A modified variational principle for gravity in modified Weyl geometry

TL;DR

This paper revises the interpretation of Weyl geometry by treating connections as tensors and introducing a new covariant derivative, leading to a more compact Riemann tensor form and a Weyl Palatini identity for gravity.

Contribution

It introduces a modified variational principle in Weyl geometry, treating connections as tensors and deriving new identities for gravitational equations.

Findings

A new form of the Riemann tensor in Weyl geometry

A Weyl version of the Palatini identity

Insights into Weyl-invariant gravity theories

Abstract

The usual interpretation of Weyl geometry is modified in two senses. First, both the additive Weyl connection and its variation are treated as (1, 2) tensors under the action of Weyl covariant derivative. Second, a modified covariant derivative operator is introduced which still preserves the tensor structure of the theory. With its help, the Riemann tensor in Weyl geometry can be written in a more compact form. We justify this modification in detail from several aspects and obtain some insights along the way. By introducing some new transformation rules for the variation of tensors under the action of Weyl covariant derivative, we find a Weyl version of Palatini identity for Riemann tensor. To derive the energy-momentum tensor and equations of motion for gravity in Weyl geometry, one naturally applies this identity at first, and then converts the variation of additive Weyl connection to those of metric tensor and Weyl gauge field. We also discuss possible connections to the current literature on Weyl-invariant extension of massive gravity and the variational principles in f(R) gravity.