The variational formalism of Carter provides a very general framework for deriving the dynamic equations of any fluid mixture and for obtaining conservation laws, using exterior calculus. In particular, this formalism is very well suited to describing superfluid systems, like laboratory superfluids or neutron star interiors, by making a clear distinction between particle velocities and the corresponding momenta (see the discussion in Section 8.3.6).

The dynamics of the system (either in relativity or in the Newtonian limit) is thus governed by a Lagrangian density , which depends on the particle 4-currents , where and are the particle number density and the 4-velocity of the constituent X, respectively. We will use Greek letters for spacetime indices with the Einstein summation convention for repeated indices. The index X runs over the different constituents in the system. Note that repeated chemical indices X will not mean summation unless explicitly specified.

The dynamic equations for a mixture of several interacting fluids (coupled by entrainment effects) can be obtained by requiring that the action integral

(where is the 4-volume element) be stationary under variations of the 4-currents . These variations are not arbitrary because they have to conserve the number of particles. In classical mechanics of point-like particles, the equations of motion can be deduced from an action integral by considering displacements of the particle trajectories. Likewise, considering variations of the 4-currents induced by displacements of the fluid-particle worldlines yield where denotes the covariant derivative. , defined by is the 4-momentum per particle associated with the 4-current , is the vorticity 2-form defined by the exterior derivative of the corresponding 4-momentum and is the (nongravitational) 4-force density acting on the constituent X. Equation (222) is the generalization to fluids of the definition introduced in classical Lagrangian mechanics for the momentum of point-like particles. This equation shows that momentum and velocity are intrinsically different mathematical objects since the former is a co-vector while the latter is a vector. The vorticity 2-form is closely analogous to the electromagnetic 2-form . Equation (221) is the covariant generalization of Euler’s equation to multi-fluid systems. The stress-energy-momentum tensor of this multi-fluid system is given by where the generalized pressure is defined by Note that so far we have made no assumptions regarding the spacetime geometry so that Equations (221), (224) and (225) are valid for both relativistic and nonrelativistic fluids. The presence of a frozen-in magnetic field and the elasticity of the solid crust can be taken into account within the same variational framework both in (special and general) relativity [73, 85] and in the Newtonian limit [73, 72].http://www.livingreviews.org/lrr-2008-10 |
This work is licensed under a Creative Commons License. Problems/comments to |