Kerr black hole formation and a dynamical approach to the Penrose inequality

Abstract

In this expository note, we present our recent result on the dynamical formation of Kerr black holes for the 3 + 1 dimensional Einstein vacuum equations and a newly dynamical approach to the Penrose inequality, based on [6]. Combining the scale-critical short-pulse mechanism with nonlinear Kerr stability, we construct Kerr-black-hole-formation spacetimes with a complete apparent horizon. We prove that this apparent horizon is smooth away from its initial spacetime point, asymptotically null, and converges to the event horizon. In particular, we show that our apparent horizon is achronal, demonstrating a monotonic formula (area increasing law) along it. Building on our analysis, without time symmetric assumption, we then put forward a new mathematical framework and prove both the dynamical Penrose inequality and the spacetime Penrose inequality for the perturbations of full subextremal Kerr spacetimes (provided that corresponding Kerr stability result holds), as well as a refinement with angular momentum under axial symmetry. Collectively, this extends Christodoulou’s celebrated trapped surface formation theorem to a black hole formation result without assuming any symmetry.