Communications in Mathematical Physics

, Volume 163, Issue 1, pp 89–112 | Cite as

The Newtonian limit for asymptotically flat solutions of the Vlasov-Einstein system

  • Alan D. Rendall


It is shown that there exist families of asymptotically flat solutions of the Einstein equations coupled to the Vlasov equation describing a collisionless gas which have a Newtonian limit. These are sufficiently general to confirm that for this matter model as many families of this type exist as would be expected on the basis of physical intuition. A central role in the proof is played by energy estimates in unweighted Sobolev spaces for a wave equation satisfied by the second fundamental form of a maximal foliation.


Neural Network Statistical Physic Complex System Wave Equation Nonlinear Dynamics 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Asano, K., Ukai, S.: On the Vlasov-Poisson limit of the Vlasov-Maxwell equation. In: Nishida, T., Mimura, M., Fujii, H. (eds.) Patterns and waves. Amsterdam: North-Holland 1986Google Scholar
  2. 2.
    Batt, J.: Global symmetric solutions of the initial value problem of stellar dynamics. J. Diff. Eq.25, 342–364 (1977)CrossRefGoogle Scholar
  3. 3.
    Cantor, M.: A necessary and sufficient condition for York data to specify an asymptotically flat spacetime. J. Math. Phys.20, 1741–1744 (1979)CrossRefGoogle Scholar
  4. 4.
    Cartan, E.: Sur les variétés à connexion affine et la théorie de la relativité généralisée. Ann. Sci. Ecole Norm. Sup.39, 325–412 (1922);41, 1–25 (1924)MathSciNetGoogle Scholar
  5. 5.
    Choquet-Bruhat, Y.: Problème de Cauchy pour le système integro differentiel d'Einstein-Liouville. Ann. Inst. Fourier (Grenoble)21, 181–201 (1971)Google Scholar
  6. 6.
    Christodoulou, D., Klainerman, S.: The global nonlinear stability of the Minkowski space. Princeton, NJ: Princeton University Press 1993Google Scholar
  7. 7.
    Dautray, R., Lions, J.-L.: Mathematical analysis and numerical methods for science and technology, Vol. 1, Berlin, Heidelberg, New York: Springer 1990Google Scholar
  8. 8.
    Degond, P.: Local existence of solutions of the Vlasov-Maxwell equations and convergence to the Vlasov-Poisson equation for infinite light velocity. Math. Meth. Appl. Sci.8, 533–558 (1986)Google Scholar
  9. 9.
    Dieudonné, J.: Foundations of modern analysis. New York: Academic Press 1969Google Scholar
  10. 10.
    Ebin, D., Marsden, J.: Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. Math.92, 102–163 (1970)Google Scholar
  11. 11.
    Ehlers, J.: The Newtonian limit of general relativity. In: Ferrarese, G. (ed.) Classical mechanics and relativity: relationship and consistency. Naples: Bibliopolis 1991Google Scholar
  12. 12.
    Friedrichs, K.O.: Eine invariante Formulierung des Newtonschen Gravitationsgesetzes und des Grenzüberganges vom Einsteinschen zum Newtonschen Gesetz. Math. Ann.98, 566–575 (1927)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Fritelli, S., Reula, O.: On the Newtonian limit of general relativity. Preprint MPA 630, GarchingGoogle Scholar
  14. 14.
    Klainerman, S.: Global existence for nonlinear wave equations. Commun. Pure Appl. Math.33, 43–101 (1980)Google Scholar
  15. 15.
    Lottermoser, M.: A convergent post-Newtonian approximation for the constraint equations in general relativity. Ann. Inst. H. Poincaré (Physique Théorique)57, 279–317 (1992)Google Scholar
  16. 16.
    Majda, A., Compressible fluid flow and systems of conservation laws in several space variables. Berlin, Heidelberg, New York: Springer 1984Google Scholar
  17. 17.
    Marsden, J., Ebin, D.G., Fischer, A.E.: Diffeomorphism groups, hydrodynamics and relativity. In: Vanstone, J.R., Proc. 13th Biennial Seminar of the Canadian Mathematical Congress. Canadian Mathematical Society, Montreal 1972Google Scholar
  18. 18.
    Rein, G., Rendall, A.D.: Global existence of solutions of the spherically symmetric Vlasov-Einstein system with small initial data. Commun. Math. Phys.150, 561–583 (1992)Google Scholar
  19. 19.
    Rein, G., Rendall, A.D.: The Newtonian limit of the spherically symmetric Vlasov-Einstein system. Commun. Math. Phys.150, 585–591 (1992)Google Scholar
  20. 20.
    Rendall, A.D.: The initial value problem for a class of general relativistic fluid bodies. J. Math. Phys.33, 1047–1053 (1992)CrossRefGoogle Scholar
  21. 21.
    Rendall, A.D.: On the definition of post-Newtonian approximations. Proc. R. Soc. Lond.438, 341–360 (1992)Google Scholar
  22. 22.
    Schaeffer, J.: The classical limit of the relativistic Vlasov-Maxwell system. Commun. Math. Phys.104, 403–421 (1986)CrossRefGoogle Scholar
  23. 23.
    Zeidler, E.: Nonlinear functional analysis and its applications, Vol. 2. Berlin, Heidelberg, New York: Springer 1990Google Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • Alan D. Rendall
    • 1
  1. 1.Max-Planck-Institut für AstrophysikGarching bei MünchenGermany

Personalised recommendations