Relativistic Dissipative Hydrodynamics

First-order dissipative hydrodynamics adds shear and bulk stress plus charge/heat transport to the ideal constitutive relations. These terms determine the universal low-$k$ attenuation and diffusion coefficients. In conventional Landau form they also make the exact equations parabolic, so their infrared accuracy must be separated from the causal status of a macroscopic initial-value formulation.

Required background. Derivative Expansion and Tensor Decomposition gives the reduced first-order basis. Sound, Shear, and Charge Modes supplies the ideal eigenvectors.

Helpful background. Sources, Linear Response, and Kubo Formulae fixes microscopic transport normalizations.

Parity-even first-order constitutive relations

In Landau frame for one charge,

$$\begin{aligned} T^{\mu\nu} &= \epsilon u^\mu u^\nu+pP^{\mu\nu} +\eta\sigma^{\mu\nu} -\zeta P^{\mu\nu}\theta +O(\partial^2),\\ J^\mu &= n\,u^\mu+\sigma_Q\mathcal V^\mu+O(\partial^2),\\ \mathcal V^\mu &= E^\mu-T\nabla_\perp^\mu(\mu/T). \end{aligned}$$

The definitions of $P^{\mu\nu}$ and $\sigma^{\mu\nu}$ are those in the hydrodynamic frame and tensor reference. In the local rest frame, the spatial viscous stress is

$$\delta T^{ij} = -\eta\left( \partial_i v_j+\partial_j v_i -\frac23\delta_{ij}\boldsymbol\nabla\!\cdot\!\mathbf v \right) -\zeta\delta_{ij}\boldsymbol\nabla\!\cdot\!\mathbf v.$$

The coefficient $\sigma_Q$ is the charge conductivity relative to energy flow. At finite density, heat and charge currents mix; a single symbol called “thermal conductivity” is convention incomplete until the frame and thermodynamic force are stated.

Shear diffusion

Linearize about homogeneous equilibrium with no sources. For a transverse velocity, energy and charge fluctuations decouple:

$$w\partial_t\mathbf v_\perp-\eta\nabla^2\mathbf v_\perp=0.$$

The Fourier convention gives

$$\omega_{\mathrm{shear}} = -iD_\eta k^2+O(k^4), \qquad D_\eta=\frac{\eta}{w}.$$

For $w>0$ and $\eta\ge0$, the pole lies in the lower half-plane. This checks the sign of the stress and the normalization of $\eta$.

Sound attenuation

For a neutral fluid, the longitudinal linear equations are

$$\partial_t\delta\epsilon+w\boldsymbol\nabla\!\cdot\!\mathbf v=0,$$ $$w\partial_t\mathbf v+\boldsymbol\nabla\delta p -\eta\nabla^2\mathbf v -\left(\zeta+\frac13\eta\right) \boldsymbol\nabla(\boldsymbol\nabla\!\cdot\!\mathbf v)=0.$$

With $\delta p=c_s^2\delta\epsilon$, their determinant is

$$\omega^2-c_s^2k^2 +i\omega k^2 \frac{\zeta+\frac43\eta}{w}=0.$$

Expanding at small $k$ gives

$$\omega_\pm = \pm c_sk -\frac{i}{2}\Gamma_s k^2 +O(k^3), \qquad \Gamma_s=\frac{\zeta+\frac43\eta}{w}.$$

Sound stability constrains the combination $\zeta+4\eta/3$. The local second law constrains shear and bulk channels separately, as the next page shows.

Charge diffusion

At $\mu=0$ in a charge-conjugation-symmetric state, charge decouples from energy and momentum. Since

$$\delta n=\chi\,\delta\mu, \qquad \mathbf j=-\sigma_Q\boldsymbol\nabla\delta\mu,$$

charge conservation gives

$$\omega_{\mathrm{charge}} = -iD_Qk^2+O(k^4), \qquad D_Q=\frac{\sigma_Q}{\chi}.$$

This is the Einstein relation. At finite density, the scalar ideal zero mode mixes energy and charge. The diffusion constant is then an eigenvalue of the susceptibility-weighted thermoelectric transport matrix, not generally $\sigma_Q/\chi_{nn}$.

Kovtun derives the neutral sound/shear results and the finite-density response structure with consistent retarded conventions Kovtun 2012, §§2.2–2.5, pp. 22–35, Open PDF.

Relativistic hydrodynamic consistency reference

This canonical table states what each framework does and does not establish. “Conditional” means that explicit coefficient, state, and regularity hypotheses are required; it never means automatic.

Framework	Variables and organization	Entropy/Onsager status	Linear stability	Characteristic causality and hyperbolicity	Nonlinear theorem status	Controlled interpretation
Ideal relativistic fluid	$T,\mu,u^\mu$; zeroth-derivative $T,J$	Entropy advected for smooth flow	Stable if thermodynamic Hessian is positive and $0\le c_s^2$	Symmetric/strongly hyperbolic and causal for a regular EOS with $0<c_s^2\le1$	Local smooth theory established; shocks require weak solutions and entropy admissibility	Long wavelengths before dissipative corrections matter
Conventional Landau first order	Same fields; algebraic viscous stress and diffusion	Nonnegative local production for $\eta,\zeta,\sigma_Q\ge0$	Rest-frame infrared modes damp; generic boosted formulation is unstable	Parabolic sectors have instantaneous support; not a causal hyperbolic PDE	No causal relativistic initial-value completion as written	Correct low-$k$ constitutive expansion, not UV evolution
Conventional Eckart first order	Velocity tied to charge flow; algebraic heat flux includes acceleration	Can satisfy local production inequality	Generic exponentially growing mode even about equilibrium	Parabolic/acausal; acceleration heat law worsens initial-value structure	No causal stable completion as written	Historical frame; not a viable exact evolution system
Müller–Israel–Stewart family	Promotes bulk, shear, and/or heat stresses to transient variables	Conditional on entropy-current ansatz and coefficient signs	Conditional; hydrodynamic plus nonhydrodynamic modes	Conditional inequalities on relaxation times and state; no guarantee from the name	Established only for particular sectors/variants and hypotheses	Transient completion whose coefficients must be specified
BRSSS	Conformal Landau-frame constitutive basis through second order; often embedded in a relaxation equation	Conformal nondissipative/dissipative basis constrained	Depends on chosen transient completion	Original gradient data alone do not define a hyperbolic PDE	Variant dependent	Universal second-order conformal data, not synonymous with MIS
DNMR	Kinetic moment closure with Knudsen and inverse-Reynolds counting; transient stresses	Inherits kinetic collision/closure assumptions	Conditional on collision model and coefficients	Conditional; not implied by moment truncation alone	Variant and sector dependent	Microscopic kinetic closure, with model-specific coefficient relations
BDNK	$T,\mu,u^\mu$ only; general-frame first-derivative $T,J$, yielding higher-time-derivative PDEs	Nonnegative production in its EFT regime under transport conditions	Conditional inequalities give rest and boosted stability	Strongly hyperbolic and causal under explicit strict inequalities	Local well-posedness proved under stated EOS, regularity, coefficient, and boundary assumptions	Causal first-order general-frame theory; not every first-order frame

The table is an evidence ceiling, not a ranking. A named implementation must still present its coefficient domain and principal symbol. The BDNK theorem row refers to the explicit nonlinear conditions of Bemfica, Disconzi, and Noronha Bemfica, Disconzi, and Noronha 2022, Theorems I–II and §§IV–V, pp. 18–24, Open PDF.

Constitutive order versus PDE order

“First order” means that $T^{\mu\nu}$ and $J^\mu$ contain one derivative. Substitution into conservation can produce two spacetime derivatives. In Landau frame the leading spatial derivatives give diffusion equations; in a suitable general frame, time-derivative terms in the constitutive tensors can instead change the principal symbol. Conversely, a second-order constitutive theory is not automatically second order in time after a particular transient completion.

The derivative expansion controls the small-$\omega$, small-$k$ series. Treating a finite truncation as an exact PDE adds a separate completion choice that must be tested; the constitutive-versus-evolution distinction is emphasized throughout Romatschke and Romatschke 2019, chs. 2–4.

The comparison below separates a formulation label from the question that must actually be tested. For this page, inspect the first column and then scan across the lower row: shared infrared transport does not fix relaxation variables, conformal closure, frame choice, or hyperbolicity.

First-order attenuation coefficients are infrared constitutive data; the exact initial-value behavior also depends on variables, frame terms, relaxation equations, and higher-order completion. Each formulation must therefore be tested for its own modes, principal symbol, coefficient domain, and EFT range. The diagram is schematic; its columns are a comparison of alternatives, not a derivation of one formulation from another.

In text: conventional Landau-frame first order gives the universal shear, sound, and charge poles but a parabolic exact truncation; transient theories add independent relaxation variables; BRSSS organizes conformal gradient data; DNMR imports a kinetic closure; and BDNK changes general-frame principal terms. Agreement at small $k$ does not identify their high-frequency PDEs.

Exercise

Derive the shear pole and check dimensions.

Solution

For $\mathbf k=k\widehat x$ and velocity $v_y$, momentum conservation gives

$$-i\omega w\,v_y+\eta k^2v_y=0.$$

Thus $\omega=-i(\eta/w)k^2$. In natural units, $[\eta]=[\text{energy density}]\,[\text{time}]$, so $\eta/w$ has dimensions of time or length and $(\eta/w)k^2$ has dimensions of inverse time. Positivity of $\eta$ places the pole below the real axis.

Where this leads

Onsager Reciprocity and Entropy Production derives the transport inequalities. Conventional Relativistic Navier–Stokes Instability and Acausality explains why these stable infrared poles do not make the conventional exact PDE causal.

References

• Bemfica, Fábio S., Marcelo M. Disconzi, and Jorge Noronha. 2022. “First-Order General-Relativistic Viscous Fluid Dynamics.” Physical Review X 12: 021044. DOI. Open PDF.

• Kovtun, Pavel. 2012. “Lectures on Hydrodynamic Fluctuations in Relativistic Theories.” Journal of Physics A: Mathematical and Theoretical 45: 473001. DOI. Open PDF.

• Romatschke, Paul, and Ulrike Romatschke. 2019. Relativistic Fluid Dynamics In and Out of Equilibrium. Cambridge University Press. DOI.