# Polyhomogeneous expansions from time symmetric initial data

## TL;DR

This paper investigates how initial data for Einstein's equations influence the asymptotic behavior of spacetime, revealing conditions under which spacetime peels or exhibits polyhomogeneous, non-peeling behavior.

## Contribution

It establishes a link between initial data properties and the presence of logarithmic terms in asymptotic expansions, providing criteria for non-peeling spacetimes.

## Key findings

- Non-vanishing Bach tensor leads to non-peeling decay rates.
- Necessary conditions for initial data to produce peeling spacetimes.
- Construction of global spacetimes with polyhomogeneous asymptotics.

## 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.