1 | In this article we use “superfluid” to refer to any system which has the ability to flow without friction. In this sense, superfluids and superconductors are viewed in the same way. When we wish to distinguish charge carrying fluids, we will call them superconductors. | |

2 | There are three space and one time dimensions that form a type of topological space known as a manifold [114]. This means that local, suitably small patches of a curved spacetime are practically the same as patches of flat, Minkowski spacetime. Moreover, where two patches overlap, the identification of points in one patch with those in the other is smooth. | |

3 | We say “combined” here because the First Law is a statement about heat and work, and says nothing about the entropy, which enters through the Second Law. Heat is not strictly equal to for all processes; they are equal for quasistatic processes, but not for free expansion of a gas into vacuum [100]. | |

4 | It is worth pointing out that we are restricting the problem somewhat by imposing particle conservation already from the outset. As we will see later, one can make good progress on less constrained problems, e.g. related to dissipation, using a slightly extended variational approach (inspired by the point particle example of Section 7). However, we feel that it is useful to first understand the details of the simpler, fully conservative, situation. | |

5 | It is important to note the difference between the vorticity formed from the momentum and the corresponding quantity in terms of the velocity. They differ because of the entrainment, and one can show that while the former is conserved along the flow, the latter is not. To avoid confusion we refer to as the “twist”. This makes some sense because when we use it in Equation (288) we have not yet associated the four-velocity with the fluid flow. |

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