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.
Wu et al. (Mon,) studied this question.