论文标题
一般相对论的拉格朗日连续理论 - 第一部分：流体动力学和弹性的变分原理和连接条件减少
General relativistic Lagrangian continuum theories -- Part I: reduced variational principles and junction conditions for hydrodynamics and elasticity
论文作者
论文摘要
我们为一般相对论的连续性理论建立了一个拉格朗日变化框架，该框架允许在相对论环境中通过对称性减少拉格朗日的过程。从相对论粒子的汉密尔顿原理的连续版本开始，我们推断出与时空协方差相关的两类减少的变分原理，这是连续理论的公理或材料协方差的公理，这与系统的特定特性有关，例如同位素同治疗。协方差假设和拉格朗日还原过程是通过使理论对给定材料和时空张量域的依赖来有效提出的，这些材料和时空张量磁场是通过连续性通过推送和倾听的操作传输的。结果表明，当用吉本斯·乌克（GHY）边界项增强时，变异配方也产生了以色列 - darmois连接条件之间的溶液之间的溶液在相对论持续图的内部和描述外部产生的重力场的溶液之间。 GHY项相对于超表面的第一个变化的表达涉及我们在本文中也得出的先前结果的一些扩展。我们详细考虑了将变分框架应用于相对论流体和相对论弹性的应用。对于后一种情况，我们的设置还允许基于相对论右cauchy-green张量或相对论的cauchy变形张量阐明相对论弹性的关系。此处开发的设置将进一步利用，以在本文的后续部分中进行建模。
We establish a Lagrangian variational framework for general relativistic continuum theories that permits the development of the process of Lagrangian reduction by symmetry in the relativistic context. Starting with a continuum version of the Hamilton principle for the relativistic particle, we deduce two classes of reduced variational principles that are associated to either spacetime covariance, which is an axiom of the continuum theory, or material covariance, which is related to particular properties of the system such as isotropy. The covariance hypotheses and the Lagrangian reduction process are efficiently formulated by making explicit the dependence of the theory on given material and spacetime tensor fields that are transported by the world-tube of the continuum via the push-forward and pull-back operations. It is shown that the variational formulation, when augmented with the Gibbons-Hawking-York (GHY) boundary terms, also yields the Israel-Darmois junction conditions between the solution at the interior of the relativistic continua and the solution describing the gravity field produced outside from it. The expression of the first variation of the GHY term with respect to the hypersurface involves some extensions of previous results that we also derive in the paper. We consider in details the application of the variational framework to relativistic fluids and relativistic elasticity. For the latter case, our setting also allows to clarify the relation between formulations of relativistic elasticity based on the relativistic right Cauchy-Green tensor or on the relativistic Cauchy deformation tensor. The setting developed here will be further exploited for modelling purpose in subsequent parts of the paper.
