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The Hoop Conjecture for Black Rings

A precise formulation of the hoop conjecture for four-dimensional spacetimes proposes that the Birkhoff invariant βfor an apparent horizon in a spacetime with mass M should satisfy β\le 4πM. The invariant βis the least maximal length of any sweepout of the 2-sphere apparent horizon by circles. An analogous conjecture in five spacetime dimensions was recently formulated, asserting that the Birkhoff invariant βfor S^1\times S^1 sweepouts of the apparent horizon should satisfy β\le (16/3)πM. Although this hoop inequality was formulated for conventional five-dimensional black holes with 3-sphere horizons, we show here that it is also obeyed by a wide variety of black rings, where the horizon instead has S^2\times S^1 topology.