Casimir Energy of a Long Wormhole Throat

TL;DR

This paper calculates the Casimir energy in a long wormhole throat and assesses its potential to stabilize the wormhole, finding that while it can slow collapse, it cannot fully stabilize the specific wormhole model.

Contribution

It provides a detailed calculation of the Casimir energy-momentum tensor in a long wormhole throat using mode sums and renormalization, exploring its stabilizing effects.

Findings

Casimir energy is negative for most observers, aiding stabilization.

The energy density is zero for null rays parallel to the throat, preventing full stabilization.

The wormhole can collapse very slowly, allowing safe traversal of its central region.

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.