Extended Gravity Theories and the Einstein-Hilbert Action

TL;DR

The paper investigates gravity theories extended beyond the Einstein–Hilbert action and shows that any Lagrangian F(R, □R, ..., □^n R) is dynamically equivalent to a scalar–tensor theory with a Brans–Dicke field (ω=0) coupled to n additional scalars. Through auxiliary-field construction and a conformal transformation, these theories are shown to be conformally equivalent to general relativity with n+1 scalar fields, with precise expressions for the conformal factor and Einstein-frame potentials. The work provides explicit examples (F=R+γ R□R and F(φ,R)=R+αR^2−8πG(ξφ^2R+∂φ^2)) illustrating the mapping to the Einstein frame and the resulting matter couplings and potentials. It also discusses limitations, boundary terms, and conditions under which the conformal mapping fails, clarifying when higher-order gravity can be recast as GR with scalars and when it cannot.

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.