Nonlinear Causality and Strong Hyperbolicity of Einstein-Israel-Stewart Theories of Transient Relativistic Fluid Dynamics

Abstract

We present the first complete analysis of nonlinear causality and local well-posedness for a very general class of bulk and shear viscous theories of relativistic transient fluid dynamics, which encompasses (i) the original Israel-Stewart theory derived from entropy-current arguments, (ii) approaches derived from kinetic theory, and (iii) resummed gradient-expansion based formulations as particular subcases. Our work establishes, for the first time, simultaneously necessary and sufficient algebraic conditions for causality, alongside sufficient conditions guaranteeing strong hyperbolicity, in the full nonlinear regime. These results are rigorously proven for both systems coupled to Einstein's equations featuring a dynamic metric and on a fixed background, with or without a cosmological constant, and include baryon conservation (in the absence of heat/diffusion currents). The conditions are purely algebraic, require no simplifying spacetime symmetry assumptions or a specific equation of state, and allow all transport coefficients to depend on the dissipative currents. We also demonstrate that the normalization, orthogonality, symmetry, and tracelessness physical constraints on the dynamical variables are properly propagated during the lifetime of the solutions. Our results provide a readily usable toolset with which one can investigate the domain of applicability of relativistic viscous fluid dynamics in numerical and phenomenological studies in heavy-ion collisions and astrophysics.