Trapped surface formation via radial boosts

Abstract

Work of Christodoulou, Klainerman, Rodnianski, Luk, An, and others has provided a number of results on the dynamical formation of trapped surfaces in vacuum solutions to the Einstein field equations. Since the stability of Minkowski spacetime as proved by Christodoulou and Klainerman implies that `small' initial data to the Einstein vacuum equations must give rise to a solution with no singularities, and hence no trapped surfaces, it has been assumed that dynamical formation of trapped surfaces requires a (hard) large-data existence result for the nonlinear hyperbolic Einstein field equations. In this paper we show, to the contrary, that dynamical trapped surface formation results qualitatively similar to those of Christodoulou and Klainerman-Rodnianski can be obtained from (classical) local, small-data existence results via a scaling we term a radial boost. In the process we also fully elucidate the short-pulse ansatz as a geometric optics ansatz by showing that, in this scaled picture, the so-called incoming shear satisfies, at highest order, a linear wave equation.