Volume 28 (2010) no. 3, pp. 1007-1031 | DOI | Zbl

[41] X. An Formation of trapped surfaces in general relativity, PhD thesis, Princeton University, ProQuest LLC, Ann Arbor, MI (2014)

[42] S. Klainerman; J. Luk; I. Rodnianski A fully anisotropic mechanism for formation of trapped surfaces in vacuum, Invent. Math., Volume 198 (2014) no. 1, pp. 1-26 | DOI | Zbl

[43] S. Klainerman; I. Rodnianski; J. Szeftel The bounded L $^{2}$ curvature conjecture, Invent. Math., Volume 202 (2015) no. 1, pp. 91-216 | DOI | Zbl

[44] J. Szeftel Sharp Strichartz estimates for the wave equation on a rough background, Ann. Sci. Éc. Norm. Supér. (4), Volume 49 (2016) no. 6, pp. 1279-1309 | DOI | Zbl

[45] J. Szeftel Parametrix for wave equations on a rough background III: space–time regularity of the phase, Astérisque (2018) no. 401, p. viii+321 | Zbl

[46] J. Szeftel Parametrix for wave equations on a rough background. I: regularity of the phase at initial time. II: construction and control at initial time, Astérisque (2023) no. 443, p. ix+275 | DOI

[47] J. Szeftel Parametrix for wave equations on a rough background: IV. Control of the error term, Astérisque (2023) no. 444, p. viii+314 | DOI

[49] J. Luk; I. Rodnianski Local propagation of impulsive gravitational waves, Commun. Pure Appl. Math., Volume 68 (2015) no. 4, pp. 511-624 | DOI | Zbl

[50] J. Luk; I. Rodnianski Nonlinear interaction of impulsive gravitational waves for the vacuum Einstein equations, Camb. J. Math., Volume 5 (2017) no. 4, pp. 435-570 | DOI | Zbl

[51] R. Penrose The geometry of impulsive gravitational waves, General Relativity (Papers in Honour of J. L. Synge), Clarendon Press, Oxford, 1972, pp. 101-115 | Zbl

[52] K. A. Khan; R. Penrose Scattering of two impulsive gravitational plane waves, Nature, Volume 229 (1971) no. 5281, pp. 185-186 | DOI

[53] J. Luk; I. Rodnianski High-frequency limits and null dust solutions in general relativity, preprint, 2020 | arXiv

[54] G. A. Burnett The high-frequency limit in general relativity, J. Math. Phys., Volume 30 (1989) no. 1, pp. 90-96 | DOI | Zbl

[55] C. Huneau; J. Luk High-frequency solutions to the Einstein equations, Class. Quantum Gravity, Volume 41 (2024), 143002 | DOI | Zbl

[56] X. An; J. Luk Trapped surfaces in vacuum arising dynamically from mild incoming radiation, Adv. Theor. Math. Phys., Volume 21 (2017) no. 1, pp. 1-120 | DOI | Zbl

[57] X. An Emergence of apparent horizon in gravitational collapse, Ann. PDE, Volume 6 (2020) no. 2, 10 | DOI | Zbl

[58] X. An A scale-critical trapped surface formation criterion: a new proof via signature for decay rates, Ann. PDE, Volume 8 (2022) no. 1, 3 | DOI | Zbl

[59] X. An; T. He Dynamics of apparent horizon and a null comparison principle, Ann. PDE, Volume 10 (2024), 15 | DOI | Zbl

[60] X. An; Q. Han Anisotropic dynamical horizons arising in gravitational collapse, preprint, 2020 | arXiv

[61] I. Rodnianski; Y. Shlapentokh-Rothman Naked singularities for the Einstein vacuum equations: the exterior solution, Ann. Math. (2), Volume 198 (2023) no. 1, pp. 231-391 | DOI | Zbl

[62] Y. Shlapentokh-Rothman Naked singularities for the Einstein vacuum equations: the interior solution, preprint, 2022 | arXiv | Zbl

[63] F. J. Tipler Singularities in conformally flat spacetimes, Phys. Lett. A, Volume 64 (1977) no. 1, pp. 8-10 | DOI

[64] C. Fefferman; C. R. Graham Conformal invariants, Astérisque (1985), pp. 95-116 Numéro Hors Série The Mathematical Heritage of Élie Cartan (Lyon, 1984) | Zbl

[65] C. Fefferman; C. R. Graham The Ambient Metric, Annals of Mathematics Studies, 178, Princeton University Press, Princeton, NJ, 2012, x+113 pages

[66] I. Rodnianski; Y. Shlapentokh-Rothman The asymptotically self-similar regime for the Einstein vacuum equations, Geom. Funct. Anal., Volume 28 (2018) no. 3, pp. 755-878 | DOI | Zbl

[67] Y. Shlapentokh-Rothman Twisted self-similarity and the Einstein vacuum equations, Commun. Math. Phys., Volume 401 (2023) no. 2, pp. 2269-2325 | DOI | Zbl

[68] L. Lehner; F. Pretorius Black strings, low viscosity fluids, and violation of cosmic censorship, Phys. Rev. Lett., Volume 105 (2010) no. 10, 101102 | DOI | Zbl

[69] X. An; X. Zhang Examples of naked singularity formation in higher-dimensional Einstein-vacuum spacetimes, Ann. Henri Poinc., Volume 19 (2018) no. 2, pp. 619-651 | DOI | Zbl

[70] A. D. Rendall Reduction of the characteristic initial value problem to the Cauchy problem and its applications to the Einstein equations, Proc. R. Soc. Lond. A, Volume 427 (1990) no. 1872, pp. 221-239 | Zbl

[71] J. Luk On the local existence for the characteristic initial value problem in general relativity, Int. Math. Res. Not. IMRN (2012) no. 20, pp. 4625-4678 | DOI | Zbl

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