The overall aim is to generalize the variational formulation described in Section 8 in such a way that
viscous “stresses” are accounted for. Because the variational foundations are the same, the number currents
play a central role (
is a constituent index as before). In addition we can introduce a number of
viscosity tensors
, which we assume to be symmetric (even though it is clear that such
an assumption is not generally correct [6]). The index
is “analogous” to the constituent
index, and represents different viscosity contributions. It is introduced in recognition of the
fact that it may be advantageous to consider different kinds of viscosity, e.g. bulk and shear
viscosity, separately. As in the case of the constituent index, a repeated index
does not imply
summation.
The key quantity in the variational framework is the master function . As it is a function of
all the available fields, we will now have
. Then a general variation leads to
As in the non-dissipative case, the variational framework suggests that the equations of motion can be written as a force-balance equation,
where the generalized forces work out to be and Finally, the stress-energy tensor becomes with the generalized pressure given by For reasons that will become clear shortly, it is useful to introduce a set of “convection vectors”. In the
case of the currents, these are taken as proportional to the fluxes. This means that we introduce
according to
In order to facilitate a similar decomposition for the viscous stresses, it is natural to choose the
conduction vector as a unit null eigenvector associated with . That is, we take
Finally, let us suppose that we choose to work in a given frame, moving with four-velocity (and
that the associated projection operator is
). Then we can use the following decompositions for the
conduction vectors:
So far the construction is quite formal, but we are now set to make contact with the physics. First we note that the above results allow us to demonstrate that
Recall that similar results were central to expressing the second law of thermodynamics in the previous two Sections 14.1 and 14.2. To see how things work out in the present formalism, let us single out the entropy fluid (with index At this point, the general formalism must be completed by some suitably simple model for the various
terms. A reasonable starting point would be to assume that each term is linear. For the chemical reactions
this would mean that we expand each according to
A detailed comparison between Carter’s formalism and the Israel–Stewart framework has been carried
out by Priou [89]. He concludes that the two models, which are both members of a large family of
dissipative models, have essentially the same degree of generality and that they are equivalent in the limit of
linear perturbations away from a thermal equilibrium state. Providing explicit relations between the main
parameters in the two descriptions, he also emphasizes the key point that analogous parameters may not
have the same physical interpretation.
In developing his theoretical framework, Carter argued in favor of an “off the peg” model for heat
conducting models [18]. This model is similar to that introduced in Section 10, and was intended as a
simple, easier to use alternative to the Israel–Stewart construction. In the particular example discussed by
Carter, he chooses to set the entrainment between particles and entropy to zero. This was done in order to
simplify the discussion. But, as a discussion by Olson and Hiscock [87] shows, it has disastrous
consequences. The resulting model violates causality in two simple model systems. As discussed by
Priou [89] and Carter and Khalatnikov [25], this breakdown emphasizes the importance of the entrainment
effect in these problems.
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