arXiv2026
In this work, we construct a theoretical framework for relativistic second-order magnetohydrodynamics based on entropy current analysis. The formalism consistently incorporates finite relaxation dynamics of the dissipative fluxes, providing a transient framework suitable for analyzing linear stability and short-wavelength causality. Utilizing linear mode analysis, we investigate the constraints imposed by causality and stability on this anisotropic system. By linearizing the theory around a homogeneous equilibrium state, we identify the magnetosonic, Alfvén, and charge-diffusion sectors, together with their coupling terms. For each sector, we derive asymptotic dispersion relations in both the long-wavelength (small-$k$) and short-wavelength (large-$k$) regimes, validating them against exact numerical roots. Our numerical analysis confirms the accuracy of these asymptotic solutions and uncovers a nontrivial angular dependence, especially near special propagation directions where the ordinary momentum expansion becomes less reliable. By evaluating the large-$k$ behavior of the propagating branches alongside the damping properties of non-hydrodynamic modes, we delineate the corresponding causality constraints. We find that the admissible linear causal domain is governed by the interplay between propagation direction, anisotropic transport coefficients, and relaxation times, with the resulting bounds being intrinsically mode-dependent. These findings provide a systematic theoretical basis for investigating linear stability and short-wavelength causality in relativistic magnetohydrodynamics beyond the first-order approximation.