Covariant formulation of f(Q) theory

TL;DR

This paper develops a covariant formulation of f(Q) gravity, addressing inconsistencies caused by the common coincident gauge, and applies it to spherically symmetric and cosmological models.

Contribution

It introduces a covariant approach to f(Q) theory, allowing for non-zero affine connections and improving the theory's consistency.

Findings

Identified limitations of the coincident gauge in f(Q) theory.

Proposed a covariant reformulation with non-vanishing affine connection.

Applied the method to spherically symmetric and cosmological spacetimes.

Abstract

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 will make the f(Q) theory inconsistent. To circumvent this problem, we reformulate 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 spherically symmetric spacetime and the homogeneous and isotropic expanding universe.