Degrees of freedom of $f(T)$ gravity

TL;DR

This paper analyzes the Hamiltonian structure of $f(T)$ gravity, revealing it has five degrees of freedom and violates local Lorentz invariance, with extra degrees of freedom interpreted as vector and scalar fields.

Contribution

It provides a Hamiltonian analysis of $f(T)$ gravity, showing the emergence of extra degrees of freedom and the loss of local Lorentz invariance compared to Teleparallel gravity.

Findings

$f(T)$ gravity has five degrees of freedom in four dimensions.

The local Lorentz invariance is broken in $f(T)$ gravity.

Extra degrees of freedom correspond to vector and scalar fields.

Abstract

We investigate the Hamiltonian formulation of $f(T)$ gravity and find that there are five degrees of freedom. The six first class constraints corresponding to the local Lorentz transformation in Teleparallel gravity become second class constraints in $f(T)$ gravity, which leads to the appearance of three extra degrees of freedom and the violation of the local Lorentz invariance in $f(T)$ gravity. In general, there are $D-1$ extra degrees of freedom for $f(T)$ gravity in $D$ dimensions, and this implies that the extra degrees of freedom correspond to one massive vector field or one massless vector field with one scalar field.