Does stability of relativistic dissipative fluid dynamics imply causality?

TL;DR

The paper analyzes causality and stability in relativistic dissipative fluid dynamics without conserved charges by performing linear stability analyses in the rest frame and in Lorentz-boosted frames. It finds that rest-frame second-order equations are stable, but a finite-wavenumber divergence of the shear-mode group velocity emerges; this acausal feature is prevented in the full theory only if an asymptotic causality condition, $\frac{\Gamma_s}{\tau_{\pi}} \le 1- c_s^2$ (equivalently $\frac{1}{b} \le \frac{D-1}{2(D-2)}(1-c_s^2)$), holds. In boosted frames, stability and causality again depend on this condition, with superluminal group velocities allowed only within finite $k$-ranges and ruled out as long as the asymptotic criterion is satisfied; the study also derives characteristic velocities showing they remain subluminal under the same constraint. Overall, second-order theories are not automatically causal and require careful tuning of $\tau_{\pi}$ relative to the sound attenuation length $\Gamma_s$; the limit $\tau_{\pi}\to 0$ (Navier-Stokes) is inherently acausal.

Abstract

We investigate the causality and stability of relativistic dissipative fluid dynamics in the absence of conserved charges. We perform a linear stability analysis in the rest frame of the fluid and find that the equations of relativistic dissipative fluid dynamics are always stable. We then perform a linear stability analysis in a Lorentz-boosted frame. Provided that the ratio of the relaxation time for the shear stress tensor, $τ_π$, to the sound attenuation length, $Γ_s = 4η/3(\varepsilon+P)$, fulfills a certain asymptotic causality condition, the equations of motion give rise to stable solutions. Although the group velocity associated with perturbations may exceed the velocity of light in a certain finite range of wavenumbers, we demonstrate that this does not violate causality, as long as the asymptotic causality condition is fulfilled. Finally, we compute the characteristic velocities and show that they remain below the velocity of light if the ratio $τ_π/Γ_s$ fulfills the asymptotic causality condition.

Figures (10)

• Figure 1: The real parts (left panel) and the imaginary parts (right panel) of the dispersion relations for the sound modes (full lines) and the nonpropagating mode (dashed line) obtained from Eq. (\ref{['eqn:rest_eq3']}). The parameters are $a=\frac{1}{4\pi}\,,\; b=6\,,\; c_{s}^{2}=\frac{1}{3}$ for the 3+1-dimensional case, $D=4$.
• Figure 2: The group velocity (\ref{['vg']}) for $a= 1/(4 \pi)\, , \;D=4\, , \; c_{s}^{2}=\frac{1}{3}$, and $b=6$ (full line), $b=2$ (dashed line), as well as $b=1.5$ (dotted line).
• Figure 3: The real parts (left panel) and the imaginary parts (right panel) of the dispersion relations for the shear modes obtained from Eq. (\ref{['eqn:rest_eq2']}). The parameters are $a=\frac{1}{4\pi}\,,\;b=6\,,\; c_{s}^{2}=\frac{1}{3}$ for the 3+1-dimensional case, $D=4$.
• Figure 4: The group velocity (\ref{['vg']}) for $D=4\, , \; b=6\,,\; c_{s}^{2}=\frac{1}{3}$, and $a= 1/(4 \pi)$ (full line), $a=1/4$ (dashed line), as well as $a=1$ (dotted line).
• Figure 5: The real parts (left panel) and the imaginary parts (right panel) of the dispersion relations for the sound modes obtained from Eq. (\ref{['eqn:rest_eq3']}). The parameters are $a=\frac{1}{4\pi}\,,\;b=1\,,\; c_{s}^{2}=\frac{1}{3}$ for the 3+1-dimensional case, $D=4$.