Brown-York Mass and the Thorne Hoop Conjecture

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 $M_{\mathrm{BY}}$ and the circumference $C$ satisfy $C<2\pi M_{\mathrm{BY}}$, 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 $M_{\mathrm{LY}}$ or the Wang-Yau mass $M_{\mathrm{WY}}$. 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.