10.1 Variational formulation of multi-fluid hydrodynamics
In the convective variational approach of hydrodynamics developed by Carter [70, 80], and recently
reviewed by Gourgoulhon [174] and Andersson & Comer [15], the dynamic equations are obtained from an
action principle by considering variations of the fluid particle trajectories. First developed in the context of
general relativity, this formalism has been adapted to the Newtonian framework in the usual
3+1 spacetime decomposition by Prix [342, 343]. As shown by Carter & Chamel in a series of
papers [74
, 75
, 76
], the Newtonian hydrodynamic equations can be written in a very concise
and elegant form in a fully-4D covariant framework. Apart from facilitating the comparison
between relativistic and nonrelativistic fluids, this approach sheds a new light on Newtonian
hydrodynamics following the steps of Elie Cartan, who demonstrated in the 1920’s that the
effects of gravitation in Newtonian theory can be expressed in geometric terms as in general
relativity.
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
].