The Hamilton equations in $f(T)$ teleparallel gravity and in New General Relativity

TL;DR

This paper derives the Hamiltonian formulation for extended teleparallel gravity theories, including $f(T)$ gravity and New General Relativity, and compares their Hamilton equations to those of other gravity models.

Contribution

It provides the Hamiltonian and Hamilton equations for $f(T)$ gravity and New General Relativity in a covariant framework, enabling direct comparison with other gravity theories.

Findings

$f(T)$ and New General Relativity reduce to TEGR.

Significant differences between $f(T)$ and $f(R)$ Hamilton equations.

Hamiltonian formulation in covariant and Weitzenb"ock gauge.

Abstract

We derive the Hamiltonian function for extended teleparallel theories of gravity in their covariant formulation. In particular, we present the Hamiltonian for $f(T)$ gravity and New General Relativity. From this, we obtain the related Hamilton equations, which are presented both in covariant formulation and Weitzenb\"ock gauge. In this framework, teleparallel equivalent to General Relativity, its $f(T)$ extension and New General Relativity can be compared. We find that $f(T)$ and New General Relativity consistently reduce to the Teleparallel Equivalent to General Relativity, while significant differences appear comparing the Hamilton equations of $f(T)$ with $f(R)$ gravity.