Background-dependent and classical correspondences between $f(Q)$ and $f(T)$ gravity

Abstract

$f(Q)$ and $f(T)$ gravity are based on fundamentally different geometric frameworks, yet they exhibit many similar properties. This article provides a comprehensive summary and comparative analysis of the various theoretical branches of torsional gravity and non-metric gravity, which arise from different choices of affine connection. We identify two types of background-dependent and classical correspondences between these two theories of gravity. The first correspondence is established through their equivalence within the Minkowski spacetime background. To achieve this, we develop the tetrad-spin formulation of $f(Q)$ gravity and derive the corresponding expression for the spin connection. The second correspondence is based on the equivalence of their equations of motion. Utilizing a metric-affine approach, we derive the general affine connection for static and spherically symmetric spacetime in $f(Q)$ gravity and compare its equations of motion with those of $f(T)$ gravity. Among others, our results reveal that, $f(T)$ solutions are not simply a subset of $f(Q)$ solutions; rather, they encompass a complex solution beyond $f(Q)$ gravity in black hole background.