The general relativistic theory of elasticity is reviewed from a Lagrangian, as opposed to Eulerian, perspective. The equations of motion and stress–energy–momentum tensor for a hyperelastic body are derived from the gauge–invariant action principle first considered by DeWitt. This action is a natural extension of the action for a single relativistic particle. The central object in the Lagrangian treatment is the Landau–Lifshitz radar metric, which is the relativistic version of the right Cauchy–Green deformation tensor. We also introduce relativistic definitions of the deformation gradient, Green strain, and first and second Piola–Kirchhoff stress tensors. A gauge-fixed description of relativistic hyperelasticity is also presented, and the nonrelativistic theory is derived in the limit as the speed of light becomes infinite.

广义相对论弹性理论是从拉格朗日而不是欧拉的角度来回顾的。超弹性体的运动方程和应力-能量-动量张量源自 DeWitt 首先考虑的规范不变作用原理。这个动作是单个相对论粒子的动作的自然延伸。拉格朗日处理的中心对象是 Landau-Lifshitz 雷达度量，它是右柯西-格林变形张量的相对论版本。我们还介绍了变形梯度、格林应变以及第一和第二 Piola-Kirchhoff 应力张量的相对论定义。还提出了相对论超弹性的规范固定描述，并在光速变得无限时在极限中导出了非相对论理论。