A recent precise formulation of the hoop conjecture in four spacetime
dimensions is that the Birkhoff invariant β (the least maximal length of
any sweepout or foliation by circles) of an apparent horizon of energy E and
area A should satisfy β≤4πE. This conjecture together with the
Cosmic Censorship or Isoperimetric inequality implies that the length ℓ of
the shortest non-trivial closed geodesic satisfies ℓ2≤πA. We have
tested these conjectures on the horizons of all four-charged rotating black
hole solutions of ungauged supergravity theories and find that they always
hold. They continue to hold in the the presence of a negative cosmological
constant, and for multi-charged rotating solutions in gauged supergravity.
Surprisingly, they also hold for the Ernst-Wild static black holes immersed in
a magnetic field, which are asymptotic to the Melvin solution. In five
spacetime dimensions we define β as the least maximal area of all
sweepouts of the horizon by two-dimensional tori, and find in all cases
examined that β(g)≤316πE, which we conjecture holds
quiet generally for apparent horizons. In even spacetime dimensions D=2N+2,
we find that for sweepouts by the product S1×SD−4, β is
bounded from above by a certain dimension-dependent multiple of the energy E.
We also find that ℓD−2 is bounded from above by a certain
dimension-dependent multiple of the horizon area A. Finally, we show that
ℓD−3 is bounded from above by a certain dimension-dependent multiple of
the energy, for all Kerr-AdS black holes.Comment: 25 page