Anisotropic Dynamical Horizons Arising in Gravitational Collapse

Abstract

In this paper, we study the evolution of the 3 + 1 dimensional Einstein vacuum equations with anisotropic characteristic initial data. In a framework set up by Christodoulou and under a general anisotropic condition introduced by Klainerman, Luk and Rodnianski, we prove that in the process of gravitational collapse, a smooth and spacelike apparent horizon (dynamical horizon) emerges from general (both isotropic and anisotropic) initial data and the corresponding black hole entropy is always strictly increasing toward the future. This extends the theory of black hole thermodynamics along the apparent horizon to anisotropic scenarios. For the proof, we combine scalecritical hyperbolic method and non-perturbative elliptic techniques. Furthermore, we construct explicit single and multi-valley anisotropic apparent horizons. They are the first mathematical examples of the anisotropic marginally outer trapped surface (MOTS) and the anisotropic apparent horizon formed in dynamics.