The relativistic fluid is a highly successful model used to describe the dynamics of many-particle, relativistic systems. It takes as input basic physics from microscopic scales and yields as output predictions of bulk, macroscopic motion. By inverting the process, an understanding of bulk features can lead to insight into physics on the microscopic scale. Relativistic fluids have been used to model systems as “small” as heavy ions in collisions, and as large as the Universe itself, with “intermediate” sized objects like neutron stars being considered along the way. The purpose of this review is to discuss the mathematical and theoretical physics underpinnings of the relativistic (multiple) fluid model. We focus on the variational principle approach championed by Brandon Carter and his collaborators, in which a crucial element is to distinguish the momenta that are conjugate to the particle number density currents. This approach differs from the “standard” text-book derivation of the equations of motion from the divergence of the stress-energy tensor in that one explicitly obtains the relativistic Euler equation as an “integrability” condition on the relativistic vorticity. We discuss the conservation laws and the equations of motion in detail, and provide a number of (in our opinion) interesting and relevant applications of the general theory.

We discuss vortex-mediated mutual friction in the two-fluid model for superfluid neutron star cores. Our discussion is based on the general formalism developed by Carter and collaborators, which makes due distinction between transport velocity and momentum for each fluid. This is essential for an implementation of the so-called entrainment effect, whereby the flow of one fluid imparts momentum in the other and vice versa. The mutual friction follows by balancing the Magnus effect that acts on the quantized neutron vortices with resistivity due to the scattering of electrons off of the magnetic field with which each vortex core is endowed. We derive the form of the macroscopic mutual friction force which is relevant for a model based on smooth-averaging over a collection of vortices. We discuss the coefficients that enter the expression for this force, and the time-scale on which the two interpenetrating fluids in a neutron star core are coupled. This discussion confirms that our new formulation accords well with previous work in this area.

We discuss the nature of the various modes of pulsation of superfluid neutron stars using comparatively simple Newtonian models. The matter in these stars is described in terms of a two-fluid model, where one fluid is the neutron superfluid, which is believed to exist in the core and inner crust of mature neutron stars, and the other fluid represents a conglomerate of all other constituents (crust nuclei, protons, electrons, etc.). In our model, we incorporate the non-dissipative interaction due to the so-called entrainment effect, whereby the momentum of one constituent (e.g. the neutrons) carries along part of the mass of the other constituent. We show that there is no independent set of propagating g-modes in a superfluid neutron star core. We also approach the toroidal pulsation modes of a slowly rotating superfluid star. We show that the superfluid equations support a new class of r-modes. Finally, the role of the entrainment effect on the superfluid mode frequencies is shown explicitly via solutions to local dispersion relations.Comment: 14 pages, MNRAS style, no figure

We develop a general formalism to treat, in general relativity, the linear oscillations of a twofluid star about static (non-rotating) configurations. Such a formalism is intended for neutron stars, whose matter content can be described, as a first approximation, by a two-fluid model: one fluid is the neutron superfluid, which is believed to exist in the core and inner crust of mature neutron stars; the other fluid is a conglomerate of all other constituents (crust nuclei, protons, electrons, etc...). We obtain a system of equations which govern the perturbations both of the metric and of the matter variables, whatever the equation of state for the two fluids. As a first application, we consider the simplified case of two non-interacting fluids, each with a polytropic equation of state. We compute numerically the quasi-normal modes (i.e. oscillations with purely outgoing gravitational radiation) of the corresponding system. When the adiabatic indices of the two fluids are different, we observe a splitting for each frequency of the analogous single fluid spectrum. The analysis also substantiates the claim that w-modes are largely due to spacetime oscillations.

Abstract. We develop a formalism that can be used to model slowly rotating superfluid Newtonian neutron stars. A simple two-fluid model is used to describe the matter, where one fluid consists of the superfluid neutrons that are believed to exist in the inner crust and core of mature neutron stars, while the other fluid is a charge neutral conglomerate of the remaining constituents (crust nuclei, core superconducting protons, electrons, etc.). We include the entrainment effect, which is a non-dissipative interaction between the two fluids whereby a momentum induced in one of the fluids will cause part of the mass of the other fluid to be carried along. The equations that describe rotational equilibria (i.e. axisymmetric and stationary configurations) are approximated using the slow-rotation approximation; an expansion in terms of the rotation rates of the two fluids where only terms up to second-order are kept. Our formalism allows the neutrons to rotate at a rate different from that of the charged constituents. For a particular equation of state that is quadratic in the two mass-densities and relative velocities of the fluids, we find an analytic solution to the slow-rotation equations. This result provides an elegant generalisation to the two-fluid problem of the standard expressions for the one-fluid polytrope E ∝ ρ 2 . The model equation of state includes entrainment and is general enough to allow for realistic values for, say, mass and radius of the star. It also includes a mixed term in the mass densities that can be related to "symmetry energy" terms that appear in more realistic equations of state. We use the analytic solution to explore how relative rotation between the two fluids, the "symmetry energy" term, and entrainment affect the neutron star's local distribution of particles, as well as global quantities as the Kepler limit, ellipticity, and moments of inertia.