Casimir energy of a long wormhole throat

We calculate the Casimir energy-momentum tensor induced in a scalar field by a macroscopic ultrastatic spherically symmetric long-throated traversable wormhole, and examine whether this exotic matter is sufficient to stabilize the wormhole itself. The Casimir energy-momentum tensor is obtained (within the R x S-2 throat) by a mode sum approach, using a sharp energy cutoff and the Abel-Plana formula; Lorentz invariance is then restored by use of a Pauli-Villars regulator. The massless conformally coupled case is found to have a logarithmic divergence (which we renormalize) and a conformal anomaly, the thermodynamic relevance of which is discussed. Provided the throat radius is above some fixed length, the renormalized Casimir energy density is seen to be negative by all timelike observers, and almost all null rays; furthermore, it has sufficient magnitude to stabilize a long-throated wormhole far larger than the Planck scale, at least in principle. Unfortunately, the renormalized Casimir energy density is zero for null rays directed exactly parallel to the throat, and this shortfall prevents us from stabilizing the ultrastatic spherically symmetric wormhole considered here. Nonetheless, the negative Casimir energy does allow the wormhole to collapse extremely slowly, its lifetime growing without bound as the throat length is increased. We find that the throat closes slowly enough that its central region can be safely traversed by a pulse of light.