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Logarithmic Newman-Penrose constants for arbitrary polyhomogeneous spacetimes

Juan Antonio Valiente Kroon

TL;DR

This work extends the Newman-Penrose framework to polyhomogeneous spacetimes by allowing logarithmic terms in asymptotic expansions and demonstrates the existence of a conserved set of logarithmic NP constants. Through a detailed analysis of polyhomogeneous expansions and the NP equations, it derives evolution equations for Psi_0 that enable the construction of 10 conserved quantities, collapsing to the standard NP constants in the non-polyhomogeneous limit. The results provide a robust generalization of NP constants applicable to generic radiating spacetimes and illuminate their behavior under slower fall-off and logarithmic corrections at null infinity. The findings have potential implications for the interpretation of conserved gravitational data at infinity and for the study of non-smooth null infinity. Future work is anticipated to address the physical meaning and possible infinite towers of such constants.

Abstract

A discussion of how to calculate asymptotic expansions for polyhomogeneous spacetimes using the Newman-Penrose formalism is made. The existence of logarithmic Newman-Penrose constants for a general polyhomogeneous spacetime (i.e. a polyhomogeneous spacetime such that $Ψ_0=Ø(r^{-3}\ln ^{N_3})$) is addressed. It is found that these constants exist for the generic case.

Logarithmic Newman-Penrose constants for arbitrary polyhomogeneous spacetimes

TL;DR

This work extends the Newman-Penrose framework to polyhomogeneous spacetimes by allowing logarithmic terms in asymptotic expansions and demonstrates the existence of a conserved set of logarithmic NP constants. Through a detailed analysis of polyhomogeneous expansions and the NP equations, it derives evolution equations for Psi_0 that enable the construction of 10 conserved quantities, collapsing to the standard NP constants in the non-polyhomogeneous limit. The results provide a robust generalization of NP constants applicable to generic radiating spacetimes and illuminate their behavior under slower fall-off and logarithmic corrections at null infinity. The findings have potential implications for the interpretation of conserved gravitational data at infinity and for the study of non-smooth null infinity. Future work is anticipated to address the physical meaning and possible infinite towers of such constants.

Abstract

A discussion of how to calculate asymptotic expansions for polyhomogeneous spacetimes using the Newman-Penrose formalism is made. The existence of logarithmic Newman-Penrose constants for a general polyhomogeneous spacetime (i.e. a polyhomogeneous spacetime such that ) is addressed. It is found that these constants exist for the generic case.

Paper Structure

This paper contains 8 sections, 1 theorem, 57 equations.

Table of Contents

  1. Introduction
  2. Notation & conventions.
  3. Solving the NP equations using polyhomogeneous expansions.
  4. Logarithmic NP constants for a generic polyhomogeneous spacetime.
  5. Conclusions
  6. Leading terms for the spin coefficients
  7. General polyhomogeneous spacetimes ($N_3\neq -\infty$).
  8. Polyhomogeneous space-times with a finite shear at ${\fam\scrfam I}$.

Key Result

Proposition 1

The quantities (5 complex in general) with $X=\max \{N_{5},3N_3+3,N_3+N_4+2\}$ are constants of motion for any polyhomogeneous spacetime.

Theorems & Definitions (1)

  • Proposition 1