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6.1 Rates-of-change and Eulerian versus Lagrangian observers

The key geometric difference between generally covariant Newtonian fluids and their general relativistic counterparts is that the former have an a priori notion of time [22Jump To The Next Citation Point23Jump To The Next Citation Point24Jump To The Next Citation Point]. Newtonian fluids also have an a priori notion of space (which can be seen explicitly in the Newtonian covariant derivative introduced earlier; cf. the discussion in [22Jump To The Next Citation Point]). Such a structure has clear advantages for evolution problems, where one needs to be unambiguous about the rate-of-change of the system. But once a problem requires, say, electromagnetism, then the a priori Newtonian time is at odds with the full spacetime covariance of electromagnetic fields. Fortunately, for spacetime covariant theories there is the so-called “3 + 1” formalism (see, for instance, [105Jump To The Next Citation Point]) that allows one to define “rates-of-change” in an unambiguous manner, by introducing a family of spacelike hypersurfaces (the “3”) given as the level surfaces of a spacetime scalar (the “1”).

Something that Newtonian and relativistic fluids have in common is that there are preferred frames for measuring changes – those that are attached to the fluid elements. In the parlance of hydrodynamics, one refers to Lagrangian and Eulerian frames, or observers. A Newtonian Eulerian observer is one who sits at a fixed point in space, and watches fluid elements pass by, all the while taking measurements of their densities, velocities, etc. at the given location. In contrast, a Lagrangian observer rides along with a particular fluid element and records changes of that element as it moves through space and time. A relativistic Lagrangian observer is the same, but the relativistic Eulerian observer is more complicated to define. Smarr and York [105] define such an observer as one who would follow along a worldline that remains everywhere orthogonal to the family of spacelike hypersurfaces.

The existence of a preferred frame for a one fluid system can be used to great advantage. In the next Section 6.2 we will use an “off-the-shelf” analysis that exploits a preferred frame to derive the standard perfect fluid equations. Later, we will use Eulerian and Lagrangian variations to build an action principle for the single and multiple fluid systems. These same variations can also be used as the foundation for a linearized perturbation analysis of neutron stars [63]. As we will see, the use of Lagrangian variations is absolutely essential for establishing instabilities in rotating fluids [44Jump To The Next Citation Point45Jump To The Next Citation Point]. Finally, we point out that multiple fluid systems can have as many notions of Lagrangian observers as there are fluids in the system.


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