For an arbitrary Tolman wormhole, unconstrained by symmetry, we shall define
the bounce in terms of a three-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.Comment: 8 pages, ReV-TeX 3.
