Symmetrization of Black Hole Horizons and the Positive Mass Theorem For Creased Spin Initial Data

Abstract

We examine aspects of hypersurfaces inside initial data for the Einstein equations. We prove that 3-dimensional initial data with an apparent horizon boundary can be perturbed to data with a $H=0$, $k=0$ boundary while preserving the dominant energy condition. This yields a reduction of the spacetime Penrose conjecture to the case of $H=0$, $k=0$ boundaries. We also give an upper bound of the spacetime Bartnik mass of apparent horizon Bartnik data satisfying a stability condition in terms of the area.Secondly, we define a new type of singularity, called a ``DEC-crease'', across a hypersurface for initial data, modeled on two spacelike slices of a spacetime meeting at a hyperbolic dihedral angle. We prove that a positive mass theorem holds for spin initial data sets containing such singularities, in any dimension. Our proof is based on Dirac-Witten spinors. At the singularity, a transmission-type boundary condition for spinors is defined and we show its ellipticity.