Local metrics of the Gaussian free field

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