RSS feeds
РусскийEnglish
Issue 4, 2025 Retrieve →
Issues Authors PACS Subscription For authors
Search →

Issues

 / 

1971

 / 

June

←  →

Reviews of topical problems


Oscillatory approach to the singular point in relativistic cosmology

, ,

Contents
1. Introduction 745
2. Kasner's Generalized Solution 746
3. Oscillatory Approach to the Singular Point 748
4. Evolution of Model Under the Influence of Two Perturbations 751
5. Evolution of Model in the Asymptotic Region of Arbitrarily Short Times 752
6. Statistical Analysis of the Evolution of the Model on Approaching the Singular Point 54
7. Construction of General Solution for a Long Era with Small Oscillations 756
8. Concluding Remarks 759
Appendices
A. Some Information from the Theory of Continuous Fractions 760
B. Refinement of the Calculations in Section 4 761
C. Homogeneous Spaces 762
D. Homogeneous Spaces of Type VIII and IX , 764
Cited Literature

Fulltext pdf (2.1 MB)
Fulltext is also available at DOI: 10.1070/PU1971v013n06ABEH004279
PACS: 98.80.Jk, 04.20.Jb, 95.30.Sf (all)
DOI: 10.1070/PU1971v013n06ABEH004279
URL: https://ufn.ru/en/articles/1971/6/d/
Citation: Belinskii V A, Lifshitz E M, Khalatnikov I M "Oscillatory approach to the singular point in relativistic cosmology" Sov. Phys. Usp. 13 745–765 (1971)
BibTexBibNote ® (generic)BibNote ® (RIS)MedlineRefWorks

Оригинал: Белинский В А, Лифшиц Е М, Халатников И М «Колебательный режим приближения к особой точке в релятивистской космологии» УФН 102 463–500 (1970); DOI: 10.3367/UFNr.0102.197011d.0463

Cited by (223) ↓ Similar articles (20)

  1. Li W Ann. PDE 11 (1) (2025)
  2. Ringström H 353 53 (2025)
  3. Li W, Van de Moortel Maxime Ann. PDE 11 (1) (2025)
  4. Van de Moortel Maxime 353 415 (2025)
  5. Lecian O M International Journal Of Topology 2 1 (2025)
  6. Acosta-Humánez P B, Morales-Ruiz Ju J, Stuchi T J SIAM J. Appl. Dyn. Syst. 24 815 (2025)
  7. Husain V, Muzammil M Phys. Rev. D 109 (4) (2024)
  8. Franco G A Ann. Henri Poincaré (2024)
  9. Giovannini M Phys. Rev. D 110 (12) (2024)
  10. Moneer E M, Alanazi M M et al Advances In Space Research 73 2753 (2024)
  11. Bruno M, Montani G Phys. Rev. D 107 (12) (2023)
  12. Conte R Phys. Scr. 98 105212 (2023)
  13. Van de Moortel Maxime Duke Math. J. 172 (15) (2023)
  14. Blacker M J, Ning S J. High Energ. Phys. 2023 (12) (2023)
  15. Chand K Gravit. Cosmol. 29 253 (2023)
  16. Moneer E M, Steklain A F et al Chaos, Solitons & Fractals 168 113152 (2023)
  17. Fournodavlos G, Rodnianski I, Speck Ja J. Amer. Math. Soc. 36 827 (2023)
  18. Cepollaro S, Chirco G et al Phys. Rev. D 108 (4) (2023)
  19. Bochner B Int. J. Mod. Phys. D 31 (13) (2022)
  20. Byalko A V Biogr. Mems Fell. R. Soc. 72 251 (2022)
  21. Alrebdi H I, Steklain A F, Zotos E E Chaos, Solitons & Fractals 158 112108 (2022)
  22. De Falco V, Borrelli W Phys. Rev. D 103 (6) (2021)
  23. Pinto-Neto N Universe 7 110 (2021)
  24. Scott S M, Whale B E Class. Quantum Grav. 38 065012 (2021)
  25. Sachs I, Schneider M, Urban M Phys. Rev. D 104 (12) (2021)
  26. Bodendorfer N, Mele F M, Münch J Class. Quantum Grav. 36 245013 (2019)
  27. Hofmann S, Schneider M, Urban M Phys. Rev. D 99 (6) (2019)
  28. Bodendorfer N, Schäfer A, Schliemann J J. High Energ. Phys. 2019 (6) (2019)
  29. Elizalde E Galaxies 6 25 (2018)
  30. Beckwith A W JHEPGC 04 236 (2018)
  31. Skugoreva M A, Toporensky A V Eur. Phys. J. C 78 (5) (2018)
  32. Singh P Comput. Sci. Eng. 20 26 (2018)
  33. Saini S, Singh P Class. Quantum Grav. 35 065014 (2018)
  34. Ali M, Husain V Phys. Rev. D 96 (4) (2017)
  35. Barrow J D, Ganguly Ch Phys. Rev. D 95 (8) (2017)
  36. Saini S, Singh P Class. Quantum Grav. 34 235006 (2017)
  37. Czuchry E, Garfinkle D et al Phys. Rev. D 95 (2) (2017)
  38. Garfinkle D Rep. Prog. Phys. 80 016901 (2017)
  39. Chugreev Yu V Phys. Part. Nuclei Lett. 14 539 (2017)
  40. Banijamali A, Solbi M Gen Relativ Gravit 49 (8) (2017)
  41. Ivashchuk V D Symmetry 9 155 (2017)
  42. Calcagni G Classical and Quantum Cosmology Graduate Texts In Physics Chapter 6 (2017) p. 261
  43. Celani Diogo C  F, Pinto-Neto N, Vitenti Sandro D  P Phys. Rev. D 95 (2) (2017)
  44. Ivashchuk V D, Melnikov V N Gravit. Cosmol. 22 166 (2016)
  45. Cataldo M, Cid A et al 57 (11) (2016)
  46. Ferronsky V I Gravitation, Inertia and Weightlessness Chapter 9 (2016) p. 281
  47. Popławski N ApJ 832 96 (2016)
  48. Barceló C, Carballo-Rubio R, Garay L Universe 2 7 (2016)
  49. Ivashchuk V D Eur. Phys. J. C 76 (5) (2016)
  50. Burlankov D E Gravit. Cosmol. 21 175 (2015)
  51. Guo Ju-Q, Joshi P S Phys. Rev. D 92 (6) (2015)
  52. Khalatnikov I M, Kamenshchik A Yu Uspekhi Fizicheskikh Nauk 185 948 (2015)
  53. [Khalatnikov I M, Kamenshchik A Yu Phys.-Usp. 58 878 (2015)]
  54. Berger B K Springer Handbook of Spacetime Springer Handbooks Chapter 21 (2014) p. 437
  55. Rennert Ju, Sloan D Class. Quantum Grav. 31 015017 (2014)
  56. Rubakov V A Uspekhi Fizicheskikh Nauk 184 137 (2014)
  57. [Rubakov V A Phys.-Usp. 57 128 (2014)]
  58. Scheel M A, Thorne K S Uspekhi Fizicheskikh Nauk 184 367 (2014)
  59. [Scheel M A, Thorne K S Phys.-Usp. 57 342 (2014)]
  60. Novello M, Bittencourt E, Salim J M Braz J Phys 44 832 (2014)
  61. Kohli I S, Haslam M C Phys. Rev. D 89 (4) (2014)
  62. Kim E J, Kawai Sh Phys. Rev. D 87 (8) (2013)
  63. Ivashchuk V D, Melnikov V N Gravit. Cosmol. 19 171 (2013)
  64. Ferronsky V I, Ferronsky S V Formation of the Solar System Chapter 10 (2013) p. 289
  65. Lecian O M, Montani G, Moriconi R Phys. Rev. D 88 (10) (2013)
  66. LUKASH VLADIMIR N, STROKOV VLADIMIR N Int. J. Mod. Phys. A 28 1350007 (2013)
  67. Lecian O M Phys. Rev. D 88 (10) (2013)
  68. Garfinkle D Class. Quantum Grav. 29 244003 (2012)
  69. Shchur L N Uspekhi Fizicheskikh Nauk 182 787 (2012)
  70. Gaspar Y, Acquaviva G Eur. Phys. J. Plus 127 (6) (2012)
  71. Ferronsky V I, Denisik S A, Ferronsky S V Astrophysics And Space Science Library Vol. Jacobi DynamicsConclusions369 Chapter 10 (2011) p. 311
  72. MARCOLLI MATILDE Int. J. Geom. Methods Mod. Phys. 08 1131 (2011)
  73. Ferronsky V I, Denisik S A, Ferronsky S V Astrophysics And Space Science Library Vol. Jacobi DynamicsIntroduction: General Principles and Approaches in Dynamics369 Chapter 1 (2011) p. 1
  74. Penrose R Phil. Trans. R. Soc. A. 369 4864 (2011)
  75. Damour T, Lecian O M Phys. Rev. D 83 (4) (2011)
  76. Kamenshchik A Yu Uspekhi Fizicheskikh Nauk 180 313 (2010)
  77. Saotome R, Akhoury R, Garfinkle D Class. Quantum Grav. 27 165019 (2010)
  78. Ivashchuk V D Gravit. Cosmol. 16 118 (2010)
  79. Abdolrahimi Sh, Shoom A A, Page D N Phys. Rev. D 82 (12) (2010)
  80. Middleton J Class. Quantum Grav. 27 225013 (2010)
  81. Ivashchuk V D, Melnikov V N Gravit. Cosmol. 15 49 (2009)
  82. Penrose R J. Phys.: Conf. Ser. 174 012001 (2009)
  83. Saha B, Rikhvitsky V Phys. Part. Nuclei 40 612 (2009)
  84. Choquet-Bruhat Y General Relativity and the Einstein Equations 1 (2008)
  85. Garfinkle D Class. Quantum Grav. 24 S295 (2007)
  86. Szydłowski M J. Cosmol. Astropart. Phys. 2007 007 (2007)
  87. GIOVANNINI MASSIMO Int. J. Mod. Phys. A 22 2697 (2007)
  88. Giovannini M Physics Letters B 634 1 (2006)
  89. Barrow J D, Clifton T Class. Quantum Grav. 23 L1 (2006)
  90. Polarski D Annalen Der Physik 518 342 (2006)
  91. Berger B K Gen Relativ Gravit 38 625 (2006)
  92. Berger B K Lecture Notes In Physics Vol. Analytical and Numerical Approaches to Mathematical RelativityComputer Simulation – a Tool for Mathematical Relativity – and Vice Versa692 Chapter 9 (2006) p. 187
  93. Lidsey Ja E Class. Quantum Grav. 23 3517 (2006)
  94. Da̧browski M P Annalen Der Physik 518 352 (2006)
  95. Clifton T, Barrow J D Class. Quantum Grav. 23 2951 (2006)
  96. Curtis J, Garfinkle D Phys. Rev. D 72 (6) (2005)
  97. Coley A A Can. J. Phys. 83 475 (2005)
  98. Novikov I D Springer Proceedings In Physics Vol. Particle Physics and the UniverseThe Internal Structure of Black Holes98 Chapter 10 (2005) p. 113
  99. Giovannini M J. Cosmol. Astropart. Phys. 2005 009 (2005)
  100. Carr B J, Coley A A Gen Relativ Gravit 37 2165 (2005)
  101. Ivashchuk V D, Melnikov V N, Singleton D Phys. Rev. D 72 (10) (2005)
  102. GARFINKLE DAVID Int. J. Mod. Phys. D 13 2261 (2004)
  103. Ivashchuk V D The Gravitational Constant: Generalized Gravitational Theories and Experiments Chapter 10 (2004) p. 205
  104. Ibáñez J, Jhingan S Phys. Rev. D 70 (6) (2004)
  105. Garfinkle D Phys. Rev. Lett. 93 (16) (2004)
  106. Harko T, Mak M K Class. Quantum Grav. 21 1489 (2004)
  107. Berger B K Class. Quantum Grav. 21 S81 (2004)
  108. Horowitz G T, Maldacena Ju J. High Energy Phys. 2004 008 (2004)
  109. Damour T International Conference on Theoretical Physics Chapter 23 (2003) p. 291
  110. Da̧browski M P, Stachowiak T, Szydłowski M Phys. Rev. D 68 (10) (2003)
  111. Novikov* I D Surveys In High Energy Physics 18 139 (2003)
  112. Garfinkle D Phys. Rev. D 65 (4) (2002)
  113. Berger B K Living Rev. Relativ. 5 (1) (2002)
  114. Cotsakis S Astrophysics And Space Science Library Vol. Modern Theoretical and Observational CosmologyCurrent Trends in Mathematical Cosmology276 Chapter 1 (2002) p. 5
  115. Coley A A Class. Quantum Grav. 19 L45 (2002)
  116. Garfinkle D Lecture Notes In Physics Vol. The Conformal Structure of Space-TimeSimulations of Generic Singularities in Harmonic Coordinates604 Chapter 18 (2002) p. 349
  117. Ivashchuk V D, Melnikov V N Class. Quantum Grav. 18 R87 (2001)
  118. Dymnikova I G, Dobosz A et al Physics Letters B 506 351 (2001)
  119. Lehner L Class. Quantum Grav. 18 R25 (2001)
  120. Obregón O, Quevedo H, Ryan M P Phys. Rev. D 65 (2) (2001)
  121. Ruban V P, Podolsky D I Phys. Rev. D 64 (4) (2001)
  122. Anninos P Living Rev. Relativ. 4 (1) (2001)
  123. Da̧browski M P Physics Letters B 474 52 (2000)
  124. Veneziano G Les Houches - Ecole D’Ete De Physique Theorique Vol. The primordial universe - L’univers primordialString Cosmology: The pre-Big Bang Scenario71 Chapter 12 (2000) p. 581
  125. Ivashchuk V D, Melnikov V N 41 6341 (2000)
  126. Kunze K E, Durrer R Class. Quantum Grav. 17 2597 (2000)
  127. Berger B K, Moncrief V Phys. Rev. D 62 (2) (2000)
  128. Gibbons G W, Rychenkova P Journal Of Geometry And Physics 32 311 (2000)
  129. Berger B K, Moncrief V Phys. Rev. D 62 (12) (2000)
  130. Buonanno A, Damour T, Veneziano G Nuclear Physics B 543 275 (1999)
  131. Berger B K Phys. Rev. D 61 (2) (1999)
  132. Giovannini M Phys. Rev. D 59 (6) (1999)
  133. Donets E E, Tentyukov M N, Tsulaia M M Phys. Rev. D 59 (6) (1999)
  134. Lazkoz R Phys. Rev. D 60 (10) (1999)
  135. Ori A Phys. Rev. Lett. 83 5423 (1999)
  136. Argyris J, Andreadis I, Ciubotariu C Chaos, Solitons & Fractals 9 1813 (1998)
  137. Anninos P Living Rev. Relativ. 1 (1) (1998)
  138. Gertsenshtein M E Russ Phys J 41 193 (1998)
  139. Berger B K, Moncrief V Phys. Rev. D 58 (6) (1998)
  140. Cotsakis S, Lemmer R L, Leach P G L Phys. Rev. D 57 4691 (1998)
  141. Anninos P Living Rev. Relativ. 1 (1) (1998)
  142. Berger B K, Garfinkle D Phys. Rev. D 57 4767 (1998)
  143. Berger B K Living Rev. Relativ. 1 (1) (1998)
  144. Weaver M, Isenberg Ja, Berger B K Phys. Rev. Lett. 80 2984 (1998)
  145. Barrow J D, Da̧browski M P Phys. Rev. D 57 7204 (1998)
  146. Foster S Class. Quantum Grav. 15 3485 (1998)
  147. Khalatnikov I M, Kamenshchik A Yu Uspekhi Fizicheskikh Nauk 168 593 (1998)
  148. Berger B K, Moncrief V Phys. Rev. D 57 7235 (1998)
  149. BERGER B K, GARFINKLE D et al Mod. Phys. Lett. A 13 1565 (1998)
  150. Khalatnikov I M, Kamenshchik A Yu Physics Reports 288 513 (1997)
  151. Kasper U, Rainer M, Zhuk A General Relativity And Gravitation 29 1123 (1997)
  152. Demaret Ja, Scheen Ch J. Phys. A: Math. Gen. 29 59 (1996)
  153. Berger B K Relativity and Scientific Computing Chapter 8 (1996) p. 152
  154. Cheb-Terrab E S, de Oliveira H P Computer Physics Communications 95 171 (1996)
  155. Barrow J D Class. Quantum Grav. 13 2965 (1996)
  156. Berger B K Class. Quantum Grav. 13 1273 (1996)
  157. Ivashchuk V D, Melnikov V N Class. Quantum Grav. 12 809 (1995)
  158. Skornyakov L A, Mysovskikh I P et al Encyclopaedia of Mathematics Chapter 6 (1995) p. 715
  159. González G, Tate R S Class. Quantum Grav. 12 1287 (1995)
  160. Graham R Chaos, Solitons & Fractals 5 1103 (1995)
  161. Ivashchuk V D, Melnikov V N Lecture Notes In Physics Vol. Birth of the Universe and Fundamental PhysicsBilliard representation for multidimensional quantum cosmology near the singularity455 Chapter 21 (1995) p. 121
  162. Kirillov A A, Melnikov V N Phys. Rev. D 52 723 (1995)
  163. Gavrilov V R, Ivashchuk V D, Melnikov V N 36 5829 (1995)
  164. Sobolev A I, Kurbatov A M et al Encyclopaedia of Mathematics Chapter 1 (1995) p. 1
  165. MacCallum M A H NATO ASI Series Vol. Deterministic Chaos in General RelativityRelativistic Cosmologies332 Chapter 11 (1994) p. 179
  166. Hobill D NATO ASI Series Vol. Deterministic Chaos in General RelativityA Brief Review of “Deterministic Chaos in General Relativity”332 Chapter 1 (1994) p. 1
  167. Berger B K Phys. Rev. D 49 1120 (1994)
  168. Francisco G, Matsas G E A NATO ASI Series Vol. Deterministic Chaos in General RelativityThe Chaoticity of the Bianchi IX Cosmological Model332 Chapter 21 (1994) p. 355
  169. Berger B K NATO ASI Series Vol. Deterministic Chaos in General RelativityThe Belinskii-Khalatnikov-Lifshitz Discrete Evolution as an Approximation to Mixmaster Dynamics332 Chapter 19 (1994) p. 329
  170. Dettmann C P, Frankel N E, Cornish N J Phys. Rev. D 50 R618 (1994)
  171. Grubišić Boro, Moncrief V Phys. Rev. D 49 2792 (1994)
  172. Berger B K Phys. Rev. D 47 3222 (1993)
  173. Berger B K, Moncrief V Phys. Rev. D 48 4676 (1993)
  174. Verdaguer E Physics Reports 229 1 (1993)
  175. Hazewinkel M Encyclopaedia of Mathematics Chapter 1 (1992) p. 1
  176. BELINSKII V A, LIFSHITZ E M et al Perspectives in Theoretical Physics (1992) p. 677
  177. Khalatnikov I M Gravitation and Modern Cosmology Chapter 8 (1991) p. 65
  178. Parnovsky S L Class. Quantum Grav. 7 571 (1990)
  179. Particle Physics and Inflationary Cosmology (1990) p. 337
  180. Qadir A, Archibald W J Nuclear Physics B - Proceedings Supplements 6 345 (1989)
  181. Liddle A R, Moorhouse R G, Henriques A B Nuclear Physics B 311 719 (1989)
  182. Hazewinkel M Encyclopaedia of Mathematics Chapter 1 (1988) p. 1
  183. Barrow J D The Early Universe Chapter 4 (1988) p. 125
  184. Fuchs H, Kasper U et al Fortschr. Phys. 36 427 (1988)
  185. Gertsenshtein M E, Melkumova E Yu Soviet Physics Journal 30 174 (1987)
  186. Barrow J D NATO ASI Series Vol. Gravitation in AstrophysicsSome Topics in Relativistic Cosmology156 Chapter 9 (1987) p. 239
  187. Henriques A B, Moorhouse R G Physics Letters B 194 353 (1987)
  188. Barrow J D Lecture Notes In Physics Vol. The Physics of Phase Space Nonlinear Dynamics and Chaos Geometric Quantization, and Wigner FunctionChaotic space-times278 Chapter 5 (1987) p. 18
  189. Barrow J D, Stein-Schabes J Physics Letters B 167 173 (1986)
  190. Ibanez J, Verdaguer E Class. Quantum Grav. 3 1235 (1986)
  191. Peresetskii A A Topics in Modern Mathematics Chapter 6 (1985) p. 173
  192. Barrow J D, Stein-Schabes J Phys. Rev. D 32 1595 (1985)
  193. Thorne K S Springer Proceedings In Physics Vol. Nonlinear Phenomena in PhysicsThe Dynamics of Spacetime Curvature: Nonlinear Aspects3 Chapter 11 (1985) p. 280
  194. Barrow J D, Sonoda D H Gen Relat Gravit 17 409 (1985)
  195. Carmeli M, Charach Ch Found Phys 14 963 (1984)
  196. Hawking S W, Luttrell Ju C Physics Letters B 143 83 (1984)
  197. Sato H 72 98 (1984)
  198. Carr B J, Verdaguer E Phys. Rev. D 28 2995 (1983)
  199. Zardecki A Phys. Rev. D 28 1235 (1983)
  200. Bogoyavlenskii O I, Novikov S P J Math Sci 21 254 (1983)
  201. Chernoff D F, Barrow J D Phys. Rev. Lett. 50 134 (1983)
  202. MacCallum M A H The Origin and Evolution of Galaxies Chapter 2 (1983) p. 9
  203. Barrow J D Physics Reports 85 1 (1982)
  204. Spokoiny B L Gen Relat Gravit 14 279 (1982)
  205. Obregón O, Ryan M P 22 623 (1981)
  206. Wheeler J A Essays in General Relativity (1980) p. 59
  207. Parnovsky S L Physica A: Statistical Mechanics And Its Applications 104 210 (1980)
  208. Grigoryan S D Mathematical Notes Of The Academy Of Sciences Of The USSR 26 610 (1979)
  209. Barrow J D, Tipler F J Physics Reports 56 371 (1979)
  210. MacCallum M Lecture Notes In Physics Vol. Physics of the Expanding UniverseThe mathematics of anisotropic spatially-homogeneous cosmologies109 Chapter 1 (1979) p. 1
  211. Gertsenshtein M E, Ingel’ L Kh, Fil’chenkov M L Soviet Physics Journal 21 841 (1978)
  212. Christodoulou D, Francaviglia M Reports On Mathematical Physics 11 377 (1977)
  213. Ginsburg W L Über Physik und Astrophysik Chapter 6 (1977) p. 139
  214. Misner Ch W Cosmology, History, and Theology Chapter 7 (1977) p. 75
  215. Bogoyavlenskii O I Theor Math Phys 27 414 (1976)
  216. Lukash V N Nuov Cim B 35 268 (1976)
  217. Lukash V N, Novikov I D et al Nuov Cim B 35 293 (1976)
  218. Hu B L Phys. Rev. D 12 1551 (1975)
  219. Goorjian P Phys. Rev. D 12 2978 (1975)
  220. Eardley D M Commun.Math. Phys. 37 287 (1974)
  221. Hu B L Phys. Rev. D 9 3263 (1974)
  222. Weinberg S Phys. Rev. D 9 3357 (1974)
  223. Misner Ch W Phys. Rev. D 8 3271 (1973)
© 1918–2025 Uspekhi Fizicheskikh Nauk
Email: ufn@ufn.ru Editorial office contacts About the journal Terms and conditions