Measurement of spin-density matrix elements in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>ρ</mml:mi><mml:mo>(</mml:mo><mml:mn>770</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math> production with a linearly polarized photon beam at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>E</mml:mi><mml:mi>γ</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>8.2</mml:mn><mml:mo>–</mml:mo><mml:mn>8.8</mml:mn><mml:mo> </mml:mo><mml:mi>GeV</mml:mi></mml:mrow></mml:math>

The GlueX experiment at Jefferson Lab studies photoproduction of mesons using linearly polarized $8.5\phantom{\rule{4pt}{0ex}}\text{GeV}$ 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 $\ensuremath{\rho}$(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 $\ensuremath{\rho}(770)$ differ by two units are negligible for $\ensuremath{-}t&lt;0.5\phantom{\rule{4pt}{0ex}}{\text{GeV}}^{2}/{c}^{2}$.