Conformal characterization of the Fefferman-Graham ambient metric

TL;DR

This paper explores the asymptotic structure of the Fefferman-Graham ambient metric, establishing conformal completion properties, the role of the obstruction tensor in even dimensions, and characterizing the metric through conformally covariant conditions.

Contribution

It introduces a conformal characterization of the ambient metric, relaxing previous conditions and linking asymptotic expansions to conformal invariants.

Findings

Every straight ambient metric admits a conformal completion with null infinity.

The asymptotic expansion relates to the homothetic horizon expansion.

The Fefferman-Graham obstruction tensor appears naturally at infinity.

Abstract

In this paper, we study the asymptotic structure of the Fefferman-Graham ambient metric. We prove that every straight ambient metric admits a conformal completion with a well-defined null infinity, and that the asymptotic expansion of the metric at infinity can be related to that at the homothetic horizon. Furthermore, in even dimensions, we show that the Fefferman-Graham obstruction tensor naturally arises in the geometry at infinity. By identifying the fundamental properties that this particular conformal extension exhibits, and analyzing their sufficiency, we arrive at the main result of the paper, namely the identification of a set of conformally covariant conditions that completely characterize the ambient metric from a conformal perspective. In particular, our result relaxes the requirement of the homothety one-form being exact.