Tolman wormholes violate the strong energy condition

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Abstract
For an arbitrary Tolman wormhole, unconstrained by symmetry, we shall define the bounce in terms of a 3-dimensional edgeless achronal spacelike hypersurface of minimal volume (zero trace for the extrinsic curvature plus a “flare-out” condition). This enables us to severely constrain the geometry of spacetime at and near the bounce and to derive general theorems regarding violations of the energy conditions—theorems that do not involve geodesic averaging but nevertheless apply to situations much more general than the highly symmetric FRW-based subclass of Tolman wormholes. [For example, even under the mildest of hypotheses, the strong energy condition (SEC) must be violated.] Alternatively, one can dispense with the minimal volume condition and define a generic bounce entirely in terms of the motion of test particles (future-pointing timelike geodesics), by looking at the expansion of their timelike geodesic congruences. One re-confirms that the SEC must be violated at or near the bounce. In contrast, it is easy to arrange for all the other standard energy conditions to be satisfied.
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