Two theorems for the gradient expansion of relativistic hydrodynamics

TL;DR

This paper proves two key theorems about the gradient expansion in relativistic hydrodynamics, showing the irrelevance of derivative ordering in certain cases and simplifying the expansion for conformal fluids.

Contribution

It provides rigorous proofs for the irrelevance of derivative ordering and the elimination of certain derivatives in the gradient expansion of relativistic hydrodynamics.

Findings

Ordering of transverse derivatives is irrelevant in non-conformal fluids

Longitudinal projection of Weyl covariant derivative can be eliminated in conformal fluids

These results simplify the mathematical structure of hydrodynamic gradient expansions

Abstract

This letter is dedicated to providing proof of two statements concerning the gradient expansion of relativistic hydrodynamics. The first statement is that \textit{the ordering of transverse derivatives is irrelevant in the gradient expansion of a non-conformal fluid}. The second statement is that \textit{the longitudinal projection of the Weyl covariant derivative can be eliminated in the gradient expansion of a conformal fluid}. This second statement does not apply to curvature tensors. In the conformal case, we know that the ordering of Weyl covariant derivatives is irrelevant in the gradient expansion.