Critical gravity from four dimensional scale invariant gravity

Ariel Edery; Yu Nakayama

Critical gravity from four dimensional scale invariant gravity

T0 review · 0 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper shows that pure scale-invariant quadratic gravity becomes critical gravity at β = 6α, with a massless graviton and logarithmic modes, without adding an Einstein term.

desk verdict A correct and clean derivation of the critical condition beta=6alpha for pure quadratic scale-invariant gravity; modest novelty, solid algebra, deserves refereeing. read the letter →

arxiv 1908.08778 v2 pith:6KVFKTNZ submitted 2019-08-23 hep-th gr-qc

Ariel Edery , Yu Nakayama This is my paper · ORCID

classification hep-thgr-qc

keywords criticalgravityscale-invariantquadraticcurvatureWeyltensormassivespin-twoghostlogarithmicmodesWaldentropydeSitterandanti-debackgrounds

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

The paper claims that critical gravity can arise from a purely quadratic, scale-invariant gravitational action with no explicit Einstein-Hilbert term or cosmological constant. For the action $S = \int d^4x \sqrt{-g}(\beta C_{\mu\nu\sigma\rho} C^{\mu\nu\sigma\rho} + \alpha R^2)$, the authors show that linearized fluctuations around de Sitter or anti-de Sitter space become critical at $\beta = 6\alpha$: the massive spin-two ghost turns into a massless graviton and logarithmic modes appear. The same condition is obtained in two independent ways, once through a conformal transformation to an Einstein-Weyl action with a massless scalar and once directly from the original quadratic action. If the claim is right, the defining features of critical gravity, including vanishing energy and Wald entropy for Schwarzschild and Kerr (A)dS black holes, are already present in a scale-invariant theory without an Einstein term.

What carries the argument

The machinery is a factorization of the linearized wave operator on maximally symmetric backgrounds. In harmonic gauge the spin-two perturbation satisfies a quartic equation that splits into two second-order wave operators, $(\bar{\Box} - 2\Lambda/3)(\bar{\Box} - 4\Lambda/3 + 4\alpha\Lambda/\beta)\tilde{h}_{\mu\nu} = 0$; the first operator is the massless graviton and the second is the massive spin-two mode. The traceless transverse field redefinition $\tilde{h}_{\mu\nu} = h_{\mu\nu} - \frac{1}{4}\bar{g}_{\mu\nu} h - \frac{3}{4\Lambda}\bar{\nabla}_{\mu}\bar{\nabla}_{\nu} h$ is what exposes the factorization. The alternative route uses a conformal transformation that maps the pure $R^2$ part to Einstein gravity with a cosmological constant and a massless scalar, leaving the Weyl-squared term untouched; the critical condition is then read off from the same factorization in that frame. The parameter dictionary between the two frames is what converts the critical condition into $\beta = 6\alpha$.

What would settle it

Compute the exact linearized propagator of the original $\beta C_{\mu\nu\sigma\rho}C^{\mu\nu\sigma\rho} + \alpha R^2$ action around anti-de Sitter space and locate its poles as a function of $\beta$; the claim predicts that the massive spin-two pole coincides with the massless pole at exactly $\beta = 6\alpha$ and splits away for any other value. A numerical or algebraic check of this pole coincidence would settle the critical condition.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the scale-invariant action $S = \int d^4x \sqrt{-g}(\beta C_{\mu\nu\sigma\rho} C^{\mu\nu\sigma\rho} + \alpha R^2)$ contains a critical line in parameter space. On a dS or AdS background, after harmonic gauge fixing and isolation of the traceless transverse spin-two perturbation $\tilde{h}_{\mu\nu}$, the linearized equation factorizes as $-\beta(\bar{\Box} - 2\Lambda/3)(\bar{\Box} - 4\Lambda/3 + 4\alpha\Lambda/\beta)\tilde{h}_{\mu\nu} = 0$. The second factor is the massive spin-two excitation, and it becomes massless exactly when $4\alpha/\beta = 2/3$, which is $\beta = 6\alpha$. At that value the two factors coincide, giving degenerate massless gravitons accompanied by logarithmic modes. The same condition is derived independently from the conformally related Einstein-Weyl action, where it appears as $\gamma = 3/(4\Lambda)$ and translates back to $\beta = 6\alpha$ through the dictionary $\gamma = 2\beta\kappa^2$, $\kappa^2 = -1/(4\alpha c_1)$, $\Lambda = -c_1/4$. The trace part of the perturbation yields a propagating massless scalar, and at the critical point the energy and Wald entropy of Schwarzschild or Kerr AdS/dS black holes vanish because both are proportional to $8\alpha - \frac{4}{3}\beta$.

Load-bearing premise

The load-bearing premise is that a de Sitter or anti-de Sitter vacuum exists and that linearizing around it in harmonic gauge, together with the conformal map between the two forms of the action, faithfully exposes the true particle content; if either step is not exact, the critical condition $\beta = 6\alpha$ may not survive.

Editorial extensions

If this is right

At $\beta = 6\alpha$ the massive spin-two ghost of the pure quadratic action becomes a massless graviton, so critical gravity exists without an explicit Einstein-Hilbert term.
The linearized theory at the critical point develops logarithmic spin-two modes, reproducing a distinctive feature of original critical gravity.
The energy and Wald entropy of Schwarzschild and Kerr (A)dS black holes vanish at $\beta = 6\alpha$, matching the zero values found in critical gravity.
With suitable boundary conditions the ghost and logarithmic modes can be removed, giving positive-energy solutions for $\beta \geq -48\alpha$ in Euclidean AdS and for $\beta < 6\alpha$ in dS.
At the critical condition the action becomes proportional to the square of the trace-free part of the Ricci tensor, linking the critical point to earlier energy results for quadratic gravity.

Reading between the lines

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

Beyond the paper's own claims, one could treat small deviations from $\beta = 6\alpha$ as a weakly coupled window in which the massless graviton carries positive energy while boundary conditions remove the remaining ghost; the paper sketches the boundary-condition mechanism but does not develop this as a phenomenological model.
The factorization mechanism suggests that analogous critical surfaces may exist for other scale-invariant curvature invariants or in other dimensions, wherever the linearized operator splits into two factors whose masses can coincide.
A direct holographic test would be to compute boundary two-point functions in the bulk theory at $\beta = 6\alpha$: the logarithmic modes should appear as logarithmic terms in the correlators, signalling a logarithmic conformal field theory on the boundary.
Because the massless scalar from the trace part arises from spontaneous breaking of the scale symmetry, it may provide a matter coupling channel through which the critical gravitational sector could be observed; computing that coupling in the Einstein frame would be a concrete next step.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Yuan — quick take on 1908.08778. The paper shows that the purely quadratic scale-invariant action beta C^2 + alpha R^2 has a critical point at beta=6alpha in dS/AdS, where the massive spin-two ghost becomes massless and the theory reduces to degenerate gravitons plus logarithmic modes. The result is real: I checked the trace contraction and the factorization in Sections 3 and 4, and the algebra is consistent. The stress-test note is right that the conformal-equivalence worry is not load-bearing. The direct linearization in Section 4 gives beta=6alpha without relying on the Einstein-frame dictionary, so even if one distrusted the conformal map the central claim stands.

What is actually new is the identification of this critical condition for the pure quadratic action, with no explicit Einstein-Hilbert term. The paper also shows two independent routes to the same condition: the R^2-to-Einstein-plus-scalar equivalence (re-derived in Section 2, not just cited) and the direct higher-derivative linearization. That is a worthwhile contribution. The zero-energy/zero-entropy at the critical point is not new — Deser-Tekin already noticed vanishing energy at the same parameter value — but the paper credits this and uses it as confirmation, which is honest.

Soft spots, in proportion. The boundary-condition discussion in Section 3 is heuristic; it imports Maldacena's and Lu-Pang-Pope's truncation arguments without adding much, and the Lorentzian-signature status remains murky. The energy/entropy section relies on external formulas from Adami et al.; fine, but it means the paper does not independently derive those charges. The conformal equivalence in Section 2 assumes phi != 0, which is satisfied by the dS/AdS vacua considered, but the equivalence is classical; that is all the linearized analysis needs. None of these undermine the main result.

The citation pattern is fair. The paper cites Deser-Tekin, the original critical gravity papers, and its own previous R^2 equivalence work — but the latter is re-derived, so no circularity burden.

Who is it for: people working on higher-derivative gravity, critical gravity, and holographic applications of log CFTs. It is not a breakthrough, but it is a solid, correct paper that clears up a gap in the literature. I would send it to peer review and expect acceptance after minor revision.

Referee Report

0 major / 4 minor

Summary. This paper considers the four-dimensional scale-invariant pure quadratic action S = ∫√(-g)(β C² + α R²) and shows that in dS or AdS backgrounds the linearized spectrum has a critical point at β=6α, at which the massive spin-two mode becomes massless and pairs with the massless graviton in a logarithmic multiplet. The result is obtained in two ways: first, by using the known equivalence of R² gravity to Einstein gravity with a cosmological constant plus a massless scalar after a conformal transformation, and second, by direct linearization of the original higher-derivative equations with a traceless-transverse field redefinition. At β=6α the energy and Wald entropy of Schwarzschild/Kerr (A)dS black holes are shown to vanish, and the boundary-condition discussion of [5,6] is invoked to argue that ghost and logarithmic modes can be removed away from the critical point.

Significance. The central claim β=6α is clean and is secured by two mutually consistent derivations; I checked the trace contraction leading to □h=0 and the factorization leading to Eq. (25), and both are correct. The paper is valuable because it extends critical gravity to a purely quadratic scale-invariant action without an explicit Einstein-Hilbert or cosmological-constant term, and it makes the parameter dictionary between the conformal-frame and original-frame descriptions explicit. The linearized calculations are presented in enough detail to be verified line by line, and the critical condition is parameter-free. The energy and entropy section is a useful consistency check with the existing critical-gravity literature, although it is not a derivation of those formulas themselves.

minor comments (4)

[Sec. 2, Eq. (8)] The definition of ψ after Eq. (8) is typeset ambiguously; it should read ψ = (√6/(2κ)) ln φ, since the canonical kinetic term then follows from κ² = -1/(4α c1).
[Sec. 3] In the paragraph after Eq. (16), 'pseudo-Rieamannian' is a typo for 'pseudo-Riemannian'.
[Sec. 5, Eq. (27)] The energy formula is imported from Ref. [19]; because the action (3) contains no explicit Einstein term, it would help the reader if one sentence explained that Eq. (27) is the appropriate limit of the conserved-charge formula of [19] for quadratic curvature actions.
[Sec. 3, Eq. (16)] The sentence beginning 'this is possible when γ ≥ -6/Λ or β ≥ -48α' refers specifically to the window 3Λ/4 ≤ m² < 0; stating this explicitly would prevent confusion with the stability condition γ ≤ 0 discussed earlier.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the critical condition β=6α is derived directly from the linearized equations in two independent ways, with no fitted parameter or self-citation chain doing the work.

full rationale

The critical condition β=6α is obtained in two independent routes, and neither reduces to an input. In the direct route, the trace contraction of Eq. (18) gives −6αΛ□h=0; with α≠0 and Λ≠0, this yields □h=0. The traceless-transverse field (20) is defined from h, and substituting it into (18) using the commutation identities (22) and (24) produces Eq. (23), which factorizes as −β(□−2Λ/3)(□−4Λ/3+4αΛ/β)h̃=0. The massive spin-two pole becomes massless exactly when 4α/β=2/3, i.e. β=6α. No parameter is fitted to the target result; the factorization is the outcome of the linearized equations, not an assumed input. The Einstein-frame route is equally self-contained: Section 2 explicitly derives the equivalence between the original quadratic action and the Einstein-Weyl action via the auxiliary field and conformal transformation, and the dictionary γ=2βκ², κ²=−1/(4αc1), Λ=−c1/4 is stated and then substituted into γ=3/(4Λ) to recover the same β=6α. The self-citations [9,11] to restricted Weyl invariance and spontaneous symmetry breaking are contextual; the paper re-derives the needed transformation and invariance statements in the text. The vanishing energy and entropy at β=6α is a direct substitution into previously known formulas, not a constraint used to derive the critical condition. The φ≠0 assumption for the conformal transformation is satisfied by the dS/AdS vacuum, and the direct higher-derivative derivation in Section 4 does not rely on that equivalence. No step equates the prediction to the input by construction, and no load-bearing conclusion depends on an unverified self-citation.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

The central derivation relies on standard linearized higher-derivative gravity around maximally symmetric backgrounds. No constants are fitted to data; the auxiliary constant c1 is arbitrary and cancels from the critical condition. No new particles or mediators are introduced.

free parameters (1)

c1 = arbitrary, cancels in critical condition
Introduced as a free constant in the auxiliary field rewriting (5); the physical dictionary between α, β and κ, Λ depends on c1, but the critical condition β=6α is independent of it.

assumptions (3)

domain assumption The background is maximally symmetric dS or AdS with Λ≠0 and curvature relations (12).
All linearizations are performed about this background; the trace equation (19) requires Λ≠0 to conclude □h=0.
domain assumption The restricted Weyl symmetry (4) is spontaneously broken by a vacuum with φ≠0, so the conformal transformation (7) is valid.
Section 2 notes the transformation is not valid for φ=0 and assumes a vacuum with φ≠0.
domain assumption α≠0 for the non-conformal case, and the linearized equations in harmonic gauge (13) preserve the gauge.
The paper explicitly assumes α≠0 before Eq. (19); the harmonic gauge condition is standard but the preservation under the field redefinition (20) is used.

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Pith. "Pith review of Critical gravity from four dimensional scale invariant gravity." pith.science (2026). https://pith.science/paper/6KVFKTNZ

@misc{pith2026190808778,
  author       = {Pith},
  title        = {Pith review of: Critical gravity from four dimensional scale invariant gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6KVFKTNZ}},
  note         = {Machine review of arXiv:1908.08778}
}

abstract

We show that a critical condition exists in four dimensional scale invariant gravity given by the pure quadratic action $\beta \,C_{\mu\nu\sigma\rho} C^{\mu\nu\sigma \rho} + \alpha \,R^2$ where $C^{\mu}_{\,\,\nu \sigma \rho}$ is the Weyl tensor, $R$ is the Ricci scalar and $\beta$ and $\alpha$ are dimensionless parameters. The critical condition in a dS or AdS background is $\beta =6 \alpha$. This leads to critical gravity where the massive spin two physical ghost becomes a massless spin two graviton. In contrast to the original work on critical gravity, no Einstein gravity with a cosmological constant is added explicitly to the higher-derivative action. The critical condition is obtained in two independent ways. In the first case, we show the equivalence between the initial action and an action containing Einstein gravity, a cosmological constant, a massless scalar field plus Weyl squared gravity. The scale invariance is spontaneously broken. The linearized Einstein-Weyl equations about a dS or AdS background yield the critical condition $\beta=6\alpha$. In the second case, we work directly with the original quadratic action. After a suitable field redefinition, where the metric perturbation is traceless and transverse, we obtain linearized equations about a dS or AdS background that yield the critical condition $\beta= 6\alpha$. As in the first case, we also obtain a propagating massless scalar field. Substituting $\beta=6\alpha$ into the energy and entropy formula for the Schwarzschild and Kerr AdS or dS black hole in higher-derivative gravity yields zero, the same value obtained in the original work on critical gravity. We discuss the role of boundary conditions in relaxing the $\beta=6\alpha$ condition.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Lu and C.N

H. Lu and C.N. Pope, Critical gravity in Four Dimensions , Phys. Rev. Lett. 106, 181302 (2011) [arXiv:1101.1971]

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