Static Traversable Wormholes and the Violation of Energy Conditions

Paul F. Rebillot, III

Raychaudhuri's equation, which relates the expansion, shear, and rate of expansion of geodesic congruences, is used to analyze for a specific case of a Morris-Thorne wormhole. This wormhole geometry is found to violate the strong and null energy conditions. With explicit values of the expansion and shear of a set of timelike geodesic congruences, we go on to examine the physical meaning of each of these quantities. The expansion describes how an infinitesimal volume element changes with proper time along a geodesic. The volume element decreases, obtains a minimum at the throat, and then increases as it comes out of the wormhole. The eigenvalues of the shear tensor describe how a sphere will deform as it enters the wormhole. As a sphere enters the wormhole, it is shown that a sphere will flatten along its two angular axes, and will elongate along an axis that is a linear combination of the length and time coordinates.