Conformal transformations in modified teleparallel theories of gravity revisited

TL;DR

This paper explores the conformal relationships between extended teleparallel gravity models, specifically $f(T,B)$ gravity, and scalar field models, revealing conditions under which these theories are equivalent and extending previous analyses.

Contribution

It extends the analysis of conformal equivalence to $f(T,B)$ gravity, identifying conditions for transforming these models into nonminimally coupled scalar field theories.

Findings

$f(T,B)$ gravity is conformally equivalent to scalar field models with nonminimal coupling.

Conditions are derived for $f(T,B)$ functions to be transformed into scalar field models.

Extended the understanding of conformal transformations beyond $f(T)$ gravity.

Abstract

It is well known that one cannot apply a conformal transformation to $f(T)$ gravity to obtain a minimally coupled scalar field model, and thus no Einstein frame exists for $f(T)$ gravity. Furthermore nonminimally coupled "teleparallel dark energy models" are not conformally equivalent to $f(T)$ gravity. However, it can be shown that $f(T)$ gravity is conformally equivalent to a teleparallel phantom scalar field model with a nonminimal coupling to a boundary term only. In this work, we extend this analysis by considering a recently studied extended class of models, known as $f(T,B)$ gravity, where $B$ is a boundary term related to the divergence of a contraction of the torsion tensor. We find that nonminimally coupled "teleparallel dark energy models" are conformally equivalent to either an $f(T,B)$ or $f(B)$ gravity model. Finally conditions on the functional form of $f(T,B)$ gravity are derived to allow it to be transformed to particular nonminimally coupled scalar field models.