Conformal transformation in $f(T)$ theories

TL;DR

This paper investigates the conformal properties of $f(T)$ theories of gravity, revealing they are not dynamically equivalent to scalar-torsion theories via conformal transformation, unlike $f(R)$ theories, and discusses observational constraints.

Contribution

It demonstrates that $f(T)$ theories cannot be transformed into scalar-torsion theories through conformal transformations, highlighting a key difference from $f(R)$ theories.

Findings

$f(T)$ theories are not conformally equivalent to scalar-torsion theories.

An additional scalar-torsion coupling term appears under conformal transformation.

Observational constraints from solar system tests are briefly discussed.

Abstract

It is well-known that $f(R)$ theories are dynamically equivalent to a particular class of scalar-tensor theories. In analogy to the $f(R)$ extension of the Einstein-Hilbert action of general relativity, $f(T)$ theories are generalizations of the action of teleparallel gravity. The field equations are always second order, remarkably simpler than $f(R)$ theories. It is interesting to investigate whether $f(T)$ theories have the similar conformal features possessed in $f(R)$ theories. It is shown, however, that $f(T)$ theories are not dynamically equivalent to teleparallel action plus a scalar field via conformal transformation, there appears an additional scalar-torsion coupling term. We discuss briefly what constraint of this coupling term may be put on $f(T)$ theories from observations of the solar system.