Birkhoff's invariant and Thorne's Hoop Conjecture

Abstract

I propose a sharp form of Thorne's hoop conjecture which relates Birkhoff's invariant $\beta$ for an outermost apparent horizon to its $ADM$ mass, $\beta \le 4\pi M_{ADM}$. I prove the conjecture in the case of collapsing null shells and provide further evidence from exact rotating black hole solutions. Since $\beta$ is bounded below by the length $l$ of the shortest non-trivial geodesic lying in the apparent horizon, the conjecture implies $l\le 4\pi M_{ADM}$. The Penrose conjecture, $\sqrt{\pi A}\le 4\pi M_{ADM}$, and Pu's theorem imply this latter consequence for horizons admitting an antipodal isometry. Quite generally, Penrose's inequality and Berger's isembolic inequality, $\sqrt{\pi A}\ge \frac{2}{\sqrt{\pi }}i$, where $i$ is the injectivity radius, imply $4c\le 2i\le 4\pi M_{ADM}$, where $c$ is the convexity radius.

References in corpus (3)

• Quasilocal mass in general relativity
• A Reformulation of the Hoop Conjecture
• Width and finite extinction time of Ricci flow

Cited by in corpus (1)

• Global embedding of the Kerr black hole event horizon into hyperbolic 3-space