## 8 The “Pull-back” Formalism for a Single Fluid

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, [363734]) 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 equation:

For reasons that will become clear shortly, it is useful to rewrite this conservation law in what may (if taken out of context) seem a rather obscure way. We introduce the dual to , i.e.
such that
This shows that acts as a volume measure which, for example, allows us to “count” the number of fluid elements. In Figure 6 we have seen that a two-form gives worldtubes. A three-form is the next higher-ranked object and it can be thought of, in an analogous way, as leading to boxes [80]. The key step here is to realize that the conservation rule is equivalent to having the three-form be closed. In differential geometry this means that
This can be shown to be equivalent to Equation (121) by contracting with .

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

where is completely antisymmetric in its indices and is a function only of the . As implied by Figure 7 the are also comoving on their respective worldlines, meaning that they are independent of the proper time , say, that parameterizes each curve:
The time part of the spacetime dependence of the is thus somewhat ad hoc; that is, if we take the flow of time to be the proper time of the worldlines ( is parallel to ), the do not change. An apparent time dependence in spacetime means that is such as to cut across fluid worldlines ( is not parallel to ), which of course have different values for the .

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:

In terms of the scalar fields , we now have particle number density currents that are automatically conserved. Thus, another way of viewing the construction is that the fundamental fluid field variables are the . In fact, the variations of can now be taken with respect to variations of the , namely
As the are scalars, the covariant derivatives can be replaced with ordinary partial derivatives, meaning that there is no metric variation to consider.

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,

where is the Lie derivative along (cf. Equation (43)). Using the fact that
means [67]
Using Equation (122) above and Equation (367) of Appendix A, we can thus infer that
This constraint guarantees that is conserved. By introducing the decomposition
we can furthermore show that
and

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 [304445] (see Section 13). Later, in Section 12, we take the limit that the speed of light becomes large so as to construct non-relativistic fluid equations. The same procedure can be used to obtain the Newtonian analogues of the formulae constructed here.

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

Using the fact that , it can be shown that
where is as in Equation (132) except is replaced with . If we set
then the last term vanishes and . Thus, we can use the arbitrary function to reduce the number of independent to three.

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

To obtain the Einstein equation (65) we must add the Einstein–Hilbert action:
where is the Ricci scalar, is the determinant of the metric of the boundary (if present) of , and is the trace of the extrinsic curvature of the boundary. This last term is added to ensure that fixing the metric on the boundary leads to a well-posed action principle [124].

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

where
The use of the letter is to remind us that this is a bulk fluid effect, which is present regardless of the number of fluids and constituents. The momentum covector is what we encountered earlier. It is dynamically, and thermodynamically, conjugate to , and its magnitude can be seen to be the chemical potential of the particles by recalling that . Correctly identifying the momentum is essential for a concise and transparent formulation of fluids, e.g. for constructing proofs about vorticity, and also for coupling in dissipative terms in the equations of motion (cf. the discussion of Section 7). For later convenience, we also introduce the momentum form defined as

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

where the boundary terms do not contribute to the field equations or stress-energy-momentum tensor (the divergence theorem implies that they become boundary terms in the action), the force density is defined in Equation (102), and the generalized pressure is defined to be
We have utilized the well-known formula [80] for the variation of the metric determinant, which can be found in Equation (370) of Appendix A. We also note that the fluid boundary term (FBT) is
so that on the boundary is fixed (i.e.  vanishes) to have a well-posed action principle.

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

and of course the Einstein equation. When , it is well-established in the relativity literature that the stress-energy-momentum tensor follows from a variation of the Lagrangian with respect to the metric, that is
When the complete set of field equations is satisfied then we see from Equation (105), which applies here, that it is automatically true that .

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).