On wormholes with arbitrarily small quantities of exotic matter

Recently several models of traversable wormholes have been proposed which require only arbitrarily small amounts of negative energy to hold them open against self-collapse. If the exotic matter is assumed to be provided by quantum fields, then quantum inequalities can be used to place constraints on the negative energy densities required. In this paper, we introduce an alternative method for obtaining constraints on wormhole geometries, using a recently derived quantum inequality bound on the null-contracted stress-energy averaged over a timelike worldline. The bound allows us to perform a simplified analysis of general wormhole models, not just those with small quantities of exotic matter. We then use it to study, in particular, the models of Visser, Kar, and Dadhich (VKD) and the models of Kuhfittig. The VKD models are constrained to be either submicroscopic or to have a large discrepancy between throat size and curvature radius. A recent model of Kuhfittig is shown to be non-traversable. This is due to the fact that the throat of his wormhole flares outward so slowly that light rays and particles, starting from outside the throat, require an infinite lapse of affine parameter to reach the throat.