On the non-linear stability of the Cosmological region of the Schwarzschild-de Sitter spacetime

TL;DR

This paper proves the non-linear stability of the Cosmological region in Schwarzschild-de Sitter spacetime near the conformal boundary using conformal Einstein equations and Gaussian gauge techniques.

Contribution

It introduces a novel perturbative approach leveraging conformal geodesics to establish stability near the conformal boundary of Schwarzschild-de Sitter spacetime.

Findings

Small perturbations lead to solutions regular at the conformal boundary.

Stability results are valid away from the intersection points of the horizon and boundary.

The method provides a foundation for future stability analysis on cosmological horizons.

Abstract

The non-linear stability of the sub-extremal Schwarzschild-de Sitter spacetime in the stationary region near the conformal boundary is analysed using a technique based on the extended conformal Einstein field equations and a conformal Gaussian gauge. This strategy relies on the observation that the Cosmological stationary region of this exact solution can be covered by a non-intersecting congruence of conformal geodesics. Thus, the future domain of dependence of suitable spacelike hypersurfaces in the Cosmological region of the spacetime can be expressed in terms of a conformal Gaussian gauge. A perturbative argument then allows us to prove existence and stability results close to the conformal boundary and away from the asymptotic points where the Cosmological horizon intersects the conformal boundary. In particular, we show that small enough perturbations of initial data for the sub-extremal Schwarzschild-de Sitter spacetime give rise to a solution to the Einstein field equations which is regular at the conformal boundary. The analysis in this article can be regarded as a first step towards a stability argument for perturbation data on the Cosmological horizons.