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Conserved Quantities for Polyhomogeneous Space-Times

J. A. Valiente Kroon

Abstract

The existence of conserved quantities with a structure similar to the Newman-Penrose quantities in a polyhomogeneous space-time is addressed. The most general form for the initial data formally consistent with the polyhomogeneous setting is found. The subsequent study is done for those polyhomogeneous space-times where the leading term of the shear contains no logarithmic terms. It is found that for these space-times the original NP quantities cease to be constants, but it is still possible to construct a set of other 10 quantities that are constant. From these quantities it is possible to obtain as a particular case a conserved quantity found by Chrusciel et al.

Conserved Quantities for Polyhomogeneous Space-Times

Abstract

The existence of conserved quantities with a structure similar to the Newman-Penrose quantities in a polyhomogeneous space-time is addressed. The most general form for the initial data formally consistent with the polyhomogeneous setting is found. The subsequent study is done for those polyhomogeneous space-times where the leading term of the shear contains no logarithmic terms. It is found that for these space-times the original NP quantities cease to be constants, but it is still possible to construct a set of other 10 quantities that are constant. From these quantities it is possible to obtain as a particular case a conserved quantity found by Chrusciel et al.

Paper Structure

This paper contains 21 sections, 92 equations.

Table of Contents

  1. Introduction.
  2. Coordinate system, null tetrad & the Newman-Penrose formalism.
  3. Initial data for the asymptotic characteristic initial value problem.
  4. Conserved Quantities.
  5. The conserved quantity of Chruściel et al.
  6. Conclusions.
  7. Acknowledgments.
  8. Polyhomogeneous expansions in the NP formalism.
  9. The relationship between $\rho$, $\sigma$ & $\Psi _0$ for a general polyhomogeneous space-time.
  10. Orders of the leading terms for a general polyhomogeneous space-time.
  11. Orders for the leading terms for polyhomogeneous space-times such that $\sigma _2=\sigma _{2,0}(\theta ,\varphi ).$
  12. The tetrad functions.
  13. Relations obtained from the Bianchi identities when $\sigma _2=\sigma _{2,0}(\theta ,\varphi )$.
  14. (Ba).
  15. (Bb).
  16. ...and 6 more sections