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arXiv · 2606.27532

Geodesic Focusing Conditions in $f(Q)$ Gravity

Chao-Qiang Geng·Chunhui Liu·Ling-Wei Luo·Jianhui Qiu
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Abstract

We study the geodesic deviation equation in symmetric teleparallel geometry (STG), where the relative acceleration is defined with respect to the STG connection. We analyze the modified Raychaudhuri equation along a geodesic congruence in $f(Q)$ gravity under the Weyl-type ansatz, together with an additional assumption under which the metric variation term along the congruence is converted into a disformation-induced acceleration term. In contrast to the purely geometrical Raychaudhuri equation obtained in general metric-affine settings, the equation derived here contains matter-source contributions through the trace equation of $f(Q)$ gravity. Different from general relativity, focusing in $f(Q)$ gravity is not automatic, and one must impose an appropriate focusing condition. We collect the model-dependent terms in the modified Raychaudhuri equation into an effective energy-momentum trace $T_{\text{eff}}$, so that the focusing condition can be written as the inequality $T\leq T_{\text{eff}}$, where $T$ is the trace of the matter energy-momentum tensor. We also apply this condition to the flat Friedmann--Lema\^{i}tre--Robertson--Walker (FLRW) background. The homogeneous and isotropic STG connection admits three branches, each characterized by a single connection function $\gamma_i$, with $i=1,2,3$. Only the first branch with the coincident gauge is compatible with the Weyl-type ansatz. We obtain the resulting effective trace $T_{\text{eff}}=T$ for any form of $f(Q)$ satisfying $f_Q>0$ in the flat FLRW universe. The focusing inequality is saturated and imposes no additional constraint on the matter content.

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Chao-Qiang Geng, Chunhui Liu, Ling-Wei Luo, Jianhui Qiu. 2026-06-25. Geodesic Focusing Conditions in $f(Q)$ Gravity. https://arxiv.org/abs/2606.27532

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