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Relativistic finite temperature multifluid hydrodynamics in a neutron star from a variational principle

We develop a relativistic multifluid dynamics appropriate for describing neutron star cores at finite temperatures based on Carter's convective variational procedure. The model includes seven fluids, accounting for both normal and superfluid/superconducting neutrons and protons, leptons (electrons and muons) and entropy. The formulation is compared to the non-variational relativistic multifluid hydrodynamics of Gusakov and collaborators and shown to be equivalent. Vortex lines and flux tubes, mutual friction, vortex pinning, heat conduction and viscosity are incorporated into the model in steps after the basic hydrodynamics is described. The multifluid system is then considered at the mesoscopic scale where the currents around individual vortex lines and flux tubes are important, and this mesoscopic theory is averaged to determine the detailed vortex line/flux tube contributions to the macroscopic "effective" theory. This matching procedure is partially successful, though obtaining full agreement between the averaged mesoscopic and macroscopic effective theory requires discarding subdominant terms. The matching procedure allow us to interpret the magnetic $H$-field inside a neutron star in a way that is consistent with condensed matter physics literature, and to clarify the difference between this interpretation and that in previous astrophysical works.