Perturbations of the asymptotic region of the Schwarzschild-de Sitter spacetime

TL;DR

This paper investigates the conformal structure of Schwarzschild-de Sitter spacetime, analyzing its asymptotic behavior and how small perturbations influence its stability and structure at the conformal boundary.

Contribution

It provides a detailed analysis of the conformal structure using extended conformal Einstein equations and demonstrates stability of the asymptotic structure under small perturbations.

Findings

Small perturbations preserve the asymptotic structure.

The conformal boundary behavior is well-understood.

Perturbed solutions exist and are stable in the future.

Abstract

The conformal structure of the Schwarzschild-de Sitter spacetime is analysed using the extended conformal Einstein field equations. To this end, initial data for an asymptotic initial value problem for the Schwarzschild-de Sitter spacetime is obtained. This initial data allows to understand the singular behaviour of the conformal structure at the asymptotic points where the horizons of the Schwarzschild-de Sitter spacetime meet the conformal boundary. Using the insights gained from the analysis of the Schwarzschild-de Sitter spacetime in a conformal Gaussian gauge, we consider nonlinear perturbations close to the Schwarzschild- de Sitter spacetime in the asymptotic region. We show that small enough perturbations of asymptotic initial data for the Schwarzschild de-Sitter spacetime give rise to a solution to the Einstein field equations which exists to the future and has an asymptotic structure similar to that of the Schwarzschild-de Sitter spacetime.