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

TL;DR

The paper analyzes background-dependent relationships between $f(Q)$ non-metricity gravity and $f(T)$ teleparallel gravity, highlighting two main correspondences: Minkowski-equivalence (ME) that maps certain flat-space solutions between the theories via a tetrad-spin formulation of $f(Q)$, and an equations-of-motion (EoMs) correspondence that aligns their field equations under symmetry. By developing the covariant $f(Q)$ framework and the tetrad-spin description of $f(T)$, the authors classify connection branches in cosmological and black-hole spacetimes, identifying when ME and EoMs correspondences hold or fail. They show that in cosmology, $f(T)$ solutions form a subset of $f(Q)$ solutions under EoMs, while in black-hole backgrounds the subset relation breaks due to the complex $f(T)$ branch, which lacks a $f(Q)$ counterpart. Overall, the work clarifies how different geometric formulations and symmetry constraints influence the interrelation of these modified gravity theories and lays groundwork for broader comparisons within General Teleparallel 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.