标题： 希尔伯特的能量-动量张量扩展 显示英文标题

标题： Hilbert's energy-momentum tensor extended

摘要： 变分导数相对于度规张量在经典场模型中用于定义物质场的希尔伯特能动张量。 在固态物理学中，基本场变量之间的本构关系是由各种各样的模型涵盖的主题。 在此背景下，高阶本构张量代替了二阶度规张量。 对于引力和电动力学的经典场模型，最近提出了类似的预度规描述与线性本构关系。 本文分析了希尔伯特能量-动量张量定义向具有广义线性本构定律的模型的扩展。 微分形式对于微分流形上积分的协变处理是必需的。 拉格朗日量、电磁流和能量-动量流都必须分别表示为扭曲的四形式、三形式和矢量值三形式。 对于形式上的任意线性映射，我们推导出一个交换变分恒等式，该恒等式允许直接变分程序而无需处理各个分量。 通过将线性映射限制为广义霍奇对偶映射（本构定律），可以在任意维度处理麦克斯韦型拉格朗日量。 作为相对于余框架场的拉格朗日量变分导数定义的希尔伯特能动张量，在微分形式中推导出来。 证明了交换变分恒等式与能量-动量张量的显式形式密切相关。 此构造应用于具有广义线性本构定律的各种场模型。 显示英文摘要

摘要： A variational derivative of a Lagrangian with regard to the metric tensor is used in classical field models to define Hilbert's energy-momentum tensor for a matter field. In solid-state physics, constitutive relationships between fundamental field variables are a topic that is covered by a broad variety of models. In this context, a constitutive tensor of higher order replaces the of the second-order metric tensor. For the classical field models of gravity and electrodynamics, a similar premetric description with a linear constitutive relation has recently presented. In this paper, we analyze the extension of the Hilbert definition of the energy-momentum tensor to models with general linear constitutive law. Differential forms are required for the covariant treatment of integrals on a differential manifold. The Lagrangian, electromagnetic current, and energy-momentum current must all be represented as twisted 4-forms, 3-forms, and vector-valued 3-forms, respectively. For an arbitrary linear map on forms, we derive a commutative variation identity that allows direct variation procedures without having to deal with the individual components. One can deal with Maxwell-type Lagrangians in any dimension by restricting the linear map to the generalized Hodge dual map (constitutive law). The Hilbert energy-momentum current, which is defined as a variation derivative of the Lagrangian with regard to a coframe field, is derived in differential form. It is demonstrated that the commutative variation identity is closely connected to the explicit form of the energy-momentum current. This construction is applied to a number of field models having a general linear constitutive law.