Introduction to Hamiltonian Formulation of General Relativity and Homogeneous Cosmologies

TL;DR

This paper provides a comprehensive pedagogical introduction to the Hamiltonian formalism of general relativity, including the ADM approach and applications to homogeneous cosmologies with various matter fields.

Contribution

It offers a detailed, accessible explanation of the Hamiltonian formulation of general relativity and its application to homogeneous universes, including Bianchi classifications and matter couplings.

Findings

Demonstrates the equivalence of Lagrangian and Hamiltonian formulations for homogeneous cosmologies.

Illustrates the ADM formalism with examples involving scalar and electromagnetic fields.

Provides a clear pedagogical pathway for advanced students to understand Hamiltonian general relativity.

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.