Geometric structure of the generic static traversable wormhole throat

Abstract

Traversable wormholes have traditionally been viewed as intrinsically topological entities in some multiply connected spacetime. Here, we show that topology is too limited a tool to accurately characterize a generic traversable wormhole: in general one needs geometric information to detect the presence of a wormhole, or more precisely to locate the wormhole throat. For an arbitrary static spacetime we shall define the wormhole throat in terms of a two-dimensional constant-time hypersurface of minimal area. (Zero trace for the extrinsic curvature plus a ``flare-out'' condition.) This enables us to severely constrain the geometry of spacetime at the wormhole throat and to derive generalized theorems regarding violations of the energy conditions, theorems that do not involve geodesic averaging but nevertheless apply to situations much more general than the spherically symmetric Morris-Thorne traversable wormhole. (For example, the null energy condition, when suitably weighted and integrated over the wormhole throat, must be violated.) The major technical limitation of the current approach is that we work in a static spacetime; this is already a quite rich and complicated system.