  We further investigate properties of the Gaussian free field (GFF) on the
metric graph associated to a discrete weighted graph (where the edges of the
latter are replaced by continuous line-segments of appropriate length) that has
been introduced by the first author. On such a metric graph, the GFF is a
random continuous function that generalises one-dimensional Brownian bridges so
that one-dimensional techniques can be used.
  In the present paper, we define and study the pseudo-metric defined on the
metric graph (and therefore also on the discrete graph itself), where the
length of a path on the metric graph is defined to be the local time at level
zero accumulated by the Gaussian free field along this path.
  We first derive a pathwise transformation that relates the GFF on the metric
graph with the reflected GFF on the metric graph via the pseudo-distance
defined by the latter. This is a generalisation of Paul L\'evy's result
relating the local time at zero of Brownian motion to the supremum of another
Brownian motion. We also compute explicitly the distribution of certain
functionals of this pseudo-metric and of the GFF. In particular, we point out
that when the boundary consists of just two points, the law of the
pseudo-distance between them depends solely on the resistance of the network
between them.
  We then discuss questions related to the scaling limit of this pseudo-metric
in the two-dimensional case, which should be the conformally invariant way to
measure distances between CLE(4) loops introduced and studied by the second
author with Wu, and by Sheffield, Watson and Wu. Our explicit laws on metric
graphs also lead to new conjectures for related functionals of the continuum
GFF on fairly general Riemann surfaces.
