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Static Traversable Wormholes and the Violation of Energy Conditions

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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.