Covariant formulation of f(Q) theory

In Symmetric Teleparallel General Relativity, gravity is attributed to the non-metricity. The so-called coincident gauge is usually taken in this theory so that the affine connection vanishes and the metric is the only fundamental variable. This gauge choice was kept in many studies on the extensions of Symmetric Teleparallel General Relativity, such as the so-called f(Q) theory. In this paper, we point out that sometimes this gauge choice conflicts with the coordinate system we selected based on symmetry. To circumvent this problem, we formulate the f(Q) theory in a covariant way with which we can find suitable non-vanishing affine connection for a given metric. We also apply this method to two important cases: the static spherically symmetric spacetime and the homogeneous and isotropic expanding universe.

Automated summary

In Symmetric Teleparallel General Relativity, gravity is attributed to non-metricity. However, the coordinate system selected based on symmetry may conflict with the choice of coincident gauge. To overcome this problem, we formulate the f(Q) theory in a covariant way, allowing us to find suitable non-vanishing affine connection.