The non-linear stability of the Schwarzschild family of black holes

Abstract

We prove the non-linear asymptotic stability of the Schwarzschild family as solutions to the Einstein vacuum equations in the exterior of the black hole region: general vacuum initial data, with no symmetry assumed, sufficiently close to Schwarzschild data evolve to a vacuum spacetime which (i) possesses a complete future null infinity ${\mathcal{I}}^{+}$ (whose past $J^{-}({\mathcal{I}}^{+})$ is moreover bounded by a regular future complete event horizon ${\mathcal{H}}^{+}$), (ii) remains close to Schwarzschild in its exterior, and (iii) asymptotes back to a member of the Schwarzschild family as an appropriate notion of time goes to infinity, provided that the data are themselves constrained to lie on a teleologically constructed codimension-$3$ "submanifold" of moduli space. This is the full nonlinear asymptotic stability of Schwarzschild since solutions not arising from data lying on this submanifold should by dimensional considerations approach a Kerr spacetime with rotation parameter $a\ne 0$, i.e. such solutions cannot satisfy (iii). The proof employs teleologically normalised double null gauges, is expressed entirely in physical space and makes essential use of the analysis in our previous study of the linear stability of the Kerr family around Schwarzschild [DHR], as well as techniques developed over the years to control the non-linearities of the Einstein equations. The present work, however, is entirely self-contained. In view of the recent [DHR19, TdCSR20] our approach can be applied to the full non-linear asymptotic stability of the subextremal Kerr family.

arXiv comment513 pages, 15 figures

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