Hamiltonian formulation of general relativity in the teleparallel geometry

TL;DR

This paper develops a Hamiltonian formulation of the teleparallel equivalent of general relativity without using the time gauge, extending the geometric framework to four dimensions and deriving the associated constraints.

Contribution

It presents a novel Hamiltonian formulation of teleparallel gravity that does not rely on the time gauge, providing new insights into its geometric and dynamical structure.

Findings

The Hamiltonian formulation differs from the ADM approach.

The primary constraints are explicitly derived.

The vector constraint is shown to follow from the Hamiltonian constraint.

Abstract

We establish the Hamiltonian formulation of the teleparallel equivalent of general relativity, without fixing the time gauge condition, by rigorously performing the Legendre transform. The time gauge condition, previously considered, restricts the teleparallel geometry to the three-dimensional spacelike hypersurface. Geometrically, the teleparallel geometry is now extended to the four-dimensional space-time. The resulting Hamiltonian formulation is different from the standard ADM formulation in many aspects, the main one being that the dynamics is now governed by the Hamiltonian constraint H_0 and a set of primary constraints. The vector constraint H_i is derived from the Hamiltonian constraint. The vanishing of the latter implies the vanishing of the vector constraint.