Determination of the Connection in Metric-Affine Gravity

Curvature is not the only geometric quantity that describes gravitational interactions; torsion and non-metricity also serve as alternative descriptions. Nevertheless, in general relativity, the connection is uniquely determined in a spacetime endowed exclusively with curvature, whereas the same does not hold for gravitational theories that involve only torsion or only non-metricity. Spherically symmetric metric solutions of f(T) gravity theory obtained in pure tetrad formulation are inconsistent in different coordinates. The covariance of the f(T) theory can only be ensured by considering spin connections. Meanwhile, different ways of determining the spin connection also give rise to different branches of covariant f(T) theory. In f(Q) gravity, a majority of studies have been conducted within the coincident gauge. Notably, in the case of a spatially flat universe, there exist three branches of connections that satisfy cosmological symmetry, with the coincident gauge being merely one of them. It is therefore worthwhile and interesting to study the determination of the connection and the relationship between the different branches in metric-affine gravity. We provide 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.