Canonical Vielbeins for General Relativity: D + 1 Decomposition and Constraint Analysis

TL;DR

This paper derives a Hamiltonian formulation of General Relativity using vielbein variables in D+1 dimensions, emphasizing Lorentz covariance, constraint algebra, and the boost generator construction.

Contribution

It provides a self-contained derivation of the Hamiltonian formulation in vielbein variables, including the Lorentz constraints and the boost generator, extending previous metric-based approaches.

Findings

Derived Lorentz- and SO(D)-covariant phase-space actions

Identified primary Lorentz constraints and constraint algebra

Constructed the boost generator in the covariant formulation

Abstract

We provide a self-contained derivation of the Hamiltonian formulation of General Relativity in vielbein variables in $d=D+1$ dimensions. Starting from the Einstein--Hilbert action in a standard metric $D+1$ decomposition, we derive Lorentz- and $\mathrm{SO}(D)$-covariant phase-space actions, identify the primary Lorentz constraints, and obtain the Hamiltonian and momentum constraints. We compute the resulting first-class constraint algebra, relate the vielbein and metric phase-space formulations, and discuss the rotation/boost decomposition. In particular, we construct the boost generator in the $\mathrm{SO}(D)$-covariant formulation, thereby recovering full local Lorentz symmetry.