On Kerr black hole formation with complete apparent horizon and a new approach toward Penrose inequality

TL;DR

This paper demonstrates the formation of Kerr black holes from specific initial data in 3+1 dimensions, introduces new methods for analyzing apparent horizons, and proves both the dynamical and spacetime Penrose inequalities without symmetry assumptions.

Contribution

It develops a novel mathematical framework combining hyperbolic and elliptic techniques to analyze Kerr black hole formation and proves key inequalities in a general setting.

Findings

Existence of dynamical Kerr black hole solutions from scale-critical initial data.

Precise characterization of apparent horizon formation and evolution.

Proof of the dynamical and spacetime Penrose inequalities without symmetry assumptions.

Abstract

Arising from admissible extended scale-critical short-pulse initial data, we show that 3+1 dimensional Einstein vacuum equations admit dynamical Kerr black hole formation solutions. Our hyperbolic arguments combine the scale-critical gravitational-collapse result by An--Luk with the recent breakthrough by Klainerman--Szeftel on proving nonlinear Kerr stability with small angular momentum, which requires us to perform various specific coordinate changes and frame transformations. Furthermore, allowing large spacetime angular momentum, with new elliptic arguments and precise leading order calculations, we also solve the apparent horizon in Kerr black hole formation spacetimes (including Klainerman--Szeftel's Kerr stability spacetimes) and conduct an exploration, detailing the emergence, evolution, asymptotics and final state of the apparent horizon. 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 in our black-hole formation spacetimes and in the perturbative regime of subextremal Kerr black holes. Collectively, without assuming any symmetry, we extend Christodoulou's celebrated trapped surface formation theorem to a black hole formation result.