  The GlueX experiment at Jefferson Lab aims to study the light meson spectrum
with an emphasis on the search for hybrid mesons. To this end, a
linearly-polarized $9\,$GeV photon beam impinges on a hydrogen target contained
within a detector with near-complete neutral and charged particle coverage. In
2018, the experiment completed its first phase of data taking in its design
configuration and the quantity of analyzed data already exceed that of previous
experiments for polarized photoproduction in this energy regime by orders of
magnitude. Polarization observables such as spin-density matrix elements
provide valuable input for the theoretical description of the production
mechanism, which will be essential for the interpretation of possible exotic
meson signals. We present results for the photoproduction of vector mesons,
focusing on the unprecedented statistical precision of the spin-density matrix
elements for the $\rho(770)$ meson.
