Variational and conformal structure of nonlinear metric-connection gravitational lagrangians

TL;DR

This paper explores the variational and conformal structures of higher order gravity theories derived from metric-connection Lagrangians, revealing new reduction methods, conformal equivalences, and implications within Weyl geometry.

Contribution

It introduces a new reduction of order method for field equations and extends the conformal equivalence theorem to Weyl geometry, highlighting novel aspects of higher order gravity theories.

Findings

A new consistent reduction method for higher order gravity field equations.

Extension of the conformal equivalence theorem to Weyl geometry.

Identification of a new 'source term' acting as stress in the extended framework.

Abstract

We examine the variational and conformal structures of higher order theories of gravity which are derived from a metric-connection Lagrangian that is an arbitrary function of the curvature invariants. We show that the constrained first order formalism when applied to these theories may lead consistently to a new method of reduction of order of the associated field equations. We show that the similarity of the field equations which are derived from appropriate actions via this formalism to those produced by Hilbert varying purely metric Lagrangians is not merely formal but is implied by the diffeomorphism covariant property of the associated Lagrangians. We prove that the conformal equivalence theorem of these theories with general relativity plus a scalar field, holds in the extended framework of Weyl geometry with the same forms of field and self-interacting potential but, in addition, there is a new `source term' which plays the role of a stress. We point out how these results may be further exploited and address a number of new issues that arise from this analysis.