Hydrodynamic Variables, Frames, and Ideal Modes

Hydrodynamics is the long-distance theory of conserved densities and any other modes whose relaxation is parametrically slow. This chapter identifies those variables, couples them to sources, separates hydrostatic information from dynamical constitutive data, and derives the ideal relativistic modes. Its central discipline is simple: temperature, chemical potential, and velocity are coordinates on the space of local states, whereas the stress tensor, currents, response functions, and pole locations are physical.

Helpful background. The Hydrodynamic Limit and Slow Variables gives the physical entry point. Hydrodynamic Effective-Theory Architecture explains why conservation, rather than canonical operator dimension, controls the expansion.

Enter this chapter

The chapter uses the site metric $g_{\mu\nu}=\operatorname{diag}(1,-1,-1,-1)$ and Fourier pair

$$\widetilde f(p)=\int \mathrm d^d x\,e^{+ip\cdot x}f(x), \qquad f(x)=\int\frac{\mathrm d^d p}{(2\pi)^d}e^{-ip\cdot x}\widetilde f(p).$$

Thus a linear mode is written $e^{-i\omega t+i\mathbf k\cdot\mathbf x}$, so stability means $\operatorname{Im}\omega\le 0$. The velocity satisfies $u^\mu u_\mu=1$, and

$$P^{\mu\nu}=u^\mu u^\nu-g^{\mu\nu}$$

is the positive rest-space metric: in the local rest frame $P^{ij}=\delta^{ij}$. These choices fix every sign in the decompositions and mode calculations that follow.

Hydrodynamic reasoning has four logically distinct steps:

1. identify the complete slow-variable set and the scale hierarchy;
2. write source-covariant conservation laws and constitutive maps;
3. quotient field-definition and equation-of-motion redundancies at a fixed derivative order;
4. test the resulting modes, thermodynamics, and domain of validity.

Skipping the first step can hide a critical mode, Goldstone field, nearly conserved charge, or integrable tower. Skipping the third can make two frames appear to describe different physics.

The chapter’s relativistic-fluid conventions and mode hierarchy follow the treatments in Kovtun 2012, §§2.1–2.4, Open PDF and Rezzolla and Zanotti 2013, chs. 2–4.

Route through the chapter

Page	Question answered	Result to carry forward
The Hydrodynamic Limit and Slow Variables	Why do conserved densities become slow?	A scale-separation and completeness test for the hydrodynamic field content
Conservation Laws and Hydrodynamic Fields	How are stress, charge, and source forces represented?	Source-covariant Ward identities and the canonical frame/tensor reference
Local Equilibrium and Hydrostatic Constraints	What does equilibrium fix before dissipation?	Thermal-vector stationarity and generating-functional constraints
Hydrodynamic Frames and Constitutive Data	Which variable definitions are conventional?	Order-by-order field redefinitions and invariant current data
Derivative Expansion and Tensor Decomposition	Which first-derivative structures are independent?	A reduced scalar/vector/tensor basis
Ideal Relativistic Hydrodynamics	What nonlinear equations follow at zeroth derivative order?	Relativistic Euler equations and their thermodynamic checks
Sound, Shear, and Charge Modes	Which ideal modes propagate or remain degenerate?	Sound speed, eigenvectors, zero-mode counting, and stability conditions

Readers mainly interested in causal dissipative evolution should still read the frame page and the ideal-mode calculation before continuing to Relativistic Dissipation, Transients, Stability, and Causality. A claim about stability or causality is otherwise liable to confuse a variable convention, a low-$k$ pole, and a property of the full initial-value problem.

A hydrodynamic theory card

Before using any constitutive equation, record:

Entry	Required statement
State	Equilibrium or controlled background, equation of state, and unbroken symmetries
Slow fields	Every exactly or parametrically conserved density, Goldstone mode, and critical variable retained
Hierarchy	$\omega\tau_{\mathrm{micro}}\ll1$, $k\ell_{\mathrm{micro}}\ll1$, and any small relaxation rate
Frame	Definitions of $T$, $\mu$, and $u^\mu$ through the retained order
Constitutive order	Gradient, amplitude, fluctuation, and inverse-Reynolds counting kept
Sources	Background metric, gauge field, anomalies, and boundary conditions
Inputs	Equation of state, susceptibilities, and transport coefficients with normalization
Checks	Ward identities, thermodynamic stability, frame translation, mode spectrum, and cutoff sensitivity
Stop rule	An omitted mode becomes slow, gradients approach the microscopic scale, or the state changes

The card is intentionally observable-facing. A named formalism is not enough: two implementations with different mode sets, matching conditions, or cutoff domains need not make the same prediction.

Review the chapter

By the end, you should be able to decompose $T^{\mu\nu}$ and $J^\mu$ relative to an arbitrary timelike $u^\mu$, derive the ideal Euler equations, translate a constitutive relation between frames without changing observables, enumerate the parity-even first-order basis, and diagonalize the ideal charged-fluid mode matrix. You should also be able to identify what those results do not establish: hydrostatics does not determine dissipative evolution, entropy advection does not survive generic viscosity, and real sound speed does not by itself prove a causal dissipative theory.

References

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

• Rezzolla, Luciano, and Olindo Zanotti. 2013. Relativistic Hydrodynamics. Oxford University Press. DOI.