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.