Hydrodynamic Frames and Constitutive Data

A constitutive relation is a map from hydrodynamic fields and sources to the physical tensors $T^{\mu\nu}$ and $J^\mu$. Away from equilibrium, derivative-order changes in the definitions of $T$, $\mu$, and $u^\mu$ move coefficients among components of those tensors. Such a hydrodynamic frame transformation is a field redefinition, not a change of the underlying state or observable.

Required background. Conservation Laws and Hydrodynamic Fields gives the stress-current decomposition. Local Equilibrium and Hydrostatic Constraints identifies the equilibrium equation-of-state surface about which frames are defined.

Helpful background. Frame-Invariant Dissipative Data develops the invariant combinations beyond this first construction.

Constitutive maps at a fixed order

Write the physical tensors through first derivative order as

$$\begin{aligned} T^{\mu\nu} &= (\epsilon+\delta\epsilon)u^\mu u^\nu +(p+\delta p)P^{\mu\nu} +u^\mu q^\nu+u^\nu q^\mu +\pi^{\mu\nu},\\ J^\mu &= (n+\delta n)u^\mu+j^\mu . \end{aligned}$$

Here $\epsilon(T,\mu)$, $p(T,\mu)$, and $n(T,\mu)$ are zeroth-order functions; every $\delta$ quantity, $q^\mu$, $j^\mu$, and $\pi^{\mu\nu}$ is $O(\partial)$. The decomposition is redundant because the same physical tensors can be parametrized by nearby fields.

Let

$$T'=T+\delta T_f, \qquad \mu'=\mu+\delta\mu_f, \qquad u'^\mu=u^\mu+\delta u_f^\mu, \qquad u_\mu\delta u_f^\mu=0,$$

with all shifts $O(\partial)$. Keeping $T^{\mu\nu}$ and $J^\mu$ fixed gives

$$\begin{aligned} \delta\epsilon' &= \delta\epsilon-\epsilon_T\delta T_f-\epsilon_\mu\delta\mu_f,\\ \delta n' &= \delta n-n_T\delta T_f-n_\mu\delta\mu_f,\\ \delta p' &= \delta p-p_T\delta T_f-p_\mu\delta\mu_f,\\ q'^\mu&=q^\mu-w\,\delta u_f^\mu,\\ j'^\mu&=j^\mu-n\,\delta u_f^\mu,\\ \pi'^{\mu\nu}&=\pi^{\mu\nu}, \end{aligned}$$

through first order, where $w=\epsilon+p$ and subscripts denote thermodynamic derivatives. Reconstructing the tensors from the primed variables verifies the transformation independently.

Landau and Eckart conventions

The Landau convention chooses

$$q^\mu_{\mathrm L}=0.$$

Starting from a regular frame, set $\delta u_f^\mu=q^\mu/w$. The resulting velocity is the timelike energy-flow eigenvector through the retained order. Existence of this perturbative choice requires $w\ne0$ and a suitable timelike eigenvector; it is not a universal nonperturbative theorem for arbitrary stress tensors.

The Eckart convention chooses

$$j^\mu_{\mathrm E}=0,$$

so $\delta u_f^\mu=j^\mu/n$. It requires a nonzero charge density and becomes singular as $n\to0$. Landau and Eckart conditions can both hold only when the invariant relative diffusion current vanishes:

$$\mathcal J^\mu := j^\mu-\frac{n}{w}q^\mu .$$

Indeed,

$$\mathcal J'^\mu = j^\mu-n\delta u_f^\mu -\frac{n}{w}\left(q^\mu-w\delta u_f^\mu\right) =\mathcal J^\mu .$$

The complete convention dictionary is the hydrodynamic frame and tensor reference.

Scalar and tensor invariants

Use $\epsilon$ and $n$ as independent thermodynamic coordinates locally, so

$$\mathrm dp=p_\epsilon\,\mathrm d\epsilon+p_n\,\mathrm dn.$$

Then

$$\mathcal B := \delta p-p_\epsilon\delta\epsilon-p_n\delta n$$

is invariant under the scalar field redefinitions above. It measures stress perpendicular to the equilibrium equation-of-state surface and becomes the bulk viscous datum once a constitutive basis is chosen. The transverse traceless tensor $\pi^{\mu\nu}$ is also invariant at first order: no scalar or vector redefinition can generate its irreducible tensor representation.

At linear order, the physical poles and residues of correlators of fixed operators $T^{\mu\nu}$ and $J^\mu$ are invariant. Individual entries called “energy correction,” “heat flow,” or “charge diffusion” are generally not.

Hydrostatic constraints on this invariant constitutive data are developed by Bhattacharyya 2014, §§2–4, Open PDF.

Order-by-order equivalence

A first-order field shift changes second-order terms. For example,

$$\eta(T')\sigma^{\mu\nu}[u'] = \eta(T)\sigma^{\mu\nu}[u] +O(\partial^2).$$

Therefore two first-order constitutive relations related by the displayed map are equivalent only through $O(\partial)$. To compare them at $O(\partial^2)$, one must transform:

• the second-order tensor basis and coefficients;
• source-dependent and curvature terms;
• the equations used to eliminate redundant derivatives;
• initial data and matching conditions;
• any noise variables or transient fields.

Applying a first-order transformation “exactly” inside a truncated PDE implicitly chooses an infinite tower of higher-order terms. Different exact completions can have different high-frequency characteristics even though they share the same hydrodynamic poles. This is why a frame-dependent PDE stability statement is not contradicted by frame invariance of low-energy observables.

Kovtun’s general-frame analysis provides an explicit linear example of stable and unstable exact completions sharing the same retained infrared transport Kovtun 2019, §§2–4, Open PDF.

Translation example

Suppose a charged fluid is given in a frame with

$$q^\mu=-\kappa_T X^\mu, \qquad j^\mu=\sigma X^\mu,$$

for one transverse thermodynamic force $X^\mu$. Moving to Landau frame uses

$$\delta u_f^\mu=-\frac{\kappa_T}{w}X^\mu.$$

The Landau-frame current becomes

$$j_{\mathrm L}^\mu = \left(\sigma+\frac{n\kappa_T}{w}\right)X^\mu = \mathcal J^\mu .$$

The coefficient multiplying $X^\mu$ has changed, but the relative charge flow with respect to energy flow has not. A comparison that quotes $\sigma$ in one frame and $j_{\mathrm L}/X$ in another without this translation compares different component conventions.

Common pitfalls

Calling the Landau velocity measurable. Detector observables couple to $T^{\mu\nu}$ and $J^\mu$. A Landau velocity is a useful reconstruction from them within a specified regime.

Comparing truncated equations after an incomplete redefinition. Transforming $u^\mu$ but not the second-order terms, sources, and initial data breaks the claimed equivalence.

Inferring that every frame is equally well behaved as an exact PDE. Low-energy observables are frame invariant order by order. Exact characteristic structure depends on the chosen higher-order completion; this distinction is central to BDNK theory.

The dashed vertical branch is the point of this page. Inspect how it leaves the constitutive-tensor box: changing frame redefines $T$, $\mu$, and $u^\mu$ order by order, while the physical stress tensor, current, and transport observables remain fixed to that order.

A frame transformation reshuffles scalar corrections and transverse energy and charge fluxes without changing $T^{\mu\nu}$ or $J^\mu$ through the retained derivative order. This equivalence is perturbative: treating two differently truncated formulations as exact PDEs can give different characteristics. The diagram is schematic and makes no claim that Landau, Eckart, and general frames have identical exact high-frequency behavior.

In text: write the same physical tensors in primed and unprimed variables, expand the field redefinition consistently, and compare invariant combinations rather than raw frame coefficients. Landau and Eckart conditions are matching conventions; neither is an observable definition of the fluid velocity.

Exercise

Show that the Landau-to-Eckart velocity difference is fixed by $\mathcal J^\mu$.

Solution

In Landau frame $q_{\mathrm L}^\mu=0$, so $\mathcal J^\mu=j_{\mathrm L}^\mu$. To reach Eckart frame choose $\delta u^\mu=j_{\mathrm L}^\mu/n$. Hence

$$u_{\mathrm E}^\mu-u_{\mathrm L}^\mu = \frac{\mathcal J^\mu}{n}+O(\partial^2).$$

Conversely, $q_{\mathrm E}^\mu=-w\mathcal J^\mu/n$. Both results reconstruct the same $T^{\mu\nu}$ and $J^\mu$ through first order.

Where this leads

Derivative Expansion and Tensor Decomposition removes frame and equation-of-motion redundancies when constructing a basis. Frame-Invariant Dissipative Data carries the translation into pole and transport comparisons.

References

• Bhattacharyya, Sayantani. 2014. “Entropy Current and Equilibrium Partition Function in Fluid Dynamics.” Journal of High Energy Physics 2014 (8): 165. DOI. Open PDF.

• Kovtun, Pavel. 2019. “First-Order Relativistic Hydrodynamics Is Stable.” Journal of High Energy Physics 2019 (10): 034. DOI. Open PDF.