Localized Penrose inequality for the Liu-Yau mass in spherical symmetry

TL;DR

This paper establishes a localized version of the Penrose inequality for Liu-Yau quasi-local mass within spherically symmetric spacetimes, providing a refined understanding of mass concentration and gravitational collapse.

Contribution

It formulates and proves a localized Penrose inequality for Liu-Yau mass under spherical symmetry, advancing the theoretical framework of quasi-local mass and gravitational collapse.

Findings

Proved a localized Penrose inequality for Liu-Yau mass.

Established the inequality specifically in spherical symmetry.

Enhanced understanding of mass concentration effects in gravitational collapse.

Abstract

For an asymptotically flat initial data, the Penrose inequality gives a lower bound of the Arnowitt-Deser-Misner total mass of a spacetime in terms of the area of certain surfaces representing black holes. This is a deep and beautiful refinement of the famous positive mass theorem and it plays an important role in the study of gravitational collapse. Gravitational collapse can also happen if sufficient mass is concentrated into a finite region. This motivates us to seek a localized version of the Penrose inequality. In this Letter, we successfully make a precise statement of this form for the Liu-Yau quasi-local mass in spherical symmetry.