In general relativity, it is difficult to localize observables such as energy, angular momentum, or center of mass in a bounded region. The difficulty is that there is dissipation. A self-gravitating system, confined by its own gravity to a bounded region, radiates some of the charges away into the environment. At a formal level, dissipation implies that some diffeomorphisms are not Hamiltonian. In fact, there is no Hamiltonian on phase space that would move the region relative to the fields. Recently, an extension of the covariant phase space has been introduced to resolve the issue. On the extended phase space, the Komar charges are Hamiltonian. They are generators of dressed diffeomorphisms. While the construction is sound, the physical significance is unclear. We provide a critical review before developing a geometric approach that takes into account dissipation in a novel way. Our approach is based on metriplectic geometry, a framework used in the description of dissipative systems. Instead of the Poisson bracket, we introduce a Leibniz bracket—a sum of a skew-symmetric and a symmetric bracket. The symmetric term accounts for the loss of charge due to radiation. On the metriplectic space, the charges are Hamiltonian, yet they are not conserved under their own flow.

中文翻译：

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引力子系统的三相几何

在广义相对论中，很难在有界区域内定位诸如能量、角动量或质心之类的可观测量。困难在于有耗散。一个自引力系统，被自身的引力限制在一个有界区域，将一些电荷辐射到环境中。在形式上，耗散意味着一些微分同胚不是哈密顿量的。事实上，相空间上没有哈密顿量可以使区域相对于场移动。最近，引入了协变相空间的扩展来解决该问题。在扩展相空间上，Komar 电荷是哈密顿量。它们是修饰微分同胚的生成器. 虽然结构完好，但物理意义尚不清楚。在开发一种以新颖的方式考虑耗散的几何方法之前，我们提供了批判性审查。我们的方法基于metriplectic几何，这是一种用于描述耗散系统的框架。代替泊松括号，我们引入了莱布尼茨括号——斜对称括号和对称括号之和。对称项说明了由于辐射引起的电荷损失。在metriplectic空间上，电荷是哈密顿量的，但它们在自己的流动下不守恒。