DYNAMICAL STRUCTURE AND DEFINITION OF ENERGY IN GENERAL RELATIVITY
DYNAMICAL STRUCTURE AND DEFINITION OF ENERGY IN GENERAL RELATIVITY
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DOI：
10.1103/physrev.116.1322
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发表时间：
1959-01-01
期刊：
影响因子：
--
通讯作者：
MISNER, CW
中科院分区：
文献类型：
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作者：
ARNOWITT, R;DESER, S;MISNER, CW
本文在形式上讨论了经典广义相对论的动力学结构和能量定义问题。在以前的文件中，所使用的技术是施温格作用原理。从一阶Palatini形式的完全爱因斯坦拉格朗日量出发，导出了消除了代数约束变量的作用量积分。这个作用具有一个“哈密顿”密度，然而，由于微分约束而消失。如果微分约束被代入作用量中，那么理论的真实的、非零的哈密顿量就出现了。通过对运动方程和约束方程的分析，明确地给出了两对动力学变量，它们分别代表引力场的两个独立自由度。理论上还有四个变量;这些变量可以任意指定，任何这样的指定都代表了坐标系的选择。结果表明，根据动力变量和任意变量获得真正的规范变量对是可能的。因此，只有在选择了一组坐标条件之后，动力学的陈述才有意义。一般来说，即使对于一个孤立的引力场，真实的哈密顿量也是与时间相关的。这样就产生了一个优选坐标系的概念，即哈密顿量守恒的坐标系。在这个特殊的坐标系中，根据物理的理由，可以用哈密顿量来定义场的能量。在这些方面，广义相对论中的情形类似于粒子力学中汉密尔顿原理的参数形式。
The problem of the dynamical structure and definition of energy for the classical general theory of relativity is considered on a formal level. As in a previous paper, the technique used is the Schwinger action principle. Starting with the full Einstein Lagrangian in first order Palatini form, an action integral is derived in which the algebraic constraint variables have been eliminated. This action possesses a" Hamiltonian" density which, however, vanishes due to the differential constraints. If the differential constraints are then substituted into the action, the true, nonvanishing Hamiltonian of the theory emerges. From an analysis of the equations of motion and the constraint equations, the two pairs of dynamical variables which represent the two independent degrees of freedom of the gravitational field are explicitly exhibited. Four other variables remain in theory; these may be arbitrarily specified, any such specification representing a choice of coordinate frame. It is shown that it is possible to obtain truly canonical pairs of variables in terms of the dynamical and arbitrary variables. Thus a statement of the dynamics is meaningful only after a set of coordinate conditions have been chosen. In general, the true Hamiltonian will be time dependent even for an isolated gravitational field. There thus arises the notion of a preferred coordinate frame, ie, that frame in which the Hamiltonian is conserved. In this special frame, on physical grounds, the Hamiltonian may be taken to define the energy of the field. In these respects the situation in general relativity is analogous to the parametric form of Hamilton's principle in particle mechanics.