Skip to main content
SpringerLink
Log in

Singularities in Cosmological Spacetimes

  • Chapter
  • First Online:
Springer Handbook of Spacetime

Part of the book series: Springer Handbooks ((SHB))

Abstract

Theorems state that gravitational collapse from generic but non-singular initial conditions results in some type of singular behavior. Here the nature of the resultant approach to the singularity is examined in spatially homogeneous, anisotropic, vacuum cosmological spacetimes . The approach to the singularity in these spacetimes is either (asymptotically) Kasner-like or Mixmaster-like. It has been conjectured that spatially inhomogeneous cosmological spacetimes approach the singularity through Kasner-like or Mixmaster-like dynamics at every spatial point. Several examples of such cosmologies are explored numerically and heuristically. The current status of a rigorous statement of this conjecture and possible approaches to a proof are discussed. This chapter will focus on singularities in cosmological spacetimes.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 269.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 349.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Abbreviations

ADM:

Arnowitt, Deser, Misner

AVTD:

asymptotically velocity term dominated

BKL:

Belinski, Khalatnikov, Lifshitz

FRW:

Friedmann–Robertson–Walker

LSF:

logarithmic scale factor

MCP:

method of consistent potentials

MSS:

minisuperspace

VTD:

velocity term dominated

References

  1. S.M. Carroll: Spacetime and Geometry (Addison Wesley, San Francisco 2004)

    Google Scholar 

  2. R.M. Wald: General Relativity (Univ. Chicago Press, Chicago 1984)

    Google Scholar 

  3. G.F.R. Ellis, M.A.H. MacCallum: A class of homogeneous cosmological models, Comm. Math. Phys. 12, 108 (1969)

    Google Scholar 

  4. M.A.H. MacCallum: Anisotropic and inhomogeneous relativistic cosmologies. In: General Relativity: An Einstein Centenary Survey, ed. by S. Hawking, W. Israel (Cambridge Univ. Press, Cambridge, 1979) pp. 533–580

    Google Scholar 

  5. M.P. Ryan Jr., L.C. Shepley: Homogeneous Relativistic Cosmologies (Princeton Univ. Press, Princeton, 1975)

    Google Scholar 

  6. C.W. Misner: Minisuperspace, magic without magic. In: J. A. Wheeler 60th Anniversary Volume, ed. by J. Klauder (W.H. Freeman, San Francisco 1972) pp. 441–473

    Google Scholar 

  7. E. Kasner: Solutions of the Einstein equations involving functions of only one variable, Trans. Am. Math. Soc. 27, 155–162 (1925)

    Google Scholar 

  8. V.A. Belinskii, E.M. Lifshitz, I.M. Khalatnikov: Oscillatory approach to the singularity point in relativisticcosmology, Sov. Phys. Usp. 13, 745–765 (1971)

    Google Scholar 

  9. B.K. Berger: Comments on the computation of Liapunov exponents for the mixmaster universe, Gen. Relativ. Gravit. 23, 1385–1402 (1991)

    Google Scholar 

  10. D.M. Eardley, E. Liang, R. Sachs: Velocity-dominated singularities in irrotational dust cosmologies, J. Math. Phys. 13, 99–107 (1972)

    Google Scholar 

  11. A. Taub: Empty space-times admitting a three-parameter group of motions, Ann. Math. 53, 472 (1951)

    Google Scholar 

  12. C.W. Misner: Mixmaster universe, Phys. Rev. Lett. 22, 1071–1074 (1969)

    Google Scholar 

  13. R.T. Jantzen: Spatially homogeneous dynamics: A unified picture, Proc. Int. School Phys. ‘Enrico Fermi’, Course 86, Varenna, Italy, 1982 (North-Holland Elsevier, Amsterdam 1986) pp. 61–147

    Google Scholar 

  14. C.W. Misner: The mixmaster cosmological metrics, NATO ASI Ser. 332, 317–328 (1994)

    Google Scholar 

  15. B.K. Berger: How to determine approximate mixmaster parameters from numerical evolution of Einstein’s equations, Phys. Rev. D 49, 1120–1123 (1994)

    Google Scholar 

  16. B.K. Berger, D. Garfinkle, E. Strasser: New algorithm for mixmaster dynamics, Class. Quantum Gravity 14, L29–L36 (1997)

    Google Scholar 

  17. D.W. Hobill, A. Burd, A.A. Coley: Deterministic Chaos in General Relativity, NATO ASI Ser. 332 (1994)

    Google Scholar 

  18. B.K. Berger, V. Moncrief: Numerical investigations of cosmological singularities, Phys. Rev. D 48, 4676–4687 (1993)

    Google Scholar 

  19. B.K. Berger: Numerical approaches to spacetime singularities, Living Rev, Relativity 5, 1 (2002)

    Google Scholar 

  20. V.A. Belinskii, I.M. Khalatnikov: Effect of scalar and vector fields on the nature of the cosmological singularity, Sov. Phys. JETP 36, 591–597 (1973)

    Google Scholar 

  21. B.K. Berger: Influence of scalar fields on the approach to the singularity in spatially inhomogeneous cosmologies, Phys. Rev. D 61, 023508 (2000)

    Google Scholar 

  22. B.K. Berger: Why solve the Hamiltonian constraint in numerical relativity?, Gen. Relativ. Gravit. 38, 625–632 (2006)

    Google Scholar 

  23. H. Ringström: Curvature blow up in Bianchi VIII and IX vacuum spacetimes, Class. Quantum Gravity 17, 713–731 (2000)

    Google Scholar 

  24. M. Weaver: Dynamics of magnetic Bianchi VI 0 cosmologies, Class. Quantum Gravity 17, 421–434 (2000)

    Google Scholar 

  25. J. Wainwright: A dynamical systems approach to Bianchi cosmologies: Orthogonal models of class A, Class. Quantum Gravity 6, 1409 (1989)

    Google Scholar 

  26. H. Ringström: The Bianchi IX attractor, Ann. Henri Poincaré 2, 405–500 (2001)

    Google Scholar 

  27. J.M. Heinzle, C. Uggla: A new proof of the Bianchi type IX attractor theorem, Class. Quantum Gravity 26, 075015 (2009)

    Google Scholar 

  28. V.A. Belinskii, E.M. Lifshitz, I.M. Khalatnikov: A general solution of the Einstein equations with a time singularity, Adv. Phys. 13, 639–667 (1982)

    Google Scholar 

  29. J.D. Barrow, F.J. Tipler: Analysis of the generic singularity studies by Belinskii, Khalatnikov, and Lifshitz, Phys. Rep. 56, 371–402 (1979)

    Google Scholar 

  30. B.K. Berger, D. Garfinkle, J.A. Isenberg, V. Moncrief, M. Weaver: The singularity in generic gravitational collapse is spacelike, local, and oscillatory, Mod. Phys. Lett. A 13, 1565–1574 (1998)

    Google Scholar 

  31. W.C. Lim, L. Andersson, D. Garfinkle, F. Pretorius: Spikes in the mixmaster regime of G 2 cosmologies, Phys. Rev. D 79, 123526 (2009)

    Google Scholar 

  32. A. Ashtekar, A. Henderson, D. Sloan: A Hamiltonian formulation of the BKL conjecture, Phys. Rev. D 83, 084024 (2011)

    Google Scholar 

  33. J.M. Heinzle, C. Uggla, N. Rohr: The cosmological billiard attractor, Adv. Theor. Math. Phys. 13, 293–407 (2009)

    Google Scholar 

  34. B. Grubišić, V. Moncrief: Asymptotic behavior of the T 3 × R Gowdy space-times, Phys. Rev. D 47, 2371–2382 (1993)

    Google Scholar 

  35. A.D. Rendall: Fuchsian methods and spacetime singularities, Class. Quantum Gravity 21, S295–S304 (2004)

    Google Scholar 

  36. R.H. Gowdy: Gravitational waves in closed universes, Phys. Rev. Lett. 27, 826 (1971)

    Google Scholar 

  37. D. Garfinkle: Numerical simulations of Gowdy spacetimes on S 2 × S 1 × R , Phys. Rev. D 60, 104010 (1999)

    Google Scholar 

  38. B.K. Berger, D. Garfinkle: Phenomenology of the Gowdy model on T 3 × R , Phys. Rev. D 57, 4767–4777 (1998)

    Google Scholar 

  39. B.K. Berger: Quantum graviton creation in a model universe, Ann. Phys. 83, 458–490 (1974)

    Google Scholar 

  40. A.D. Rendall, M. Weaver: Manufacture of Gowdy spacetimes with spikes, Class. Quantum Gravity 18, 2959–2975 (2001)

    Google Scholar 

  41. H. Ringström: Asymptotic expansions close to the singularity in Gowdy spacetimes, Class. Quantum Gravity 21, S305–S322 (2004)

    Google Scholar 

  42. B.K. Berger, P.T. Chruściel, J.A. Isenberg, V. Moncrief: Global foliations of vacuum spacetimes with T 2 isometry, Ann. Phys. 260, 117–148 (1997)

    Google Scholar 

  43. B.K. Berger, J.A. Isenberg, M. Weaver: Oscillatory approach to the singularity in vacuum spacetimes with T 2 isometry, Phys. Rev. D 64, 084006 (2001)

    Google Scholar 

  44. B.K. Berger: Hunting local mixmaster dynamics in spatially inhomogeneous cosmologies, Class. Quantum Gravity 21, S81–S96 (2004)

    Google Scholar 

  45. J.M. Heinzle, C. Uggla, W.C. Lim: Spike Oscillations, Phys. Rev. D 86, 104049 (2012)

    Google Scholar 

  46. T. Damour, M. Henneaux, A.D. Rendall, M. Weaver: Kasner-like behaviour for subcritical Einstein-matter systems, Ann. Henri Poincare 3, 1049–1111 (2002)

    Google Scholar 

  47. E. Ames, F. Beyer, J. Isenberg, P.G. LeFloch: Quasilinear hyperbolic fuchsiansystems and AVTD behavior in T 2 -symmetric vacuum spacetimes, Annal. Henri Poincaré 14, 1445–1523 (2012)

    Google Scholar 

  48. V. Moncrief: Reduction of Einstein’s equations for vacuum space-times with spacelike U ( 1 ) isometry groups, Ann. Phys. 167, 118–142 (1986)

    Google Scholar 

  49. B.K. Berger, V. Moncrief: Numerical evidence that the singularity in polarized U ( 1 ) symmetric cosmologies on T 3 × R is velocity dominated, Phys. Rev. D 57, 7235–7240 (1998)

    Google Scholar 

  50. B.K. Berger, V. Moncrief: Evidence for an oscillatory singularity in generic U ( 1 ) symmetric cosmologies on T 3 × R , Phys. Rev. D 58, 1–8 (1998)

    Google Scholar 

  51. Y. Choquet-Bruhat, J. Isenberg, V. Moncrief: Topologically general U ( 1 ) symmetric Einstein spacetimes with AVTD behavior, Nuovo Cim. B 119, 625–638 (2004)

    Google Scholar 

  52. B.K. Berger, V. Moncrief: Evidence for an oscillatory singularity in generic U ( 1 ) symmetric cosmologies on T 3 × R , Phys. Rev. D 58, 064023 (1998)

    Google Scholar 

  53. D. Garfinkle: Numerical simulations of generic collapse, Phys. Rev. Lett. 93, 161101 (2004)

    Google Scholar 

  54. L. Andersson, A.D. Rendall: Quiescent cosmological singularities, Commun. Math. Phys. 218, 479–511 (2001)

    Google Scholar 

  55. C. Uggla, H. van Elst, J. Wainwright, G.F.R. Ellis: The past attractor in inhomogeneous cosmology, Phys. Rev. D 68, 103502 (2003)

    Google Scholar 

  56. C. Gundlach, J.M. Martin-Garcia: Critical phenomena in gravitational collapse, Living Rev, Relativity 10, 5 (2007)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. 2131 Chateau PL, CA 94550, Livermore, California, USA

    Beverly K. Berger

Authors
  1. Beverly K. Berger
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Beverly K. Berger .

Editor information

Editors and Affiliations

  1. Department of Physics, Pennsylvania State University, 16802, University Park, PA, USA

    Abhay Ashtekar

  2. Institute for Foundational Studies Hermann Minkowski, Montreal, Quebec, Canada

    Vesselin Petkov

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Berger, B.K. (2014). Singularities in Cosmological Spacetimes. In: Ashtekar, A., Petkov, V. (eds) Springer Handbook of Spacetime. Springer Handbooks. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-41992-8_21

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-41992-8_21

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-41991-1

  • Online ISBN: 978-3-642-41992-8

  • eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)

Publish with us

Policies and ethics

Search

Navigation