Actions for Gravity, with Generalizations: A Review

TL;DR

This review explores new classical actions and generalizations of gravity in various dimensions, clarifying their relations and introducing detailed Hamiltonian analyses and potential higher-dimensional formulations.

Contribution

It provides a detailed constraint analysis of the Hamiltonian formulation and discusses the relations between different gravity actions, including potential higher-dimensional generalizations.

Findings

Detailed Hamiltonian constraint analysis of Hilbert-Palatini action in 3+1 dimensions

Canonical transformation between Ashtekar and ADM Hamiltonians in 2+1 dimensions

Discussion on the possibility of higher-dimensional Ashtekar formulations

Abstract

Related to the classical Ashtekar Hamiltonian, there have been discoveries regarding new classical actions for gravity in (2+1)- and (3+1)-dimensions, and also generalizations of Einstein's theory of gravity. In this review, I will try to clarify the relations between the new and old actions for gravity, and also give a short introduction to the new generalizations. The new results/treatments in this review are: 1. A more detailed constraint analysis of the Hamiltonian formulation of the Hilbert- Palatini Lagrangian in (3+1)-dimensions. 2. The canonical transformation relating the Ashtekar- and the ADM-Hamiltonian in (2+1)-dimensions is given. 3. There is a discussion regarding the possibility of finding a higher dimensional Ashtekar formulation. There are also two clarifying figures (in the beginning of chapter 2 and 3, respectively) showing the relations between different action-formulations for Einstein gravity in (2+1)- and (3+1)-dimensions.