Extended Gravity Theories and the Einstein-Hilbert Action

TL;DR

This paper explores the deep connections between high-order gravity theories, scalar-tensor formulations, and the Einstein-Hilbert action, revealing their conformal equivalences and scalar field representations.

Contribution

It demonstrates that high-order gravity theories are dynamically equivalent to multi-scalar Brans-Dicke models and conformally related to Einstein-Hilbert action with multiple scalar fields.

Findings

High-order gravity theories correspond to scalar-tensor models with multiple scalar fields.

These theories are conformally equivalent to Einstein-Hilbert action with scalar fields.

Clarifies the extent of conformal equivalence between extended gravity and general relativity.

Abstract

I discuss the relation between arbitrarily high-order theories of gravity and scalar-tensor gravity at the level of the field equations and the action. I show that $(2n+4)$-order gravity is dynamically equivalent to Brans-Dicke gravity with an interaction potential for the Brans-Dicke field and $n$ further scalar fields. This scalar-tensor action is then conformally equivalent to the Einstein-Hilbert action with $n+1$ scalar fields. This clarifies the nature and extent of the conformal equivalence between extended gravity theories and general relativity with many scalar fields.