In this section the equations of motion and the stress-energy-momentum tensor for a one-component,
general relativistic fluid are obtained from an action principle. Specifically a so-called “pull-back” approach
(see, for instance, [36, 37, 34]) is used to construct a Lagrangian displacement of the number density
four-current
, whose magnitude
is the particle number density. This will form the basis for the
variations of the fundamental fluid variables in the action principle.
As there is only one species of particle considered here, is conserved, meaning that once a number
of particles
is assigned to a particular fluid element, then that number is the same at each point of the fluid
element’s worldline. This would correspond to attaching a given number of particles (i.e.
,
, etc.)
to each of the worldlines in Figure 7
. Mathematically, one can write this as a standard particle-flux conservation
equation4:
The main reason for introducing the dual is that it is straightforward to construct a particle number density three-form that is automatically closed, since the conservation of the particle number density current should not – speaking from a strict field theory point of view – be a part of the equations of motion, but rather should be automatically satisfied when evaluated on a solution of the “true” equations.
This can be made to happen by introducing a three-dimensional “matter” space – the left-hand part of
Figure 7 – which can be labelled by coordinates
, where
. For each
time slice in spacetime, there will be the same configuration in the matter space; that is, as
time goes forward, the fluid particle positions in the matter space remain fixed even as the
worldlines form their weavings in spacetime. In this sense we are “pushing forward” from the
matter space to spacetime (cf. the previous discussion of the Lie derivative). The three-form
can be pulled back to its three-dimensional matter space by using the mappings
. This
construction then guarantees a three-form that is automatically closed on spacetime, namely
Because the matter space indices are three-dimensional and the closure condition involves four spacetime
indices, and also the are scalars on spacetime (and thus two covariant differentiations commute), the
construction does indeed produce a closed three-form:
Let us introduce the Lagrangian displacement on spacetime for the particles, to be denoted . This is
related to the variation
via another push-forward from the matter space into spacetime,
The Lagrangian variation resulting from the Lagrangian displacement is given by
from which it follows that which is entirely consistent with the pull-back construction. We also find that These formulae and their Newtonian analogues have been adroitly used by Friedman and Schutz in their establishment of the so-called Chandrasekhar–Friedman–Schutz (CFS) instability [30, 44 At first glance, there appears to be a glaring inconsistency between the pull-back construction and the
Lagrangian variation, since the latter seems to have four independent components, but the former
clearly has three. In fact, there is a gauge freedom in the Lagrangian variation that can be
used to reduce the number of independent components. Take Equation (132) and substitute
With a general variation of the conserved four-current in hand, we can now use an action principle to
derive the equations of motion and the stress-energy-momentum tensor. The central quantity in the analysis
is the so-called “master” function , which is a function of the scalar
. For this single fluid
system, it is such that
corresponds to the local thermodynamic energy density. In the action
principle, the master function is the Lagrangian density for the fluid, i.e. for a spacetime region
the
fluid action is
We should point out that our consideration of a master function of the form is based, in
part, on the assumptions that the matter is locally isotropic, meaning that there are no locally preferred
directions (such as in a neutron star crust) or another covector
to form the additional scalar
(as would be the case with coupling to electromagnetism, say). A term that could
be added is of the form
for an arbitrary scalar field
. Unlike the previous two
possible additional terms, it would not affect the equations of motion, since
by
construction, and an integration by parts generates a boundary term. Our point of view is
that the master function is fixed by the local microphysics of the matter; (cf. the discussion in
Section 5.2).
An unconstrained variation of is with respect to
and the metric
, and allows the four
components of
to be varied independently. It takes the form
If the variation of the four-current was left unconstrained, the equations of motion for the fluid deduced
from the above variation of would require, incorrectly, that the momentum covector
should vanish
in all cases. This reflects the fact that the variation of the conserved four-current must be constrained,
meaning that not all components of
can be treated as independent. In terms of the constrained
Lagrangian displacement of Equation (132
), a first-order variation of the general relativity plus fluid
Lagrangian yields
At this point we can return to the view that is the fundamental field for the fluid. The principle of
least action implies that the coefficients of
and
and the boundary terms must vanish. Thus, the
equations of motion consist of the original conservation condition from Equation (121
), the Euler equation
Let us now recall the discussion of the point particle. There we pointed out that only the fully
conservative form of Newton’s Second Law follows from an action, i.e. external or dissipative forces are
excluded. However, we argued that a well-established form of Newton’s Second Law is known that allows for
external and/or dissipative forces (cf. Equation (120)). There is thus much purpose in using the particular
symbol
in Equation (152
). We may take the
to be the relativistic analogue of the
left-hand-side of Equation (120
) in every sense. In particular, when dissipation and/or external
“forces” act in a general relativistic setting, they are incorporated via the right-hand-side of
Equation (152
).
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