Tolman wormholes violate the strong energy condition

TL;DR

This paper proves that Tolman wormholes, regardless of symmetry, necessarily violate the strong energy condition at the bounce, using geometric and geodesic-based definitions of the bounce.

Contribution

It introduces general theorems showing that the SEC must be violated at the bounce of Tolman wormholes without relying on symmetry assumptions.

Findings

SEC violation is unavoidable at the bounce.

Theorems apply to general, non-symmetric Tolman wormholes.

Other energy conditions can still be satisfied.

Abstract

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.