Polyhomogeneous expansions from time symmetric initial data

Abstract

We make use of Friedrich's construction of the cylinder at spatial infinity to relate the logarithmic terms appearing in asymptotic expansions of components of the Weyl tensor to the freely specifiable parts of time symmetric initial data sets for the Einstein field equations. Our analysis is based on the assumption that a particular type of formal expansions near the cylinder at spatial infinity corresponds to the leading terms of actual solutions to the Einstein field equations. In particular, we show that if the Bach tensor of the initial conformal metric does not vanish at the point at infinity then the most singular component of the Weyl tensor decays near null infinity as $O({\tilde{r}}^{-3}\ln \tilde{r})$ so that spacetime will not peel. We also provide necessary conditions on the initial data which should lead to a peeling spacetime. Finally, we show how to construct global spacetimes which are candidates for non-peeling polyhomogeneous) asymptotics.

29 Pages, 2 Figures

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Cited by in corpus (10)

• The Case Against Smooth Null Infinity I: Heuristics and Counter-Examples
• A comparison of Ashtekar's and Friedrich's formalisms of spatial infinity
• Peeling in Generalized Harmonic Gauge
• High Order Asymptotic Expansions of a Good-Bad-Ugly Wave Equation
• The good-bad-ugly system near spatial infinity on flat spacetime
• Staticity and regularity for zero rest-mass fields near spatial infinity on flat spacetime
• Asymptotics of spin-0 fields and conserved charges on n-dimensional Minkowski spaces
• Hyperboloidal approach for linear and non-linear wave equations in FLRW spacetimes
• Bondi Mass, Memory Effect and Balance Law of Polyhomogeneous Spacetime
• The Null-Timelike Boundary Problems of Linear Wave Equations in Asymptotically Anti-de Sitter Space