Birkhoff's invariant and Thorne's Hoop Conjecture

TL;DR

This paper proposes a precise version of Thorne's hoop conjecture linking Birkhoff's invariant of black hole horizons to their mass, proves it for specific cases, and explores related geometric inequalities.

Contribution

It introduces a sharp form of Thorne's hoop conjecture involving Birkhoff's invariant and provides proofs and evidence for its validity in certain scenarios.

Findings

Proves the conjecture for collapsing null shells.

Provides evidence from exact rotating black hole solutions.

Derives geometric inequalities related to black hole horizons.

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, $\sqrt{\pi A} \ge {2\over\sqrt\pi} i$, where $i$ is the injectivity radius, imply $ 4c \le 2 i \le 4 \pi M_{ADM}$, where $c$ is the convexity radius.