Free data at spacelike $\mathscr{I}$ and characterization of Kerr-de Sitter in all dimensions

TL;DR

This paper characterizes the free data at spacelike infinity for asymptotically Einstein metrics with a cosmological constant, providing a geometric framework for understanding Kerr-de Sitter spacetimes across all dimensions.

Contribution

It introduces a geometric definition of free data at spacelike infinity, applicable in all dimensions, and characterizes Kerr-de Sitter metrics via this data.

Findings

Rescaled Weyl tensor relates to free data when conformally flat.

Free data can be extracted from Weyl tensor differences in general cases.

Characterization of Kerr-de Sitter metrics through geometric data at infinity.

Abstract

We study the free data in the Fefferman-Graham expansion of asymptotically Einstein metrics with non-zero cosmological constant. We prove that if $\mathscr{I}$ is conformally flat, the rescaled Weyl tensor at $\mathscr{I}$ agrees up to a constant with the free data at $\mathscr{I}$ , namely the traceless part of the $n$-th order coefficient of the expansion. In the non-conformally flat case, the rescaled Weyl tensor is generically divergent at $\mathscr{I}$ but one can still extract the free data in terms of the difference of the Weyl tensors of suitably constructed metrics, in full generality when the spacetime dimension $D$ is even and provided the so-called obstruction tensor at $\mathscr{I}$ is identically zero when $D$ is odd. These results provide a geometric definition of the data, particularly relevant for the asymptotic Cauchy problem of even dimensional Einstein metrics with positive $\Lambda$ and also for the odd dimensional analytic case irrespectively of the sign of $\Lambda$. We establish a Killing initial data equation at spacelike $\mathscr{I}$ in all dimension for analytic data. These results are used to find a geometric characterization of the Kerr-de Sitter metrics in all dimensions in terms of its geometric data at null infinity.