Hydrodynamic Variables, Frames, and Ideal Modes

Hydrodynamics describes variables that remain slow after microscopic excitations relax. Conservation laws determine their evolution, and constitutive relations connect them to measurable currents and stress; a frame choice only parametrizes that description.

Enter this chapter

Begin with The Hydrodynamic Limit and Slow Variables to identify the slow variables and scale separation of a fluid.

Helpful background. The Hydrodynamic Limit and Slow Variables and Hydrodynamic Effective-Theory Architecture develop the definitions used by the chapter’s opening pages.

Topics

1. The Hydrodynamic Limit and Slow Variables — Why do conserved densities and parametrically slow modes admit a hydrodynamic description at long wavelengths and late times?
2. Conservation Laws and Hydrodynamic Fields — How are energy, momentum, charge, and entropy represented by local hydrodynamic fields and conservation equations in relativistic notation?
3. Local Equilibrium and Hydrostatic Constraints — What is local equilibrium, which stationary source configurations define hydrostatics, and what constitutive data are fixed before entropy production is considered?
4. Hydrodynamic Frames and Constitutive Data — How are constitutive relations organized, and why are temperature, velocity, and chemical-potential definitions frame conventions rather than observables?
5. Derivative Expansion and Tensor Decomposition — How are independent scalar, vector, and tensor constitutive structures classified at a fixed derivative order after equations of motion and frame redundancies are removed?
6. Ideal Relativistic Hydrodynamics — How do local equilibrium and isotropy determine ideal relativistic stress and charge currents, and what nonlinear conservation equations follow?
7. Sound, Shear, and Charge Modes — Which sound, transverse, entropy, and charge modes follow from linearized ideal conservation, and how do thermodynamic stability conditions enter their speeds and residues?

How the ideas fit together

Conservation equations alone do not close a fluid theory. Local equilibrium and an equation of state supply the ideal constitutive relations; linearizing the resulting equations yields propagating sound and nondissipative transverse or charge sectors Rezzolla and Zanotti 2013, chs. 2–4.

Away from equilibrium, temperature, chemical potential, and velocity can be redefined by derivative corrections. This changes the coefficients used to represent the stress tensor and current, but not those physical tensors at consistently retained order. Hydrodynamic frames are therefore choices of variables, not different microscopic fluids.

Where to go next

Continue to Relativistic Dissipation, Transients, Stability, and Causality to connect dissipative stresses to long-wavelength attenuation.

References

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