论文标题
哈密顿的一般相对论和均质宇宙学的表述简介
Introduction to Hamiltonian Formulation of General Relativity and Homogeneous Cosmologies
论文作者
论文摘要
我们对高级本科和研究生级别的一般相对性的哈密顿形式主义进行了教学介绍。在涵盖了数学先决条件以及时空的$ 3+1 $分配之后，我们开始讨论一般相对论的Arnowitt-Deser-Misner（ADM）形式主义（AMD）形式主义（一种汉密尔顿的方法）。然后，我们开始简要介绍（但不一定是各向同性）宇宙的简短但独立的介绍，并讨论相关的比安奇分类。我们首先研究了它们在拉格朗日配方中的动态，然后是哈密顿式配方，以显示两种方法的等效性。 We present a variety of examples to illustrate the ADM formalism: (i) free & massless scalar field coupled to homogeneous (in particular, Bianchi IX) universe, (ii) scalar field with a potential term coupled to Bianchi IX universe, (iii) electromagnetic field coupled to gravity in general, and (iv) electromagnetic field coupled to Bianchi IX universe.
We give a pedagogical introduction to the Hamiltonian formalism of general relativity at an advanced undergraduate and graduate levels. After covering the mathematical pre-requisites as well as the $3+1$-decomposition of spacetime, we proceed to discuss the Arnowitt-Deser-Misner (ADM) formalism (a Hamiltonian approach) of general relativity. Then we proceed to give a brief but self-contained introduction to homogeneous (but not necessarily isotropic) universes and discuss the associated Bianchi classification. We first study their dynamics in the Lagrangian formulation, followed by the Hamiltonian formulation to show the equivalence of both approaches. We present a variety of examples to illustrate the ADM formalism: (i) free & massless scalar field coupled to homogeneous (in particular, Bianchi IX) universe, (ii) scalar field with a potential term coupled to Bianchi IX universe, (iii) electromagnetic field coupled to gravity in general, and (iv) electromagnetic field coupled to Bianchi IX universe.
