Traversable wormholes with arbitrarily small energy condition violations

Traversable wormholes necessarily require violations of the averaged null energy condition, this being the definition of "exotic matter." However, the theorems which guarantee the energy condition violation are remarkably silent when it comes to making quantitative statements regarding the "total amount" of energy condition violating matter in the spacetime. We develop a suitable measure for quantifying this notion and demonstrate the existence of spacetime geometries containing traversable wormholes that are supported by arbitrarily small quantities of exotic matter.