  We consider the metric growth in Liouville quantum gravity (LQG) for $\gamma
\in (0,2)$. We show that a process associated with the trace of the free field
on the boundary of a filled LQG ball is stationary, for every $\gamma \in
(0,2)$. The infinitesimal version of this stationarity combined with an
explicit expression of the generator of the evolution of the trace field
$(h_t)$ provides a formal invariance equation that a measure on trace fields
must satisfy. When considering a modified process corresponding to an evolution
of LQG surfaces, we prove that the invariance equation is satisfied by an
explicit $\sigma$-finite measure on trace fields. This explicit measure on
trace fields only corresponds to the pure gravity case. On the way to prove
this invariance, we retrieve the specificity of both $\gamma = \sqrt{8/3}$ and
of the LQG dimension $d_{\gamma} = 4$. In this case, we derive an explicit
expression of the (nonsymmetric) Dirichlet form associated with the process
$(h_t)$ and construct dynamics associated with its symmetric part.
