Stability of the expanding region of Kerr-de Sitter spacetimes and smoothness at the conformal boundary

TL;DR

This paper provides a new proof of the nonlinear stability of the expanding region of Kerr-de Sitter spacetimes, demonstrating smoothness at the conformal boundary using a modified harmonic gauge and analyzing the asymptotic behavior.

Contribution

It introduces a new gauge for analyzing Kerr-de Sitter stability, simplifying the asymptotic analysis and establishing smoothness at the conformal boundary with mild singularities.

Findings

Proves nonlinear stability of Kerr-de Sitter spacetime's expanding region.

Shows the conformally rescaled metric is smooth at the future boundary.

Identifies mild singularities in the Fefferman-Graham expansion at timelike infinity.

Abstract

We give a new proof of the recent result by Fournodavlos-Schlue on the nonlinear stability of the expanding region of Kerr-de Sitter spacetimes as solutions of the Einstein vacuum equations with positive cosmological constant. Our gauge is a modification of a generalized harmonic gauge introduced by Ringstr\"om in which the asymptotic analysis becomes particularly simple. Due to the hyperbolic character of our gauge, our stability result is local near points on the conformal boundary. We show furthermore that, in yet another gauge, the conformally rescaled metric is smooth down to the future conformal boundary, with the coefficients of its Fefferman-Graham type asymptotic expansion featuring a mild singularity at future timelike infinity of the black hole.