There are many different ways of constructing general relativistic fluid equations. Our purpose here is not to review all possible methods, but rather to focus on a couple: (i) an “off-the-shelve” consistency analysis for the simplest fluid a la Eckart , to establish some key ideas, and then (ii) a more powerful method based on an action principle that varies fluid element world lines. The ideas behind this variational approach can be traced back to Taub  (see also ). Our description of the method relies heavily on the work of Brandon Carter, his students, and collaborators [19, 36, 37, 28, 29, 67, 90, 91]. We prefer this approach as it utilizes as much as possible the tools of the trade of relativistic fields, i.e. no special tricks or devices will be required (unlike even in the case of our “off-the-shelve” approach). One’s footing is then always made sure by well-grounded, action-based derivations. As Carter has always made clear: When there are multiple fluids, of both the charged and uncharged variety, it is essential to distinguish the fluid momenta from the velocities, in particular in order to make the geometrical and physical content of the equations transparent. A well-posed action is, of course, perfect for systematically constructing the momenta.
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