The Brown-York mass and the Thorne hoop conjecture

TL;DR

This paper investigates the Thorne hoop conjecture in spherical symmetry, demonstrating that the Brown-York mass provides a precise criterion for gravitational collapse, unlike other mass definitions.

Contribution

It establishes a specific inequality involving the Brown-York mass and circumference that predicts collapse, highlighting its unique suitability over Liu-Yau and Wang-Yau masses.

Findings

Brown-York mass yields a clear collapse criterion in spherical symmetry.

Liu-Yau and Wang-Yau masses do not satisfy the same inequality.

The result challenges the extension of the hoop conjecture beyond spherical symmetry.

Abstract

The Thorne hoop conjecture is an attempt to make precise the notion that gravitational collapse occurs if enough energy is compressed into a small enough volume, with the `size' being defined by the circumference. We can make a precise statement of this form, in spherical symmetry, using the Brown-York mass as our measure of the energy. Consider a spherical 2-surface in a spherically symmetric spacetime. If the Brown-York mass, $M_{BY}$, and the circumference, $C$, satisfy $C < 2\pi M_{BY}$, then the system must either have emerged from a white hole or will collapse into a black hole. We show that no equivalent result can hold true using either the Liu-Yau mass, $M_{LY}$ or the Wang-Yau mass, $M_{WY}$. This forms a major obstacle to any attempt to establish a Thorne-type hoop theorem in the general case based on either the Liu-Yau or the Wang-Yau mass.