Classification of Kerr-de Sitter-like spacetimes with conformally flat $\mathscr{I}$ in all dimensions

TL;DR

This paper classifies Kerr-de Sitter-like spacetimes with conformally flat infinity across all dimensions, establishing a correspondence with Kerr-Schild-de Sitter spacetimes and explicitly constructing these metrics from initial data.

Contribution

It extends the classification of Kerr-de Sitter-like spacetimes to all dimensions and provides explicit constructions from initial data at null infinity.

Findings

Kerr-Schild-de Sitter spacetimes correspond to Kerr-de Sitter-like class with conformally flat infinity

Explicit construction of all metrics in this class as limits or extensions of Kerr-de Sitter

Analysis of limits based on asymptotic data rather than direct metric properties

Abstract

Using asymptotic characterization results of spacetimes at conformal infinity, we prove that Kerr-Schild-de Sitter spacetimes are in one-to-one correspondence with spacetimes in the Kerr-de Sitter-like class with conformally flat $\mathscr{I}$. Kerr-Schild-de Sitter are spacetimes of Kerr-Schild form with de Sitter background that solve the $(\Lambda>0)$-vacuum Einstein equations and admit a smooth conformal compactification sharing $\mathscr{I}$ with the background metric. Kerr-de Sitter-like metrics with conformally flat $\mathscr{I}$ are a generalization of the Kerr-de Sitter metrics, defined originally in four spacetime dimensions and extended here to all dimensions in terms of their initial data at null infinity. We explicitly construct all metrics in this class as limits or analytic extensions of Kerr-de Sitter. The structure of limits is inferred from corresponding limits of the asymptotic data, which appear to be hard to guess from the spacetime metrics.