  We construct traversable wormholes by starting with simple four-dimensional
classical solutions respecting the null energy condition and containing a pair
of oppositely charged black holes connected by a non-traversable wormhole. We
then consider the perturbative back-reaction of bulk quantum fields in
Hartle-Hawking states. Our geometries have zero cosmological constant and are
asymptotically flat except for a cosmic string stretching to infinity that is
used to hold the black holes apart. Another cosmic string wraps the
non-contractible cycle through the wormhole, and its quantum fluctuations
provide the negative energy needed for traversability. Our setting is closely
related to the non-perturbative construction of Maldacena, Milekhin, and Popov
(MMP), but the analysis is complementary. In particular, we consider cases
where back-reaction slows, but fails to halt, the collapse of the wormhole
interior, so that the wormhole is traversable only at sufficiently early times.
For non-extremal backgrounds, we find the integrated null energy along the
horizon of the classical background to be exponentially small, and thus
traversability to be exponentially fragile. Nevertheless, if there are no
larger perturbations, and for appropriately timed signals, a wormhole with
mouths separated by a distance $d$ becomes traversable with a minimum transit
time $t_{\text{min transit}} = d + \text{logs}$. Thus $\frac{t_{\text{min
transit}}}{d}$ is smaller than for the eternally traversable MMP wormholes by
more than a factor of 2, and approaches the value that, at least in higher
dimensions, would be the theoretical minimum. For contrast we also briefly
consider a `cosmological wormhole' solution where the back-reaction has the
opposite sign, so that negative energy from quantum fields makes the wormhole
harder to traverse.
