  We introduce the concept of a local metric of the Gaussian free field (GFF)
$h$, which is a random metric coupled with $h$ in such a way that it depends
locally on $h$ in a certain sense. This definition is a metric analog of the
concept of a local set for $h$. We establish general criteria for two local
metrics of the same GFF $h$ to be bi-Lipschitz equivalent to each other and for
a local metric to be a.s. determined by $h$. Our results are used in subsequent
works which prove the existence, uniqueness, and basic properties of the
$\gamma$-Liouville quantum gravity (LQG) metric for all $\gamma \in (0,2)$, but
no knowledge of LQG is needed to understand this paper.
