Linear stability analysis in inhomogeneous equilibrium configurations

We propose a novel method to find local plane-wave solutions of the linearized equations of motion of relativistic hydrodynamics in inhomogeneous equilibrium configurations, i.e., when a fluid in equilibrium is rigidly moving with nonzero thermal vorticity. Our method is based on extending the conserved currents to the tangent bundle, using a type of Wigner transformation. The Wigner-transformed conserved currents can then be Fourier-transformed into the cotangent bundle to obtain the dispersion relations for the space-time dependent eigenfrequencies. We show that the connection between the stability of hydrodynamics and the evolution of plane waves is not as straightforward as in the homogeneous case, namely, it is restricted to the equilibrium-preserving directions in the cotangent bundle. We apply this method to Mueller-Israel-Stewart (MIS) theory and show that the interplay between the bulk viscous pressure and the shear-stress tensor with acceleration and rotation leads to novel modes, as well as modifications of the already known ones. We conclude that, within the domain of applicability, i.e., when boundary effects are negligible and the vorticity is not too large, MIS theory is stable and causal, with the same stability and causality conditions as for homogeneous equilibrium configurations.