Geometric constraints on nonsingular, momentarily static, axisymmetric systems in general relativity

This paper attempts to examine the relationship between system size and gravitational collapse for the case of axial symmetry. The approach here is to construct noncollapsing systems, with momentarily static matter interiors and static vacuum exteriors, and to find limitations on the validity of the construction. Specifically, the exteriors are static, axisymmetric, asymptotically flat, vacuum geometries, described by Weyl solutions of the Einstein field equations. These solutions have singular sources (naked singularities, except for the Schwarzschild solution); here, regions of the Weyl solutions containing the singularities are replaced by momentarily static material bodies. These are described by axisymmetric solutions of Brill's time-symmetric initial-value equation with non-negative energy density, joining smoothly to the Weyl geometries at the bodies' boundaries. The consistency requirements of such a construction limit the choice of surfaces in the exterior geometry suitable as matter/vacuum boundaries; general constraints on the boundary location and geometry are derived here. For the explicit examples of the $\Gamma$ metric and the Bach-Weyl ring metric as exteriors, these constraints forbid the boundary surface to be arbitrarily near the Weyl singularity. The "hoop conjecture" demands, roughly, that the largest circumference of the boundary surface of such a noncollapsing system always exceed a limit of the order of the system's mass. The specific examples studied here are all consistent with the hoop conjecture, but they show that the boundary constraints derived in this paper are not in general related to boundary surface size and thus that these constraints do not embody the hoop conjecture.