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

ABSTRACT: In this talk, I will discuss a strategy to prove the non-linear stability of the subextremal Schwarzschild-de Sitter spacetime in the stationary region near the conformal boundary. This technique is based on the extended conformal Einstein field equations and a conformal Gaussian gauge.

The key observation is 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 proving 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.