First-order relativistic hydrodynamics is stable

TL;DR

The paper demonstrates that first-order relativistic viscous hydrodynamics can be linearly stable in a broad class of frames outside the traditional Landau–Lifshitz and Eckart choices. By analyzing the most general one-derivative constitutive relations and performing a linear stability analysis in the uncharged case, it identifies a finite region in the transport-coefficient space where all perturbations decay and causality is preserved. The work shows that stability hinges on nonzero relaxation-like coefficients (e.g., $\varepsilon_1$, $\theta$, $\pi_1$) and provides explicit inequalities—particularly in shear and sound channels—and demonstrates that conformal theories admit a simplified stability criterion. Overall, stable, physical relativistic hydrodynamics can be achieved without introducing extra dynamical fields, suggesting a viable ultraviolet regulator via frame choice rather than extended theories like Israel–Stewart.

Abstract

We study linearized stability in first-order relativistic viscous hydrodynamics in the most general frame. There is a region in the parameter space of transport coefficients where the perturbations of the equilibrium state are stable. This defines a class of stable frames, with the Landau-Lifshitz frame falling outside the class. The existence of stable frames suggests that viscous relativistic fluids may admit a sensible hydrodynamic description in terms of temperature, fluid velocity, and the chemical potential only, i.e. in terms of the same hydrodynamic variables as non-relativistic fluids. Alternatively, it suggests that the Israel-Stewart and similar constructions may be unnecessary for a sensible relativistic hydrodynamic theory.

Figures (5)

• Figure 1: Phase velocity $c_v(\phi)$ for waves with a linear dispersion relation in a moving fluid, shown for different angles $0{\leqslant}\phi{\leqslant}\pi$ of the wave vector with respect to ${\bf v}_0$. Each colour corresponds to a given value of $\phi$. In the plot, the wave speed in the fluid at rest is taken $c_0=\frac{1}{2}$.
• Figure 2: Real and imaginary parts of the shear channel eigenfrequencies, shown for $v_0=0.9$ and $\theta/\eta=2$, for different angles $0\leqslant\phi\leqslant\pi/2$ of the wave vector ${\bf k}$ with respect to ${\bf v}_0$. Each colour corresponds to a given value of $\phi$ (blue corresponds to $\phi=0$, purple to $\phi=\pi/2$), $k\equiv |{\bf k}|$, and $w_0\equiv(\epsilon_0{+}p_0)$. At small $k$, the gapless and the gapped modes (\ref{['eq:wshear-gapless']}), (\ref{['eq:wshear-gapped']}) are clearly visible. At large $k$, the modes follow a linear dispersion relation $\omega = c_{\rm shear}(\phi)k$, with the velocity $c_{\rm shear}(\phi)$ determined by Eq. (\ref{['eq:cs-shear']}). The dashed lines denote the light cone $\omega=\pm k$.
• Figure 3: Constraints on the transport coefficients $\theta$ and $\varepsilon_1$ obtained by demanding that the sound-channel modes for the fluid at rest are stable and causal. For illustrative purposes, we have taken $\varepsilon_2=0$, $\pi_1/\gamma_s = 3/v_s^2$. The stability region is shaded with a colour corresponding to a given value of $v_s$. The stability region is larger for smaller $v_s$. Left: the region where all modes are stable. Right: the region where all modes are stable and the short-wavelength modes are causal, $\lim_{k\to\infty}|\omega(k)/k|<1$. In the right plot, the origin $\varepsilon_1=\theta=0$ is always outside the stability region.
• Figure 4: Sound velocities $c_s$ and sound damping coefficients $\Gamma_s$ for the two branches of sound in a moving fluid, shown as functions of the angle $\phi$ between ${\bf k}$ and ${\bf v}_0$. The damping coefficient $\Gamma_s$ is plotted in units of $\gamma_s/(\epsilon_0 {+}p_0)$. Each colour corresponds to a given branch, with $c_s$ shown by solid curves, and $\Gamma_s$ by dashed curves. For illustrative purposes, we have taken the speed of sound for the fluid at rest to be $v_s=1/2$.
• Figure 5: Real and imaginary parts of the sound channel eigenfrequencies, shown for $v_0=0.9$, and different angles $0\leqslant\phi\leqslant\pi/2$ of the wave vector ${\bf k}$ with respect to ${\bf v}_0$. Each colour corresponds to a given value of $\phi$ (blue corresponds to $\phi=0$, purple to $\phi=\pi/2$), $k\equiv |{\bf k}|$, and $w_0\equiv(\epsilon_0{+}p_0)$. For illustrative purposes, the speed of sound is taken as $v_s=0.5$, and the values of the transport coefficients were taken as follows: $v_s^2\varepsilon_1/\gamma_s = 3$, $\theta/\gamma_s=4$, $\varepsilon_2=0$, $\pi_1/\gamma_s = 3/v_s^2$. The dashed lines denote the light cone $\omega=\pm k$. The values of $\varepsilon_1$ and $\theta$ are inside the stability region of Fig. \ref{['fig:soundstability-1']} (right), near the boundary of the stability region.