Relativistic Fluids and Kinetic Theory

Relativistic matter admits two complementary descriptions. Continuum theory follows a few local fields and closes their conservation laws with thermodynamics; kinetic theory follows a distribution on the mass shell and recovers currents and stresses as moments. This chapter develops both from the same covariant conservation principles, identifies when the perfect-fluid reduction is justified, and extends Euler flow to weak solutions when compression forms shocks.

Helpful background. Hilbert Stress–Energy from Metric Variation supplies the common stress-tensor language; The Einstein Equation supplies gravitational coupling; Relativistic Matter and Compact Objects gives the volume map; and Events, Intervals, and Minkowski Geometry fixes four-velocity and mass-shell kinematics.

From local isotropy to evolution

The opening page, Perfect Fluids and Stress–Energy, asks what an observer-independent statement of local isotropy permits. A unit timelike velocity $u^a$ and its orthogonal projector $h^{ab}=g^{ab}+u^a u^b$ split the tensor into

$$T^{ab}=\varepsilon u^a u^b+p h^{ab}.$$

This formula is a constitutive restriction, not a coordinate definition. A boosted observer sees momentum density and anisotropic spatial components even though the rest-frame stress is isotropic. Heat flux, shear stress, magnetic stress, and velocity dispersion mark controlled departures from the perfect model.

Relativistic Euler Equations then projects $\nabla_aT^{ab}=0$ along and orthogonal to $u^a$. The result separates comoving energy balance from acceleration by a spatial pressure gradient. Number conservation and a thermodynamic relation close the smooth system. The derivation also exposes a crucial relativistic fact: the inertial density is $\varepsilon+p$. Andersson and Comer develop this projection-first route in Andersson and Comer 2021, § 5.2, Eqs. (5.11)–(5.25).

Thermodynamics controls the sound cone

Thermodynamics, Sound Speed, and Causality connects the fundamental energy function to the acoustic characteristics. For fixed entropy per baryon and specified composition,

$$c_s^2=\left(\frac{\partial p}{\partial\varepsilon}\right)_{s,Y_i}.$$

Linearizing about a local rest state gives $\omega^2=c_s^2k^2$. Negative $c_s^2$ is a mechanical instability; $c_s^2>1$ places the ideal-fluid sound cone outside the metric light cone. The inequalities $0\leq c_s^2\leq1$ are therefore indispensable, but they do not replace full convexity, susceptibility, or multi-mode characteristic checks. Composition that equilibrates rapidly and composition frozen on the wave timescale generally produce different derivatives.

This page also clarifies which familiar conditions are independent. The dominant energy condition constrains values of $p$ and $\varepsilon$; causal sound propagation constrains a derivative; positive heat capacity constrains a thermal direction. An equation of state can pass one test and fail another. The local sound-wave derivation is given in Andersson and Comer 2021, § 7.2, Eqs. (7.20)–(7.34).

When momentum space cannot be hidden

Relativistic Boltzmann Theory lifts the description from spacetime to the future mass shell. The scalar distribution $f(x,p)$ obeys a Liouville equation with a collision term. Its first moments give the number current and stress tensor, and collision invariants explain why their divergences vanish. Local equilibrium makes those moments perfect-fluid-like; away from equilibrium the unresolved hierarchy contains heat flux, anisotropic stress, and nonthermal structure. The mass-shell and collision constructions are checked against Andréasson 2011, § 1.1 and § 2, while the moment and entropy identities follow Rocha et al. 2024, § V.A, Eqs. (76)–(83).

Einstein–Vlasov Matter takes the collisionless limit and allows the distribution to gravitate. It is the natural model when particles free-stream on macroscopic scales or when several momentum streams occupy one position. A regular phase-space distribution can represent overlapping streams because velocity remains an independent coordinate; this does not guarantee regularity of an exactly cold delta-function limit or exclude gravitational collapse. Its momentum moments change the metric through $G_{ab}=8\pi G T_{ab}$. The coupled formulation and local initial-value statement are given in Andréasson 2011, § 2, Eqs. (25)–(34) and Theorem 1, with the general local Cauchy theorem in Rendall 2002, § 1, Theorem 1.1.

Shocks complete the continuum picture

Smooth Euler evolution is not the whole theory. Compression can steepen into a discontinuity, after which differential equations hold only in the distributional sense. Shocks, Weak Solutions, and Entropy Conditions derives the covariant Rankine–Hugoniot relations and explains why an entropy inequality is needed to reject expansion shocks. Rarefactions, contacts, and shocks form the elementary wave pattern of a Riemann problem.

The same page turns the theory into a falsifiable numerical test. It specifies continuum equations, initial states, equation of state, boundary conditions, an exact relativistic shock-tube reference, positivity and entropy checks, three-level refinement, error norms, and common failure modes. The emphasis is methodological: a sharp-looking profile is not validation. Conserved jump residuals, comparison with an exact solution, and resolution dependence are independent evidence. Martí and Müller give the exact Riemann construction in Martí and Müller 2015, § 8.5, Eqs. (179)–(200) and the canonical blast benchmark in §§ 6.3–6.3.1, Table 3.

Choosing the right description

The chapter’s models are organized by resolved scale, not by prestige. A perfect fluid is efficient when microscopic relaxation is fast and local isotropy is accurate. Dissipative hydrodynamics is needed when small nonequilibrium fluxes matter but a moment expansion remains controlled. Boltzmann theory is needed when the mean free path or angular structure cannot be eliminated. Vlasov theory is appropriate when collisions are negligible but self-gravity is not. Weak solutions are required whenever a continuum model forms discontinuities.

Three checks should accompany any choice. First, identify the conserved currents and the constitutive inputs. Second, compare the mean free path, relaxation time, and wave scales with the macroscopic problem. Third, test a limit in which the answer is known: the local rest frame for tensor projections, Newtonian Euler or Vlasov–Poisson at low speeds, equilibrium moments for fluid closure, or an exact Riemann solution for shocks. These checks keep a convenient approximation from becoming an unexamined assumption.

Review the chapter

A collisionless stellar system develops two counter-streaming populations at one radius. Decide whether dust, a perfect fluid, or Vlasov matter can represent it without discarding essential information.

Solution

Dust and a perfect fluid assign one timelike velocity to each event, so neither can represent two resolved streams without introducing separate components. A Vlasov distribution can have two peaks in momentum at the same position and evolves both along the geodesic flow. Its combined stress tensor may be anisotropic even if each stream is cold. A two-fluid model could also work if interactions and separate closures are specified, but a single perfect fluid cannot.

References

• Andersson, Nils, and Gregory L. Comer. “Relativistic Fluid Dynamics: Physics for Many Different Scales.” Living Reviews in Relativity 24, 3 (2021). DOI.
• Andréasson, Håkan. “The Einstein–Vlasov System/Kinetic Theory.” Living Reviews in Relativity 14, 4 (2011). DOI.
• Martí, José María, and Ewald Müller. “Grid-Based Methods in Relativistic Hydrodynamics and Magnetohydrodynamics.” Living Reviews in Computational Astrophysics 1, 3 (2015). DOI.
• Rendall, Alan D. “The Einstein–Vlasov System.” In The Einstein Equations and the Large Scale Behavior of Gravitational Fields, 231–250. Birkhäuser, 2004. Preprint.
• Rocha, Gabriel S., David Wagner, Gabriel S. Denicol, Jorge Noronha, and Dirk H. Rischke. “Theories of Relativistic Dissipative Fluid Dynamics.” Entropy 26, 189 (2024). DOI. Open preprint.