Conformal characterization of the Fefferman-Graham ambient metric
Marc Mars, Gabriel Sánchez-Pérez
TL;DR
The work investigates the Fefferman-Graham ambient metric, proving that straight ambient metrics admit a conformal completion with a well-defined null infinity and that in even dimensions the obstruction tensor manifests at infinity. It develops a conformal characterization using three covariant conditions tied to a bifurcate conformal Killing horizon and a conformal factor, and shows that these conditions uniquely determine the ambient metric (relaxing the requirement that the homothety one-form be exact). By constructing geodesic Killing gauge and adapted Rácz–Wald coordinates, the authors connect the FG expansion at the horizon to that at null infinity, revealing that the ambient-free data aligns with gravitational radiation data. The results pave the way for a broader geometric understanding of asymptotically flat spacetimes and the role of conformal data in radiation, with planned future work to develop a complete existence-uniqueness theory from null infinity data and to further explore conformal freedom.
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.