### 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 [22, 23, 24]. 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 [22]). 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, [105]) 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 [44, 45]. Finally, we point out that
multiple fluid systems can have as many notions of Lagrangian observers as there are fluids in the
system.