Hamilton's equations in the covariant teleparallel equivalent of general relativity

TL;DR

This paper derives Hamilton's equations for the covariant teleparallel equivalent of general relativity (TEGR), introducing a new formalism that includes Lorentz matrices and their momenta, enabling a covariant Hamiltonian analysis.

Contribution

It presents the first covariant Hamiltonian formulation of TEGR using Lorentz matrices, expanding the mathematical tools for analyzing this gravity theory.

Findings

Hamilton's equations for covariant TEGR are explicitly derived.

New relations between tetrad and metric conjugate momenta are established.

The formalism accommodates both Weitzenbock gauge and covariant approaches.

Abstract

We present Hamilton's equations for the teleparallel equivalent of general relativity (TEGR), which is a reformulation of general relativity based on a curvatureless, metric compatible, and torsionful connection. For this, we consider the Hamiltonian for TEGR obtained through the vector, antisymmetric, symmetric and trace-free, and trace irreducible decomposition of the phase space variables. We present the Hamiltonian for TEGR in the covariant formalism for the first time in the literature, by considering a spin connection depending on Lorentz matrices. We introduce the mathematical formalism necessary to compute Hamilton's equations in both Weitzenbock gauge and covariant formulation, where for the latter we must introduce new fields: Lorentz matrices and their associated momenta. We also derive explicit relations between the conjugate momenta of the tetrad and the conjugate momenta for the metric that are traditionally defined in GR, which are important to compare both formalisms.