Local univalence versus stability and causality in hydrodynamic models

Abstract

Our main objective is to compare the analytic properties of hydrodynamic series with the stability and causality conditions applied to hydrodynamic modes. Analyticity, in this context, implies that the hydrodynamic series behaves as a univalent or single-valued function. Stability and causality adhere to physical constraints where hydrodynamic modes neither exhibit exponential growth nor travel faster than the speed of light. Through an examination of various hydrodynamic models, such as the Muller-Israel-Stewart (MIS) and the first-order hydro models like the BDNK (Bemfica-Disconzi-Noronha-Kovtun) model, we observe no new restrictions stemming from the analyticity limits in the shear channel of these models. However, local univalence is maintained in the sound channel of these models despite the global divergence of the hydrodynamic series. Notably, differences in the sound equations between the MIS and BDNK models lead to distinct analyticity limits. The MIS model's sound mode remains univalent at high momenta within a specific transport range. Conversely, in the BDNK model, the univalence of the sound mode extends to intermediate momenta across all stable and causal regions. Generally, the convergence radius is independent of univalence and the given dispersion relation predominantly influences their correlation. For second-order frequency dispersions, the relationship is precise, i.e. within the convergence radius, the hydro series demonstrates univalence. However, with higher-order dispersions, the hydro series is locally univalent within certain transport regions, which may fall within or outside the stable and causal zones.