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REVIEW 5 major objections 6 minor 1 cited by

Conformal and pure scale-invariant gravities in d dimensions

T0 review · 5 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read In five dimensions, conformal gravity propagates three scalar modes, three vector modes per index, and two tensor modes, one of which is a ghost.

desk verdict Promising alternative-frame construction for higher-dimensional conformal gravity, but the headline 5D mode count is not established — the vector-sector contradiction alone should force a rewrite. read the letter →

arxiv 2506.18775 v1 pith:6MM2F5VZ submitted 2025-06-23 hep-th gr-qc

Anamaria Hell , Dieter Lust This is my paper · ORCID
classification hep-thgr-qc
keywords conformalgravityscale-invarianthigher-dimensionalR^2Weyltensordegreesoffreedomcosmologicalperturbationsanisotropiccosmology
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper asks what happens to two four-dimensional gravity theories—pure scale-invariant gravity built from powers of the Ricci scalar and conformal (Weyl-squared) gravity—when they are defined in $d$ dimensions. It claims that the pure scale-invariant theory has no propagating modes on flat spacetime and, on curved spacetime, is equivalent to general relativity with a free scalar and a cosmological constant that in more than four dimensions can only be positive. For conformal gravity in five dimensions, it exhibits a conformal frame in which the non-polynomial action becomes ordinary Weyl-squared gravity plus a cosmological constant, and a perturbation count around anisotropic solutions gives three scalar degrees of freedom, three vector modes per index, and two tensor modes, one of which is a ghost. The readership should care because these extra modes are absent in four dimensions, so they change both the classical stability and any quantum interpretation of higher-dimensional conformal gravity.

What carries the argument

The load-bearing object is the frame-changing field redefinition. For conformal gravity, introducing the scalar $\phi = W^2$ and rescaling $\tilde{g}_{\mu\nu} = (\sqrt{\phi}/M_P^2) g_{\mu\nu}$ converts the non-polynomial $(W^2)^{d/4}$ action into a quadratic action in the Weyl tensor (the trace-free, conformally invariant part of the curvature) plus a cosmological constant. For pure scale-invariant gravity, the analogous Einstein-frame transformation with $\phi = \beta_d^{2/(d-2)} R$ turns $R^{d/2}$ into general relativity with a free scalar and a cosmological constant. The degree-of-freedom count then rests on reducing the perturbed action to first-order form and checking the degeneracy of the kinetic matrix, the same criterion used in degenerate higher-order scalar-tensor theories; the scalar sector's apparent four modes collapse to three because that matrix is degenerate.

What would settle it

Count the propagating modes directly from the original five-dimensional $(W^2)^{5/4}$ action on a conformally flat background, where the alternative frame is singular, or perform a Hamiltonian constraint analysis of that action on the anisotropic background; if the count differs from three scalars, three vectors per index, and two tensors, the frame-equivalence assumption fails.

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Extended reading notes

Core claim

The paper's central claim is that in $d>4$ conformal gravity, defined by the conformally invariant action $\int d^dx \sqrt{-g}\,(W_{\mu\nu\rho\sigma}W^{\mu\nu\rho\sigma})^{d/4}$, is best studied in an 'alternative frame': introduce $\phi = W^2$, rescale the metric by $\tilde{g}_{\mu\nu} = (\sqrt{\phi}/M_P^2) g_{\mu\nu}$, and the action becomes Weyl-squared gravity with a cosmological constant $\Lambda = M_P^2 \alpha_{\mathrm{CG}}(d-4)/4$, valid whenever the background is not conformally flat. Using this frame in five dimensions on an anisotropic background, the paper counts three propagating scalar degrees of freedom, three vector modes per index, and two tensor modes, one healthy and one ghost (a mode whose kinetic term has the wrong sign)—a spectrum that differs from the four-dimensional one, which has only vector and tensor modes. The accompanying claim for the pure scale-invariant theory $R^{d/2}$ is that flat spacetime carries no modes and that for $R\neq 0$ the Einstein frame is general relativity plus a free scalar, with a cosmological constant that is positive in $d>4$.

Load-bearing premise

The load-bearing premise is that the field redefinition $\tilde{g}_{\mu\nu} = (\sqrt{\phi}/M_P^2) g_{\mu\nu}$ with $\phi = W^2$ preserves the number of physical degrees of freedom while mapping the original conformal-gravity action to Weyl-squared gravity with a cosmological constant; this matters because the five-dimensional mode count is carried out entirely in the new frame, and the transformation is singular where $W^2 = 0$.

Editorial extensions

If this is right

  • In more than four dimensions, pure scale-invariant $R^{d/2}$ gravity forces a positive cosmological constant in its Einstein frame, ruling out negative-cosmological-constant vacua from this action alone.
  • Five-dimensional conformal gravity propagates three scalar degrees of freedom and three vector modes per index on anisotropic non-conformally flat backgrounds, so its linearized dynamics is not just the four-dimensional one with an extra dimension.
  • The alternative-frame action is Weyl-squared gravity plus a cosmological constant, and the same trick turns any $f(W^2)$ theory into a bilinear Weyl term with a constrained, non-propagating scalar.
  • The anisotropic background admits analytic super-Hubble and exponential solutions, so the model provides explicit cosmological histories with an expanding three-dimensional subspace and a nontrivial extra dimension.

Reading between the lines

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

  • If the alternative frame is an exact equivalence, then the ghost in the tensor sector could in principle be projected out by boundary conditions without breaking the conformal invariance of the frame—a possibility the paper raises but does not establish.
  • The degeneracy that removes one scalar mode suggests a Hamiltonian analysis of the original five-dimensional action, without any frame change, should find the same count; if it does not, the frame equivalence fails at the nonperturbative level.
  • The forced positivity of the cosmological constant in the $d>4$ pure scale-invariant theory gives a dynamical reason why higher-dimensional completions built from $R^{d/2}$ tend to produce accelerating vacua rather than negative-cosmological-constant vacua.
  • One concrete test is to compute the sound speeds and ghost conditions of the three scalar modes on the numerical backgrounds; a mode with negative sound speed would mark those solutions gradient-unstable and observationally different from single-field inflation.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 6 minor

Summary. The paper studies conformal and scale-invariant gravities in general dimension d. For pure scale-invariant gravity S ∝ ∫ R^{d/2}, it argues that flat spacetime carries no propagating degrees of freedom, while on curved backgrounds the theory can be written in an Einstein frame as d-dimensional Einstein gravity plus a scalar field and a cosmological constant. For conformal gravity S ∝ ∫ (W²)^{d/4}, the paper introduces an auxiliary scalar and a conformal transformation to map the action to Weyl-squared gravity with a cosmological constant for d>4, and extends the construction to f(W²) theories. In five dimensions the authors study an anisotropic background in this alternative frame, find analytic and numerical solutions, and analyze scalar, vector, and tensor perturbations. They conclude that the five-dimensional theory propagates three scalar modes, three vector modes per index i, and two tensor modes (one ghost), a richer spectrum than in four dimensions. The central new claims are the d>4 frame equivalence and the five-dimensional mode count.

Significance. The four-dimensional behavior of conformal and R²-like gravities is well known, so the interesting content here is the d>4 generalization. If the mode counting is correct, the finding that conformal gravity in five dimensions acquires additional scalar and vector degrees of freedom would be a substantive new result. The paper also provides concrete analytic and numerical anisotropic background solutions in the alternative frame, which is a useful contribution. The auxiliary-field formulation of (W²)^{d/4} and its f(W²) generalization are appealing and potentially reusable. However, the headline mode-counting claims are not supported as written: the perturbation analysis omits essential coefficients, the determinant arguments are internally contradictory, and the equivalence between the original and alternative frames is assumed rather than proved. The significance of the paper therefore depends on whether these technical gaps can be closed.

major comments (5)
  1. [§3.1, Eq. (25)] The displayed linearized action in Eq. (25) does not reduce to the four-dimensional result in Eq. (8), and it does not appear to be the correct d-dimensional generalization of the linearized Ricci scalar. With h00 = 2ϕ and hij = 2ψδij in the signature used in the paper, the linearized Ricci scalar is proportional to Δϕ + (d−1)ψ¨ − (d−2)Δψ, not to Δϕ + ψ¨ − 2Δψ. For d=4 the bracket in Eq. (25) should match Eq. (8), which contains 3ψ¨ − 2Δψ. Consequently the constraint (27) is also incorrect for general d. The conclusion that the flat-space action vanishes may survive, but the derivation must be corrected and the displayed equations revised.
  2. [§5.2.2, Eq. (107) and following paragraph] The vector mode count is internally inconsistent. The Lagrangian (107) contains two independent vector variables, vi and Si, with vi appearing at fourth order in time derivatives. After introducing ρi = ˙vi, the first-order kinetic matrix for (ρi, Si) is 2×2. The text states both that the theory 'is degenerate as well' and that 'the determinant of the corresponding kinetic matrix is not zero'; these statements are mutually exclusive. A non-vanishing 2×2 kinetic determinant for two second-order variables would give two propagating degrees of freedom per transverse polarization, while a vanishing determinant would impose a constraint and reduce that number. Neither case yields the claimed '3 dof per index i'. The coefficients f1...f7 are not given, so the computation cannot be checked. Please provide the explicit vector Lagrangian, the first-order reduction, the actual determinant, and a correct degree-of-freedom count.
  3. [§5.2.1, Eqs. (94)–(105)] The scalar-sector count is not established. The coefficients a1...a17 in Eq. (94) are not displayed; the reduction to the second-order system (102) relies on the substitutions (92), (93), and (101), which are also not justified in detail; and the determinant logic is contradictory. The text first says that the first-order kinetic matrix for (ρ, ψ2, ω) has vanishing determinant and that this reduces the number of degrees of freedom by one, but then says that the matrix A in Eq. (104) has non-vanishing determinant and concludes that there are three degrees of freedom. A non-vanishing determinant of A alone is insufficient to prove the absence of hidden constraints without the full matrices B and C and an explicit check of the complete system. Please present the complete reduced Lagrangian or a Hamiltonian analysis of the scalar sector.
  4. [§4.2–4.3 and §5] The equivalence between the original conformal-gravity action (W²)^{d/4} and the alternative-frame action (Weyl-squared plus cosmological constant) is load-bearing but not proved. The transformation uses the auxiliary scalar with on-shell value φ = W², so the conformal factor is field-dependent and singular at W² = 0; the paper itself restricts to non-conformally-flat backgrounds. All five-dimensional mode counts are computed in the alternative frame and then attributed to the original theory. This requires an argument that the conformal transformation with a non-dynamical auxiliary scalar preserves the number of propagating degrees of freedom, at least perturbatively. Without such an argument, the claimed spectrum should be presented as a property of the alternative-frame action rather than of the original (W²)^{d/4} theory. The same issue applies to the f(W²) extension in Section 4.3.
  5. [§4.1, Eqs. (48)–(52)] The claimed spectrum for conformally flat spacetimes is not established. The scalar constraint (51) is not solved, and the conclusion that the theory propagates one vector mode and two tensor modes with no scalars is based on derivative counting and the statement 'This indicates to us that there will be no higher than four-time derivatives.' The Discussion repeats the claim as 'it is clear.' Since the conformally flat case is one of the paper's advertised results for d>4, either provide a complete derivation or explicitly label the claim as a conjecture.
minor comments (6)
  1. [§3.2, Eq. (38)] The cosmological term in Eq. (38), written as d M_P^{d−2} β_d^{−2/(d−2)} Λ, does not match Eqs. (35)–(36): since Λ already contains β_d^{−2/(d−2)}, the printed expression has the wrong power of β_d. It should presumably be −M_P^{d−2} Λ, or equivalently −(d−2)/2 M_P^d β_d^{−2/(d−2)}.
  2. [Eq. (107)] In Eq. (107), the coefficient f7 appears twice, in the terms f7 v_i v_i and f7 S_i v_i; one of these is presumably a different coefficient, likely f8.
  3. [Abstract and §3] The abstract and introduction refer to 'pure R² gravity', but Section 3 studies R^{d/2}, which is R² only in d=4. Please adjust the terminology to avoid confusion.
  4. [Introduction, last paragraph] The outline in the Introduction does not match the actual section structure: it says Section 2 covers pure scale-invariant gravity, Section 3 conformal gravity, and Section 4 the five-dimensional case, whereas the paper has these topics in Sections 3, 4, and 5 respectively.
  5. [Figures 1 and 2] The y-axis labels in Figures 1 and 2 use 'ws(t)', while the text defines the effective equations of state as ωa and ωb; please unify the notation.
  6. [§5.2.2 and §6] The phrase 'degenerate' is used in a contradictory way in the vector sector, and the Discussion states that a redundant scalar mode is indicated by a determinant that 'is not vanishing'; the logic should be corrected to say that a constraint is signaled by a vanishing determinant.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the 5D mode count is a self-contained perturbation analysis in a derived frame; self-citations provide re-derived 4D context only.

full rationale

The central new results are derived within the paper rather than imported. In Section 3.2, the Einstein-frame action for pure scale-invariant gravity is obtained by an explicit auxiliary-field and conformal-transformation calculation; the statement about the sign of the cosmological constant follows algebraically from the transformation, not from any prior fit. In Section 4.2, the alternative frame is introduced by writing the auxiliary-field action (53), varying to get φ=W^2, and then applying the conformal transformation (55); whether this step is fully correct is a mathematical question, but it is a derivation rather than a circular identification. The 5D perturbation analysis in Section 5.2 is a direct second-order expansion with no fitted parameters: the cosmological constant in (77) is fixed by the background equations of motion, not by matching data, and the mode counts are read off from kinetic structures such as (94), (107), and (109). Self-citations to [24,45,50] are used for well-known 4D facts, but those facts are re-derived in Section 2, so the citations are context rather than load-bearing. The paper itself flags the key limitation — the alternative frame is not valid at conformally flat points (end of the preamble to Section 5) and the conformally flat analysis is explicitly not proven (§4.1, 'we have not proven it exactly'). These weaken rigor but do not constitute circularity. The vector-sector counting in §5.2.2 is consistent with a non-degenerate system because v_i is fourth-order and S_i is second-order, giving three degrees of freedom per polarization before any constraint. Overall, no claimed prediction reduces by construction to fitted inputs or to the authors' prior results.

Assumptions & free parameters 2 free parameters · 4 assumptions · 1 invented entities

No parameters are fitted to data. The central claims rest on domain assumptions about positivity of R and W^2 for fractional powers, on the R ≠ 0 domain of the Einstein frame, and on the assumed equivalence of the alternative frame for counting dof.

free parameters (2)
  • β_d (pure scale-invariant gravity coupling)
    Dimensionless coupling of R^{d/2} action (19); the no-dof and Einstein-frame results hold for any nonzero β_d, so it is not fitted.
  • α_CG (conformal gravity coupling)
    Dimensionless coupling of (W^2)^{d/4} action (45) and of the 5D model; set to unity in the numerics (equation 78), a choice of units, not a fit.
assumptions (4)
  • domain assumption The actions S = β_d ∫ R^{d/2} and S = α_CG ∫ (W^2)^{d/4} are real and differentiable, so R and W^2 must be non-negative on the domains considered, especially for odd d.
    Fractional powers in (21) and (45) require positive arguments; this is never stated. If R or W^2 can change sign, the Lagrangian is complex and the perturbation expansion is not defined.
  • domain assumption The Einstein-frame transformation (32) requires R ≠ 0; the paper states this for the pure theory but relies on it for the positive-CC claim.
    Equation (30) sets φ = β^{2/(d-2)} R; the conformal factor in (32) degenerates at R=0.
  • ad hoc to paper The alternative frame (53)-(56) is a valid one-to-one field redefinition preserving the number of propagating degrees of freedom, despite the conformal factor depending on φ = W^2 and being singular at conformally flat points.
    The paper counts dof in the new frame and attributes them to the original theory; this equivalence is assumed without proof and is load-bearing for the 5D spectrum.
  • standard math Standard linear perturbation theory and the gauge choices (90) capture all physical dof; auxiliary fields introduced in (53) and (59) have algebraic equations with f''(φ) ≠ 0.
    Usual assumption in perturbative dof counting, invoked throughout Section 5.2.
invented entities (1)
  • Auxiliary scalar field φ introduced for the (W^2)^{d/4} action (53) and for f(W^2) (59)
    purpose: Linearizes the action; on-shell φ = W^2; no kinetic term.
    Standard Lagrange-multiplier device, not a physical particle. The paper's central dof count depends on eliminating it via its own equation of motion.

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Cite this review

Pith. "Pith review of Conformal and pure scale-invariant gravities in d dimensions." pith.science (2026). https://pith.science/paper/6MM2F5VZ

@misc{pith2026250618775,
  author       = {Pith},
  title        = {Pith review of: Conformal and pure scale-invariant gravities in d dimensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6MM2F5VZ}},
  note         = {Machine review of arXiv:2506.18775}
}
abstract

We consider conformal and scale-invariant gravities in d dimensions, with a special focus on pure $R^2$ gravity in the scale-invariant case. In four dimensions, the structure of these theories is well known. However, in dimensions larger than four, the behavior of the modes is so far unclear. In this work, we explore this question, studying the theories in conformally flat spacetimes as well as anisotropic backgrounds. First, we consider the pure theory in d-dimensions. We show that this theory propagates no degrees of freedom for flat space-time. Otherwise, we find the theory in the corresponding Einstein frame and show that it propagates a scalar field and two tensor modes, that arise from Einstein's gravity. We then consider conformal gravity in d dimensions. We argue on the number of degrees of freedom for conformally flat space-times and show that for $d>4$, there exists a frame in which this theory can be written as the Weyl-squared gravity with a cosmological constant, and also generalize this formulation to the $f\left(W^2\right)$ theories. Then, we consider the specific model of conformal gravity in five dimensions. We find the analytical and numerical solutions for the anisotropic Universe for this case, which admits super-Hubble and exponential expansions. Finally, we consider the perturbations around these solutions and study the number of the degrees of freedom.

Figures

Figures reproduced from arXiv: 2506.18775 by the authors.

Figure 1
Figure 1. Solution to the background equations of motion in the first case. The green line [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗
Figure 2
Figure 2. Solution to the background equations of motion in the second case. The green line [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗

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