Casimir Energy of a Long Wormhole Throat

Abstract

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 stabilise the wormhole itself. The Casimir energy-momentum tensor is obtained (within the $\mathbb{R}\times S_2$ throat) by a mode sum approach, using a sharp energy cut-off 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 renormalise) and a conformal anomaly, the thermodynamic relevance of which is discussed. Provided the throat radius is above some fixed length, the renormalised Casimir energy-density is seen to be negative by all timelike observers, and almost all null rays; furthermore, it has sufficient magnitude to stabilise a long-throated wormhole far larger than the Planck scale, at least in principle. Unfortunately, the renormalised Casimir energy-density is zero for null rays directed exactly parallel to the throat, and this shortfall prevents us from stabilising 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.