Convergence of the hydrodynamic gradient expansion in relativistic kinetic theory

TL;DR

This paper proves that in relativistic kinetic theory, the hydrodynamic gradient expansion converges within a finite radius, establishing bounds on shear viscosity based on the non-hydrodynamic sector's gap and scattering properties.

Contribution

It provides a rigorous proof of the convergence of hydrodynamic series and bounds on shear viscosity in relativistic kinetic theory.

Findings

Hydrodynamic dispersion relations have a finite convergence radius.

The radius of convergence for shear waves is at least half the gap size.

A lower bound on the scattering cross-section ensures a gapped non-hydrodynamic sector.

Abstract

We rigorously prove that, in any relativistic kinetic theory whose non-hydrodynamic sector has a finite gap, the Taylor series of all hydrodynamic dispersion relations has a finite radius of convergence. Furthermore, we prove that, for shear waves, such radius of convergence cannot be smaller than $1/2$ times the gap size. Finally, we prove that the non-hydrodynamic sector is gapped whenever the total scattering cross-section (expressed as a function of the energy) is bounded below by a positive non-zero constant. These results, combined with well-established covariant stability criteria, allow us to derive a rigorous upper bound on the shear viscosity of relativistic dilute gases.