Carter's Multifluid Variational Formalism

• Carter's multifluid variational formalism is a relativistic, covariant framework that unifies hydrodynamics, thermodynamics, and entrainment via a scalar master function.
• It employs convective variations and matter-space pull-backs to enforce current conservation and derive Euler-type equations along with stress-energy tensors.
• The formalism extends to handle dissipative processes and astrophysical applications, enabling causal heat transport and analysis of dark-sector models.

Carter’s multifluid variational formalism is a relativistic, covariant action principle for systems composed of several interacting fluid constituents, each represented by its own current $n_x^a$, with the spacetime metric $g_{ab}$ and a scalar master function $\Lambda$ supplying the fundamental dynamical data. The formalism packages hydrodynamics, thermodynamics, entrainment, and stress–energy conservation into a single variational structure: $\Lambda$ depends only on scalar invariants formed from the currents and the metric, and constrained convective variations yield the conjugate momenta, Euler-type equations, and stress tensor. In later developments the same framework has been extended to dissipative channels, causal heat transport, superfluid hydrodynamics, radiation hydrodynamics, compact-star structure, and interacting dark-sector models (Osano et al., 2018, Andersson et al., 2013).

1. Covariant kinematics and matter spaces

At its core, the formalism assigns to each constituent $x$ a timelike current $n_x^a$, often written as $n_x^a = n_x u_x^a$ with $u_x^a u^{x}_a = -1$. Different constituents need not share the same congruence of worldlines, so relative flows are native to the theory rather than perturbative corrections. The master function is a scalar under diffeomorphisms and therefore depends only on combinations such as

$$n_x^2 = -g_{ab}n_x^a n_x^b, \qquad n_{xy}^2 = -g_{ab}n_x^a n_y^b.$$

This already anticipates the two defining features of the formalism: covariance and the possibility of entrainment through cross-scalars (Andersson et al., 2013).

Carter’s “convective” construction enforces current conservation geometrically by introducing, for each constituent, a three-dimensional matter space $\mathcal{B}_x$ with coordinates $g_{ab}$0, together with a matter-space volume form $g_{ab}$1. Pull-back to spacetime produces a three-form

$g_{ab}$2

and the current is its Hodge dual,

$g_{ab}$3

Closure of the pulled-back three-form,

$g_{ab}$4

implies

$g_{ab}$5

This is the characteristic pull-back implementation of conserved fluxes in the conservative theory (Andersson et al., 2013).

Variations are not free variations of $g_{ab}$6. They are generated by constituent-dependent Lagrangian displacements $g_{ab}$7, with

$g_{ab}$8

Because the label fields are Lie-dragged along the flow, the variation automatically respects the conservation constraints. This distinguishes Carter’s formalism from approaches that postulate conservation equations separately from the action (Andersson et al., 2013).

2. Master function, conjugate momenta, and Euler equations

For an isotropic multifluid, the matter action is

$g_{ab}$9

Differentiation of $\Lambda$0 with respect to the currents defines the conjugate momentum covectors,

$\Lambda$1

Using the scalar dependence of $\Lambda$2, these take the standard multifluid form

$\Lambda$3

with

$\Lambda$4

The diagonal coefficients $\Lambda$5 encode self-couplings, while the symmetric off-diagonal coefficients $\Lambda$6 encode entrainment (Andersson et al., 2013).

The unconstrained variation of the matter Lagrangian density can be written as

$\Lambda$7

After imposing the convective constraints, the action variation becomes

$\Lambda$8

where the generalized force density is

$\Lambda$9

The Euler–Lagrange equations for the conservative theory are therefore

$\Lambda$0

They are the multifluid relativistic generalization of Euler’s equation, written in terms of canonical momenta and their vorticity two-forms rather than a single velocity field (Andersson et al., 2013).

Variation with respect to the metric yields the stress–energy tensor in Carter form,

$\Lambda$1

When the current conservation laws and Euler equations hold, this tensor satisfies

$\Lambda$2

Electromagnetic coupling can be incorporated by adding the Maxwell sector and charge currents, in which case the fluid equations acquire Lorentz-force and resistive terms while the total stress tensor becomes $\Lambda$3 (Osano et al., 2018).

3. Entrainment, equilibrium thermodynamics, and gauge structure

Entrainment is the defining nondissipative coupling of the theory. Because $\Lambda$4 depends on cross-scalars $\Lambda$5, the momentum of one constituent need not be parallel to its own current. In physical terms, “the momentum of one species carries with it mass current of the other species,” and relative velocities enter directly through

$\Lambda$6

In a comoving state all four-velocities are parallel, while genuinely multifluid states are multi-velocity configurations; in cosmological applications, such relative flows can suggest anisotropic backgrounds unless isotropization occurs (Osano et al., 2018).

A detailed thermodynamic interpretation of the formalism emerges by studying equilibrium with a heat bath. In the thermal frame $\Lambda$7, one identifies

$\Lambda$8

as the temperature and thermodynamic chemical potentials. The internal-energy density $\Lambda$9 obeys

$x$0

while a further Legendre transform gives

$x$1

In this sense the master function is not merely a formal Lagrangian density; it is a thermodynamic potential with a precise equilibrium interpretation (Gavassino et al., 2019).

The same analysis shows that equilibrium requires vanishing entropy flux and vanishing normal currents in the thermal frame, together with the standard chemical-equilibrium condition

$x$2

for reactions

$x$3

A notable consequence is that the standard notions of affinity and reaction coordinate remain intact in the presence of superfluid currents (Gavassino et al., 2019).

An important technical refinement is the existence of a gauge freedom in the decomposition of the normal momenta. In equilibrium, the equation of state depends on fewer state variables than the full phenomenological model suggests, so spatial parts of the normal momenta may be shifted without changing $x$4, $x$5, or the perfect-limit hydrodynamics. The superfluid momenta are gauge invariant, and the physically meaningful superfluid entrainment data are encoded in the matrix $x$6. This gauge freedom is what allows one to simplify the perfect multifluid by choosing, for example, a gauge with no entropy entrainment in the thermal frame (Gavassino et al., 2019).

4. Dissipative generalizations, causality, and hyperbolicity

A persistent misconception is that Carter’s framework is intrinsically restricted to conservative flows. The dissipative extension instead relaxes the assumption that each matter-space volume form depends only on its own matter-space coordinates. Allowing

$x$7

breaks closure of the pulled-back three-forms and hence allows

$x$8

The resulting equations of motion become

$x$9

where $n_x^a$0 are resistive forces and $n_x^a$1 are dissipative stresses. In the most general case the individual dissipative stress tensors need not be spacetime symmetric, although the total stress–energy tensor remains symmetric (Andersson et al., 2013).

The same logic supports causal dissipative superfluid hydrodynamics. A later three-current model based on Carter’s approach promotes the quasi-particle current $n_x^a$2 to an independent field alongside the particle current $n_x^a$3 and entropy current $n_x^a$4. In the nondissipative limit it reduces to the relativistic two-fluid model of Carter, Khalatnikov and Gusakov; near equilibrium it can be rewritten in Israel–Stewart form and yields telegraph-type equations for bulk viscosity and heat conduction, with finite relaxation times and four bulk-viscosity coefficients (Gavassino et al., 2021).

Thermodynamic stability imposes stringent algebraic constraints on the multifluid constitutive matrices. In the pressure–momenta representation, the Hessian $n_x^a$5, the entrainment matrix $n_x^a$6, and the difference matrix $n_x^a$7 must be positive definite. The linear sound speeds then follow from

$n_x^a$8

and the positivity conditions imply subluminal propagation. In this sense, thermodynamic stability implies linear causality, and the positive definiteness of the entrainment matrix is a necessary stability condition (Gavassino, 2022).

Near equilibrium, the same thermodynamic structure also controls the PDE character of the theory. In the absence of macroscopic magnetic fields and spins, the Onsager–Casimir principle enforces symmetry of the principal matrices of the linearized hydrodynamic system. Carter’s multifluid theory can then be written in symmetric-hyperbolic form in the linear regime, with a positive-definite information current furnishing the natural norm for linear stability (Gavassino, 2022).

5. Dual formulations and geometric reinterpretations

Although the convective current-based formulation is canonical, Carter’s theory also admits an equivalent pressure–momenta representation. In that dual language one starts from a generalized pressure $n_x^a$9 and writes

$n_x^a = n_x u_x^a$0

Lorentz covariance implies that $n_x^a = n_x u_x^a$1 depends on the momenta only through scalar combinations $n_x^a = n_x u_x^a$2, and one obtains

$n_x^a = n_x u_x^a$3

This representation is particularly effective in stability theory and in the analysis of constitutive matrices (Gavassino, 2022).

The same duality underlies a precise mapping between GENERIC heat conduction and Carter’s multifluid theory. In that construction two momentum covectors $n_x^a = n_x u_x^a$4 and $n_x^a = n_x u_x^a$5 generate the entropy and particle currents through a $n_x^a = n_x u_x^a$6 mobility matrix, and the stress tensor takes the Carter form

$n_x^a = n_x u_x^a$7

The mapping is exact and non-perturbative. A particularly striking consequence is that, in the limit of infinite heat conductivity, GENERIC heat conduction reduces to the relativistic two-fluid model for superfluidity (Gavassino, 2023).

In the single-fluid limit, Carter’s canonical viewpoint also supports a Hamiltonian and geometric reformulation. Barotropic fluid elements admit a canonical one-form $n_x^a = n_x u_x^a$8, the Poincaré–Cartan integral invariant yields Kelvin’s theorem, and barotropic flows are conformally geodesic. Contrary to earlier assumptions, the same Hamiltonian structure can accommodate perfectly conducting magnetofluids via the Bekenstein–Oron description of ideal magnetohydrodynamics (Markakis et al., 2016). A more recent mathematical development reformulates the Lichnerowicz–Carter equations as the intersection

$n_x^a = n_x u_x^a$9

inside an infinite-dimensional symplectic manifold and identifies a five-dimensional origin of the formalism (Nekrasov et al., 31 Dec 2025).

6. Astrophysical and cosmological realizations

The formalism has become a general-purpose framework for relativistic multifluid astrophysics. In neutron-star core modeling at finite temperature, one implementation includes seven fluids: normal neutrons, normal protons, electrons, muons, superfluid neutrons, superconducting protons, and entropy. Vortex lines and flux tubes, mutual friction, vortex pinning, heat conduction, and viscosity are then incorporated in stages, and the resulting variational hydrodynamics is shown to be equivalent to the non-variational formalism of Gusakov and collaborators (Rau et al., 2020).

Relativistic radiation hydrodynamics also fits naturally into Carter’s scheme when matter particles, photons, matter entropy, and radiation entropy are treated as four currents. In that setting the matter–radiation interaction is encoded through generalized forces and entropy production, and the theory reduces in appropriate limits either to a bulk-viscous multifluid or to a heat-conducting multifluid, with thermal conductivity

$u_x^a u^{x}_a = -1$0

This use of the variational formalism exposes explicit Onsager and positivity constraints on the microscopic coefficients (Gavassino et al., 2020).

Cosmological dark-sector models provide another application. One construction treats dark matter and dark energy as distinct fluids, each with its own particle and entropy flux, and writes the master function schematically as

$u_x^a u^{x}_a = -1$1

The cross terms $u_x^a u^{x}_a = -1$2 and $u_x^a u^{x}_a = -1$3 encode entrainment between dark matter and dark energy, and the formalism makes it possible to discuss whether such coupling could affect structure growth, the late Integrated Sachs–Wolfe effect, or CMB anisotropies (Osano et al., 2018).

The framework has also been pushed into relativistic stellar perturbation theory. A general-relativistic treatment of adiabatic tidal deformations for compact stars composed of an arbitrary number of interacting fluids uses Carter’s variational formalism throughout. A distinctive feature of that analysis is nondissipative mutual entrainment, yet the final result is that entrainment leaves adiabatic tidal responses unchanged and therefore produces no measurable effect on the inspiral gravitational-wave signal before internal-mode resonances are excited (Carlier et al., 31 Mar 2026).

Taken together, these developments establish Carter’s multifluid variational formalism not merely as a convenient notation for relativistic hydrodynamics, but as a unifying covariant structure in which conservation laws, thermodynamic conjugacy, entrainment, dissipation, causal transport, and astrophysical microphysics can be treated within a single action-based language (Andersson et al., 2013).