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Unified Form of the Initial Value Conditions

  • James W. York
Chapter
Part of the Astrophysics and Space Science Library book series (ASSL, volume 367)

Abstract

In this paper both the initial value problem and the conformal thin sandwich problem are written in a unified way.

Keywords

Conformal Transformation Extrinsic Curvature Riemann Tensor Constant Mean Curvature Lapse Function 
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.

Notes

Acknowledgement

The author thanks Michael Brown for his invaluable assistance in preparing the manuscript.

References

  1. 1.
    A. Anderson and J.W. York, “Hamiltonian time evolution for general relativity,” Phys. Rev. Letters, 81, 1154-1157 (1998).MathSciNetADSzbMATHCrossRefGoogle Scholar
  2. 2.
    R. Arnowitt, S. Deser, and C.W. Misner, “The Dynamics of General Relativity”, pp. 227–265 in L. Witten (ed.), Gravitation (Wiley, New York, 1962).Google Scholar
  3. 3.
    A. Ashtekar, “New Hamiltonian formulation of general relativity,” Phys. Rev. D 36, 1587–1602 (1987).MathSciNetADSCrossRefGoogle Scholar
  4. 4.
    Y. Choquet-Bruhat, Théorèm d’existence pour certains systèms d’équations aux derivées partielles non linéaires,” Acta. Math. 88, 141–255 (1955).CrossRefGoogle Scholar
  5. 5.
    Y. Choquet-Bruhat, Seminar, Princeton University, October, 1972.Google Scholar
  6. 6.
    Y. Choquet-Bruhat and C. DeWitt-Morette, Analysis, Manifolds, and Physics, (North Holland, Amsterdam, 1977).zbMATHGoogle Scholar
  7. 7.
    Y. Choquet-Bruhat, J. Isenberg, and J.W. York, “Asymptotically euclidean solutions of the Einstein constraints,” Phys. Rev. D 61, 084034 (2000).MathSciNetADSCrossRefGoogle Scholar
  8. 8.
    Y. Choquet-Bruhat and T. Ruggeri, “Hyperbolic form of the Einstein equations,” Commun. Math. Phys. 89, 269–283 (1983).MathSciNetADSzbMATHCrossRefGoogle Scholar
  9. 9.
    Y. Choquet-Bruhat and J.W. York, “Well posed reduced systems for the Einstein equation,” Banach Center Publications, I, 41, 119–131 (1997).MathSciNetGoogle Scholar
  10. 10.
    E. Cotton, “Sur les variétés à trois dimensions,” Ann. Fac. d. Sc. Toulouse (II) 1, 385–438, (1899).MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    S. Deser, “Covariant decompositions of symmetric tensors and the gravitational Cauchy problem,” Ann. Inst. Henri Poincaré A7, 149–188 (1967).Google Scholar
  12. 12.
    P.A.M. Dirac, “The theory of gravitation in Hamiltonian form,” Proc. Roy. Soc. Lond. A246, 333–343 (1958).MathSciNetADSGoogle Scholar
  13. 13.
    P.A.M. Dirac, “Fixation of coordinates in the Hamiltonian theory of gravitation,” Phys. Rev. 114, 924–930 (1959).MathSciNetADSzbMATHCrossRefGoogle Scholar
  14. 14.
    D. Hilbert, “Die Grundlagen der Physik,” Konigl. Gesell. d. Wiss. Göttingen, Nachr., Math.-Phys. Kl., 395–407 (1915).Google Scholar
  15. 15.
    J.A. Isenberg, private communication (1973).Google Scholar
  16. 16.
    A. Lichnerowicz, “L’intégration des équations de la gravitation problème des n corps,” J. Math. Pures Appl. 23, 37–63, (1944).MathSciNetzbMATHGoogle Scholar
  17. 17.
    N.S. O’Murchadha, private communication (1972).Google Scholar
  18. 18.
    H. Pfeiffer, Ph.D. thesis, Cornell University, 2003.Google Scholar
  19. 19.
    H. Pfeiffer and J.W.York, “Extrinsic curvature and the Einstein constraints,” Phys. Rev. D 67, 044022 (2003).MathSciNetADSCrossRefGoogle Scholar
  20. 20.
    J.A. Schouten, Ricci Calculus, Springer-Verlag (Berlin, 1954).zbMATHCrossRefGoogle Scholar
  21. 21.
    C. Teitelboim, “Quantum mechanics of the gravitational field,” Phys. Rev. D 28, 297–311 (1983).MathSciNetADSCrossRefGoogle Scholar
  22. 22.
    F.J. Tipler and J.E. Marsden, “Maximal hypersurfaces and foliations of constant mean curvature in general relativity,” Phys. Reports 66, 109–139 (1980).MathSciNetADSCrossRefGoogle Scholar
  23. 23.
    A. Trautman, “Conservation Laws in General Relativity,” pp. 169–198 in L. Witten (ed.), Gravitation (Wiley, New York, 1962).Google Scholar
  24. 24.
    J.W. York, “Role of conformal three-geometry in the dynamics of gravitation,” Phys. Rev. Letters 28, 1082–1085 (1972).ADSCrossRefGoogle Scholar
  25. 25.
    J.W. York, “Conformal thin sandwich problem,” Phys. Rev. Letters 82, 1350–1353 (1999).MathSciNetADSzbMATHCrossRefGoogle Scholar
  26. 26.
    J.W. York, “Decomposition of symmetric tensors,” Ann. Inst. Henri Poincaré, A21, 319–332 (1974). It should be noted that this method is the correct one for decomposing metric perturbations.MathSciNetADSGoogle Scholar
  27. 27.
    J.W. York, “Kinematics and Dynamics of General Relativity,” pp. 83–126 in L. Smarr (ed.), Sources of Gravitational Radiation, (Cambridge, 1979).Google Scholar
  28. 28.
    J.W. York, private communication to S. Teukolsky and H. Pfeiffer (2001).Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Department of PhysicsNorth Carolina State UniversityRaleighUSA

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