Manifestly covariant variational principle for gauge theories of gravity

TL;DR

This paper introduces a covariant variational principle for gauge theories of gravity, ensuring manifest covariance throughout calculations, and applies it to Weyl gauge theory and extended Weyl gauge theory, revealing new insights into their structure.

Contribution

It develops a manifestly covariant variational framework for gauge theories of gravity, applicable to various symmetry groups, and clarifies relationships between different covariant forms of variational derivatives.

Findings

Explicit covariant expressions for variational derivatives and Noether's theorems.

Application to Weyl gauge theory and extended Weyl gauge theory.

Analysis of gauge field strengths set to zero before or after variation.

Abstract

A variational principle for gauge theories of gravity is presented, which maintains manifest covariance under the symmetries to which the action is invariant, throughout the calculation of the equations of motion and conservation laws. This is performed by deriving explicit manifestly covariant expressions for the Euler--Lagrange variational derivatives and Noether's theorems for a generic action of the form typically assumed in gauge theories of gravity. The approach is illustrated by application to two scale-invariant gravitational gauge theories, namely Weyl gauge theory (WGT) and the recently proposed `extended' Weyl gauge theory (eWGT), where the latter may be considered as a novel gauging of the conformal group, but the method can be straightforwardly applied to other theories with smaller or larger symmetry groups. The approach also enables one easily to establish the relationship between manifestly covariant forms of variational derivatives obtained when one or more of the gauge field strengths is set to zero either before or after the variation is performed. This is illustrated explicitly for both WGT and eWGT in the case where the translational gauge field strength (or torsion) is set to zero before and after performing the variation, respectively.