本文提出 IS 算法系统构造有限密度带电流体至三阶梯度的相对论本构关系，并通过参考系不变系数在 Landau 系中导出剪切、声和扩散模式色散关系。
摘要：
水的运动由 Navier–Stokes 方程支配，连续性方程对其形成补充，以确保局域质量守恒。在本工作中，我们针对弯曲 d 维时空中具有守恒荷的流体，通过梯度展开构造这些方程的相对论推广。我们采用一般流体动力学参考系方法，并引入 Irreducible-Structure（IS，不可约结构）算法，该算法基于膨胀标量以及剪切张量和涡度张量的导数。通过这种方法，我们系统地生成直至指定阶数的所有允许梯度，并导出带电流体最全面的本构关系，其精度达到三阶梯度。这些本构关系被表述为适用于普通、非共形以及共形不变的带电流体。此外，我们在梯度展开至三阶的范围内，考察了非共形带电流体输运系数对流体动力学参考系的依赖。本构关系标量、矢量和张量部分的参考系依赖，以基本流体动力学变量的场重定义来表示。由于其非线性特征，处理这些参考系依赖具有挑战性。然而，在线性区域，这些高阶变换变得易于处理，从而能够识别出一组参考系不变系数。使用这些系数的一个优势是可以在任意选定参考系中研究线性运动方程，因此我们将这一方法应用于 Landau 参考系。随后，这些线性方程在动量空间中求解，得到非共形带电流体剪切、声和扩散模式的色散关系，并以参考系不变的输运系数表示。
英文摘要：
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.
Two theorems for the gradient expansion of relativistic hydrodynamics
这封信是专门提供证明的两个声明有关的梯度展开的相对论流体力学。第一种说法是\textit{the ordering of transverse derivatives is irrelevant in the gradient expansion of a non-conformal fluid}。第二种说法是\textit{the longitudinal projection of the Weyl covariant derivative can be eliminated in the gradient expansion of a conformal fluid}。这第二个陈述不适用于曲率张量。在共形的情况下，我们知道Weyl协变导数的排序在梯度展开中是无关的。
