Introduction to Hamiltonian Formulation of General Relativity and Homogeneous Cosmologies

Abstract

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.

Same as published version, 108 pages, 13 figures

References in corpus (5)

• 3+1 Formalism and Bases of Numerical Relativity
• Classical and Quantum Features of the Mixmaster Singularity
• A Hamiltonian Formulation of the BKL Conjecture
• Hypersurface-deformation algebroids and effective space-time models
• Hamiltonian General Relativity and the Belinskii, Khalatnikov, Lifshitz Conjecture

Cited by in corpus (4)

• Black holes and covariance in effective quantum gravity: A solution without Cauchy horizons
• Herglotz Action for Homogeneous Cosmology
• Multiway Junction Conditions: Booklets and Webs
• Strong Gravitational Lensing by Compact Object without Cauchy Horizons in Effective Quantum Gravity