Does stability of relativistic dissipative fluid dynamics imply causality?

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(ε + 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.

Relativistic magnetohydrodynamics

“…In conventional hydrodynamics, higher-order terms are required to render the theory causal [28] (see e.g. [2,29] for more recent discussions). We expect that a causal formulation of MHD will involve higher-order relaxation times as well as the electric field dynamics.…”

Nonlinear waves in second order conformal hydrodynamics

“…In [28] the authors suggested that the problem of acausality and instability are correlated in relativistic dissipative hydrodynamics.…”

“…We followed closely the study performed in [28] for the Israel-Stewart theory where the dispersion relation (13) and the decomposition of its modes in real and imaginary parts were considered in the large and small wavenumber limits. In [28] the authors suggested that the problem of acausality and instability are correlated in relativistic dissipative hydrodynamics.…”

“…To study the stability and causality properties of (10), we decompose the roots of (15) in two components as in [21,28]:…”

“…The simplest extension of the well-known Navier-Stokes equations to relativistic fluids is plagued with instabilities and acausal signal propagation in the resulting equations [26][27][28], and, thus, they are not usually employed in numerical simulations. Currently, most fluiddynamical simulations of the QGP employ a set of relaxation-type equations similar to those derived by Israel and Stewart (IS) [29] to close the conservation laws.…”