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 (9)

• A rotating black ring in five dimensions
• Entropy of Non-Extreme Charged Rotating Black Holes in String Theory
• Charged Rotating Black Holes in Four-Dimensional Gauged and Ungauged Supergravities
• Positivity of Quasilocal Mass
• Quasilocal mass in general relativity
• Isometric embeddings into the Minkowski space and new quasi-local mass
• Collapsing Shells and the Isoperimetric Inequality for Black Holes
• A Reformulation of the Hoop Conjecture
• Width and finite extinction time of Ricci flow

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• More about Birkhoff's Invariant and Thorne's Hoop Conjecture for Horizons
• Black Hole Formation at the Correspondence Point
• Future Boundaries and the Black Hole Information Paradox
• The Hoop Conjecture for Black Rings
• Investigations on hoop conjecture for horizonless spherical charged stars