Article,
More about Birkhoff's Invariant and Thorne's Hoop Conjecture for Horizons
(Apr 22, 2011)
Abstract
A recent precise formulation of the hoop conjecture in four spacetime
dimensions is that the Birkhoff invariant \$\beta\$ (the least maximal length of
any sweepout or foliation by circles) of an apparent horizon of energy \$E\$ and
area \$A\$ should satisfy \$4 E\$. This conjecture together with the
Cosmic Censorship or Isoperimetric inequality implies that the length \$\ell\$ of
the shortest non-trivial closed geodesic satisfies \$\ell^2 A\$. 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 \$\beta\$ as the least maximal area of all
sweepouts of the horizon by two-dimensional tori, and find in all cases
examined that \$ \beta(g) 16 \pi3 E\$, which we conjecture holds
quiet generally for apparent horizons. In even spacetime dimensions \$D=2N+2\$,
we find that for sweepouts by the product \$S^1 S^D-4\$, \$\beta\$ is
bounded from above by a certain dimension-dependent multiple of the energy \$E\$.
We also find that \$\ell^D-2\$ is bounded from above by a certain
dimension-dependent multiple of the horizon area \$A\$. Finally, we show that
\$\ell^D-3\$ is bounded from above by a certain dimension-dependent multiple of
the energy, for all Kerr-AdS black holes.
