Hydrodynamic Gradient Expansion Diverges beyond Bjorken Flow

TL;DR

This paper demonstrates that the hydrodynamic gradient expansion diverges factorially in general nonlinear flows beyond Bjorken flow, highlighting limitations of the expansion and proposing a cutoff method to address divergence.

Contribution

It introduces a simple method to analyze convergence of the hydrodynamic gradient expansion in nonlinear flows and shows divergence beyond Bjorken flow, extending previous results.

Findings

Gradient expansion diverges factorially in nonlinear flows

Bjorken flow is a special case with convergent expansion

Momentum space cutoff can remove divergence

Abstract

The gradient expansion is the fundamental organising principle underlying relativistic hydrodynamics, yet understanding its convergence properties for general nonlinear flows has posed a major challenge. We introduce a simple method to address this question in a class of fluids modelled by Israel-Stewart--type relaxation equations. We apply it to (1+1)-dimensional flows and provide numerical evidence for factorially divergent gradient expansions. This generalises results previously only obtained for (0+1)-dimensional comoving flows, notably Bjorken flow. We also demonstrate that the only known nontrivial case of a convergent hydrodynamic gradient expansion at the nonlinear level relies on Bjorken flow symmetries and becomes factorially divergent as soon as these are relaxed. Finally, we show that factorial divergence can be removed using a momentum space cutoff, which generalises a result obtained earlier in the context of linear response.