Characterization of gravitational radiation at infinity with a cosmological constant

TL;DR

The paper develops a covariant, gauge-invariant framework to characterize gravitational radiation at conformal infinity $\mathscr{J}$ for any cosmological constant $\Lambda$. Building on the rescaled Weyl tensor and the asymptotic Bel-Robinson tensor, it defines the asymptotic supermomentum and decomposes it into a density and a flux (the super-Poynting vector) to measure tidal-energy transport. For $\Lambda=0$ the criterion reduces to the well-known News condition, while for $\Lambda>0$ radiation absence is equivalent to the commutativity of the Fefferman–Graham electric part $D_{\alpha\beta}$ and Cotton–York part $C_{\alpha\beta}$, and for $\Lambda<0$ it requires a family of observers tangent to $\mathscr{J}$ to have vanishing transversal flux or, equivalently, a functional dependence between $D_{\alpha\beta}$ and $C_{\alpha\beta}$. The results provide a fully covariant, gauge-independent definition of radiation at infinity that agrees with exact solutions and integrates Friedrich’s initial-boundary data framework for $\Lambda>0$.

Abstract

The existence or absence of gravitational radiation escaping from the spacetime at $\mathscr{J}$ is characterized in the presence of a cosmological constant $Λ$ of any sign. To that end, the properties of the asymptotic super-momentum are used. When $Λ=0$, the characterization is equivalent to that based on the News tensor. For $Λ\neq 0$, it provides the first reliable definition of existence of radiation at $\mathscr{J}$, and it gives fully satisfactory results in known exact solutions.

Theorems & Definitions (4)

• Theorem 1: No radiation on $\Delta\subset\mathscr{J}$, case $\Lambda =0$
• Theorem 2: No radiation on $\Delta\subset\mathscr{J}$, case $\Lambda >0$
• Theorem 3: No radiation on $\Delta\subset \mathscr{J}$, case $\Lambda<0$
• Remark 1