On the non-linear stability of the cosmological region of the Schwarzschild-de Sitter spacetime. (English) Zbl 1526.83009
Summary: 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 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.
MSC:
| 83C57 | Black holes |
| 30F45 | Conformal metrics (hyperbolic, Poincaré, distance functions) |
| 28C20 | Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) |
| 70S15 | Yang-Mills and other gauge theories in mechanics of particles and systems |
| 53C22 | Geodesics in global differential geometry |
| 53C07 | Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) |
| 35B35 | Stability in context of PDEs |
| 14J70 | Hypersurfaces and algebraic geometry |
| 65M75 | Probabilistic methods, particle methods, etc. for initial value and initial-boundary value problems involving PDEs |
| 83C25 | Approximation procedures, weak fields in general relativity and gravitational theory |
Keywords:
conformal Einstein field equations; conformal Gaussian gauge; conformal geodesics; Weyl connection; Schwarzschild-de Sitter spacetime; non-linear stability; partial differential equationsReferences:
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