Geodesic Deviation Equation In $f(Q)$-Gravity

TL;DR

This paper explores the geodesic deviation equation within the framework of $f(Q)$-gravity, extending classical relations like Mattig and Dyer-Roeder to this modified gravity context, relevant for understanding cosmic structure and light propagation.

Contribution

It formulates the GDE in $f(Q)$-gravity, deriving generalized relations and equations that extend standard cosmological tools to this modified gravity theory.

Findings

Derived the GDE in $f(Q)$-gravity.

Obtained the generalized Mattig relation.

Presented an equivalent Dyer-Roeder equation for $f(Q)$-gravity.

Abstract

In the present paper we study the Geodesic Deviation Equation (GDE) in the modified $f(Q)$-gravity theories. The formulation of GDE in General Relativity in the case of the homogeneous and isotropic Friedman-Lema\^{i}tre-Robertson-Walker (FLRW) spacetime is briefly discussed and then extended in modified $f(Q)$-gravity using its covariant counterpart. The generalised Mattig relation is obtained. Finally, an equivalent expression to the Dyer-Roeder equation in General Relativity in the case of $f(Q)$-gravity is presented.