  The GlueX experiment at Jefferson Lab studies photoproduction of mesons using
linearly polarized $8.5\,\text{GeV}$ photons impinging on a hydrogen target
which is contained within a detector with near-complete coverage for charged
and neutral particles. We present measurements of spin-density matrix elements
for the photoproduction of the vector meson $\rho$(770). The statistical
precision achieved exceeds that of previous experiments for polarized
photoproduction in this energy range by orders of magnitude. We confirm a high
degree of $s$-channel helicity conservation at small squared four-momentum
transfer $t$ and are able to extract the $t$-dependence of natural and
unnatural-parity exchange contributions to the production process in detail. We
confirm the dominance of natural-parity exchange over the full $t$ range. We
also find that helicity amplitudes in which the helicity of the incident photon
and the photoproduced $\rho(770)$ differ by two units are negligible for
$-t<0.5\,\text{GeV}^{2}/c^{2}$.
