The null energy condition in dynamic wormholes

We extend previous proofs that violations of the null energy condition (NEC) are a generic and universal feature of traversable wormholes to completely non-symmetric time-dependent wormholes. We show that the analysis can be phrased purely in terms of local geometry at and near the wormhole throat, and 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. Introduction: The fact that traversable wormholes are accompanied by unavoidable violations of the null energy condition (NEC) is perhaps one of the most central features of wormhole physics [1–3]. The original proof of the necessity for NEC violations at or near the throat of a traversable wormhole was limited to the static spherically symmetric Morris-Thorne wormhole [1], though it was rapidly realized that NEC violations typically occurred in at least some explicit examples of static non-symmetric [4] and spherically-symmetric timedependent [5] wormholes. A considerably more general proof of the necessity of NEC violations was provided by the topological censorship theorem of Friedman, Schleich, and Witt [6] though this theorem requires many technical assumptions concerning asymptotic flatness and causality conditions that limit its applicability. We have recently adopted a different strategy and sought to develop new general theorems concerning energy condition violations at and near the throat of traversable wormholes by focusing attention only on the local behaviour of the geometry at and near the throat, and discarding all assumptions about symmetry, asymptotic behaviour, and causal properties. (This strategy was inspired by the fact that there are many classes of objects that we would meaningfully wish to call wormholes that possess either trivial topology [3] or do not necessarily possess asymptotically flat regions [7].) Such a strategy first requires that we develop robust and general definitions of what it means to be a wormhole throat. Recently, we have succeeded in developing such definitions and theorems, first for the static but completely non-symmetric case [8,9], and secondly (as reported in this Letter) for the completely general time-dependent non-symmetric wormhole. Our central results are these: (1) A time-dependent traversable wormhole will have two throats, one throat being associated with each direction of travel through the wormhole. These two throats coalesce into one in the case of a static traversable wormhole but it is important to keep them distinguished in the time-dependent case. (2) For each one of these throats the NEC is either violated or on the verge of being violated on the throat itself. (This is easy to prove.) (3) For each one of these throats there will be an open region (topologically open in one of the null hypersurfaces passing through the throat) whose closure intersects the throat such that the NEC is violated throughout the open region. (Proving this requires a little more analysis.) (4) For each one of these throats there will be an open interval such that the transverse averaged NEC (transverse averaged over the throat after it is pushed out in the appropriate null direction) is violated throughout the open interval. (Proving this requires a little more analysis.) The proofs are sketched below, and are based on an extension of an idea due to Page [10] whereby wormhole throats are viewed as anti-trapped surfaces in spacetime. Further technical details may be found in [11], which also extends the present analysis to spacetimes including nonzero torsion. We note that there are a number of published papers which claim to construct wormholes without NEC violations. These claims are most often based on semantic confusion, though sometimes actual computational errors have crept in, and we shall discuss the situation more fully in our conclusions. Basic Definitions: Null geodesic congruences. Consider a compact two-dimensional spacelike hypersurface (denoted Σ) embedded in (3 + 1)-dimensional spacetime. (More precisely, a compact two dimensional orientable manifold that is embedded in spacetime in a two-sided and spacelike manner.) At each point on the hypersurface there are two null vectors orthogonal to the hypersurface, and these two null vectors can be extended to two null geodesic congruences (vector fields l±) that are well defined on an open neighborhood of the hypersurface [12,13]. Coordinates on the hypersurface will be denoted x, while the null geodesic congruences can be used to set up null coordinates u± on the orthogonal hyperplanes. The ± labels are often denoted “ingoing” and “outgoing” though these labels are prejudicial and in the case of non-trivial topology are actually meaningless, the key point is that there are two possible null directions to travel along. For each one of these null geodesic congruences one can define expansion, shear, and vorticity, in a manner completely analogous to standard textbook discus-