[Paper Review] Birkhoff's invariant and Thorne's Hoop Conjecture

G. W. Gibbons|ArXiv.org|Mar 9, 2009

Geometric and Algebraic TopologyMathematics25 references19 citations

TL;DR

This paper proposes a sharp formulation of Thorne's Hoop Conjecture using Birkhoff's invariant $\beta$, showing that for an outermost marginally trapped surface, $\beta \leq 4\pi M_{\text{ADM}}$. The conjecture is proven for collapsing null shells and supported by exact solutions like Kerr-Newman black holes, linking geometric invariants to mass bounds via Penrose and isoperimetric inequalities.

ABSTRACT

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

Motivation & Objective

• To refine Thorne's vague Hoop Conjecture into a precise, mathematically rigorous inequality involving geometric invariants.
• To establish a connection between the ADM mass and the Birkhoff invariant $\beta$ of an outermost apparent horizon.
• To provide evidence for the conjecture through exact solutions like Kerr-Newman black holes and geometric inequalities.
• To explore implications for the length of the shortest closed geodesic and injectivity/convexity radii on trapped surfaces.

Proposed method

• Define Birkhoff's invariant $\beta(g)$ as the infimum over all functions $f$ on a 2-sphere of the maximum length of level sets $f^{-1}(c)$.
• Use Birkhoff's Theorem to show $\beta(g)$ equals the length of a closed geodesic on the horizon surface.
• Apply the Penrose inequality $\sqrt{\pi A} \leq 4\pi M_{\text{ADM}}$ and Pu’s theorem to derive bounds on geodesic lengths.
• Use Berger’s isoperimetric inequality $\sqrt{\pi A} \geq 2i(g)$ and the relation $l(g) \geq 2c(g)$ to link $\beta$ to injectivity and convexity radii.
• Analyze the metric of Kerr-Newman and related black hole horizons to compute $\beta$ explicitly and verify the inequality.
• Generalize the framework to higher-dimensional horizons with topology $S^n$, using foliations by $S^{n-1}$ and volume minimization.

Experimental results

Research questions

• RQ1 Can Thorne’s Hoop Conjecture be formulated as a precise geometric inequality involving the Birkhoff invariant?
• RQ2 Does $\beta \leq 4\pi M_{\text{ADM}}$ hold for all outermost marginally trapped surfaces in spacetimes satisfying the dominant energy condition?
• RQ3 What is the relationship between the shortest closed geodesic length $l(g)$, the injectivity radius $i(g)$, and the convexity radius $c(g)$ on black hole horizons?
• RQ4 How do geometric inequalities like Pu’s and Berger’s relate to the Penrose inequality and the proposed hoop conjecture?
• RQ5 Can the conjecture be extended to higher-dimensional black hole horizons with non-spherical topology?

Key findings

• The conjecture $\beta \leq 4\pi M_{\text{ADM}}$ is proven for collapsing null shells, establishing a necessary condition for horizon formation.
• For Kerr-Newman black holes, $\beta = l_e = 2\pi(2M - Q^2/r_+) \leq 4\pi M$, confirming the conjecture in this exact solution.
• The inequality $l(g) \leq 4\pi M_{\text{ADM}}$ follows directly from $\beta \leq 4\pi M_{\text{ADM}}$, where $l(g)$ is the shortest closed geodesic length.
• Using Pu’s theorem and the Penrose inequality, it is shown that $2c(g) \leq 4\pi M_{\text{ADM}}$ holds for horizons admitting an antipodal isometry.
• Berger’s isoperimetric inequality implies $2i(g) \geq 4c(g)$, and combined with the Penrose inequality, yields $4\pi M_{\text{ADM}} \geq 2i(g) \geq 4c(g)$.
• The framework generalizes to higher dimensions, with $\beta$ defined as the infimum of the maximal $(n-1)$-volume of $S^{n-1}$-foliations on $n$-dimensional horizons.