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

TL;DR

The paper tackles the problem of secular growth in solving the initial-value problem for relativistic hydrodynamics with dissipation by formulating a covariant, order-by-order gradient-expansion resummation that preserves conservation laws. The authors demonstrate the method on diffusion to all orders and show that at first order it reproduces BDNK as an intermediate step, with the full solution obtained by computing higher-order corrections that cancel transient, non-hydrodynamic modes; the framework is then extended to second order and argued to generalize to arbitrary gradient order. The key contributions are a detailed all-orders diffusion example, a first-order hydrodynamics formulation that clarifies the role of frame choices, and a scalable higher-order scheme with explicit resummation rules and currents, all without introducing extra degrees of freedom or initial data. The approach yields a causal, well-posed evolution for relativistic fluids, with direct applicability to quark-gluon plasma modeling and neutron-star mergers, and provides a path to integrating fluctuations and gravity within the same EFT mindset.

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.