SciPost Submission Page
Introduction to Hamiltonian Formulation of General Relativity and Homogeneous Cosmologies
by Rishabh Jha
This Submission thread is now published as
Submission summary
| Submission information |
| Preprint Link: |
scipost_202303_00015v2
(pdf)
|
| Date accepted: |
2023-07-18 |
| Date submitted: |
2023-06-30 00:08 |
| Submitted by: |
Jha, Rishabh |
| Submitted to: |
SciPost Physics Lecture Notes |
| Ontological classification |
| Academic field: |
Physics |
| Specialties: |
- Gravitation, Cosmology and Astroparticle Physics
- Mathematical Physics
|
| Approach: |
Theoretical |
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.
