Causal Viscosity and Heat Conduction

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Relativistic dissipation cannot be added safely by copying instantaneous Fourier and Navier–Stokes laws into a covariant tensor. Those first-order laws make heat and momentum diffuse with infinite mathematical speed and can destabilize equilibrium in relativistic settings. Transient theories instead promote bulk pressure, heat flux, and shear stress to dynamical variables that relax over finite times. The resulting telegraph-like modes can be causal and stable, but only when the full coefficient set satisfies appropriate inequalities.

Required background. Hilbert Stress–Energy from Metric Variation supplies stress decomposition and conservation; Differential Equations supplies diffusion, telegraph equations, stability, and hyperbolicity.

Helpful background. The Einstein Equation supplies gravitational coupling; Relativistic Matter and Compact Objects gives the material context; and Computational Relativity supplies stiff relaxation methods.

Dissipative stress and frame choice

Assume a near-equilibrium, single-fluid state with small dissipative corrections and positive enthalpy density $w=\varepsilon+p$ . Relative to a chosen unit velocity $u^a$ , define $h^{ab}=g^{ab}+u^au^b$ and write

T^{ab}=\varepsilon u^au^b+(p+\Pi)h^{ab} +2u^{(a}q^{b)}+\pi^{ab},

where $q^au_a=0$ , $\pi^{ab}u_b=0$ , and $\pi^a{}_a=0$ . The scalar $\Pi$ is bulk viscous pressure, $q^a$ is heat or energy flux, and $\pi^{ab}$ is shear stress.

The decomposition depends on how $u^a$ is defined. In the Landau frame it is the timelike eigenvector of $T^{ab}$ , so energy flux vanishes and diffusion appears in particle currents. In the Eckart frame it follows a chosen particle current, so $q^a$ can be nonzero. Observable tensors agree only when variables and constitutive coefficients are transformed consistently; “the heat flux” without a frame is incomplete.

Why first-order laws are insufficient

The relativistic first-order constitutive laws have schematic form

\Pi=-\zeta\theta, \qquad q^a=-\kappa h^{ab}(\nabla_bT+Ta_b), \qquad \pi^{ab}=-2\eta\sigma^{ab},

with nonnegative bulk viscosity $\zeta$ , thermal conductivity $\kappa$ , and shear viscosity $\eta$ . Here $D=u^a\nabla_a$ , $\theta=\nabla_au^a$ , $a^a=Du^a$ , and

\sigma^{ab}=h^{ac}h^{bd}\nabla_{(c}u_{d)}-\frac13h^{ab}\theta

is the spatial, symmetric, trace-free velocity gradient. The acceleration term in $q^a$ is essential. In a static metric with lapse $N=\sqrt{-g_{tt}}$ and static matter, $a_i=\partial_i\ln N$ . Zero heat flux therefore requires

\partial_iT+T\partial_i\ln N=0, \qquad TN=\text{constant}.

This is the gravitational equilibrium temperature law: the local temperature need not be spatially uniform. The constitutive laws generate nonnegative entropy near equilibrium, but they respond instantaneously to gradients.

Consider a transverse velocity perturbation $v_y(t,x)$ in homogeneous flat matter. Momentum conservation with Navier–Stokes shear gives

(\varepsilon+p)\partial_t v_y-\eta\partial_x^2v_y=0.

This is a diffusion equation. Its Green function is nonzero at arbitrarily large distance for any $t\gt0$ , so it has no finite signal cone. The transverse mode displayed here decays in the matter rest frame; acausality does not mean that this particular mode grows there. Conventional relativistic Navier–Stokes theory nevertheless lacks stability in every inertial frame: boosting the equilibrium exposes unstable modes. See Rocha et al. 2024, preprint §§ II.G and III.B, Eqs. (34)–(35).

Here “first order” must not be mistaken for a theorem of acausality. The modern BDNK construction uses the freedom to define out-of-equilibrium temperature, chemical potential, and velocity in a general hydrodynamic frame. Its stress tensor and baryon current remain first order in gradients, yet suitable inequalities on the transport and frame coefficients make the coupled Einstein–fluid system causal and strongly hyperbolic without adding independent relaxation variables. Equilibrium stability requires further conditions. See Bemfica, Disconzi, and Noronha 2022, §§ IV–V, Theorems I–II, and § VII, Eq. (48). Derivative order alone does not decide causality; the equations’ principal part, coefficient domain, and variable choice do. The relaxation model below makes the finite-speed mechanism especially transparent.

Relaxation restores finite characteristic speeds

Israel–Stewart-type theories replace algebraic response with evolution, schematically

\tau_\Pi D\Pi+\Pi=-\zeta\theta,

\tau_q h^a{}_bDq^b+q^a =-\kappa h^{ab}(\nabla_bT+Ta_b),

\tau_\pi\Delta^{ab}{}_{cd}D\pi^{cd}+\pi^{ab} =-2\eta\sigma^{ab},

where $\Delta^{ab}{}_{cd}=h^a{}_{(c}h^b{}_{d)}-h^{ab}h_{cd}/3$ projects onto spatial, symmetric, trace-free tensors. Second-order expansion, vorticity, and cross-coupling terms complete the chosen formulation; the displayed equations alone do not specify a full Israel–Stewart model. The positive relaxation times encode how quickly fluxes approach their constitutive values. The entropy-current construction is developed in Rocha et al. 2024, preprint § IV.C, Eqs. (62)–(71).

For the transverse perturbation, take

(\varepsilon+p)\partial_t v_y+\partial_x\pi^{xy}=0, \qquad \tau_\pi\partial_t\pi^{xy}+\pi^{xy}=-\eta\partial_xv_y.

Eliminating $\pi^{xy}$ gives

\tau_\pi(\varepsilon+p)\partial_t^2v_y +(\varepsilon+p)\partial_tv_y -\eta\partial_x^2v_y=0.

The high-frequency shear characteristic speed is

v_{\rm sh}^2=\frac{\eta}{\tau_\pi(\varepsilon+p)}.

It is finite, and causality of this shear subsystem requires $v_{\rm sh}^2\leq1$ . This check also shows that relaxation time cannot be sent to zero at fixed viscosity while retaining a causal high-frequency theory.

There are two modes rather than the diffusion equation’s one. Substituting $v_y\propto e^{-i\omega t+ikx}$ and writing $\nu_\eta=\eta/w$ gives

\tau_\pi\omega^2+i\omega-\nu_\eta k^2=0, \qquad \omega_\pm=\frac{-i\pm\sqrt{4\tau_\pi\nu_\eta k^2-1}}{2\tau_\pi}.

For small $k$ , one mode has $\omega=-i\nu_\eta k^2+O(k^4)$ , while the other has $\omega=-i/\tau_\pi+O(k^2)$ . Both decay for $\eta,\tau_\pi,w\gt0$ . The second mode describes initial relaxation of an independently specified shear stress; eliminating it prematurely produces the instantaneous diffusion approximation.

Stability, limits, and constitutive scope

For frequencies small compared with $\tau_\pi^{-1}$ , the relaxation equation gives $\pi^{ab}\simeq-2\eta\sigma^{ab}$ and recovers Navier–Stokes behavior as a late-time approximation. At high frequency the additional nonhydrodynamic mode matters. A stable causal theory must place that mode in the decaying half-plane and keep every combined sound, heat, bulk, and shear characteristic inside the null cone.

Positive $\eta$ , $\zeta$ , $\kappa$ , and relaxation times are necessary but not sufficient. Even the simple longitudinal sector with bulk and shear relaxation but no heat flux has the characteristic speed

v_{\rm L}^2=c_s^2+\frac{\zeta}{w\tau_\Pi} +\frac{4\eta}{3w\tau_\pi}.

This follows by combining the principal parts of energy conservation, longitudinal momentum conservation, and the two stress-relaxation equations. Its bound $v_{\rm L}^2\leq1$ is stronger than the shear bound alone; compare Rocha et al. 2024, preprint § II.G, Eq. (28). It is still only a condition for this linearized model. Heat coupling, equation-of-state susceptibilities, and nonlinear states can impose additional conditions. A large-wavenumber group velocity by itself is not a general proof of causal propagation; the full initial-value problem must have a causal domain of dependence.

The entropy current in a transient theory contains terms quadratic in dissipative variables. Its divergence is nonnegative to the order of the construction when the evolution equations hold. That perturbative derivation defines a near-equilibrium domain; large $q^a$ or $\pi^{ab}$ can invalidate truncation before a numerical instability appears.

At material interfaces or in numerical evolution, relaxation sources can be stiff when $\tau$ is shorter than the resolved time step. Implicit–explicit methods can handle the timescale but do not repair acausal coefficients. A useful check compares the full dispersion relation with its hydrodynamic small-wavenumber limit and its causal large-wavenumber characteristic speeds.

Exercises

Derive the transverse telegraph equation and identify its Navier–Stokes limit.

Solution

Differentiate momentum conservation in time and use the spatial derivative of the relaxation equation to eliminate $\partial_t\partial_x\pi^{xy}$ . The result is

\tau_\pi(\varepsilon+p)\partial_t^2v_y +(\varepsilon+p)\partial_tv_y -\eta\partial_x^2v_y=0.

For $\tau_\pi\nu_\eta k^2\ll1$ and times long compared with $\tau_\pi$ , the transient mode has decayed and the second time derivative is subleading. Dividing by $\varepsilon+p$ gives the shear diffusion equation with diffusivity $\nu_\eta=\eta/(\varepsilon+p)$ . A late time alone does not justify using diffusion at arbitrarily short wavelengths. The approximation loses the finite high-frequency signal cone.

References

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