A variation of the chemical potential covectors that leaves the metric fixed takes the form
The sound speed analysis is local, but its results are seen globally in the analysis of modes of oscillation of a fluid body. For a neutron star, the full spectrum of modes is quite impressive (see McDermott et al. ): polar (or spheroidal) f-, p-, and g-modes, and the axial (or toroidal) r-modes. Epstein  was the first to suggest that there should be even more modes in superfluid neutron stars because the superfluidity allows the neutrons to move independently of the protons. Mendell  developed this idea further by using an analogy with coupled pendulums. He argued that the new modes should feature a counter-motion between the neutrons and protons, i.e. as the neutrons move out radially, say, the protons will move in. This is in contrast to ordinary fluid motion that would have the neutrons and protons move in more or less “lock-step”. Analytical and numerical studies [70, 74, 38, 5] have confirmed this basic picture and the new modes of oscillation are known as superfluid modes.
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