Stable evolution of relativistic hydrodynamics order-by-order in gradients

TL;DR

This paper develops a systematic, covariant framework for solving relativistic hydrodynamics through a gradient expansion, effectively managing secular growth and clarifying the emergence of BDNK equations without extra initial data.

Contribution

It introduces a covariant resummation scheme for relativistic hydrodynamics that handles secular growth and explicitly derives equations at first and second order in the gradient expansion.

Findings

The scheme can be applied at any order in the gradient expansion.

BDNK equations emerge naturally at first order in the Landau frame.

Non-hydrodynamic modes cancel out in the final solutions.

Abstract

We provide a systematic framework for solving the initial value problem for relativistic hydrodynamics formulated as a gradient expansion. Secular growth is handled by a suitable covariant resummation scheme, which reorganises the degrees of freedom at each order in the expansion while preserving the sum. Our scheme can be applied to any order in the gradient expansion; we provide the explicit formulation at first and second orders. When working to first order, we find that the BDNK equations of motion emerge as an intermediate step in a calculation performed in the Landau frame. We show that non-hydrodynamic modes appear only in such intermediate calculations and cancel when evaluating solutions to the required order. Our procedure does not introduce any other fields or require any additional initial data beyond those appearing in the theory of ideal fluids.