Causality and stability in relativistic viscous non-resistive magneto-fluid dynamics

Abstract

We investigate the causality and the stability of the relativistic viscous non-resistive magneto-hydrodynamics in the framework of the Israel-Stewart (IS) second-order theory, and also within a modified IS theory which incorporates the effect of magnetic fields in the relaxation equations of the viscous stress. We compute the dispersion relation by perturbing the fluid variables around their equilibrium values. In the ideal magnetohydrodynamics limit, the linear dispersion relation yields the well-known propagating modes: the Alfvén and the magneto-sonic modes. In the presence of bulk viscous pressure, the causality bound is found to be independent of the magnitude of the magnetic field. The same bound also remains true, when we take the full non-linear form of the equation using the method of characteristics. In the presence of shear viscous pressure, the causality bound is independent of the magnitude of the magnetic field for the two magneto-sonic modes. The causality bound for the shear-Alfvén modes, however, depends both on the magnitude and the direction of the propagation. For modified IS theory in the presence of shear viscosity, new non-hydrodynamic modes emerge but the asymptotic causality condition is the same as that of IS. In summary, although the magnetic field does influence the wave propagation in the fluid, the study of the stability and asymptotic causality conditions in the fluid rest frame shows that the fluid remains stable and causal given that they obey certain asymptotic causality condition.

Relativistic non-resistive viscous magnetohydrodynamics from the kinetic theory: a relaxation time approach

“…In ref. [66] we showed that this new theory of second-order relativistic MHD is causal and stable under linear perturbation. In this paper, we derive the RMHD equations for the non-resistive case using Chapman-Enskog expansion of the single-particle distribution function within relaxation time approximation (RTA).…”

Modeling general-relativistic plasmas with collisionless moments and dissipative two-fluid magnetohydrodynamics

“…Different from the original formulation of Israel & Stewart (1979), the approach of Denicol et al (2012) allows for a systematic power counting expansion in terms of inverse Reynolds and Knudsen numbers, whose asymptotic regime agrees with the Chapman-Enskog expansion of the Boltzmann equation (Chapman & Cowling 1990;Denicol et al 2012). The consistent derivation from the Boltzmann-Vlasov equations results in generalized Israel-Stewart-like equations that are likely to possess a causal regime for some range of values of transport coefficients and dissipative fluxes (Biswas et al 2020), and those equations can account for anisotropic pressures as well as heat conduction and bulk viscosities. However, they are formally only valid in the regime of strong and weak collisionality.…”

General-relativistic hydrodynamics of non-perfect fluids: 3+1 conservative formulation and application to viscous black-hole accretion

“…In this way, it was found that stability implies causality and vice-versa, but only if the relaxation times are larger than a certain timescale. These studies were recently extended to include heat flow (Brito & Denicol 2020) and a magnetic field (Biswas et al 2020).…”