Frame Hydrodynamics Overview

• Frame hydrodynamics is a framework where the choice of reference frame defines constitutive decompositions and variable evolution in nonequilibrium systems.
• It encompasses formulations like relativistic, mixed-frame radiation, and kinetic models that affect stability, causality, and conservation laws.
• Recent advances, including second-order theories and Density Frame approaches, demonstrate practical solutions for robust dissipative and transient behavior.

Frame hydrodynamics denotes hydrodynamic theories in which the choice of frame is part of the constitutive, kinetic, geometric, or order-parameter content of the model. In relativistic dissipative fluids, the frame specifies how $u^\mu$, $T$, and $\mu$ are defined away from equilibrium; in mixed-frame radiation hydrodynamics, radiation moments are evolved in the inertial frame while emissivity and opacity are specified in the comoving frame; in biaxial nematics, the hydrodynamic variable is an orthonormal frame $p=(n_1,n_2,n_3)\in SO(3)$; and in rotating quantum hydrodynamics or spherically symmetric gravitation, the fundamental equations are written in a non-inertial or foliation-adapted local frame (Minami et al., 2012, Noronha et al., 2021, He et al., 2024, Li et al., 2022, Trukhanova, 2016, Jai-akson et al., 6 Jan 2026). Across these settings, the common theme is that the frame is not merely a change of notation: it fixes which variables are evolved, how constitutive terms are organized, and which stability, causality, or conservation properties are manifest.

1. Relativistic hydrodynamic frame as constitutive data

In relativistic hydrodynamics, the frame ambiguity arises because out of equilibrium the split of conserved currents into “ideal” and “dissipative” parts is not unique. A general decomposition is

$$T^{\mu\nu} = (\varepsilon+\mathcal A)u^\mu u^\nu + (P+\Pi)\Delta^{\mu\nu} + \pi^{\mu\nu} + \mathcal Q^\mu u^\nu + \mathcal Q^\nu u^\mu ,$$

$J^\mu = (n+\mathcal N)u^\mu + J^\mu ,$

with $\mathcal A$, $\Pi$, $\mathcal Q^\mu$, $\pi^{\mu\nu}$, $T$0, and $T$1 frame-sensitive off-equilibrium pieces (Noronha et al., 2021). The familiar Landau frame sets the transverse energy flow to zero, while the Eckart frame aligns the velocity with a conserved particle current. General-frame formulations do not impose either condition a priori.

A central correction to older terminology is that, in the linear regime around global equilibrium, the Landau–Eckart distinction is not fundamentally a choice of local rest frame but rather a choice of gross variables. In Mori’s projection-operator derivation, the natural slow variables are the conserved densities $T$2, whereas the Eckart variable $T$3 is not itself an independent conserved slow mode; its slow part is a projected combination of momentum and scalar conserved-density fluctuations (Minami et al., 2012). This directly supports the statement that frame choice is dynamical and constitutive, not merely kinematical.

A common misconception is that “frame” only means a preferred velocity convention. The relativistic literature instead treats frame choice as a prescription for identifying hydrodynamic fields from a nonequilibrium state. This matters because different choices reshuffle energy-density correction, charge-density correction, energy flux, diffusion current, bulk pressure, and shear stress, while leaving the full $T$4 and $T$5 unchanged. A plausible implication is that frame hydrodynamics is best understood as a theory of constitutive decompositions rather than a theory of coordinate transformations.

2. General-frame thermodynamics and generalized Gibbs structure

A major development was the formulation of relativistic thermodynamics in an arbitrary frame. In that construction, the inverse-temperature four-vector is decomposed as

$T$6

with $T$7, so the equilibrium state itself may carry a spacelike component relative to the chosen $T$8 (Ván et al., 2011). The equilibrium momentum density satisfies

$T$9

and the generalized Gibbs relation becomes

$\mu$0

This replaces the Eckart form $\mu$1 by a frame-sensitive version in which momentum density contributes explicitly.

The corresponding nonequilibrium entropy flux retains the classical-looking form

$\mu$2

but the thermal force contains not only the temperature gradient and acceleration, but also $\mu$3 and $\mu$4. The resulting first-order constitutive laws are

$\mu$5

and the theory is argued to be generically stable under the usual thermodynamic stability conditions and nonnegative transport coefficients (Ván et al., 2011).

This thermodynamic line of work reframed a long-standing controversy. The standard view associated first-order relativistic dissipation with inevitable instability, but the general-frame analysis argued that instability is tied to how local equilibrium and heat flow are formulated, not to first order as such. The limitation stated in the same work is equally important: causality is not established in the strong Israel–Stewart sense. This suggests a durable distinction within frame hydrodynamics between thermodynamic stability and fully hyperbolic causal formulation.

3. Kinetic derivations, matching conditions, and the limits of frame freedom

Kinetic theory sharpened the frame problem by showing that conservation and frame choice must be implemented simultaneously at the level of the collision model. In the extended relaxation-time approximation, the Boltzmann equation relaxes not to $\mu$6, but to a nearby equilibrium distribution $\mu$7, with

$\mu$8

This separates the hydrodynamic frame from the thermodynamic frame and restores macroscopic conservation “by construction” when the relaxation time depends on particle energy (Dash et al., 2021). The explicit first-order derivation in that work is carried out in the Landau frame, but the formalism is presented as frame-aware rather than conceptually Landau-restricted.

A second kinetic result made the limitation of frame freedom explicit. In first-order relativistic hydrodynamics derived from a momentum-dependent relaxation-time approximation,

$\mu$9

the crucial coefficients multiplying $p=(n_1,n_2,n_3)\in SO(3)$0, $p=(n_1,n_2,n_3)\in SO(3)$1, and $p=(n_1,n_2,n_3)\in SO(3)$2 vanish for momentum-independent RTA, $p=(n_1,n_2,n_3)\in SO(3)$3, in every frame. The conclusion is stated sharply: a general hydrodynamic frame by itself is not enough; momentum dependence of the microscopic interaction rate is imperative for producing the field corrections that generate causal and stable nonhydrodynamic behavior. In the examples studied, shear-channel causality and longitudinal stability require conditions such as $p=(n_1,n_2,n_3)\in SO(3)$4 (Biswas et al., 2022).

The particle frame supplies a third kinetic lesson. A stable first-order particle-frame theory was derived from the relativistic Boltzmann equation with the constraints

$p=(n_1,n_2,n_3)\in SO(3)$5

rather than Eckart’s condition $p=(n_1,n_2,n_3)\in SO(3)$6. Its constitutive structure removes the acceleration term from the heat force and yields linear stability of equilibrium, while acausality is explicitly left unresolved (0709.3645). This suggests that the admissible frame conditions are constrained by microscopic dynamics, not selected solely by phenomenological convenience.

4. Second-order and transient general-frame theories

Second-order frame hydrodynamics extends the first-order constitutive ambiguity into a transient sector. A general-frame Israel–Stewart-type construction promotes all frame-sensitive dissipative variables to dynamical fields: $p=(n_1,n_2,n_3)\in SO(3)$7 Its nonequilibrium entropy current contains quadratic terms in all of these variables, and the resulting relaxation equations introduce additional transport coefficients $p=(n_1,n_2,n_3)\in SO(3)$8, $p=(n_1,n_2,n_3)\in SO(3)$9, and $$T^{\mu\nu} = (\varepsilon+\mathcal A)u^\mu u^\nu + (P+\Pi)\Delta^{\mu\nu} + \pi^{\mu\nu} + \mathcal Q^\mu u^\nu + \mathcal Q^\nu u^\mu ,$$0 for $$T^{\mu\nu} = (\varepsilon+\mathcal A)u^\mu u^\nu + (P+\Pi)\Delta^{\mu\nu} + \pi^{\mu\nu} + \mathcal Q^\mu u^\nu + \mathcal Q^\nu u^\mu ,$$1, $$T^{\mu\nu} = (\varepsilon+\mathcal A)u^\mu u^\nu + (P+\Pi)\Delta^{\mu\nu} + \pi^{\mu\nu} + \mathcal Q^\mu u^\nu + \mathcal Q^\nu u^\mu ,$$2, and $$T^{\mu\nu} = (\varepsilon+\mathcal A)u^\mu u^\nu + (P+\Pi)\Delta^{\mu\nu} + \pi^{\mu\nu} + \mathcal Q^\mu u^\nu + \mathcal Q^\nu u^\mu ,$$3, alongside the usual $$T^{\mu\nu} = (\varepsilon+\mathcal A)u^\mu u^\nu + (P+\Pi)\Delta^{\mu\nu} + \pi^{\mu\nu} + \mathcal Q^\mu u^\nu + \mathcal Q^\nu u^\mu ,$$4, $$T^{\mu\nu} = (\varepsilon+\mathcal A)u^\mu u^\nu + (P+\Pi)\Delta^{\mu\nu} + \pi^{\mu\nu} + \mathcal Q^\mu u^\nu + \mathcal Q^\nu u^\mu ,$$5, and $$T^{\mu\nu} = (\varepsilon+\mathcal A)u^\mu u^\nu + (P+\Pi)\Delta^{\mu\nu} + \pi^{\mu\nu} + \mathcal Q^\mu u^\nu + \mathcal Q^\nu u^\mu ,$$6 (Noronha et al., 2021). In this framework, not fixing a frame means that the corresponding frame-sensitive sectors become dynamical variables with their own relaxation equations.

Two limiting relations organize the subject. First, the first-order truncation reduces to Bemfica–Disconzi–Noronha–Kovtun theory. Second, the limit $$T^{\mu\nu} = (\varepsilon+\mathcal A)u^\mu u^\nu + (P+\Pi)\Delta^{\mu\nu} + \pi^{\mu\nu} + \mathcal Q^\mu u^\nu + \mathcal Q^\nu u^\mu ,$$7 with vanishing mixed $$T^{\mu\nu} = (\varepsilon+\mathcal A)u^\mu u^\nu + (P+\Pi)\Delta^{\mu\nu} + \pi^{\mu\nu} + \mathcal Q^\mu u^\nu + \mathcal Q^\nu u^\mu ,$$8-parameters reproduces Mueller–Israel–Stewart theory in the Landau frame, where $$T^{\mu\nu} = (\varepsilon+\mathcal A)u^\mu u^\nu + (P+\Pi)\Delta^{\mu\nu} + \pi^{\mu\nu} + \mathcal Q^\mu u^\nu + \mathcal Q^\nu u^\mu ,$$9 and $J^\mu = (n+\mathcal N)u^\mu + J^\mu ,$0 (Noronha et al., 2021). The spectral consequence is that the general-frame second-order theory carries more nonhydrodynamic modes than either MIS or BDNK.

A distinct second-order program constructs a causal Eckart-frame theory by systematic gradient expansion in curved spacetime. There the constitutive sectors are organized as scalar $J^\mu = (n+\mathcal N)u^\mu + J^\mu ,$1, vector $J^\mu = (n+\mathcal N)u^\mu + J^\mu ,$2, and tensor $J^\mu = (n+\mathcal N)u^\mu + J^\mu ,$3 in the Eckart frame, and the second-order equations contain explicit curvature terms such as

$J^\mu = (n+\mathcal N)u^\mu + J^\mu ,$4

$J^\mu = (n+\mathcal N)u^\mu + J^\mu ,$5

and

$J^\mu = (n+\mathcal N)u^\mu + J^\mu ,$6

Linearized propagation speeds remain finite provided the relaxation times satisfy inequalities of the form

$J^\mu = (n+\mathcal N)u^\mu + J^\mu ,$7

in the notation used there (Lahiri, 2019). This establishes a specifically Eckart-frame second-order causal sector rather than a frame-independent transient theory.

5. Density Frame, BDNK, and numerical realizations

A recent numerical realization of frame hydrodynamics is the Density Frame for one-dimensional relativistic viscous flows. Its defining condition is

$J^\mu = (n+\mathcal N)u^\mu + J^\mu ,$8

so the local hydrodynamic variables are determined directly by the conserved densities on a spatial slice (Bhambure et al., 2024). The constitutive relation is written only for the spatial stress,

$J^\mu = (n+\mathcal N)u^\mu + J^\mu ,$9

with

$\mathcal A$0

Because time derivatives in the constitutive relation are eliminated using the ideal equations of motion, the resulting PDE system is first order in time, requires no auxiliary fields, and has no non-hydrodynamic modes (Bhambure et al., 2024).

The same study compares Density Frame evolution with QCD kinetic theory, BDNK, and the MIS/DNMR code MUSIC. The stated outcome is excellent agreement with QCD kinetic theory within the regime of applicability of hydrodynamics, with well-behaved and robust results near the boundary of applicability (Bhambure et al., 2024). By contrast, the paper describes BDNK as causal and strongly hyperbolic, but also notes that in localized one-dimensional tests BDNK can display oscillations near the causality edge, while Density Frame solutions can exhibit exponentially small tails outside strict Lorentz-causal support.

This comparison clarifies a broader divide within frame hydrodynamics. One route, exemplified by BDNK, uses general first-order constitutive freedom to obtain causal and strongly hyperbolic equations. Another, exemplified by Density Frame, uses a density-based choice of variables to preserve a first-order-in-time evolution with no auxiliary sectors. The two approaches solve different parts of the same structural problem.

6. Mixed-frame and kinetic formulations outside relativistic viscous fluids

In multigroup radiation hydrodynamics, a mixed-frame formulation evolves radiation moments in the inertial or lab frame while keeping emissivity and absorption opacity in the comoving frame of the gas. The evolved radiation variables are $\mathcal A$1, $\mathcal A$2, and $\mathcal A$3 in the lab frame, while $\mathcal A$4, $\mathcal A$5, and $\mathcal A$6 are specified in the comoving frame (He et al., 2024). The motivation is that lab-frame moment equations are manifestly conservative, while matter interaction coefficients are simplest in the comoving frame, where they are isotropic. The resulting scheme conserves total energy and momentum to machine precision, uses a sparse Jacobian matrix of size $\mathcal A$7, and inverts the local matter–radiation coupling with $\mathcal A$8 complexity (He et al., 2024).

A different kinetic use of frame hydrodynamics appears in viscous shallow-water equations. There the forcing is represented through a shifted equilibrium,

$\mathcal A$9

and the pressure can be split consistently between equilibrium and a force-like contribution through a reference pressure $\Pi$0 (Hosseini et al., 10 May 2025). The model is designed to recover the full viscous shallow-water hydrodynamics with no errors in the dissipation rates, allows independent bulk-viscosity control, and interprets forcing as relaxation toward a shifted equilibrium rather than as a conventional additive source term (Hosseini et al., 10 May 2025). This suggests a kinetic analogue of frame freedom: the macroscopic equations are fixed, but the partition of pressure and force between equilibrium and shifted target state is not unique.

These mixed and kinetic formulations broaden the subject beyond relativistic constitutive theory. Here “frame” refers neither to Landau nor to Eckart, but to the assignment of variables or coefficients to inertial, comoving, or shifted kinetic states. The common structural point is again that conservation, accuracy, and dissipation depend on how that assignment is made.

7. Frame fields, rotating frames, and geometro-hydrodynamics

In biaxial nematic liquid crystals, frame hydrodynamics is literal: the orientational variable is an orthonormal frame field

$\Pi$1

rather than a single director or a $\Pi$2-tensor (Li et al., 2022). The hydrodynamic PDE couples $\Pi$3 to an incompressible velocity field, preserves the orthonormality constraints automatically in the reformulated equations, and satisfies an exact energy dissipation law. In $\Pi$4 and $\Pi$5, smooth solutions are locally well posed and satisfy a Beale–Kato–Majda-type blow-up criterion; in $\Pi$6, global weak solutions exist and are smooth away from finitely many singular times (Li et al., 2022). A subsequent result proves uniqueness of those global weak solutions in $\Pi$7 by weaker energy estimates in a Littlewood–Paley framework and by exploiting the rotational derivatives on $\Pi$8, cancellation relations, and the dissipative structure of the biaxial frame system (Li et al., 2022).

In rotating quantum hydrodynamics, the frame is a rotating reference frame built directly into the many-particle Pauli–Schrödinger equation. The continuity equation keeps its inertial-frame form, but the momentum equation acquires inertial force densities corresponding to Coriolis, centrifugal, and Euler terms, together with a spin-rotation force term, while the spin evolution equation acquires the torque $\Pi$9 (Trukhanova, 2016). For electric-dipole systems, the rotating-frame polarization-current dynamics leads to dipole-inertial waves, including the two-dimensional dispersion law

$\mathcal Q^\mu$0

The rotating frame is therefore not a perturbative add-on but part of the microscopic definition of the hydrodynamic variables.

A geometric extension of the idea appears in spherically symmetric spacetime. Using the rigging technique, one constructs a foliation-adapted frame $\mathcal Q^\mu$1 and dual coframe $\mathcal Q^\mu$2 on spherical hypersurfaces, with the special property that in the constant-radius foliation the vertical vector is the Kodama vector and the quasi-local Misner–Sharp energy becomes the natural energy density variable (Jai-akson et al., 6 Jan 2026). In that frame the Einstein equations take the form of a gravitational Euler equation and a gravitational Young–Laplace equation, and spacetime is interpreted as a concentric stack of “gravitational bubbles” carrying Misner–Sharp energy density and geometric pressure (Jai-akson et al., 6 Jan 2026). The same structure extends to Lovelock gravity through generalized energy and pressure variables.

Taken together, these developments show that frame hydrodynamics is not a single formalism but a family of theories in which the frame enters at the level of constitutive decomposition, kinetic closure, order-parameter geometry, non-inertial reference structure, or foliation-adapted gravitational dynamics. What unifies them is the claim, explicit in several of the cited works, that frame choice is part of the physical and mathematical content of the theory rather than a superficial rewriting.