Diffeomorphism Symmetry in the Lagrangian Formulation of Gravity

TL;DR

This paper derives the diffeomorphism transformation laws for metric and connection in both second and first order formulations of gravity, clarifying their independent and interconnected roles in the Lagrangian framework.

Contribution

It provides a detailed derivation of diffeomorphism transformations for metric and connection in both formulations, highlighting differences in their gauge variations.

Findings

Transformation laws for the metric and connection are explicitly derived.

Differences between metric and Palatini formulations are clarified.

Independent derivations of connection transformations in Palatini formulation are presented.

Abstract

Starting from a knowledge of certain identities in the Lagrangian description, the diffeomorphism transformations of metric and connection are obtained for both the second order (metric) and the first order (Palatini) formulations of gravity. The transformation laws of the connection and the metric are derived independently in the Palatini formulation in contrast to the metric formulation where the gauge variation of the connection is deduced from the gauge variation of the metric.