On existence of a possible Lorentz invariant modified gravity in Weitzenb\"{o}ck spacetime

TL;DR

This paper investigates whether Lorentz-invariant modified gravity theories can be constructed within Weitzenböck spacetime, concluding that only $f(R)$ gravity with the Ricci scalar computed via Weitzenböck's connection maintains Lorentz invariance.

Contribution

The paper demonstrates that alternative torsion-based modified gravity theories like $f(T)$ and $f(T, abla T)$ cannot be Lorentz invariant, emphasizing the uniqueness of $f(R)$ gravity in this context.

Findings

$f(R)$ gravity with Ricci scalar in Weitzenböck spacetime is Lorentz invariant.

$f(T)$ and $f(T, abla T)$ theories do not respect Lorentz symmetry.

Lorentz invariance constrains the form of viable modified gravity theories in Weitzenböck spacetime.

Abstract

Modified gravity which was constructed by torsion scalar $T$, namely $f(T)$ doesn't respect Lorentz symmetry. As an attempt to make a new torsion based modified gravity with Lorentz invarianve, recently $f(T,\mathcal{B})$ introduced where $B=2\nabla_{\mu}T^{\mu}$ \citep{Bahamonde:2015zma}. We would argue, even when all is constructed and done in a self-consistent form, if you handle them properly,we observe that there is no Lorentz invariant teleparallel equivalent of $f(R)$ gravity. All we found is that the $f(R)$ gravity in which $R$ must be computed in Weitzenb\"{o}ck spacetime, using Weitzenb\"{o}ck's connection, nor Levi-Civita connections is the only possible Lorentz invariant type of modified gravity. Consequently, $f(T)$ gravity can not obey Lorentz symmetry not only in its orthodoxica form but even in this new framework $f(T,\mathcal{B})$.