Advertisement

Annales Henri Poincaré

, Volume 16, Issue 12, pp 2919–2954 | Cite as

Conformal Parameterizations of Slices of Flat Kasner Spacetimes

  • David MaxwellEmail author
Article
First Online:

Abstract

The Kasner metrics are among the simplest solutions of the vacuum Einstein equations, and we use them here to examine the conformal method of finding solutions of the Einstein constraint equations. After describing the conformal method’s construction of constant mean curvature (CMC) slices of Kasner spacetimes, we turn our attention to non-CMC slices of the smaller family of flat Kasner spacetimes. In this restricted setting we obtain a full description of the construction of certain U n-1 symmetric slices, even in the far-from-CMC regime. Among the conformal data sets generating these slices we find that most data sets construct a single flat Kasner spacetime, but that there are also far-from-CMC data sets that construct one-parameter families of slices. Although these non-CMC families are analogues of well-known CMC one-parameter families, they differ in important ways. Most significantly, unlike the CMC case, the condition signaling the appearance of these non-CMC families is not naturally detected from the conformal data set itself. In light of this difficulty, we propose modifications of the conformal method that involve a conformally transforming mean curvature.

Keywords

Constraint Equation Fundamental Form Conformal Factor Conformal Killing Constant Mean Curvature 
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.
Download to read the full article text

References

  1. 1.
    Allen P.T., Clausen A., Isenberg J.: Near-constant mean curvature solutions of the Einstein constraint equations with non-negative Yamabe metrics. Class. Quantum Gravity 25(7), 075009–075015 (2008)MathSciNetCrossRefADSGoogle Scholar
  2. 2.
    Foures-Bruhat Y.: Théorème d’existence pour certains systèmes d’équations aux dérivées partielles non linéaires. Acta Mathematica 88(1), 141–225 (1952)MathSciNetCrossRefADSzbMATHGoogle Scholar
  3. 3.
    Holst M., Nagy G., Tsogtgerel G.: Rough solutions of the Einstein constraints on closed manifolds without near-CMC conditions. Commun. Math. Phys. 288(2), 547–613 (2009)MathSciNetCrossRefADSzbMATHGoogle Scholar
  4. 4.
    Isenberg J., Moncrief V.: A set of nonconstant mean curvature solutions of the Einstein constraint equations on closed manifolds. Class. Quantum Gravity 13(7), 1819–1847 (1996)MathSciNetCrossRefADSzbMATHGoogle Scholar
  5. 5.
    Isenberg J.: Constant mean curvature solutions of the Einstein constraint equations on closed manifolds. Class. Quantum Gravity 12(9), 2249–2274 (1995)MathSciNetCrossRefADSzbMATHGoogle Scholar
  6. 6.
    Kasner E.: Geometrical theorems on Einstein’s cosmological equations. Am. J. Math. 43(4), 217–221 (1921)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Lichnerowicz A.: L’intégration des équations de la gravitation relativiste et le problème des n corps. Journal de Mathématiques Pures et Appliquées. Neuvième Série 23, 37–63 (1944)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Maxwell D.: A class of solutions of the vacuum Einstein constraint equations with freely specified mean curvature. Math. Res. Lett. 16(4), 627–645 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Maxwell D.: A model problem for conformal parameterizations of the Einstein constraint equations. Commun. Math. Phys. 302(3), 697–736 (2011)MathSciNetCrossRefADSzbMATHGoogle Scholar
  10. 10.
    Maxwell, D.: The conformal method and the conformal thin-sandwich method are the same. arXiv:1402.5585 (2014)
  11. 11.
    Ó Murchadha N., York J.W.: Initial-value problem of general relativity. I. General formulation and physical interpretation. Phys. Rev. D. 10(2), 428–436 (1974)MathSciNetCrossRefADSGoogle Scholar
  12. 12.
    Pfeiffer H.P., York J.W.: Extrinsic curvature and the Einstein constraints. Phys. Rev. D. 67(4), 044022–044028 (2003)MathSciNetCrossRefADSGoogle Scholar
  13. 13.
    York J.W.: Conformally invariant orthogonal decomposition of symmetric tensors on Riemannian manifolds and the initial-value problem of general relativity. J. Math. Phys. 14(4), 456–464 (1973)MathSciNetCrossRefADSzbMATHGoogle Scholar
  14. 14.
    York J.W.: Conformal “thin-sandwich” data for the initial-value problem of general relativity. Phys. Rev. Lett. 82(7), 1350–1353 (1999)MathSciNetCrossRefADSzbMATHGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of Alaska FairbanksFairbanksUSA

Personalised recommendations