Skip to main content

Abstract

Rigorous results on solutions of the Einstein-Vlasov system are surveyed. After an introduction to this system of equations and the reasons for studying it, a general discussion of various classes of solutions is given. The emphasis is on presenting important conceptual ideas, while avoiding entering into technical details. Topics covered include spatially homogeneous models, static solutions, spherically symmetric collapse and isotropic singularities.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Andréasson, H., Rein, G. and Rendall, A.D. 2003 On the Einstein-Vlasov system with hyperbolic symmetry. Math. Proc. Camb. Phil. Soc. 134, 529–549.

    Article  MATH  Google Scholar 

  2. Anguige, K. 2000 Isotropic cosmological singularities 3: The Cauchy problem for the inhomogeneous conformal Einstein-Vlasov equations. Ann. Phys. (NY) 282, 395–419.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  3. Binney, J., Tremaine, S. 1987 Galactic dynamics. Princeton University Press, Princeton.

    MATH  Google Scholar 

  4. Börner, G. 1993 The early universe. Facts and fiction. Springer, Berlin.

    Google Scholar 

  5. Choquet-Bruhat, Y. 1971 Problème de Cauchy pour le système intégro différentiel d’Einstein-Liouville. Ann. Inst. Fourier 21, 181–201.

    Article  MathSciNet  MATH  Google Scholar 

  6. Chruściel, P.T. and Rendall, A.D. 1995 Strong cosmic censorship in vacuum space-times with compact locally homogeneous Cauchy surfaces. Ann. Phys. (NY) 242, 349–385.

    Article  ADS  MATH  Google Scholar 

  7. Ehlers, J. 1973 Survey of general relativity theory. In: Israel, W. (ed.) Relativity, Astrophysics and Cosmology. Reidel, Dordrecht.

    Google Scholar 

  8. Friedrich, H., Rendall, A.D. 2000 The Cauchy problem for the Einstein equations. In B.G. Schmidt (ed) Einstein’s Field Equations and Their Physical Implications. Lecture Notes in Physics 540. Springer, Berlin.

    Google Scholar 

  9. Hawking, S.W., Ellis, G.F.R. 1973 The large-scale structure of space-time. Cambridge University Press, Cambridge.

    Book  MATH  Google Scholar 

  10. Holm, D.D., Marsden, J.E., Ratiu, T. and Weinstein, A. 1985 Nonlinear stability of fluid and plasma equilibria. Phys. Rep. 123, 1–116.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  11. Kichenassamy, S. 1996 Nonlinear wave equations. Marcel Dekker, New York.

    MATH  Google Scholar 

  12. Kichenassamy, S., Rendall, A.D. 1998 Analytic description of singularities in Gowdy spacetimes. Class. Quantum Gray. 15, 1339–1355.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  13. Olabarrieta, I., Choptuik, M.W. 2002 Critical phenomena at the threshold of black hole formation for collisionless matter in spherical symmetry. Phys. Rev. D65, 024007.

    MathSciNet  ADS  Google Scholar 

  14. Rein, G. 1994 Static solutions of the spherically symmetric Vlasov-Einstein system. Math. Proc. Camb. Phil. Soc. 115, 559–570.

    Article  MathSciNet  MATH  Google Scholar 

  15. Rein, G. 1999 Static shells for the Vlasov-Poisson and Vlasov-Einstein systems. Indiana University Math. J. 48, 335–346.

    MathSciNet  MATH  Google Scholar 

  16. Rein, G. 2000 Stationary and static stellar dynamic models with axial symmetry. Nonlinear Analysis; Theory, Methods and Applications 41, 313–344.

    MathSciNet  ADS  MATH  Google Scholar 

  17. Rein, G. 2002 Stability of spherically symmetric steady states in galactic dynamics against general perturbations. Arch. Rat. Mech. Anal. 161, 27–42.

    Article  MathSciNet  MATH  Google Scholar 

  18. Rein, G., Rendall, A. D. 1992 Global existence of solutions of the spherically symmetric Vlasov-Einstein system with small initial data. Commun. Math. Phys. 150:561–583

    Article  MathSciNet  ADS  MATH  Google Scholar 

  19. Rein, G., Rendall, A.D. 2000 Compact support of spherically symmetric equilibria in non-relativistic and relativistic galactic dynamics. Math. Proc. Camb. Phil. Soc. 128, 363–380.

    Article  MathSciNet  MATH  Google Scholar 

  20. Rein, G., Rendall, A.D. and Schaeffer, J. 1995 A regularity theorem for solutions of the spherically symmetric Vlasov-Einstein system. Commun. Math. Phys. 168, 467–478.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  21. Rein, G., Rendall, A.D. and Schaeffer, J. 1998 Critical collapse of collisionless matter — a numerical investigation. Phys. Rev. D58, 044007

    ADS  Google Scholar 

  22. Rendall, A.D. 1992 Cosmic censorship and Vlasov equation. Class. Quantum Graw matter. 9, L99–L104.

    Article  MathSciNet  ADS  Google Scholar 

  23. Rendall, A.D. 1995 Global properties of locally spatially homogeneous cosmological models with matter. Math. Proc. Camb. Phil. Soc. 118, 511–526.

    Article  MathSciNet  MATH  Google Scholar 

  24. Rendall, A.D. 1997 An introduction to the Einstein-Vlasov system. Banach Centre Publications 41, 35–68.

    MathSciNet  Google Scholar 

  25. Rendall, A.D. 2002 Cosmological models and centre manifold theory. Gen. Rel. Gray. 34, 1277–1294.

    Article  MathSciNet  MATH  Google Scholar 

  26. Rendall, A.D. 2002 Theorems on existence and global dynamics for the Einstein equations. Living Reviews in Relativity 5, 6.

    MathSciNet  ADS  Google Scholar 

  27. Rendall, A.D., Tod, K.P. 1999 Dynamics of spatially homogeneous solutions of the Einstein-Vlasov equations which are locally rotationally symmetric. Class. Quantum Gray. 16, 1705–1726.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  28. Rendall, A.D., Uggla, C. 2000 Dynamics of spatially homogeneous locally rotationally symmetric solutions of the Einstein-Vlasov equations. Class. Quantum Gray. 17, 4697–4714.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  29. Rendall, A.D., Blow-up for solutions of hyperbolic PDE and spacetime singularities. In Depauw, N., Robert, D. and Saint Raymond, X. (eds.) Journées Equations aux Dérivées Partielles, Nantes, 2000. CNRS, Nantes.

    Google Scholar 

  30. Ringström, H. 2000 Curvature blow up in Bianchi VIII and IX vacuum spacetimes. Class. Quantum Gray. 17, 713–731.

    Article  ADS  MATH  Google Scholar 

  31. Schaeffer, J. 1999 A class of counterexamples to Jeans’ theorem for the Vlasov Einstein system. Commun. Math. Phys. 204, 313–327.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  32. Swanson, D.C., 1989 Plasma Waves. Academic Press, Boston.

    Book  Google Scholar 

  33. Wainwright, J., Ellis, G.F.R. 1997 Dynamical systems in cosmology. Cambridge University Press, Cambridge.

    Book  Google Scholar 

  34. Wald, R.M. 1984 General Relativity. Chicago University Press, Chicago.

    MATH  Google Scholar 

  35. Wolansky, G. 2001 Static solutions of the Vlasov-Einstein system. Arch. Rat. Mech. Anal. 156, 205–230.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2004 Springer Basel AG

About this paper

Cite this paper

Rendall, A.D. (2004). The Einstein-Vlasov System. In: Chruściel, P.T., Friedrich, H. (eds) The Einstein Equations and the Large Scale Behavior of Gravitational Fields. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7953-8_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-7953-8_6

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9634-4

  • Online ISBN: 978-3-0348-7953-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics