Relativistic dynamics of superfluid-superconducting mixtures in the presence of topological defects and an electromagnetic field with application to neutron stars

The relativistic dynamic equations are derived for a superfluid-superconducting mixture coupled to an electromagnetic field. For definiteness, and bearing in mind possible applications of our results to neutron stars, it is assumed that the mixture is composed of superfluid neutrons, superconducting protons, and normal electrons. We analyze the proton superconductivity of both types I and II and allow for the possible presence of neutron and proton vortices (or magnetic domains in the case of type-I proton superconductivity). The derived equations neglect all dissipative effects except for the mutual friction dissipation and are valid for arbitrary temperatures (i.e., they do not imply that all nucleons are paired), which is especially important for magnetar conditions. It is demonstrated that these general equations can be substantially simplified for typical neutron stars, for which a kind of magneto-hydrodynamic approximation is justified. Our results are compared to the nonrelativistic formulations existing in the literature, and a number of discrepancies are found. In particular, it is shown that, generally, the electric displacement D does not coincide with the electric field E, contrary to what is stated in previous works. The relativistic framework developed here is easily extendable to account for more sophisticated microphysics models, and it provides the necessary basis for realistic modeling of neutron stars.