Third-order relativistic fluid dynamics at finite density in a general hydrodynamic frame

Abstract

The motion of water is governed by the Navier-Stokes equations, which are complemented by the continuity equation to ensure local mass conservation. In this work, we construct the relativistic generalization of these equations through a gradient expansion for a fluid with conserved charge in a curved $d$-dimensional spacetime. We adopt a general hydrodynamic frame approach and introduce the Irreducible-Structure (IS) algorithm, which is based on derivatives of both the expansion scalar and the shear and vorticity tensors. By this method, we systematically generate all permissible gradients up to a specified order and derive the most comprehensive constitutive relations for a charged fluid, accurate to third-order gradients. These constitutive relations are formulated to apply to ordinary, non-conformal, and conformally invariant charged fluids. Furthermore, we examine the hydrodynamic frame dependence of the transport coefficients for a non-conformal charged fluid up to the third order in the gradient expansion. The frame dependence of the scalar, vector, and tensor parts of the constitutive relations is obtained in terms of the field redefinitions of the fundamental hydrodynamic variables. Managing these frame dependencies is challenging due to their non-linear character. However, in the linear regime, these higher-order transformations become tractable, enabling the identification of a set of frame-invariant coefficients. An advantage of employing these coefficients is the possibility of studying the linear equations of motion in any chosen frame and, hence, we apply this approach to the Landau frame. Subsequently, these linear equations are solved in momentum space, yielding dispersion relations for shear, sound, and diffusive modes for a non-conformal charged fluid, expressed in terms of the frame-invariant transport coefficients.