Causality in Strong Shear Flows

Abstract

It is well known that the standard transport equations violate causality when gradients are large or when temporal variations are rapid. We derive a modified set of transport equations that satisfy causality. These equations are obtained from the underlying Boltzmann equation. We use a simple model for particle collisions which enables us to derive moment equations non-perturbatively, i.e. without making the usual assumption that the distribution function deviates only slightly from its equilibrium value. We apply the model to two problems: particle diffusion and viscous transport. In both cases we show that signals propagate at a finite speed and therefore that the formalism obeys causality. When the velocity gradient is large on the scale of a mean free path, the viscous shear stress is suppressed relative to the prediction of the standard diffusion approximation. The shear stress reaches a maximum at a finite value of the shear amplitude and then decreases as the velocity gradient increases. In the case of a steady Keplerian accretion disk with hydrodynamic turbulent viscosity, the stress-limit translates to an upper bound on the Shakura-Sunyaev $\alpha$-parameter, namely $\alpha<0.07$. The limit on $\alpha$ is much stronger in narrow boundary layers where the velocity shear is larger than Keplerian.