# Linking the ADM formulation to other Hamiltonian formulations of general relativity

## TL;DR

This paper establishes explicit connections between various Hamiltonian formulations of general relativity, including ADM, Palatini, and Ashtekar-Barbero variables, across different dimensions and gauge choices.

## Contribution

It provides explicit mappings and relations among phase-space variables, symplectic structures, and constraints for multiple Hamiltonian formulations of GR.

## Key findings

- Derived ADM from Palatini formulations in various dimensions.
- Connected Ashtekar-Barbero variables with ADM formulation.
- Clarified gauge fixing effects on Hamiltonian structures.

## Abstract

We obtain the Arnowitt-Deser-Misner formulation of general relativity in $n$ dimensions ($n \geq 3$) from its either $SO(n-1,1)$ [$SO(n)$] or $SO(n-1)$ Palatini Hamiltonian formulations and vice versa [we recall that $SO(n-1,1)$ [$SO(n)$] requires no gauge fixing whereas $SO(n-1)$ involves the time gauge]. Similarly, the Hamiltonian formulation of general relativity in terms of Ashtekar-Barbero variables can also be directly obtained from the Arnowitt-Deser-Misner Hamiltonian formulation and vice versa, which is an alternative approach to the way followed by Barbero. We give the relevant maps among the phase-space variables and relate the corresponding symplectic structures and the first-class constraints.