Dynamics of Apparent Horizon and a Null Comparison Principle

TL;DR

This paper studies the global behavior of apparent horizons in gravitational collapse, introducing a null comparison principle and proving their smooth, asymptotically null nature with implications for physical laws.

Contribution

It introduces the null comparison principle and provides new proofs for physical laws along the apparent horizon, advancing understanding of horizon dynamics in gravitational collapse.

Findings

Apparent horizon is smooth and asymptotically null.

Null comparison principle guarantees horizon's piecewise spacelike or null nature.

Validated physical laws along the apparent horizon.

Abstract

This paper investigates the global dynamics of the apparent horizon. We present an approach to establish its existence and its long-term behaviors. Our apparent horizon is constructed by solving the marginally outer trapped surface (MOTS) along each incoming null hypersurface. Based on the nonlinear hyperbolic estimates established in [24] by Klainerman-Szeftel under polarized axial symmetry, we prove that the corresponding apparent horizon is smooth, asymptotically null and converging to the event horizon eventually. To further address the local achronality of the apparent horizon, a new concept, called the null comparison principle, is introduced in this paper. For three typical scenarios of gravitational collapse, our null comparison principle is tested and verified, which guarantees that the apparent horizon must be piecewise spacelike or piecewise null. In addition, we also validate and provide new proofs for several physical laws along the apparent horizon.