Quantum backreaction and stability of topological wormholes

We investigate the quantum stability of a timelike topological wormhole with a simple geometry $M_2 \times S^2$, supported classically by anisotropic fluid. We compute the one-loop quantum backreaction generated by the vacuum fluctuations of a minimally coupled, massive scalar field propagating on the wormhole background. Using dimensional regularization we renormalize the one-loop energy-momentum tensor and identify the necessary gravitational counterterms. We then solve the semiclassical Einstein equations to linear order in $\hbar$ for both time-dependent and static metric {\it Ans\"{a}tze}. Depending on the choice of finite counterterms, the quantum effects can induce either negative or positive angular pressure, which will tend to destabilize or stablize the wormhole, respectively. We also show that a classically traversable wormhole will remain traversable when the quantum backreaction is taken into account. It would be of interest to investigate whether these conclusions remain true when the equations of semiclassical gravity are self-consistently solved.