Formulating viscous hydrodynamics for large velocity gradients

Abstract

Viscous corrections to relativistic hydrodynamics, which are usually formulated for small velocity gradients, have recently been extended from Navier-Stokes formulations to a class of treatments based on Israel-Stewart equations. Israel-Stewart treatments, which treat the spatial components of the stress-energy tensor ${\ensuremath{\tau}}_{\mathit{ij}}$ as dynamical objects, introduce new parameters, such as the relaxation times describing nonequilibrium behavior of the elements ${\ensuremath{\tau}}_{\mathit{ij}}$. By considering linear response theory and entropy constraints, we show how the additional parameters are related to fluctuations of ${\ensuremath{\tau}}_{\mathit{ij}}$. Furthermore, the Israel-Stewart parameters are analyzed for their ability to provide stable and physical solutions for sound waves. Finally, it is shown how these parameters, which are naturally described by correlation functions in real time, might be constrained by lattice calculations, which are based on path-integral formulations in imaginary time.