BKL singularity (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "BKL singularity" in English language version.

refsWebsite

Global rank English rank

44doi.org

2^nd place

42harvard.edu

18^th place

17^th place

19semanticscholar.org

11^th place

8^th place

16arxiv.org

69^th place

59^th place

7nih.gov

4^th place

5books.google.com

3^rd place

3jstor.org

26^th place

20^th place

1linas.org

low place

1einstein-online.info

low place

1psu.edu

207^th place

136^th place

1youtube.com

9^th place

13^th place

1archive.org

6^th place

1worldcat.org

5^th place

1ams.org

451^st place

277^th place

ams.org

mathscinet.ams.org

Taub 1951. Taub, A.H. (1951). "Empty space-times admitting a three parameter group of motions". Annals of Mathematics. Second Series. 53 (3): 472–490. Bibcode:1951AnMat..53..472T. doi:10.2307/1969567. ISSN 0003-486X. JSTOR 1969567. MR 0041565.

archive.org

Misner, Thorne & Wheeler 1973, p. 564. Misner, Charles W.; Thorne, Kip S.; Wheeler, John Archibald (1973). Gravitation. San Francisco: W.H. Freeman. ISBN 978-0-7167-0334-1.^{[dead link]}

arxiv.org

Damour & Henneaux 2000. Damour, Thibault; Henneaux, Marc (2000). "Chaos in superstring cosmology". Physical Review Letters. 85 (5): 920–923. arXiv:hep-th/0003139. Bibcode:2000PhRvL..85..920D. doi:10.1103/PhysRevLett.85.920. PMID 10991439. S2CID 10752273.
Damour et al. 2001. Damour, Thibault; Henneaux, Marc; Julia, Bernard; Nicolai, Hermann (2001). "Hyperbolic Kac–Moody algebras and chaos in Kaluza-Klein models". Physics Letters B. 509 (3–4): 323–330. arXiv:hep-th/0103094. Bibcode:2001PhLB..509..323D. doi:10.1016/S0370-2693(01)00498-1. S2CID 16964946.
Henneaux, Persson & Spindel 2008. Henneaux, Marc; Persson, Daniel; Spindel, Philippe (2008). "Spacelike Singularities and Hidden Symmetries of Gravity". Living Reviews in Relativity. 11 (1): 1. arXiv:0710.1818. Bibcode:2008LRR....11....1H. doi:10.12942/lrr-2008-1. PMC 5255974. PMID 28179821.
Ashtekar, Henderson & Sloan 2011 Ashtekar, Abhay; Henderson, Adam; Sloan, David (2011). "Hamiltonian formulation of the Belinskii-Khalatnikov-Lifshitz conjecture". Physical Review D. 83 (8): 084024. arXiv:1102.3474. Bibcode:2011PhRvD..83h4024A. doi:10.1103/PhysRevD.83.084024. S2CID 119888235.
Berger 2002 Berger, Beverly K (2002). "Numerical Approaches to Spacetime Singularities". Living Reviews in Relativity. 5 (1): 1. arXiv:gr-qc/0201056. Bibcode:2002LRR.....5....1B. doi:10.12942/lrr-2002-1. PMC 5256073. PMID 28179859.
Garfinkle 2004. Garfinkle, David (2004). "Numerical Simulations of Generic Singularities". Phys. Rev. Lett. 93 (16): 161101. arXiv:gr-qc/0312117. Bibcode:2004PhRvL..93p1101G. doi:10.1103/PhysRevLett.93.161101. PMID 15524970. S2CID 37917240.
Berger & Moncrief 1993. Berger, Beverly; Moncrief, Vincent (1993). "Numerical investigation of cosmological singularities". Physical Review D. 48 (10): 4676–4687. arXiv:gr-qc/9307032. Bibcode:1993PhRvD..48.4676B. doi:10.1103/PhysRevD.48.4676. PMID 10016121. S2CID 450414.
Berger et al. 1998. Berger, Beverly; Garfinkle, David; Isenberg, James; Moncrief, Vincent; Weaver, Marsha (1998). "The singularity in generic gravitational collapse is spacelike, local, and oscillatory". Modern Physics Letters A. 13 (19): 1565–1573. arXiv:gr-qc/9805063. Bibcode:1998MPLA...13.1565B. doi:10.1142/S0217732398001649. S2CID 118889710.
Weaver, Isenberg & Berger 1998. Weaver, Marsha; Isenberg, James; Berger, Beverly K. (1998). "Mixmaster behavior in inhomogeneous cosmological spacetimes". Physical Review Letters. 80 (14): 2984–2987. arXiv:gr-qc/9712055. Bibcode:1998PhRvL..80.2984W. doi:10.1103/PhysRevLett.80.2984. S2CID 119467400.
Andersson & Rendall 2001. Andersson, Lars; Rendall, Alan D. (2001). "Quiescent cosmological singularities". Communications in Mathematical Physics. 218 (3): 479–511. arXiv:gr-qc/0001047. Bibcode:2001CMaPh.218..479A. doi:10.1007/s002200100406. S2CID 16167683.
Damour et al. 2002. Damour, Thibault; Henneaux, Marc; Rendall, Alan D.; Weaver, Marsha (2002). "Kasner-like behaviour for subcritical Einstein-matter systems". Annales Henri Poincaré. 3 (6): 1049–1111. arXiv:gr-qc/0202069. Bibcode:2002AnHP....3.1049D. doi:10.1007/s000230200000. S2CID 17092394.
Berger & Moncrief 1998. Berger, Beverly K.; Moncrief, Vincent (1998). "Numerical evidence that the singularity in polarized U(1) symmetric cosmologies on T³ × R is velocity dominated". Physical Review D. 57 (12): 7235–7240. arXiv:gr-qc/9801078. Bibcode:1998PhRvD..57.7235B. doi:10.1103/PhysRevD.57.7235. S2CID 119539705.
Berger & Moncrief 2000. Berger, Beverly K.; Moncrief, Vincent (2000). "Signature for local Mixmaster dynamics in U(1) symmetric cosmologies". Physical Review D. 62 (123501): 123501. arXiv:gr-qc/0006071. Bibcode:2000PhRvD..62l3501B. doi:10.1103/PhysRevD.62.123501. S2CID 10389665.
Garfinkle 2007. Garfinkle, David (2007). "Numerical simulations of general gravitational singularities". Classical and Quantum Gravity. 24 (12): 295–306. arXiv:0808.0160. Bibcode:2007CQGra..24S.295G. doi:10.1088/0264-9381/24/12/S19. S2CID 16899122.
Saotome, Akhoury & Garfinkle 2010. Saotome, Ryo; Akhoury, Ratindranath; Garfinkle, David (2010). "Examining gravitational collapse with test scalar fields". Classical and Quantum Gravity. 27 (165019): 165019. arXiv:1004.3569. Bibcode:2010CQGra..27p5019S. doi:10.1088/0264-9381/27/16/165019. S2CID 55229333.
Bini, Cherubini & Jantzen 2007. Bini, Donato; Cherubini, Christian; Jantzen, Robert (2007). "The Lifshitz-Khalatnikov Kasner index parametrization and the Weyl tensor". Il Nuovo Cimento B. 122 (5): 521–536. arXiv:0710.4902. Bibcode:2007NCimB.122..521B. doi:10.1393/ncb/i2007-10396-4. S2CID 15854390.

books.google.com

The convention used by BKL is the same as in the Landau & Lifshitz (1988) book. The Latin indices run through the values 0, 1, 2, 3; Greek indices run through the space values 1, 2, 3. The metric g_ik has the signature (+ − − −); γ_αβ = −g_αβ is the 3-dimensional space metric tensor. BKL use a system of units, in which the speed of light and the Einstein gravitational constant are equal to 1. Landau, Lev D.; Lifshitz, Evgeny M. (1988). Classical Theory of Fields. Course of Theoretical Physics. Vol. 2 (7th ed.). Moscow: Nauka. ISBN 978-5-02-014420-0.
Landau & Lifshitz 1988, Ch. 97 Landau, Lev D.; Lifshitz, Evgeny M. (1988). Classical Theory of Fields. Course of Theoretical Physics. Vol. 2 (7th ed.). Moscow: Nauka. ISBN 978-5-02-014420-0.
Landau & Lifshitz 1988, Ch. 117, Problem 3. Landau, Lev D.; Lifshitz, Evgeny M. (1988). Classical Theory of Fields. Course of Theoretical Physics. Vol. 2 (7th ed.). Moscow: Nauka. ISBN 978-5-02-014420-0.
Landau & Lifshitz 1988, Ch. 109. Landau, Lev D.; Lifshitz, Evgeny M. (1988). Classical Theory of Fields. Course of Theoretical Physics. Vol. 2 (7th ed.). Moscow: Nauka. ISBN 978-5-02-014420-0.
Landau & Lifshitz 1988, Ch. 103–105. Landau, Lev D.; Lifshitz, Evgeny M. (1988). Classical Theory of Fields. Course of Theoretical Physics. Vol. 2 (7th ed.). Moscow: Nauka. ISBN 978-5-02-014420-0.

doi.org

In a more precise calculation, a slowly changing logarithmic term appears in the sine argument, and a multiplier appears in front the exponent in the expression for с(ξ), see Belinsky, Khalatnikov & Lifshitz 1970, Appendix B. Belinsky, Vladimir A.; Khalatnikov, Isaak M.; Lifshitz, Evgeny M. (1970). "Колебательный режим приближения к особой точке в релятивистской космологии". Uspekhi Fizicheskikh Nauk. 102 (11): 463–500. Bibcode:1970UsFiN.102..463B. doi:10.3367/ufnr.0102.197011d.0463.; English translation in Belinskii, V.A.; Khalatnikov, I.M.; Lifshitz, E.M. (1970). "Oscillatory Approach to a Singular Point in the Relativistic Cosmology". Advances in Physics. 19 (80): 525–573. Bibcode:1970AdPhy..19..525B. doi:10.1080/00018737000101171.
Squared H terms in eq. 103 result only in small (≈1/ξ) corrections in σ and φ. Calculation with cubic terms leads to appearance of a weak dependence of A, B from ξ that can be presented as an appearance of logarithmic phases in the oscillating factors in eq. 115. These calculations for the case ρ = 0 are given in Belinsky & Khalatnikov (1970, Appendix B) (cf. the analogous situation for homogeneous models, Belinsky, Khalatnikov & Lifshitz (1970, Appendix B)). Belinsky, Vladimir A.; Khalatnikov, Isaak M. (1970). "General solution of the gravitational equations with a physical oscillatory singularity". JETP. 59: 314. Bibcode:1971JETP...32..169B. Belinsky, Vladimir A.; Khalatnikov, Isaak M.; Lifshitz, Evgeny M. (1970). "Колебательный режим приближения к особой точке в релятивистской космологии". Uspekhi Fizicheskikh Nauk. 102 (11): 463–500. Bibcode:1970UsFiN.102..463B. doi:10.3367/ufnr.0102.197011d.0463.; English translation in Belinskii, V.A.; Khalatnikov, I.M.; Lifshitz, E.M. (1970). "Oscillatory Approach to a Singular Point in the Relativistic Cosmology". Advances in Physics. 19 (80): 525–573. Bibcode:1970AdPhy..19..525B. doi:10.1080/00018737000101171.
The regular decomposition of the general solution of Einstein equations contains (in addition to the four three-dimensional functions) three independent functions of two coordinates (cf. Petrov 1969, Ch. 40; Lifshitz & Khalatnikov (1963, Appendix A)) Petrov, Alexey Z. (1969). Einstein Spaces. Oxford: Pergamon Press. ISBN 978-0-08-012315-8. Lifshitz, Evgeny M.; Khalatnikov, Isaak M. (1963). "Проблемы релятивистской космологии". Uspekhi Fizicheskikh Nauk. 80 (7): 391–438. doi:10.3367/UFNr.0080.196307d.0391.; English translation in Lifshitz, E.M.; Khalatnikov, I.M. (1963). "Problems in the Relativistic Cosmology". Advances in Physics. 12 (46): 185. Bibcode:1963AdPhy..12..185L. doi:10.1080/00018736300101283.
Belinsky, Khalatnikov & Lifshitz 1970 Belinsky, Vladimir A.; Khalatnikov, Isaak M.; Lifshitz, Evgeny M. (1970). "Колебательный режим приближения к особой точке в релятивистской космологии". Uspekhi Fizicheskikh Nauk. 102 (11): 463–500. Bibcode:1970UsFiN.102..463B. doi:10.3367/ufnr.0102.197011d.0463.; English translation in Belinskii, V.A.; Khalatnikov, I.M.; Lifshitz, E.M. (1970). "Oscillatory Approach to a Singular Point in the Relativistic Cosmology". Advances in Physics. 19 (80): 525–573. Bibcode:1970AdPhy..19..525B. doi:10.1080/00018737000101171.
Demaret, Henneaux & Spindel 1985. Demaret, Jacques; Henneaux, Marc; Spindel, Philipe (1985). "Non-oscillatory behaviour in vacuum Kaluza–Klein cosmologies". Physics Letters. 164B (1–3): 27–29. Bibcode:1985PhLB..164...27D. doi:10.1016/0370-2693(85)90024-3.
Demaret et al. 1986. Demaret, Jacques; Hanquin, Jean-Luc; Henneaux, Marc; Spindel, Philipe; Taormina, Anna (1986). "The fate of the Mixmaster behaviour in vacuum inhomogeneous Kaluza-Klein cosmological models". Physics Letters B. 175 (2): 129–132. Bibcode:1986PhLB..175..129D. doi:10.1016/0370-2693(86)90701-X.
Demaret, de Rop & Henneaux 1989. Demaret, Jacques; de Rop, Yves; Henneaux, Marc (1989). "Are Kaluza-Klein models of the Universe chaotic?". International Journal of Theoretical Physics. 28 (9): 1067–1079. Bibcode:1989IJTP...28.1067D. doi:10.1007/BF00670349. S2CID 121065247.
Damour & Henneaux 2000. Damour, Thibault; Henneaux, Marc (2000). "Chaos in superstring cosmology". Physical Review Letters. 85 (5): 920–923. arXiv:hep-th/0003139. Bibcode:2000PhRvL..85..920D. doi:10.1103/PhysRevLett.85.920. PMID 10991439. S2CID 10752273.
Damour et al. 2001. Damour, Thibault; Henneaux, Marc; Julia, Bernard; Nicolai, Hermann (2001). "Hyperbolic Kac–Moody algebras and chaos in Kaluza-Klein models". Physics Letters B. 509 (3–4): 323–330. arXiv:hep-th/0103094. Bibcode:2001PhLB..509..323D. doi:10.1016/S0370-2693(01)00498-1. S2CID 16964946.
Damour, Henneaux & Nicolai 2003. Damour, Thibault; Henneaux, Marc; Nicolai, Hermann (2003). "Cosmological billiards". Classical and Quantum Gravity. 20 (9): R145–R200. CiteSeerX 10.1.1.262.7733. doi:10.1088/0264-9381/20/9/201. S2CID 250877925.
Kac 1983. Kac, Victor G. (1983). Infinite dimensional Lie algebras: an introduction. Progress in Mathematics. Vol. 44. Basel: Birkhäuser. p. 245. doi:10.1007/978-1-4757-1382-4. ISBN 9780817631185.
Henneaux, Persson & Spindel 2008. Henneaux, Marc; Persson, Daniel; Spindel, Philippe (2008). "Spacelike Singularities and Hidden Symmetries of Gravity". Living Reviews in Relativity. 11 (1): 1. arXiv:0710.1818. Bibcode:2008LRR....11....1H. doi:10.12942/lrr-2008-1. PMC 5255974. PMID 28179821.
Lifshitz & Khalatnikov 1963 Lifshitz, Evgeny M.; Khalatnikov, Isaak M. (1963). "Проблемы релятивистской космологии". Uspekhi Fizicheskikh Nauk. 80 (7): 391–438. doi:10.3367/UFNr.0080.196307d.0391.; English translation in Lifshitz, E.M.; Khalatnikov, I.M. (1963). "Problems in the Relativistic Cosmology". Advances in Physics. 12 (46): 185. Bibcode:1963AdPhy..12..185L. doi:10.1080/00018736300101283.
Hawking 1965. Hawking, Stephen W. (1965). "Occurrence of Singularities in Open Universes". Physical Review Letters. 15 (17): 689–690. Bibcode:1965PhRvL..15..689H. doi:10.1103/PhysRevLett.15.689.
Hawking & Ellis 1968. Hawking, Stephen W.; Ellis, G.F.R. (1968). "The Cosmic Black-Body Radiation and the Existence of Singularities in Our Universe". Astrophysical Journal. 152: 25. Bibcode:1968ApJ...152...25H. doi:10.1086/149520.
Geroch 1966. Geroch, Robert P. (1966). "Singularities in Closed Universes". Physical Review Letters. 17 (8): 445–447. Bibcode:1966PhRvL..17..445G. doi:10.1103/PhysRevLett.17.445.
Ashtekar, Henderson & Sloan 2011 Ashtekar, Abhay; Henderson, Adam; Sloan, David (2011). "Hamiltonian formulation of the Belinskii-Khalatnikov-Lifshitz conjecture". Physical Review D. 83 (8): 084024. arXiv:1102.3474. Bibcode:2011PhRvD..83h4024A. doi:10.1103/PhysRevD.83.084024. S2CID 119888235.
Barrow & Tipler 1979. Barrow, John D.; Tipler, Frank J. (1979). "Analysis of the generic singularity studies by Belinskii, Khalatnikov, and Lifshitz". Physics Reports. 56 (7): 371–402. Bibcode:1979PhR....56..371B. doi:10.1016/0370-1573(79)90097-8.
Barrow & Tipler 1981. Barrow, John D.; Tipler, Frank J. (1981). "Generic singularity studies revisited". Physics Letters. 82A (9): 441–445. doi:10.1016/B978-0-08-036364-6.50050-8.
Berger 2002 Berger, Beverly K (2002). "Numerical Approaches to Spacetime Singularities". Living Reviews in Relativity. 5 (1): 1. arXiv:gr-qc/0201056. Bibcode:2002LRR.....5....1B. doi:10.12942/lrr-2002-1. PMC 5256073. PMID 28179859.
Garfinkle 2004. Garfinkle, David (2004). "Numerical Simulations of Generic Singularities". Phys. Rev. Lett. 93 (16): 161101. arXiv:gr-qc/0312117. Bibcode:2004PhRvL..93p1101G. doi:10.1103/PhysRevLett.93.161101. PMID 15524970. S2CID 37917240.
Berger & Moncrief 1993. Berger, Beverly; Moncrief, Vincent (1993). "Numerical investigation of cosmological singularities". Physical Review D. 48 (10): 4676–4687. arXiv:gr-qc/9307032. Bibcode:1993PhRvD..48.4676B. doi:10.1103/PhysRevD.48.4676. PMID 10016121. S2CID 450414.
Berger et al. 1998. Berger, Beverly; Garfinkle, David; Isenberg, James; Moncrief, Vincent; Weaver, Marsha (1998). "The singularity in generic gravitational collapse is spacelike, local, and oscillatory". Modern Physics Letters A. 13 (19): 1565–1573. arXiv:gr-qc/9805063. Bibcode:1998MPLA...13.1565B. doi:10.1142/S0217732398001649. S2CID 118889710.
Weaver, Isenberg & Berger 1998. Weaver, Marsha; Isenberg, James; Berger, Beverly K. (1998). "Mixmaster behavior in inhomogeneous cosmological spacetimes". Physical Review Letters. 80 (14): 2984–2987. arXiv:gr-qc/9712055. Bibcode:1998PhRvL..80.2984W. doi:10.1103/PhysRevLett.80.2984. S2CID 119467400.
Andersson & Rendall 2001. Andersson, Lars; Rendall, Alan D. (2001). "Quiescent cosmological singularities". Communications in Mathematical Physics. 218 (3): 479–511. arXiv:gr-qc/0001047. Bibcode:2001CMaPh.218..479A. doi:10.1007/s002200100406. S2CID 16167683.
Damour et al. 2002. Damour, Thibault; Henneaux, Marc; Rendall, Alan D.; Weaver, Marsha (2002). "Kasner-like behaviour for subcritical Einstein-matter systems". Annales Henri Poincaré. 3 (6): 1049–1111. arXiv:gr-qc/0202069. Bibcode:2002AnHP....3.1049D. doi:10.1007/s000230200000. S2CID 17092394.
Berger & Moncrief 1998. Berger, Beverly K.; Moncrief, Vincent (1998). "Numerical evidence that the singularity in polarized U(1) symmetric cosmologies on T³ × R is velocity dominated". Physical Review D. 57 (12): 7235–7240. arXiv:gr-qc/9801078. Bibcode:1998PhRvD..57.7235B. doi:10.1103/PhysRevD.57.7235. S2CID 119539705.
Berger & Moncrief 2000. Berger, Beverly K.; Moncrief, Vincent (2000). "Signature for local Mixmaster dynamics in U(1) symmetric cosmologies". Physical Review D. 62 (123501): 123501. arXiv:gr-qc/0006071. Bibcode:2000PhRvD..62l3501B. doi:10.1103/PhysRevD.62.123501. S2CID 10389665.
Garfinkle 2007. Garfinkle, David (2007). "Numerical simulations of general gravitational singularities". Classical and Quantum Gravity. 24 (12): 295–306. arXiv:0808.0160. Bibcode:2007CQGra..24S.295G. doi:10.1088/0264-9381/24/12/S19. S2CID 16899122.
Saotome, Akhoury & Garfinkle 2010. Saotome, Ryo; Akhoury, Ratindranath; Garfinkle, David (2010). "Examining gravitational collapse with test scalar fields". Classical and Quantum Gravity. 27 (165019): 165019. arXiv:1004.3569. Bibcode:2010CQGra..27p5019S. doi:10.1088/0264-9381/27/16/165019. S2CID 55229333.
Kasner 1921. Kasner, Edward (1921). "Geometrical Theorems on Einstein's Cosmological Equations". American Journal of Mathematics. 43 (4): 217–221. Bibcode:1921AmJM...43..217K. doi:10.2307/2370192. JSTOR 2370192.
Rugh 1994. Rugh, Svend E. (1994). "Chaos in the Einstein Equations -- characterization and importance?". Deterministic Chaos in General Relativity. Kananaskis, Alberta, Canada: Springer Science+Business Media New York. pp. 359–422. doi:10.1007/978-1-4757-9993-4. ISBN 978-1-4757-9995-8.
Bini, Cherubini & Jantzen 2007. Bini, Donato; Cherubini, Christian; Jantzen, Robert (2007). "The Lifshitz-Khalatnikov Kasner index parametrization and the Weyl tensor". Il Nuovo Cimento B. 122 (5): 521–536. arXiv:0710.4902. Bibcode:2007NCimB.122..521B. doi:10.1393/ncb/i2007-10396-4. S2CID 15854390.
Nelson 1981. Nelson, Robert A. (1981). "Coordinate-dependent 3 + 1 formulation of the General Relativity equations for a perfect fluid". General Relativity and Gravitation. 13 (6): 569–580. Bibcode:1981GReGr..13..569N. doi:10.1007/BF00757243. S2CID 122196359.
Khalatnikov & Lifshitz 1970. Khalatnikov, I.M.; Lifshitz, E.M. (1970). "General Cosmological Solution of the Gravitational Equations with a Singularity in Time". Physical Review Letters. 24 (2): 76. Bibcode:1970PhRvL..24...76K. doi:10.1103/PhysRevLett.24.76.
Belinsky, Khalatnikov & Lifshitz 1970, Appendix C. Belinsky, Vladimir A.; Khalatnikov, Isaak M.; Lifshitz, Evgeny M. (1970). "Колебательный режим приближения к особой точке в релятивистской космологии". Uspekhi Fizicheskikh Nauk. 102 (11): 463–500. Bibcode:1970UsFiN.102..463B. doi:10.3367/ufnr.0102.197011d.0463.; English translation in Belinskii, V.A.; Khalatnikov, I.M.; Lifshitz, E.M. (1970). "Oscillatory Approach to a Singular Point in the Relativistic Cosmology". Advances in Physics. 19 (80): 525–573. Bibcode:1970AdPhy..19..525B. doi:10.1080/00018737000101171.
Lifshitz & Khalatnikov 1963, Appendix C. Lifshitz, Evgeny M.; Khalatnikov, Isaak M. (1963). "Проблемы релятивистской космологии". Uspekhi Fizicheskikh Nauk. 80 (7): 391–438. doi:10.3367/UFNr.0080.196307d.0391.; English translation in Lifshitz, E.M.; Khalatnikov, I.M. (1963). "Problems in the Relativistic Cosmology". Advances in Physics. 12 (46): 185. Bibcode:1963AdPhy..12..185L. doi:10.1080/00018736300101283.
Taub 1951. Taub, A.H. (1951). "Empty space-times admitting a three parameter group of motions". Annals of Mathematics. Second Series. 53 (3): 472–490. Bibcode:1951AnMat..53..472T. doi:10.2307/1969567. ISSN 0003-486X. JSTOR 1969567. MR 0041565.
Khalatnikov et al. 1985 Khalatnikov, Isaak M.; Lifshitz, E.M.; Khanin, K.M.; Shchur, L.N.; Sinai, Ya.G. (1985). "On the stochasticity in relativistic cosmology". Journal of Statistical Physics. 38 (1–2): 97–114. Bibcode:1985JSP....38...97K. doi:10.1007/BF01017851. S2CID 120521834.
Chernoff & Barrow 1983. Chernoff, David F.; Barrow, J.D. (1983). "Chaos in the Mixmaster Universe". Physical Review Letters. 50 (2): 134–137. Bibcode:1983PhRvL..50..134C. doi:10.1103/PhysRevLett.50.134.
Belinsky, Khalatnikov & Lifshitz 1970, Appendix A. Belinsky, Vladimir A.; Khalatnikov, Isaak M.; Lifshitz, Evgeny M. (1970). "Колебательный режим приближения к особой точке в релятивистской космологии". Uspekhi Fizicheskikh Nauk. 102 (11): 463–500. Bibcode:1970UsFiN.102..463B. doi:10.3367/ufnr.0102.197011d.0463.; English translation in Belinskii, V.A.; Khalatnikov, I.M.; Lifshitz, E.M. (1970). "Oscillatory Approach to a Singular Point in the Relativistic Cosmology". Advances in Physics. 19 (80): 525–573. Bibcode:1970AdPhy..19..525B. doi:10.1080/00018737000101171.
Einstein & Rosen 1937. Einstein, Albert; Rosen, Nathan (1937). "On gravitational waves". Journal of the Franklin Institute. 223 (1): 43–54. Bibcode:1937FrInJ.223...43E. doi:10.1016/S0016-0032(37)90583-0.
Bondi, Pirani & Robinson 1959. Bondi, Herman; Pirani, F.A.E.; Robinson, I. (1959). "Gravitational Waves in General Relativity. III. Exact Plane Waves". Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences. 251 (1267): 519–533. Bibcode:1959RSPSA.251..519B. doi:10.1098/rspa.1959.0124. JSTOR 100727. S2CID 122766998.
Misner 1969. Misner, Ch. W. (1969). "Mixmaster Universe". Physical Review Letters. 22 (20): 1071–1074. Bibcode:1969PhRvL..22.1071M. doi:10.1103/PhysRevLett.22.1071.

einstein-online.info

Garfinkle, David (2007). "Of singularities and breadmaking". Einstein Online. Band 03. Max Planck Institute for Gravitational Physics. 03–1014. Retrieved 2020-10-15.

harvard.edu

ui.adsabs.harvard.edu

In a more precise calculation, a slowly changing logarithmic term appears in the sine argument, and a multiplier appears in front the exponent in the expression for с(ξ), see Belinsky, Khalatnikov & Lifshitz 1970, Appendix B. Belinsky, Vladimir A.; Khalatnikov, Isaak M.; Lifshitz, Evgeny M. (1970). "Колебательный режим приближения к особой точке в релятивистской космологии". Uspekhi Fizicheskikh Nauk. 102 (11): 463–500. Bibcode:1970UsFiN.102..463B. doi:10.3367/ufnr.0102.197011d.0463.; English translation in Belinskii, V.A.; Khalatnikov, I.M.; Lifshitz, E.M. (1970). "Oscillatory Approach to a Singular Point in the Relativistic Cosmology". Advances in Physics. 19 (80): 525–573. Bibcode:1970AdPhy..19..525B. doi:10.1080/00018737000101171.
Squared H terms in eq. 103 result only in small (≈1/ξ) corrections in σ and φ. Calculation with cubic terms leads to appearance of a weak dependence of A, B from ξ that can be presented as an appearance of logarithmic phases in the oscillating factors in eq. 115. These calculations for the case ρ = 0 are given in Belinsky & Khalatnikov (1970, Appendix B) (cf. the analogous situation for homogeneous models, Belinsky, Khalatnikov & Lifshitz (1970, Appendix B)). Belinsky, Vladimir A.; Khalatnikov, Isaak M. (1970). "General solution of the gravitational equations with a physical oscillatory singularity". JETP. 59: 314. Bibcode:1971JETP...32..169B. Belinsky, Vladimir A.; Khalatnikov, Isaak M.; Lifshitz, Evgeny M. (1970). "Колебательный режим приближения к особой точке в релятивистской космологии". Uspekhi Fizicheskikh Nauk. 102 (11): 463–500. Bibcode:1970UsFiN.102..463B. doi:10.3367/ufnr.0102.197011d.0463.; English translation in Belinskii, V.A.; Khalatnikov, I.M.; Lifshitz, E.M. (1970). "Oscillatory Approach to a Singular Point in the Relativistic Cosmology". Advances in Physics. 19 (80): 525–573. Bibcode:1970AdPhy..19..525B. doi:10.1080/00018737000101171.
The regular decomposition of the general solution of Einstein equations contains (in addition to the four three-dimensional functions) three independent functions of two coordinates (cf. Petrov 1969, Ch. 40; Lifshitz & Khalatnikov (1963, Appendix A)) Petrov, Alexey Z. (1969). Einstein Spaces. Oxford: Pergamon Press. ISBN 978-0-08-012315-8. Lifshitz, Evgeny M.; Khalatnikov, Isaak M. (1963). "Проблемы релятивистской космологии". Uspekhi Fizicheskikh Nauk. 80 (7): 391–438. doi:10.3367/UFNr.0080.196307d.0391.; English translation in Lifshitz, E.M.; Khalatnikov, I.M. (1963). "Problems in the Relativistic Cosmology". Advances in Physics. 12 (46): 185. Bibcode:1963AdPhy..12..185L. doi:10.1080/00018736300101283.
Belinsky, Khalatnikov & Lifshitz 1970 Belinsky, Vladimir A.; Khalatnikov, Isaak M.; Lifshitz, Evgeny M. (1970). "Колебательный режим приближения к особой точке в релятивистской космологии". Uspekhi Fizicheskikh Nauk. 102 (11): 463–500. Bibcode:1970UsFiN.102..463B. doi:10.3367/ufnr.0102.197011d.0463.; English translation in Belinskii, V.A.; Khalatnikov, I.M.; Lifshitz, E.M. (1970). "Oscillatory Approach to a Singular Point in the Relativistic Cosmology". Advances in Physics. 19 (80): 525–573. Bibcode:1970AdPhy..19..525B. doi:10.1080/00018737000101171.
Demaret, Henneaux & Spindel 1985. Demaret, Jacques; Henneaux, Marc; Spindel, Philipe (1985). "Non-oscillatory behaviour in vacuum Kaluza–Klein cosmologies". Physics Letters. 164B (1–3): 27–29. Bibcode:1985PhLB..164...27D. doi:10.1016/0370-2693(85)90024-3.
Demaret et al. 1986. Demaret, Jacques; Hanquin, Jean-Luc; Henneaux, Marc; Spindel, Philipe; Taormina, Anna (1986). "The fate of the Mixmaster behaviour in vacuum inhomogeneous Kaluza-Klein cosmological models". Physics Letters B. 175 (2): 129–132. Bibcode:1986PhLB..175..129D. doi:10.1016/0370-2693(86)90701-X.
Demaret, de Rop & Henneaux 1989. Demaret, Jacques; de Rop, Yves; Henneaux, Marc (1989). "Are Kaluza-Klein models of the Universe chaotic?". International Journal of Theoretical Physics. 28 (9): 1067–1079. Bibcode:1989IJTP...28.1067D. doi:10.1007/BF00670349. S2CID 121065247.
Damour & Henneaux 2000. Damour, Thibault; Henneaux, Marc (2000). "Chaos in superstring cosmology". Physical Review Letters. 85 (5): 920–923. arXiv:hep-th/0003139. Bibcode:2000PhRvL..85..920D. doi:10.1103/PhysRevLett.85.920. PMID 10991439. S2CID 10752273.
Damour et al. 2001. Damour, Thibault; Henneaux, Marc; Julia, Bernard; Nicolai, Hermann (2001). "Hyperbolic Kac–Moody algebras and chaos in Kaluza-Klein models". Physics Letters B. 509 (3–4): 323–330. arXiv:hep-th/0103094. Bibcode:2001PhLB..509..323D. doi:10.1016/S0370-2693(01)00498-1. S2CID 16964946.
Henneaux, Persson & Spindel 2008. Henneaux, Marc; Persson, Daniel; Spindel, Philippe (2008). "Spacelike Singularities and Hidden Symmetries of Gravity". Living Reviews in Relativity. 11 (1): 1. arXiv:0710.1818. Bibcode:2008LRR....11....1H. doi:10.12942/lrr-2008-1. PMC 5255974. PMID 28179821.
Lifshitz & Khalatnikov 1963 Lifshitz, Evgeny M.; Khalatnikov, Isaak M. (1963). "Проблемы релятивистской космологии". Uspekhi Fizicheskikh Nauk. 80 (7): 391–438. doi:10.3367/UFNr.0080.196307d.0391.; English translation in Lifshitz, E.M.; Khalatnikov, I.M. (1963). "Problems in the Relativistic Cosmology". Advances in Physics. 12 (46): 185. Bibcode:1963AdPhy..12..185L. doi:10.1080/00018736300101283.
Hawking 1965. Hawking, Stephen W. (1965). "Occurrence of Singularities in Open Universes". Physical Review Letters. 15 (17): 689–690. Bibcode:1965PhRvL..15..689H. doi:10.1103/PhysRevLett.15.689.
Hawking & Ellis 1968. Hawking, Stephen W.; Ellis, G.F.R. (1968). "The Cosmic Black-Body Radiation and the Existence of Singularities in Our Universe". Astrophysical Journal. 152: 25. Bibcode:1968ApJ...152...25H. doi:10.1086/149520.
Geroch 1966. Geroch, Robert P. (1966). "Singularities in Closed Universes". Physical Review Letters. 17 (8): 445–447. Bibcode:1966PhRvL..17..445G. doi:10.1103/PhysRevLett.17.445.
Ashtekar, Henderson & Sloan 2011 Ashtekar, Abhay; Henderson, Adam; Sloan, David (2011). "Hamiltonian formulation of the Belinskii-Khalatnikov-Lifshitz conjecture". Physical Review D. 83 (8): 084024. arXiv:1102.3474. Bibcode:2011PhRvD..83h4024A. doi:10.1103/PhysRevD.83.084024. S2CID 119888235.
Barrow & Tipler 1979. Barrow, John D.; Tipler, Frank J. (1979). "Analysis of the generic singularity studies by Belinskii, Khalatnikov, and Lifshitz". Physics Reports. 56 (7): 371–402. Bibcode:1979PhR....56..371B. doi:10.1016/0370-1573(79)90097-8.
Berger 2002 Berger, Beverly K (2002). "Numerical Approaches to Spacetime Singularities". Living Reviews in Relativity. 5 (1): 1. arXiv:gr-qc/0201056. Bibcode:2002LRR.....5....1B. doi:10.12942/lrr-2002-1. PMC 5256073. PMID 28179859.
Garfinkle 2004. Garfinkle, David (2004). "Numerical Simulations of Generic Singularities". Phys. Rev. Lett. 93 (16): 161101. arXiv:gr-qc/0312117. Bibcode:2004PhRvL..93p1101G. doi:10.1103/PhysRevLett.93.161101. PMID 15524970. S2CID 37917240.
Berger & Moncrief 1993. Berger, Beverly; Moncrief, Vincent (1993). "Numerical investigation of cosmological singularities". Physical Review D. 48 (10): 4676–4687. arXiv:gr-qc/9307032. Bibcode:1993PhRvD..48.4676B. doi:10.1103/PhysRevD.48.4676. PMID 10016121. S2CID 450414.
Berger et al. 1998. Berger, Beverly; Garfinkle, David; Isenberg, James; Moncrief, Vincent; Weaver, Marsha (1998). "The singularity in generic gravitational collapse is spacelike, local, and oscillatory". Modern Physics Letters A. 13 (19): 1565–1573. arXiv:gr-qc/9805063. Bibcode:1998MPLA...13.1565B. doi:10.1142/S0217732398001649. S2CID 118889710.
Weaver, Isenberg & Berger 1998. Weaver, Marsha; Isenberg, James; Berger, Beverly K. (1998). "Mixmaster behavior in inhomogeneous cosmological spacetimes". Physical Review Letters. 80 (14): 2984–2987. arXiv:gr-qc/9712055. Bibcode:1998PhRvL..80.2984W. doi:10.1103/PhysRevLett.80.2984. S2CID 119467400.
Andersson & Rendall 2001. Andersson, Lars; Rendall, Alan D. (2001). "Quiescent cosmological singularities". Communications in Mathematical Physics. 218 (3): 479–511. arXiv:gr-qc/0001047. Bibcode:2001CMaPh.218..479A. doi:10.1007/s002200100406. S2CID 16167683.
Damour et al. 2002. Damour, Thibault; Henneaux, Marc; Rendall, Alan D.; Weaver, Marsha (2002). "Kasner-like behaviour for subcritical Einstein-matter systems". Annales Henri Poincaré. 3 (6): 1049–1111. arXiv:gr-qc/0202069. Bibcode:2002AnHP....3.1049D. doi:10.1007/s000230200000. S2CID 17092394.
Berger & Moncrief 1998. Berger, Beverly K.; Moncrief, Vincent (1998). "Numerical evidence that the singularity in polarized U(1) symmetric cosmologies on T³ × R is velocity dominated". Physical Review D. 57 (12): 7235–7240. arXiv:gr-qc/9801078. Bibcode:1998PhRvD..57.7235B. doi:10.1103/PhysRevD.57.7235. S2CID 119539705.
Berger & Moncrief 2000. Berger, Beverly K.; Moncrief, Vincent (2000). "Signature for local Mixmaster dynamics in U(1) symmetric cosmologies". Physical Review D. 62 (123501): 123501. arXiv:gr-qc/0006071. Bibcode:2000PhRvD..62l3501B. doi:10.1103/PhysRevD.62.123501. S2CID 10389665.
Garfinkle 2007. Garfinkle, David (2007). "Numerical simulations of general gravitational singularities". Classical and Quantum Gravity. 24 (12): 295–306. arXiv:0808.0160. Bibcode:2007CQGra..24S.295G. doi:10.1088/0264-9381/24/12/S19. S2CID 16899122.
Saotome, Akhoury & Garfinkle 2010. Saotome, Ryo; Akhoury, Ratindranath; Garfinkle, David (2010). "Examining gravitational collapse with test scalar fields". Classical and Quantum Gravity. 27 (165019): 165019. arXiv:1004.3569. Bibcode:2010CQGra..27p5019S. doi:10.1088/0264-9381/27/16/165019. S2CID 55229333.
Kasner 1921. Kasner, Edward (1921). "Geometrical Theorems on Einstein's Cosmological Equations". American Journal of Mathematics. 43 (4): 217–221. Bibcode:1921AmJM...43..217K. doi:10.2307/2370192. JSTOR 2370192.
Bini, Cherubini & Jantzen 2007. Bini, Donato; Cherubini, Christian; Jantzen, Robert (2007). "The Lifshitz-Khalatnikov Kasner index parametrization and the Weyl tensor". Il Nuovo Cimento B. 122 (5): 521–536. arXiv:0710.4902. Bibcode:2007NCimB.122..521B. doi:10.1393/ncb/i2007-10396-4. S2CID 15854390.
Nelson 1981. Nelson, Robert A. (1981). "Coordinate-dependent 3 + 1 formulation of the General Relativity equations for a perfect fluid". General Relativity and Gravitation. 13 (6): 569–580. Bibcode:1981GReGr..13..569N. doi:10.1007/BF00757243. S2CID 122196359.
Belinsky & Khalatnikov 1966. Belinsky, Vladimir A.; Khalatnikov, I.M. (1966). "A general solution of the gravitational equations with a simultaneous fictitious singularity". JETP. 49 (3): 1000. Bibcode:1966JETP...22..694B.
Khalatnikov & Lifshitz 1970. Khalatnikov, I.M.; Lifshitz, E.M. (1970). "General Cosmological Solution of the Gravitational Equations with a Singularity in Time". Physical Review Letters. 24 (2): 76. Bibcode:1970PhRvL..24...76K. doi:10.1103/PhysRevLett.24.76.
Belinsky, Khalatnikov & Lifshitz 1970, Appendix C. Belinsky, Vladimir A.; Khalatnikov, Isaak M.; Lifshitz, Evgeny M. (1970). "Колебательный режим приближения к особой точке в релятивистской космологии". Uspekhi Fizicheskikh Nauk. 102 (11): 463–500. Bibcode:1970UsFiN.102..463B. doi:10.3367/ufnr.0102.197011d.0463.; English translation in Belinskii, V.A.; Khalatnikov, I.M.; Lifshitz, E.M. (1970). "Oscillatory Approach to a Singular Point in the Relativistic Cosmology". Advances in Physics. 19 (80): 525–573. Bibcode:1970AdPhy..19..525B. doi:10.1080/00018737000101171.
Lifshitz & Khalatnikov 1963, Appendix C. Lifshitz, Evgeny M.; Khalatnikov, Isaak M. (1963). "Проблемы релятивистской космологии". Uspekhi Fizicheskikh Nauk. 80 (7): 391–438. doi:10.3367/UFNr.0080.196307d.0391.; English translation in Lifshitz, E.M.; Khalatnikov, I.M. (1963). "Problems in the Relativistic Cosmology". Advances in Physics. 12 (46): 185. Bibcode:1963AdPhy..12..185L. doi:10.1080/00018736300101283.
Taub 1951. Taub, A.H. (1951). "Empty space-times admitting a three parameter group of motions". Annals of Mathematics. Second Series. 53 (3): 472–490. Bibcode:1951AnMat..53..472T. doi:10.2307/1969567. ISSN 0003-486X. JSTOR 1969567. MR 0041565.
Khalatnikov et al. 1985 Khalatnikov, Isaak M.; Lifshitz, E.M.; Khanin, K.M.; Shchur, L.N.; Sinai, Ya.G. (1985). "On the stochasticity in relativistic cosmology". Journal of Statistical Physics. 38 (1–2): 97–114. Bibcode:1985JSP....38...97K. doi:10.1007/BF01017851. S2CID 120521834.
Chernoff & Barrow 1983. Chernoff, David F.; Barrow, J.D. (1983). "Chaos in the Mixmaster Universe". Physical Review Letters. 50 (2): 134–137. Bibcode:1983PhRvL..50..134C. doi:10.1103/PhysRevLett.50.134.
Belinsky, Khalatnikov & Lifshitz 1970, Appendix A. Belinsky, Vladimir A.; Khalatnikov, Isaak M.; Lifshitz, Evgeny M. (1970). "Колебательный режим приближения к особой точке в релятивистской космологии". Uspekhi Fizicheskikh Nauk. 102 (11): 463–500. Bibcode:1970UsFiN.102..463B. doi:10.3367/ufnr.0102.197011d.0463.; English translation in Belinskii, V.A.; Khalatnikov, I.M.; Lifshitz, E.M. (1970). "Oscillatory Approach to a Singular Point in the Relativistic Cosmology". Advances in Physics. 19 (80): 525–573. Bibcode:1970AdPhy..19..525B. doi:10.1080/00018737000101171.
Belinsky & Khalatnikov 1970 Belinsky, Vladimir A.; Khalatnikov, Isaak M. (1970). "General solution of the gravitational equations with a physical oscillatory singularity". JETP. 59: 314. Bibcode:1971JETP...32..169B.
Einstein & Rosen 1937. Einstein, Albert; Rosen, Nathan (1937). "On gravitational waves". Journal of the Franklin Institute. 223 (1): 43–54. Bibcode:1937FrInJ.223...43E. doi:10.1016/S0016-0032(37)90583-0.
Bondi, Pirani & Robinson 1959. Bondi, Herman; Pirani, F.A.E.; Robinson, I. (1959). "Gravitational Waves in General Relativity. III. Exact Plane Waves". Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences. 251 (1267): 519–533. Bibcode:1959RSPSA.251..519B. doi:10.1098/rspa.1959.0124. JSTOR 100727. S2CID 122766998.
Misner 1969. Misner, Ch. W. (1969). "Mixmaster Universe". Physical Review Letters. 22 (20): 1071–1074. Bibcode:1969PhRvL..22.1071M. doi:10.1103/PhysRevLett.22.1071.

jstor.org

Kasner 1921. Kasner, Edward (1921). "Geometrical Theorems on Einstein's Cosmological Equations". American Journal of Mathematics. 43 (4): 217–221. Bibcode:1921AmJM...43..217K. doi:10.2307/2370192. JSTOR 2370192.
Taub 1951. Taub, A.H. (1951). "Empty space-times admitting a three parameter group of motions". Annals of Mathematics. Second Series. 53 (3): 472–490. Bibcode:1951AnMat..53..472T. doi:10.2307/1969567. ISSN 0003-486X. JSTOR 1969567. MR 0041565.
Bondi, Pirani & Robinson 1959. Bondi, Herman; Pirani, F.A.E.; Robinson, I. (1959). "Gravitational Waves in General Relativity. III. Exact Plane Waves". Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences. 251 (1267): 519–533. Bibcode:1959RSPSA.251..519B. doi:10.1098/rspa.1959.0124. JSTOR 100727. S2CID 122766998.

linas.org

Equation 74 was known already to Gauss, and an equation of type eq. 73c was considered in this connection by Rodion Kuzmin (see Gauss–Kuzmin distribution). Further information on the chaotic behaviour and entropy of continued fractions in Linas Vepstas. 2008. Entropy of Continued Fractions (Gauss-Kuzmin Entropy)