Abstract

Initial data for solutions of Einstein’s gravitational field equations cannot be chosen freely: the data must satisfy the four Einstein constraint equations. We first discuss the geometric origins of the Einstein constraints and the role the constraint equations play in generating solutions of the full system. We then discuss various ways of obtaining solutions of the Einstein constraint equations, and the nature of the space of solutions.

Keywords

constraint equations Einstein equations conformal method quasi-spherical thin sandwich 

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References

  1. [1]
    R. Abraham, J.E. Marsden, and T. Ratiu. Manifolds, tensor analysis, and applications. Springer-Verlag, 1988.MATHCrossRefGoogle Scholar
  2. [2]
    A. Anderson and J.W. York Jr. Hamiltonian time evolution for general relativity. Phys. Rev. Lett., 81:1154–1157, 1998. grqc/9807041.MathSciNetADSMATHCrossRefGoogle Scholar
  3. [3]
    A. Anderson and J.W. York. Fixing Einstein’s equations. Phys. Rev. Lett., pages 82:4384–4387, 1999.MathSciNetADSMATHCrossRefGoogle Scholar
  4. [4]
    L. Andersson and P.T. Chruściel. On asymptotic behavior of solutions of the constraint equations in general relativity with “hyperboloidal boundary conditions”. Dissert. Math., 355:1–100, 1996.Google Scholar
  5. [5]
    L. Andersson and P.T. Chruściel, and H. Friedrich. On the regularity of solutions to the Yamabe equations and the existence of smooth hyperboloidal initial data for Einstein’s field equations. Comm. Math. Phys.,149:587–612, 1992.MathSciNetADSMATHCrossRefGoogle Scholar
  6. [6]
    J. Arms, J. E. Marsden, and V. Moncrief. The structure of the space of solutions of Einstein’s equations II: Several Killing fields and the Einstein-Yang-Mills equations. Annals Phys., 144(1):81–106, 1982.MathSciNetADSMATHCrossRefGoogle Scholar
  7. [7]
    R. Arnowitt, S. Deser, and C. Misner. The dynamics of general relativity. In L. Witten, editor, Gravitation,pages 227–265. Wiley, N.Y., 1962.Google Scholar
  8. [8]
    R.F. Baierlein, D.H. Sharp, and J.A. Wheeler. Phys. Rev.,126:1, 1962.MathSciNetADSCrossRefGoogle Scholar
  9. [9]
    R. Bartnik. Phase space for the Einstein equations. gr-qc/0402070.Google Scholar
  10. [10]
    R. Bartnik. The existence of maximal hypersurfaces in asymptotically flat space-times. Commun. Math. Phys.,94:155–175, 1984.MathSciNetADSMATHCrossRefGoogle Scholar
  11. [11]
    R. Bartnik. The regularity of variational maximal surfaces. Acta Math.,161:145–181, 1988.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    R. Bartnik. Remarks on cosmological spacetimes and constant mean curvature surfaces. Commun. Math. Phys., 117:615–624, 1988.MathSciNetADSMATHCrossRefGoogle Scholar
  13. [13]
    R. Bartnik. New definition of quasilocal mass. Phys. Rev. Lett., 62(20):2346–2348, May 1989.MathSciNetADSCrossRefGoogle Scholar
  14. [14]
    R. Bartnik. Quasi-spherical metrics and prescribed scalar curvature. J. Diff. Geom., 37:31–71, 1993.MathSciNetMATHGoogle Scholar
  15. [15]
    R. Bartnik. Einstein equations in the null quasi-spherical gauge. Class. Quant. Gravity, 14:2185–2194, 1997. gr-qc/9611045.MathSciNetADSMATHCrossRefGoogle Scholar
  16. [16]
    R. Bartnik. Energy in general relativity. In Shing-Tung Yau, editor, Tsing Hua Lectures on Analysis and Geometry, pages 5–28. International Press, 1997.Google Scholar
  17. [17]
    R. Bartnik and G. Fodor. On the restricted validity of the thin sandwich conjecture. Phys. Rev. D, 48:3596–3599, 1993. gr-qc/9304004.MathSciNetADSCrossRefGoogle Scholar
  18. [18]
    R. Bartnik and A. Norton. Numerical methods for the Einstein equations in null quasi-spherical coordinates. SIAM J. Sci. Comp., 22:917–950, 2000. gr-qc/9904045.MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    R. Bartnik and L. Simon. Spacelike hypersurfaces with prescribed boundary values and mean curvature. Commun. Math. Phys., 87:131–152, 1982.MathSciNetADSMATHCrossRefGoogle Scholar
  20. [20]
    R. Beig, P.T. Chruściel and R. Schoen. KIDs are non-generic. gr-qc/0403042.Google Scholar
  21. [21]
    E.P. Belasco and H.C. Ghanian. Initial conditions in general relativity: lapse and shift formulation. J. Math. Phys., 10:1503, 1969.ADSCrossRefGoogle Scholar
  22. [22]
    A. Besse. Einstein manifolds. Springer, 1987.MATHGoogle Scholar
  23. [23]
    H. Bondi. Gravitational waves in general relativity. Nature,186:535, 1960.ADSMATHCrossRefGoogle Scholar
  24. [24]
    D. Brill. On spacetimes without maximal surfaces. In H. Ning, editor, Proc. Third Marcel Grossman meeting. North Holland, 1982.Google Scholar
  25. [25]
    D. Brill and M. Cantor. The Laplacian on asymptotically flat manifolds and the specification of scalar curvature. Composit. Math., 43:317, 1981.MathSciNetMATHGoogle Scholar
  26. [26]
    M. Cantor. The existence of non-trivial asymptotically flat initial data for vacuum spacetimes. Commun. Math. Phys., 57:83, 1977.MathSciNetADSMATHCrossRefGoogle Scholar
  27. [27]
    Y. Choquet-Bruhat. Spacelike submanifolds with constant mean extrinsic curvature of a Lorentzian manifold. Ann. d. Scuola Norm. Sup. Pisa, 3:361–376, 1976.MathSciNetMATHGoogle Scholar
  28. [28]
    Y. Choquet-Bruhat. Non-strict and strict hyperbolic systems for the Einstein equations. In Ancon and J. Vaillant, editors, Partial Differential Equations and Mathematical Physics. Dekker, 2003.Google Scholar
  29. [29]
    Y. Choquet-Bruhat and R. Geroch. Global aspects of the Cauchy problem in general relativity. Commun. Math. Phys., 14:329–335, 1969.MathSciNetADSMATHCrossRefGoogle Scholar
  30. [30]
    Y. Choquet-Bruhat, J. Isenberg, and J. York. Einstein constraints on asymptotically Euclidean manifolds. Phys. Rev. D, 61:084034, 2000.MathSciNetADSCrossRefGoogle Scholar
  31. [31]
    Y. Choquet-Bruhat and J.W. York. The Cauchy problem. In A. Held, editor, General Relativity and Gravitation - the Einstein Centenary,chapter 4, pages 99–160. Plenum, 1979.Google Scholar
  32. [32]
    P.T. Chruściel and E. Delay. Existence of non-trivial vacuum asymptotically simple spacetimes. Class. Quant. Gray., 19:L71–L79, 2002.CrossRefGoogle Scholar
  33. [33]
    P.T. Chruściel and E. Delay. On mapping properties of general relativistic constraints operators in weighted function spaces, with applications. Mémoires de la Société Mathématique de France 94:1–103, 2003. gr-qc/0301073.Google Scholar
  34. [34]
    P.T. Chruściel, J. Isenberg and D. Pollack. Initial data engineering. gr-qc/0403066.Google Scholar
  35. [35]
    G. Cook. Private communication. 2003.Google Scholar
  36. [36]
    J. Corvino. Scalar curvature deformation and a gluing construction for the Einstein constraint equations. Commun. Math. Phys., 214:137–189, 2000.MathSciNetADSMATHCrossRefGoogle Scholar
  37. [37]
    J. Corvino and R. Schoen. On the asymptotics for the vacuum constraint equations. gr-qc/0301071.Google Scholar
  38. [38]
    S. Dain. Trapped surfaces as boundaries for the constraint equations, 2003. grqc/0308009.Google Scholar
  39. [39]
    Th. de Donder. Théorze des champs gravitiques. Gauthier-Villars, Paris, 1926.Google Scholar
  40. [40]
    D.M. DeTurck. The Cauchy problem for Lorentz metrics with prescribed Ricci curvature. Comp. Math., 48:327–349, 1983.MathSciNetMATHGoogle Scholar
  41. [41]
    A.E. Fischer, J.E. Marsden, and V. Moncrief. The structure of the space of solutions of Einstein’s equations I. One Killing field. Ann. Inst. H. Poincaré,33:147–194, 1980.MathSciNetMATHGoogle Scholar
  42. [42]
    A.E. Fischer and J.E. Marsden. Topics in the dynamics of general relativity. In J. Ehlers, editor, Structure of Isolated Gravitating Systems, pages 322–395. 1979.Google Scholar
  43. [43]
    Y. Fourés-Bruhat. Théorème d’existence pour certains systèmes d’équations aux dérivées partielles non linéaires. Acta Math.,88:141–225, 1952.MathSciNetMATHCrossRefGoogle Scholar
  44. [44]
    H. Friedrich. On the hyperbolicity of Einstein’s and other gauge field equations. Commun. Math. Phys., 100:525–543, 1985.MathSciNetADSMATHCrossRefGoogle Scholar
  45. [45]
    D. Garfinkle. Numerical simulations of generic singularities. Unpublished. 2003.Google Scholar
  46. [46]
    C. Gerhardt. H-surfaces in Lorentzian manifolds. Commun. Math. Phys., 89:523–553, 1983.MathSciNetADSMATHCrossRefGoogle Scholar
  47. [47]
    R. Geroch. Energy extraction. Ann. N.Y. Acd. Sci., 224:108–117, 1973.ADSCrossRefGoogle Scholar
  48. [48]
    G. Huisken and T. Ilmanen. The inverse mean curvature flow and the Riemannian Penrose inequality. J. Diff. Geom., 59:353–438, 2001.MathSciNetMATHGoogle Scholar
  49. [49]
    G. Huisken and S.-T. Yau. Definition of center of mass for isolated physical systems and unique foliations by stable spheres with constant mean curvature. Invent. Math., 124:281–311, 1996.MathSciNetADSMATHCrossRefGoogle Scholar
  50. [50]
    J. Isenberg. Constant mean curvature solutions of the Einstein constraint equations on closed manifolds. Class. Quant. Gray.,12:2249, 1995.MathSciNetADSMATHCrossRefGoogle Scholar
  51. [51]
    J. Isenberg. Near constant mean curvature solutions of the Einstein constraint equations with non-negative Yamabe metrics. Unpublished. 2003.Google Scholar
  52. [52]
    J. Isenberg, R. Mazzeo, and D. Pollack. Gluing and wormholes for the Einstein constraint equations. Comm. Math. Phys., 231:529–568, 2002.MathSciNetADSMATHCrossRefGoogle Scholar
  53. [53]
    J. Isenberg, R. Mazzeo, and D. Pollack. On the topology of vacuum spacetimes. Ann. Inst. H. Ponicaré,4:369–383, 2003. gr-qc/0206034.MathSciNetMATHGoogle Scholar
  54. [54]
    J. Isenberg and V. Moncrief. A set of nonconstant mean curvature solutions of the Einstein constraint equations on closed manifolds. Class. Quant. Gray.,13:1819, 1996.MathSciNetADSMATHCrossRefGoogle Scholar
  55. [55]
    J. Isenberg and N.Ό Murchadha. Non-CMC conformal data sets which do not produce solutions of the Einstein constraint equations. gr-qc/0311057.Google Scholar
  56. [56]
    J. Isenberg and J. Park. Asymptotically hyperbolic non-constant mean curvature solu-tions of the Einstein constraint equations. Class. Quant. Gray., 14:A189, 1997.MathSciNetADSMATHCrossRefGoogle Scholar
  57. [57]
    V. Iyer and R.M. Wald. Some properties of Noether charge and a proposal for dynamical black hole entropy. Phys. Rev., D50:846–864, 1994. gr-qc/9403028.MathSciNetADSGoogle Scholar
  58. [58]
    F. John. Partial Differential Equations. Springer, 4th edition, 1982.Google Scholar
  59. [59]
    C. Lanczos. EM vereinfachendes Koordinatensystem für die Einsteinschen Gravitationsgleichungen. Phys. Zeit., 23:537–539, 1922.MATHGoogle Scholar
  60. [60]
    L. Lindblom and M.A. Scheel. Dynamical gauge conditions for the Einstein evolution equations. Phys. Rev., 7:124005, 2003. gr-qc/0301120.MathSciNetGoogle Scholar
  61. [61]
    D. Maxwell. Solutions of the Einstein constraint equations with apparent horizon boundary, 2003. gr-qc/0307117.Google Scholar
  62. [62]
    C.W. Misner, K.S. Thorne, and J.A. Wheeler. Gravitation. Freeman, 1973.Google Scholar
  63. [63]
    V. Moncrief. Spacetime symmetries and linearization stability of the Einstein equations I. J. Math. Phys., 16(3):493–498, March 1975.MathSciNetADSMATHCrossRefGoogle Scholar
  64. [64]
    V. Moncrief. Spacetime symmetries and linearization stability of the Einstein equations II. J. Math. Phys., 17(10):1893–1902 1976.MathSciNetADSCrossRefGoogle Scholar
  65. [65]
    R. Racke. Lectures on Nonlinear Evolution Equations. Friedrich Vieweg, 1992.MATHGoogle Scholar
  66. [66]
    R.K. Sachs. Gravitational waves in general relativity, VIII. Waves in asymptotically flat space-time. Proc. Roy. Soc. Lond. A, A270:103–126, 1962.MathSciNetADSGoogle Scholar
  67. [67]
    R. Schoen and S.-T. Yau. Proof of the positive mass theorem. Comm. Math. Phys.,65:45–76, 1979.MathSciNetADSMATHCrossRefGoogle Scholar
  68. [68]
    J. Sharples. Spacetime initial data and quasi-spherical coordinates. PhD thesis, University of Canberra, 2001.Google Scholar
  69. [69]
    Y. Shi and L.F. Tam. Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature. J. Diff. Geom.,62:79–125, 2002. math.DG/0301047.MathSciNetMATHGoogle Scholar
  70. [70]
    L. Simon. Seminar on Minimal Submanifolds, chapter Survey Lectures on Minimal Submanifolds, pages 3–52. Annals of Math. Studies 103. Princeton UP, 1983.Google Scholar
  71. [71]
    B. Smith and G. Weinstein. On the connectedness of the space of initial data for the Einstein equations. Electron. Res. Announc. Amer. Math. Soc., 6:52–63, 2000.MathSciNetMATHCrossRefGoogle Scholar
  72. [72]
    B. Smith and G. Weinstein. Quasiconvex foliations and asymptotically flat metrics of non-negative scalar curvature. Comm. Analysis and Geom., 2004. to appear.Google Scholar
  73. [73]
    M. Spillane. The Einstein equations in the null quasi-spherical gauge. PhD thesis, Australian National University, 1994.Google Scholar
  74. [74]
    M.E. Taylor. Partial Differential Equations III, volume 117 of Applied Mathematical Sciences. Springer, 1996.Google Scholar
  75. [75]
    R.M. Wald. General Relativity. University of Chicago Press, 1984.MATHGoogle Scholar
  76. [76]
    J.W. York. Conformal “Thin-Sandwich” data for the initial-value problem of general relativity. Phys. Rev. Lett., 82:1350–1353, 1999.MathSciNetADSMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2004

Authors and Affiliations

  • Robert Bartnik
    • 1
  • Jim Isenberg
    • 2
  1. 1.School of Mathematics and StatisticsUniversity of CanberraAustralia
  2. 2.Department of Mathematics and Institute for Theoretical ScienceUniversity of OregonEugeneUSA

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