Geodesic Deviation Equation in $f(T)$ gravity

TL;DR

This paper develops the geodesic deviation equation within $f(T)$ gravity, a teleparallel gravity theory, by deriving equivalent equations to those in $f(R)$ gravity and extending to the modified Mattig relation.

Contribution

It introduces a novel formulation of the geodesic deviation equation in $f(T)$ gravity, bridging teleparallel and modified gravity frameworks.

Findings

Derived the GR equivalent equations for $f(T)$ gravity.

Formulated the geodesic deviation equation in $f(T)$ gravity.

Extended the analysis to modify the Mattig relation.

Abstract

In this work, we show that it is possible to study the notion of geodesic deviation equation in $f(T)$ gravity, in spite of the fact that in teleparallel gravity there is no notion of geodesics, and the torsion is responsible for the appearance of gravitational interaction. In this regard, we obtain the GR equivalent equations for $f(T)$ gravity which are in the modified gravity form such as $f(R)$ gravity. Then, we obtain the GDE within the context of this modified gravity. In this way, the obtained geodesic deviation equation will correspond to the $f(T)$ gravity. Eventually, we extend the calculations to obtain the modification of Matting relation.