Construction of Cauchy data for the dynamical formation of apparent horizons and the Penrose Inequality

Abstract: Based on scale critical initial data, we construct smooth asymptotically flat Cauchy initial data for the Einstein vacuum system that does not contain Marginally Outer Trapped Surfaces (MOTS) but whose future evolution contains a trapped region, which itself is bounded by an apparent horizon (a smooth hypersurface foliated by MOTS). Although the long time behaviour of these solutions is unknown, a statement of Kerr Stability would yield a dynamical, scale critical, non-spherically symmetric class of vacuum examples for the conjectures of Weak Cosmic Censorship and Final State. Owing to estimates obtained for the ADM mass of the data and the area of the MOTS foliating the apparent horizon, this construction yields a dynamical setting in which to test the conjectured spacetime Penrose Inequality. We show that the inequality holds in an open region in the future of the initial data, which itself can be controlled by the parameters of the initial data.