The spatial stress is a two-index, symmetric tensor, and the only objects that can be used to carry the
indices are the four-velocity and the metric
. Furthermore, because the spatial stress must also be
symmetric, the only possibility is a linear combination of
and
. Given that
, we
find
Given a relation , there are four independent fluid variables. Because of this the equations of
motion are often understood to be
, which follows immediately from the Einstein equations
and the fact that
. To simplify matters, we take as equation of state a relation of the form
where
is the particle number density. The chemical potential
is then given by
Let us now get rid of the free index of in two ways: first, by contracting it with
and
second, by projecting it with
(letting
). Recalling the fact that
we have the
identity
However, we should not be too quick to think that this is the only way to understand .
There is an alternative form that makes the perfect fluid have much in common with vacuum
electromagnetism. If we define
To demonstrate the role of as the vorticity, consider a small region of the fluid where the
time direction
, in local Minkowski coordinates, is adjusted to be the same as that of the
fluid four-velocity so that
. Equation (105
) and the antisymmetry then
imply that
can only have purely spatial components. Because the rank of
is two,
there are two “nulling” vectors, meaning their contraction with either index of
yields
zero (a condition which is true also for vacuum electromagnetism). We have arranged already
that
be one such vector. By a suitable rotation of the coordinate system the other one
can be taken as
, thus implying that the only non-zero component of
is
. “MTW” [80
] points out that such a two-form can be pictured geometrically as a
collection of oriented worldtubes, whose walls lie in the
and
planes. Any
contraction of a vector with a two-form that does not yield zero implies that the vector pierces the
walls of the worldtubes. But when the contraction is zero, as in Equation (105
), the vector
does not pierce the walls. This is illustrated in Figure 6
, where the red circles indicate the
orientation of each world-tube. The individual fluid element four-velocities lie in the centers of the
world-tubes. Finally, consider the closed contour in Figure 6
. If that contour is attached to
fluid-element worldlines, then the number of worldtubes contained within the contour will not
change because the worldlines cannot pierce the walls of the worldtubes. This is essentially the
Kelvin–Helmholtz theorem on conservation of vorticity. From this we learn that the Euler equation is an
integrability condition which ensures that the vorticity two-surfaces mesh together to fill out
spacetime.
As we have just seen, the form Equation (105) of the equations of motion can be used to discuss the
conservation of vorticity in an elegant way. It can also be used as the basis for a derivation of other known
theorems in fluid mechanics. To illustrate this, let us derive a generalized form of Bernoulli’s theorem. Let
us assume that the flow is invariant with respect to transport by some vector field
. That is, we have
Given that we have just inferred the equations of motion from the identity that , we now
emphatically state that while the equations are correct the reasoning is severely limited. In fact, from a field
theory point of view it is completely wrong! The proper way to think about the identity is that the
equations of motion are satisfied first, which then guarantees that
. There is no clearer way to
understand this than to study the multi-fluid case: Then the vanishing of the covariant divergence
represents only four equations, whereas the multi-fluid problem clearly requires more information (as there
are more velocities that must be determined). We have reached the end of the road as far as the
“off-the-shelf” strategy is concerned, and now move on to an action-based derivation of the fluid equations
of motion.
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