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Extended Gravity Theories and the Einstein-Hilbert Action

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David Wands

General Relativity and Quantum Cosmology · Submitted 23 Jul 1993 · 25 references

Pith's verdict: A correct, modest generalization: the Teyssandier–Tourrenc trick pushed to (2n+4)-order gravity, with the on-shell versus action-level distinction stated honestly.

T0 · AI review

How this review was made: An AI referee (selfhosted/deepseek-v4.1-flash) read the paper on November 11, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

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arXiv IDgr-qc/9307034 (version 1)
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CitationDavid Wands. "Extended Gravity Theories and the Einstein-Hilbert Action." arXiv:gr-qc/9307034 (1993). https://pith.science/paper/YOD7J2YM
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The paper

The paper claims

Wands shows that gravity theories with arbitrarily many derivatives of the Ricci scalar are, at the level of their classical field equations, ordinary scalar-tensor theories in disguise, and that this scalar-tensor form can be conformally r

The reading

Wands asks whether gravity theories built from ever-higher derivatives of the curvature scalar are genuinely new physics or general relativity wearing a disguise. His answer is a two-step dictionary. First, a Lagrangian $F(R,\Box R,\dots,\Box^n R)$ is rewritten, by introducing $n+1$ auxiliary fields $\phi_i$ that the equations of motion force to equal $\Box^i R$, as an ordinary second-order scalar-tensor theory: Brans-Dicke gravity with coupling $\omega=0$, plus $n$ further scalar fields interacting through a potential. Second, that scalar-tensor action is conformally rescaled into the Einstein-Hilbert action with $n+1$ scalar fields. The distinction the paper insists on is that the second step is an equivalence of actions, while the first is only an equivalence of classical field equations, so the original higher-order theory never acquires a conformal action of its own. If the construction holds, a whole tower of higher-derivative gravities studied for inflation and for quantum corrections becomes tractable with second-order tools.

What carries the argument

The load-bearing device is the auxiliary-field substitution: replace $F(\Box^i R)$ by $F(\phi_i)+\sum_j (\partial F/\partial\phi_j)(\Box^j R-\phi_j)$, a Legendre-style move that keeps the field equations unchanged while making the dependence on $R$ linear after integration by parts. Two conditions carry the argument: the Hessian $F_{jk}$ must be invertible so that the $\phi_k$ equations really imply $\phi_i=\Box^i R$, and the composite $G_D\Phi=\sum_j \Box^j(\partial F/\partial\phi_j)$ must be positive so that it can serve as a gravitational coupling and as the conformal factor. The paper illustrates the machinery on $F=R+\gamma R\Box R$ (sixth-order gravity, giving Brans-Dicke plus one scalar $\sigma$ with potential $V\propto\sigma(G\Phi-1)$) and on $R+\alpha R^2$ with a non-minimally coupled inflaton, recovering the known $F(R)$ case $\Omega^2=(dF/dR)^{2/(D-2)}$ as the $n=0$ limit.

Extended reading notes

Core claim

The central claim is that $(2n+4)$-order gravity — any theory whose Lagrangian depends on $R$, $\Box R$, and up to $n$ iterated d'Alembertians of $R$, with the highest-derivative term appearing linearly and multiplied by a function of $R$ alone — is dynamically equivalent to Brans-Dicke gravity with $\omega=0$ coupled through an interaction potential to $n$ additional scalar fields. The proof proceeds by trading the higher derivatives for independent fields $\phi_i$, showing their equations of motion enforce $\phi_i=\Box^i R$ whenever the matrix $F_{jk}=\partial^2 F/\partial\phi_j\partial\phi_k$ is non-degenerate, and then integrating by parts so that the Ricci scalar appears only linearly, multiplied by the composite scalar $G_D\Phi=\sum_j \Box^j(\partial F/\partial\phi_j)$. That scalar-tensor action is in turn conformally equivalent, with $\Omega^2=(G_D\Phi)^{2/(D-2)}$, to the Einstein-Hilbert action plus $n+1$ scalar fields, one of which has a standard kinetic term and the rest non-standard ones. Wands is explicit that the conformal step is exact at the level of the action, boundary terms included, whereas the higher-order-to-scalar-tensor step is only a dynamical equivalence valid along classical trajectories.

Load-bearing premise

The action-level rewriting assumes boundary terms can be dropped during integration by parts, even though the paper itself states that the correct boundary term for higher-order gravity is not known and that only the field-equation-level equivalence is actually established.

Editorial extensions

If this is right

  • Every $(2n+4)$-order gravity theory with non-degenerate Hessian can be analysed with second-order equations, so the standard cosmological and post-Newtonian toolkits apply directly to it.
  • The conformal Einstein frame of such a theory contains $n+1$ scalar fields, giving a systematic accounting of the extra degrees of freedom that higher-derivative terms introduce.
  • Quantum fluctuations generated during inflation can legitimately be computed in the conformal Einstein frame of the equivalent scalar-tensor theory, because there the action, not merely the field equations, is transformed.
  • Theories whose Hessian matrix is degenerate, such as the derivative-coupled model with $F_1=F_1(\varphi)$ discussed at the end, have no conformal Einstein frame at all, which marks out a real boundary of the equivalence.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The non-degeneracy condition on $F_{jk}$ reads like a classification principle: it splits higher-order gravities into those with a scalar-tensor shadow and those without, and the second class may be where genuinely new behaviour hides.
  • Because the higher-order action has no conformal partner, entropy and boundary-term arguments borrowed from scalar-tensor or Einstein gravity cannot be transported back to the higher-derivative theory; the paper's own remark about not knowing the correct boundary term makes this a live obstruction rather than a technicality.
  • The construction suggests a practical recipe for model builders: write down the desired inflaton potential in the Einstein frame with $n+1$ scalars and invert the dictionary to find which higher-order Lagrangian generates it.
Original 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.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,611 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.08 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.50 · compares with prior results 0.18 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

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.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 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. contribution 0.79 0.79 0.07 asserted 0.52
s1 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. result 0.99 0.91 0.62 asserted 1.00
s2 This scalar-tensor action is then conformally equivalent to the Einstein-Hilbert action with $n+1$ scalar fields. result 0.99 0.78 0.26 asserted 1.00
s3 This clarifies the nature and extent of the conformal equivalence between extended gravity theories and general relativity with many scalar fields. interpretation 1.00 0.64 0.26 asserted 0.97

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References
How the literature uses it

44 arXiv papers cite this in 56 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 42 sentences without a confident label

Uses its method or tool 3 papers
Cites it as support 2 papers
Cites it as background 9 papers
  • The first class of theories, in general, can be mapped to a scalar tensor theory [ref] , and are subject to constraints from both solar system tests of gravity ( [ref] ) and cosmology ( [ref] ).

    Unsuccessful cosmology with Modified Gravity Models · Introduction · 0.83

  • Another source of problems is that it is rather non-trivial to satisfy laboratory and Solar system tests in $f(R)$ gravity since it formally represents the limiting case $ω=0$ of scalar-tensor gravity where $ω$ is the Brans-Dicke parameter [ref] , though sometimes this limit should be taken carefully (see [ref] for recent reconsideration).

    Disappearing cosmological constant in f(R) gravity · Introduction · 0.85

  • There has been a lot of investigations on perturbation theory in generalized gravity, where the Ricci scalar and a scalar field are non-minimally coupled via an arbitrary function [ref] : [equation] But most of them are restricted to special cases with only one degree of freedom, although it was believed that there should be two degrees of freedom in general [ref] .

    1/R Correction to Gravity in the Early Universe · Inflation in Generalized Gravity · 0.97

Discussion

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