More about Birkhoff's Invariant and Thorne's Hoop Conjecture for Horizons

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 $\beta \le 4\pi E$. This conjecture together with the Cosmic Censorship or Isoperimetric inequality implies that the length $\ell$ of the shortest non-trivial closed geodesic satisfies ${\ell }^2\le \pi A$. 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 $\beta (g)\le \frac{16\pi }{3}E$, which we conjecture holds quiet generally for apparent horizons. In even spacetime dimensions $D=2N+2$, we find that for sweepouts by the product $S^1\times S^{D-4}$, $\beta$ 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.

25 pages

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• Analytical studies on the hoop conjecture in charged curved spacetimes
• Future Boundaries and the Black Hole Information Paradox
• Investigations on hoop conjecture for horizonless spherical charged stars
• The Hoop Conjecture for Black Rings
• Apparent Horizons with Nontrivial Topology and the Hyperhoop Conjecture in Six-Dimensional Space-Times
• Testing generalized spacetimes for black holes using the Hod function representation of the hoop conjecture
• A unified hoop conjecture for black holes and horizonless compact stars