Does stability of relativistic dissipative fluid dynamics imply causality?

TL;DR

This paper analyzes the stability and causality of relativistic dissipative fluid dynamics, showing that stability is maintained under certain conditions and that superluminal group velocities do not violate causality if specific ratios are satisfied.

Contribution

It establishes a causality condition relating relaxation time and sound attenuation length, ensuring stability and causal consistency in relativistic dissipative fluid equations.

Findings

Relativistic dissipative fluid equations are always stable in the fluid rest frame.

Stability in boosted frames depends on the ratio of relaxation time to sound attenuation length.

Superluminal group velocities do not imply causality violation if the asymptotic causality condition holds.

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, $\tau_\pi$, to the sound attenuation length, $\Gamma_s = 4\eta/3(\varepsilon+P)$, 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 $\tau_\pi/\Gamma_s$ fulfills the asymptotic causality condition.