Relativistic Hydrodynamics 2025, 11 - Causal Theory

CIT is complicated by the algebraic nature of constitutive equations underlying it, which describes the instantaneous reaction of TD fluxes. The consequences of this is found in the heat-flux law or "Fourier Law": q = -κ∇T. It implies a temperature diffusion, ∂_tT = χ_t∇²T which is a parabolic DEG. It is diagonalizable with divergent eigenvalues. The heat flux changes instantly with changes to temperature gradients. In any realistic physical theory with finite propagation speeds, the relaxation time lapses before the flux has concluded. Dissipative processes then have to make do without relaxation times. The parabolic nature of the Newtonian heat-diffusion becomes evident in combination with the constitutive equations. For a non-perfect fluid at rest in flat spacetime with zero viscous pressure and heat flux, the characteristic equations reduce to (e+p)a_μ+D_μp - 2D_ν(ησ^ν_μ) - 2ηa^νσ_μν = 0 u^ν∇_νe + Θ(e+p) - 2σ^μνσ_μν = 0 For flow only along the x-dimension and perturbation affecting only energy density and fluid velocity, retaining only the linear terms in the perturbed form, what's left is a parabolic evolution for perturbation. ∂_tδu - η/(e₀+p₀)∂²δu = 0 + O(δ²)

Second-order theories are constructed to extend the space of variables by treating dissipative quantities and describing the evolution of the extended fluxes. The Israel-Stewart formulation states that the classical entropy current is too simple, and instead supposes that S^μsρu^μ + R^μ/T R^μ = q^μ - (β₀Π² + β₁q_νq^ν + β₂π_αβπ^αβ)u^μ/2 + α₀Πq^μ + α₁π^μνq_ν From this, a fully expanded expression for T∇_μS^μ extended forces with expressions neglected in the original formulation, involving the gradients of the TD coefficients α_i, β_i. The former can't be ignored a priori if the fluid is non-homogenous. The relaxation times τ_i are obtained by imposition of additional linear relations.

At equilibrium state, introduce the perturbations within "Cowling approximation" (Not including perturbations in space-time metric).

Solving the system will reveal whether characteristic velocities exist for the system.

In case the system is not actually described by a hyperbolic system of equations, the entropy current vector can't necessarily be extended as in the Israel-Stewart formulation. To ensure that this approach is valid, Rational Extended Thermodynamics is constructed to be relativistically covariant, follow the entropy principle and be hyperbolic. It sees constant mass density currents and energy-momentum tensors, as well as ∇_μF^μνσ = I^νσ, where I is the production density tensor and continues these consequences into the formulation of its constitutive functions for S, I and F.