GROUPOID SYMMETRY AND CONSTRAINTS IN GENERAL RELATIVITY
GROUPOID SYMMETRY AND CONSTRAINTS IN GENERAL RELATIVITY
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DOI：
10.1142/s0219199712500617
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发表时间：
2013-02-01
影响因子：
1.6
通讯作者：
Weinstein, Alan
中科院分区：
文献类型：
--
作者：
Blohmann, Christian;Barbosa Fernandes, Marco Cezar;Weinstein, Alan
当真空Einstein方程转化为Hamilton演化方程时，初值位于Cauchy超曲面S上的黎曼度量流形MS的余切丛中。正如在所有具有对称性的拉格朗日场论中一样，初始数据必须满足约束。但是，与规范理论不同的是，广义相对论的约束并不是作为任何哈密顿群作用量的动量出现的。本文证明了广义相对论约束之间的括号关系与时空中类空超曲面之间的类空同态构成的广群的李代数胚中的括号关系是一致的。约束本身的定义与爱因斯坦方程密切相关，而我们的广群则完全不受爱因斯坦方程的影响。我们讨论了一些困难，使这样的连接。在附录中，我们发展了函数空间学的一些方面，这是我们处理函数空间的基本框架。
When the vacuum Einstein equations are cast in the form of Hamiltonian evolution equations, the initial data lie in the cotangent bundle of the manifold MS of Riemannian metrics on a Cauchy hypersurface S. As in every Lagrangian field theory with symmetries, the initial data must satisfy constraints. But, unlike those of gauge theories, the constraints of general relativity do not arise as momenta of any Hamiltonian group action. In this paper, we show that the bracket relations among the constraints of general relativity are identical to the bracket relations in the Lie algebroid of a groupoid consisting of diffeomorphisms between space-like hypersurfaces in spacetimes. A direct connection is still missing between the constraints themselves, whose definition is closely related to the Einstein equations, and our groupoid, in which the Einstein equations play no role at all. We discuss some of the difficulties involved in making such a connection. In an appendix, we develop some aspects of diffeology, the basic framework for our treatment of function spaces.