  We study a group field theory (GFT) for quantum gravity coupled to four
massless scalar fields, using these matter fields to define a (relational)
coordinate system. We exploit symmetries of the GFT action, in particular under
shifts in the values of the scalar fields, to derive a set of classically
conserved currents, and show that the same conservation laws hold exactly at
the quantum level regardless of the choice of state. We propose a natural
interpretation of the conserved currents which implies that the matter fields
always satisfy the Klein--Gordon equation in GFT. We then observe that in our
matter reference frame, the same conserved currents can be used to extract all
components of an effective GFT spacetime metric. Finally, we apply this
construction to the simple example of a spatially flat homogeneous and
isotropic universe, where we derive an effective Friedmann equation directly
from this metric. The Friedmann equation displays a bounce and a late-time
limit equivalent to general relativity with a single scalar field. Our proposal
goes substantially beyond the GFT literature in which only specific geometric
quantities such as the total volume or volume perturbations could be defined,
opening up the possibility to study more general geometries as emerging from
GFT.
