Challenges in Solving Chiral Hydrodynamics*

We show that the widely known theory of ideal chiral hydrodynamics, as derived from chiral kinetic theory, is acausal and its initial-value problem is ill-posed. Having an ill-posed initial-value problem means that the solution of the partial differential equations does not exist or is not unique, which implies that the system cannot be solved and numerical simulations cannot be performed. Therefore, such theory cannot be used to determine how the chiral anomaly affects the hydrodynamic evolution of the quark-gluon plasma formed in ultrarelativistic heavy-ion collisions. We also demonstrate that the standard first-order theory of viscous chiral hydrodynamics is acausal. We show that these fundamental issues can be cured by using different definitions (frames) for the hydrodynamic fields.