The hoop conjecture for black rings

A precise formulation of the hoop conjecture for four-dimensional spacetimes proposes that the Birkhoff invariant β for an apparent horizon in a spacetime with mass M should satisfy β⩽4πM. The invariant β is 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 β for S1×S1 sweepouts of the apparent horizon should satisfy β⩽163πM. 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 S2×S1 topology.