Linear response hydrodynamics of a relativistic dissipative fluid with spin

We formulate a Lagrangian hydrodynamics including shear and bulk viscosity in the presence of spin density, and investigate it using the linear response functional formalism. The result is a careful accounting of all sound and vortex interactions close to local equilibrium. In particular, we demonstrate that the mixing of sound waves and vortices via polarization, first observed in the ideal fluid limit, extends to the shear mode once dissipative effects are included. This provides a realization within Lagrangian hydrodynamics of the symmetric shear polarization contribution recently advocated from transport and Zubarev approaches as well as phenomenological considerations. Once causal relaxational dynamics is included, this effect, seemingly puzzling because it results in a nondissipative coupling between a transient mode to an equilibrium quantity, can be understood as a competition between the Israel-Stewart and the polarization relaxation timescale, and a breakdown of local Markovianity. We close by discussing phenomenological implications of these results.