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.

It is by now apparent that topology is too crude a tool to accurately characterize a generic traversable wormhole. In two earlier papers we developed a complete characterization of generic but static traversable wormholes, and in the present paper extend the discussion to arbitrary time-dependent (dynamical) wormholes. A local definition of a wormhole throat, free from assumptions about asymptotic flatness, symmetries, future and past null infinities, embedding diagrams, topology, and even time dependence is developed that accurately captures the essence of what a wormhole throat is, and where it is located. Adapting and extending a suggestion due to Page, we define a wormhole throat to be a marginally anti-trapped surface, that is, a closed two-dimensional spatial hypersurface such that one of the two future-directed null geodesic congruences orthogonal to it is just beginning to diverge. Typically a dynamic wormhole will possess two such throats, corresponding to the two orthogonal null geodesic congruences, and these two throats will not coincide (though they do coalesce into a single throat in the static limit). The divergence property of the null geodesics at the marginally anti-trapped surface generalizes the “flare-out” condition for an arbitrary wormhole. We derive theorems regarding violations of the null energy condition (NEC) at and near these throats and find that, even for wormholes with arbitrary time dependence, the violation of the NEC is a generic property of wormhole throats. We also discuss wormhole throats in the presence of fully antisymmetric torsion and find that the energy condition violations cannot be dumped into the torsion degrees of freedom. Finally by means of a concrete example we demonstrate that even temporary suspension of energy-condition violations is incompatible with the flare-out property of dynamic throats.

We extend previous proofs that violations of the null energy condition are a generic and universal feature of traversable wormholes to completely nonsymmetric time-dependent wormholes. We show that the analysis can be phrased purely in terms of local geometry at and near the wormhole throat, and we do not have to make any technical assumptions about asymptotic flatness or other global properties. A key aspect of the analysis is the demonstration that time-dependent wormholes have two throats, one for each direction through the wormhole, and that the two throats coalesce only for the case of a static wormhole.

We present the first results of a self-consistent solution of the semiclassical Einstein field equations corresponding to a Lorentzian wormhole coupled to a quantum scalar field. The specific solution presented here represents a wormhole connecting two asymptotically spatially flat regions. In general, the diameter of the wormhole throat, in units of the Planck length, can be arbitrarily large, depending on the values of the scalar coupling $\xi$ and the boundary values for the shape and redshift functions. In all cases we have considered, there is a fine structure in the form of Planck-scale oscillations or ripples superimposed on the solutions.