Skip to main content
SpringerLink
Log in

3+1 Decomposition of Einstein Equation

  • Chapter
  • First Online:
3+1 Formalism in General Relativity

Part of the book series: Lecture Notes in Physics ((LNP,volume 846))

  • 3984 Accesses

Abstract

The fundamental equation for general relativity, the Einstein equation, is decomposed orthogonally with respect to a 3+1 foliation of spacetime. Then we introduce spatial coordinates on the hypersurfaces forming the foliation, thereby introducing the famous shift vector. This enables one to turn the 3+1 Einstein equation into a system of partial-differential equations. This system can be formulated as a Cauchy problem with constraints and we discuss briefly the known existence and uniqueness results regarding it. Finally we discuss the ADM Hamiltonian approach to general relativity, which is based on the 3+1 decomposition.

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 16.99
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

Notes

  1. 1.

    It is polynomial in the derivatives of \(\gamma_{kl}\) and involves at most rational fractions in \(\gamma_{kl}\) (to get the inverse metric \(\gamma^{kl}\)).

  2. 2.

    we use the same notation as that defined by Eq. (5.87).

References

  1. Wheeler, J.A.: Geometrodynamics and the issue of the final state. In: DeWitt, C., DeWitt, B.S. (eds) Relativity Groups and Topology, pp. 316. Gordon and Breach, New York (1964)

    Google Scholar 

  2. Fourés-Bruhat, Y., Choquet-Bruhat, Y.: Sur l’Intégration des Équations de la Relativité Générale. J. Ration. Mech. Anal. 5, 951 (1956)

    MATH  Google Scholar 

  3. Darmois, G.: Les équations de la gravitation einsteinienne, Mémorial des Sciences Mathématiques 25, Gauthier-Villars, Paris (1927)

    Google Scholar 

  4. Lichnerowicz, A.: Sur certains problèmes globaux relatifs au système des équations d’Einstein. Hermann, Paris; Actual. Sci. Ind. 833, (1939)

    Google Scholar 

  5. Lichnerowicz, A.: L’intégration des équations de la gravitation relativiste et le problème des n corps. J. Math. Pures Appl. 23, 37 (1944); reprinted in Lichnerowicz, A.: Choix d’\(\oe\)uvres mathématiques, p. 4. Hermann, Paris (1982)

    Google Scholar 

  6. Fourés-Bruhat, Y., Choquet-Bruhat, Y.: Sur l’intégration des équations d’Einstein. C. R. Acad. Sci. Paris 226, 1071 (1948)

    MathSciNet  MATH  Google Scholar 

  7. Arnowitt, R., Deser, S., Misner, C.W.: The dynamics of general relativity. In: Witten, L. (ed.) Gravitation: an introduction to current research, p. 227. Wiley, New York (1962) http://arxiv.org/abs/gr-qc/0405109

  8. York, J.W.: Velocities and momenta in an extended elliptic form of the initial value conditions. Nuovo Cim. B119, 823 (2004)

    MathSciNet  ADS  Google Scholar 

  9. Anderson, A., York, J.W.: Hamiltonian time evolution for general relativity. Phys. Rev. Lett. 81, 1154 (1998)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  10. Fischer, A.E., Marsden, J.: The Einstein equation of evolution—a geometric approach. J. Math. Phys. 13, 546 (1972)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  11. Fischer, A., Marsden, J.: The initial value problem and the dynamical formulation of general relativity. In: Hawking, S.W., Israel, W. (eds) General Relativity: An Einstein Centenary Survey, pp. 138. Cambridge University Press, Cambridge (1979)

    Google Scholar 

  12. Wald, R.M.: General relativity. University of Chicago Press, Chicago (1984)

    MATH  Google Scholar 

  13. Courant, R., Hilbert, D.: Methods of mathematical physics; vol. II: partial differential equations. Interscience, New York (1962)

    MATH  Google Scholar 

  14. Knapp, A.M., Walker, E.J., Baumgarte, T.W.: Illustrating stability properties of numerical relativity in electrodynamics. Phys. Rev. D 65, 064031 (2002)

    Article  ADS  Google Scholar 

  15. Baumgarte, T.W., Shapiro, S.L.: Numerical relativity and compact binaries. Phys. Rep. 376, 41 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  16. Fourés-Bruhat, Y., (Choquet-Bruhat, Y).: Théorème d’existence pour certains systèmes d’équations aux dérivées partielles non linéaires. Acta Mathematica 88, 141 (1952) http://fanfreluche.math.univ-tours.fr/

  17. Bartnik, R., Isenberg, J.: The Constraint equations, in Ref. [33], p. 1

    Google Scholar 

  18. Choquet-Bruhat, Y., York, J.W.: The Cauchy problem. In: Held, A. (eds) General Relativity and Gravitation one hundred Years after the Birth of Albert Einstein, pp. 99. Plenum Press, New York (1980)

    Google Scholar 

  19. Choquet-Bruhat, Y., Geroch, R.: Global aspects of the Cauchy problem in general relativity. Commun. Math. Phys. 14, 329 (1969)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  20. Klainerman, S., Nicoló, F.: On the local and global aspects of the Cauchy problem in general relativity. Class. Quantum Grav. 16, R73 (1999)

    Article  ADS  MATH  Google Scholar 

  21. Andersson, L.: The global existence problem in general relativity, in Ref. [33], p. 71

    Google Scholar 

  22. Rendall, A.D.: Theorems on existence and global dynamics for the Einstein equations. Living Rev. Relativ. 8, 6 (2005) http://www.livingreviews.org/lrr-2005-6

  23. Choquet-Bruhat, Y.: General relativity and Einstein’s equations. Oxford University Press, New York (2009)

    Google Scholar 

  24. Dirac, P.A.M.: The theory of gravitation in Hamiltonian form. Proc. Roy. Soc. Lond. A 246, 333 (1958)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  25. Dirac, P.A.M.: Fixation of coordinates in the Hamiltonian theory of gravitation. Phys. Rev. 114, 924 (1959)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  26. Deser, S.: Some remarks on Dirac’s contributions to general relativity. Int. J. Mod. Phys. A 19S1, 99 (2004)

    Article  MathSciNet  Google Scholar 

  27. Regge, T., Teitelboim, C.: Role of surface integrals in the Hamiltonian formulation of general relativity. Ann. Phys. (N.Y.) 88, 286 (1974)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  28. Misner, C.W., Thorne, K.S., Wheeler, J.A.: Gravitation. Freeman, New York (1973)

    Google Scholar 

  29. Poisson, E.: A relativist’s toolkit, the mathematics of black-hole mechanics. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  30. Henneaux, M.: Hamiltonian formalism of general relativity, lectures at Institut Henri Poincaré, Paris (2006), http://www.luth.obspm.fr/IHP06/

  31. Schäfer, G.: Equations of motion in the ADM formalism, lectures at Institut Henri Poincaré, Paris (2006), http://www.luth.obspm.fr/IHP06/

  32. Deruelle, N.: General relativity: a primer, lectures at Institut Henri Poincaré, Paris (2006), http://www.luth.obspm.fr/IHP06/

  33. Chruściel, P.T., Friedrich, H. (eds): The Einstein equations and the large scale behavior of gravitational fields—50 years of the Cauchy problem in general relativity. Birkhäuser Verlag, Basel (2004)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Lab. Univers et Théories (LUTH) UMR 8102 du CNRS, Observatoire de Paris, Université Paris Diderot, place Jules Janssen 5, 92190, Meudon, France

    Éric Gourgoulhon

Authors
  1. Éric Gourgoulhon
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Gourgoulhon, É. (2012). 3+1 Decomposition of Einstein Equation. In: 3+1 Formalism in General Relativity. Lecture Notes in Physics, vol 846. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24525-1_5

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-24525-1_5

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-24524-4

  • Online ISBN: 978-3-642-24525-1

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

Publish with us

Policies and ethics

Search

Navigation