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Gravitational waves in general relativity: XIV. Bondi expansions and the ``polyhomogeneity'' of \Scri

Piotr T. Chrusciel, Malcolm A. H. MacCallum, David B. Singleton

TL;DR

This paper extends the Bondi–Sachs framework to polyhomogeneous spacetimes, showing that metric expansions with logarithmic terms are formally compatible with Einstein equations and amenable to a Bondi–Sachs type evolution. It establishes a hierarchical propagation of expansion coefficients, preserves the Bondi mass loss law, and identifies constants of motion associated with leading log terms, while clarifying the behavior of Weyl curvature at Scri and the status of Newman–Penrose constants. The results imply that polyhomogeneous Scri can describe radiating spacetimes without contradictions to isolation conditions and that Bondi coordinates remain available, with the asymptotic symmetry group collapsing to the BMS group. Collectively, the work broadens the admissible asymptotic structures in general relativity and clarifies their physical and geometrical implications for gravitational radiation.

Abstract

The structure of polyhomogeneous space-times (i.e., space-times with metrics which admit an expansion in terms of $r^{-j}\log^i r$) constructed by a Bondi--Sachs type method is analysed. The occurrence of some log terms in an asymptotic expansion of the metric is related to the non--vanishing of the Weyl tensor at Scri. Various quantities of interest, including the Bondi mass loss formula, the peeling--off of the Riemann tensor and the Newman--Penrose constants of motion are re-examined in this context.

Gravitational waves in general relativity: XIV. Bondi expansions and the ``polyhomogeneity'' of \Scri

TL;DR

This paper extends the Bondi–Sachs framework to polyhomogeneous spacetimes, showing that metric expansions with logarithmic terms are formally compatible with Einstein equations and amenable to a Bondi–Sachs type evolution. It establishes a hierarchical propagation of expansion coefficients, preserves the Bondi mass loss law, and identifies constants of motion associated with leading log terms, while clarifying the behavior of Weyl curvature at Scri and the status of Newman–Penrose constants. The results imply that polyhomogeneous Scri can describe radiating spacetimes without contradictions to isolation conditions and that Bondi coordinates remain available, with the asymptotic symmetry group collapsing to the BMS group. Collectively, the work broadens the admissible asymptotic structures in general relativity and clarifies their physical and geometrical implications for gravitational radiation.

Abstract

The structure of polyhomogeneous space-times (i.e., space-times with metrics which admit an expansion in terms of ) constructed by a Bondi--Sachs type method is analysed. The occurrence of some log terms in an asymptotic expansion of the metric is related to the non--vanishing of the Weyl tensor at Scri. Various quantities of interest, including the Bondi mass loss formula, the peeling--off of the Riemann tensor and the Newman--Penrose constants of motion are re-examined in this context.

Paper Structure

This paper contains 4 sections, 6 theorems, 145 equations.

Table of Contents

  1. Introduction
  2. The Bondi--Sachs characteristic initial value problem
  3. Geometric Interpretation
  4. Existence of Bondi coordinates

Key Result

Proposition 2.1

: Given any sequence $\{N_i\}_{i=0}^\infty$, $N_0 = 0$, there exists a sequence $\{{\tilde{N}}_i\}_{i=0}^\infty$ with ${\tilde{N}}_0 = 0$, ${\tilde{N}}_1 = N_1$, ${\tilde{N}}_ i \ge N_i$, such that for all $h_{ab} \in {\cal A}^{\{{\tilde{N}}_i\}} \cap C^0(\bar{\cal N} )$ satisfying $\lim_{r\rightarr

Theorems & Definitions (6)

  • Proposition 2.1
  • Proposition 2.2
  • Proposition 3.1
  • Proposition 3.2
  • Proposition .1
  • Proposition .2