Henneaux-Teitelboim Form of the Generalized Unimodular Gravity Action

Dmitry Nesterov; Ksenia Lyamkina

REVIEW 3 major objections 4 minor 18 references

Henneaux-Teitelboim Form of the Generalized Unimodular Gravity Action

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper derives an alternative action for generalized unimodular gravity, equivalent to the original one, that makes a hidden spatial nonlocality explicit and shows the metric-sector gauge symmetry is not full diffeomorphism invariance.

desk verdict A plausible but thinly supported extension of Henneaux-Teitelboim to GUMG, with the load-bearing constraint-algebra step deferred to a companion paper. read the letter →

arxiv 2412.16139 v2 pith:3IP6IE7C submitted 2024-12-20 hep-th gr-qc

Dmitry Nesterov , Ksenia Lyamkina This is my paper · ORCID

classification hep-thgr-qc PACS 04.20.Cv04.20.Fy04.60.-m

keywords generalizedunimodulargravityspatialnonlocalitytimereparameterizationconstraintalgebracosmologicalconstantperfectfluidgaugesymmetrycanonicalaction

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 tries to establish that generalized unimodular gravity (GUMG) can be rewritten in a new action form, analogous to the covariant action known for unimodular gravity, without losing physical equivalence. The proposed action combines the scalar-curvature term with a cosmological-constant field $\Lambda$ and a term built from a spatially nonlocal operator $\hat E$. If the construction is correct, it reproduces the original GUMG dynamics, including the perfect-fluid equation of state $p=W\rho$, while making explicit that the theory carries a genuinely nonlocal spatial structure and that its gauge symmetry is narrower than full spacetime diffeomorphism invariance. A careful reader would care because the nonlocality is claimed to be present already in the original formulation, not manufactured by the new variables, and the new form gives a cleaner starting point for cosmological and quantum studies.

What carries the argument

The load-bearing mechanism has three cooperating parts. The first is a homogeneous time parameterization that introduces the auxiliary clock $\tau_0$, its momentum $\pi_0$, and the multiplier $\lambda_0$, preserving the constraint rank. The second is the spatially nonlocal operator $\hat E$, whose kernel is a delta function plus rank-one corrections built from the barotropic index $W$ and the background spatial volume; it reshapes the functionally incomplete Hamiltonian constraint into a functionally complete one, and its inverse (10) is used to pass to the simpler action (9). The third is the first-class combination $P_I=\pi_0+F H_\perp+U_0^n H_n$, formed with an on-shell shift $U_0^n$, which generates the canonical gauge algebra together with the transverse spatial diffeomorphisms.

What would settle it

For a nonconstant $W$ model such as $F(\sqrt\gamma)=\sqrt\gamma^{1/2}$, compute the full Poisson-bracket matrix of the constraints in the parameterized action (20), including $\int\epsilon(\pi_0+F H_\perp+U_0^n H_n)$. If the rank of that matrix is larger than the original set $(H_n,(WFH_\perp)_{,m})$, or if closure forces new on-shell conditions, the claimed equivalence to the original GUMG action fails.

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Extended reading notes

Core claim

On its own terms, the paper's central claim is that the action $S_{\mathrm{alt}}[g,\Lambda,V]=\int dt\,dx\,\sqrt{|g|}({}^dR(g)-\Lambda)+\int dt\,dx\,\partial_\mu V^\mu \hat E(F\sqrt{\gamma}\,\Lambda)$, with the nonlocal kernel (7), is an equivalent alternative description of GUMG. The derivation passes through a homogeneous time parameterization in which the new first-class constraint $\int\epsilon P_I$ with $P_I=\pi_0+F H_\perp+U_0^n H_n$ generates the canonical gauge algebra; in this basis the Hamiltonian constraint becomes $\pi_0+\hat E F H_\perp=0$. The operator $\hat E$ is needed because the secondary constraint freezes only the average-free part of the weighted Hamiltonian density, so rebuilding a functionally complete constraint spreads it over the spatial volume. On shell the redefined field $\Lambda_0=\hat E F\sqrt{\gamma}\,\Lambda$ is an unfixed spacetime constant, while $\Lambda$ itself behaves as a spatially nonlocal constant, $\Lambda\sim\sqrt{\gamma}^{-1}F^{-1}(W^{-1}/\overline{W^{-1}})c$. The field equations acquire a perfect-fluid source with energy density $\Lambda$ and pressure $W\Lambda$, recovering $p=W\rho$; in contrast to unimodular gravity, the resulting action is not fully diffeomorphism-invariant.

Load-bearing premise

The equivalence rests on the assumption that the homogeneous time parameterization preserves the number of independent gauge symmetries and that the proposed combination $P_I=\pi_0+F H_\perp+U_0^n H_n$ is first-class; the detailed proof is deferred to a companion paper.

Editorial extensions

If this is right

For nonconstant $W$, the on-shell cosmological-constant field $\Lambda$ is genuinely spatially nonlocal, so the energy density and pressure of the effective perfect fluid vary across a spatial slice even though the integration constant $c$ is fixed.
The metric-sector gauge symmetry of GUMG is only a homogeneous time reparameterization; the average-free part of the new symmetry acts in the auxiliary sector, so the alternative action is not invariant under the full diffeomorphism group.
For the w-GUMG subfamily ($W\equiv w$), $\hat E$ becomes the identity and the nonlocality disappears; in the unimodular limit $W=-1$, full diffeomorphism invariance is restored.
The action places the cosmological constant on the same footing as a dynamical field with a free integration constant, the property that makes the covariant unimodular formulation attractive for studying quantum properties.
The construction is presented as extendable to other restricted gravity theories whose secondary constraints have the same gradient form.

Reading between the lines

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

Extension: for a nonconstant $W$, cosmological perturbations in this formulation should inherit nonlocal integral terms, and computing the scalar power spectrum for a specific $F(\sqrt\gamma)$ would show whether such terms are observable.
Extension: the averages and volume $V$ that define $\hat E$ and its inverse presuppose a compact spatial slice; on noncompact slices an infrared prescription is needed, and the physical content may depend on that choice.
Extension: the same operator construction should apply to any field theory whose secondary constraints are gradients of a weighted Hamiltonian density; building the analogous nonlocal action for another restricted system would be a direct test of the method's generality.

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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

The genuinely new thing here is the spatially nonlocal operator E-hat and the claim that GUMG, unlike UMG, does not regain full diffeomorphism invariance in the HT-type formulation. That is a real conceptual point, and the w-GUMG limit correctly reduces to known results. The paper is clearly written for a letter, and the authors know the Dirac machinery well. But the central equivalence rests on a step that is asserted, not shown: that the homogeneous time parameterization of (16) preserves the constraint rank and that the new first-class constraint is integral epsilon P_I. The stress-test note lands. Equations (18)-(19) are in tension with the local form of (20) for nonconstant W. The bracket of C[epsilon] with H_n contains a term proportional to epsilon, n, so unless the smearing is homogeneous, the local pi0 + F Hperp constraint does not have the first-class property the action (20) needs. The paper's own footnote 9 says the rank property 'may be checked' and defers to [13]. That is the load-bearing step. If the constraint algebra of (20) is not the stated first-class algebra, the Legendre reduction to S_alt (6) does not follow, and the paper's main result is unproven. The derivations of (21) and the dynamics in Section IV are fine conditional on that algebra, and the nonlocality discussion is interesting, but they do not supply the missing proof. The paper honestly flags the GR branch and the W=0, W=-1 exceptions, which is good. The citation pattern looks fair, and the authors are not overclaiming. Still, as a referee I would not accept this as a standalone letter without the companion's analysis, and the companion is not available in this submission. For a reader who cares about GUMG or HT formulations, this is worth knowing about, but it is not a closed result. My recommendation: send it to peer review, but with the clear expectation that the constraint analysis must be either included or substantially summarized, not merely deferred. If the companion checks out, this would be a solid contribution. As it stands, the letter is a promissory note with an interesting stake.

Referee Report

3 major / 4 minor

Summary. This paper proposes an alternative, Henneaux-Teitelboim-type action for generalized unimodular gravity (GUMG), given in Eq. (6) as S_alt = ∫dt dx sqrt(|g|)(R - Λ) + ∫dt dx ∂_μ V^μ Ê F √γ Λ, with Ê a spatially nonlocal operator defined by the kernel (7). The derivation starts from the canonical action (16), introduces a homogeneous time parameterization, then adds average-free auxiliary fields to arrive at the parameterized action (20). After a constraint-basis rearrangement and Legendre reduction, the paper obtains the covariant action (6) and its equivalent form (9). The paper shows that the dynamics reproduce the GUMG perfect-fluid equation of state p = Wρ, that the cosmological-constant field becomes an on-shell spacetime constant via the nonlocal operator, and that the gauge symmetry in the gravitational sector is a homogeneous time reparameterization, not full diffeomorphism invariance. The exceptional cases W = 0, W = -1 and the GR branch H⊥ = 0 are mentioned but not analyzed in detail, with reference to the companion paper [13].

Significance. If the equivalence claim is correct, the paper provides a new covariant formulation of GUMG that makes an implicit spatial nonlocality explicit and clarifies the gauge structure of the theory. This is potentially useful for further studies of restricted gravity theories and for quantum considerations. The paper has the merit of presenting the nonlocal operator Ê and its inverse explicitly, and of deriving the equation of state and the on-shell behavior of Λ without fitting any parameters. However, the central derivation is not fully self-contained: the rank-preservation and gauge-accessibility properties of the parameterized action (20) are asserted rather than proven, and the key simplifications in the constraint-basis rearrangement are delegated to the companion paper [13]. The strong reliance on [13] makes the letter difficult to evaluate on its own.

major comments (3)

[Section III, Eq. (20) and Eqs. (18)-(19)] The rank-preservation claim for the constraint set of the parameterized action (20) is not established in the manuscript. For nonconstant W, the Poisson bracket {∫ε(π0+F H⊥), ∫η^n H_n} in Eq. (19) contains the term -∫ ε,n η^n W^{-1} T, which does not vanish on the constraint surface unless the smearing ε is homogeneous. The action (20) is written with local auxiliary fields τ0(t,x), π0(t,x), λ0(t,x), so the Hamiltonian constraint is naturally smeared by the local multiplier λ0(x). Footnote 9 asserts that the rank may be checked after substituting ε → ε̄, ε′ → ε̄′, but this is not a demonstration and does not address the status of the local constraint in the action as printed. Since the subsequent Legendre transformation to the nonlocal action (6) relies on this constraint structure, the equivalence claim is not yet supported.
[Section III, between Eq. (20) and Eq. (21)] The reduction of the secondary constraint to π0,m = 0 is stated without proof: 'The latter looks awkward but simplifies to π0,m = 0 [13].' This step is essential for the inversion π0 = -Ê F H⊥ and for the final form of the action (6). Without a derivation or a precise statement of the conditions under which it holds, the reader cannot verify the equivalence of (20) and (21). Please provide the computation or at least a self-contained appendix.
[Section III and footnote 5] The paper explicitly excludes the GR branch H⊥ = 0 and the cases W = 0 and W = -1 from the main analysis, yet claims that the new parameterized description correctly describes the GR branch (footnote 5). No argument is given for this claim, and the reduction to the original GUMG action on the GR branch is not demonstrated. In the exceptional case W = -1 (UMG), the paper appeals to the Henneaux-Teitelboim result, but the behavior of the new action for W = 0 is only sketched. Since these branches are part of the GUMG family, the equivalence claim should either cover them or explicitly state the conditions under which it applies.

minor comments (4)

[Abstract and Section I] The term 'functionally incomplete symmetry' is used without definition; it should be defined in the introduction or in a footnote where it first appears.
[Eq. (7) and Eq. (10)] The notation V = ∫1 for the background spatial volume is terse; spell it out as V = ∫_{t=const} dx 1, or define it explicitly in the text.
[Section III, after Eq. (20)] The phrase 'universally valid homogeneous time parameterization' and the subsequent addition of 'average-free parts' is not clearly delineated; a step-by-step outline of the two-step construction would improve readability.
[Throughout] The references to [13] are frequent and not specific; for example, 'The rank property may be checked using (19)' and 'simplifies to π0,m = 0 [13]'. Provide section or equation numbers in [13] for these claims so that the reader can locate the supporting arguments.

Circularity Check

1 steps flagged · score 4.0 of 10

No definitional circularity in the operator construction, but the central equivalence claim relies on an unproven constraint-algebra step deferred to the authors' companion paper [13].

self citation load bearing [Section III, around Eq. (20) and the following paragraph; also Section V and Eqs. (18)-(19)]
"For the general GUMG model, this parameterization yields the equivalent parameterized action in the form ... (20) ... It can be verified that the rank of the matrix of Poisson brackets of constraints does not change, the gauge τ0−t=0 is accessible, and the reduction with respect to (π0+F H⊥, τ0−t) reproduce (16). ... The form of the new first-class constraint ... can be guessed from general principles. This is P_I = π0+F H⊥+ U^n_0 H_n ... The new first-class constraint ∫ ε P_I and the transverse diffeomorphisms ... generate the canonical gauge algebra of the model."

The claimed equivalence of the proposed action to GUMG rests on two load-bearing facts: (i) that the homogeneous time parameterization (20) preserves the constraint-bracket rank, and (ii) that ∫ ε P_I is a first-class generator of the gauge algebra with U^n_0 an on-shell multiplier solution. Neither is demonstrated in the letter for general nonconstant W. The letter says the rank can be 'verified' using (19), but (19) itself shows that the local constraint π0+F H⊥ has non-vanishing brackets with H_n for W≠const unless the smearing ε is homogeneous, while the printed action (20) contains local λ0(t,x); the resolution is not shown. The Conclusions defer all 'detailed discussion' and 'complete analysis of the gauge structure' to the companion paper [13], by the same author.

full rationale

The construction of the nonlocal operator (7) is not circular in the definitional sense: the paper defines E so that the constraint set π0+E F H⊥=0, π0,m=0 is equivalent to the original GUMG constraints, and the Legendre transformation from (20) to (21) and then to (6) is an explicit invertible rearrangement. The equation-of-state result p=Wρ is a reproduction of the known GUMG relation, not a prediction obtained from fitted parameters. The main circularity burden is the load-bearing self-citation: the rank-preservation and first-class property of P_I, which are essential for the equivalence of (20) to (16) and hence for the whole alternative action, are asserted rather than proven and are deferred to the same-author companion paper [13]. Because the explicit action and several consistency checks are given in the letter, the central claim still has independent content, but the derivation is not fully self-contained at its pivotal step.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new physical particles or forces. The main new mathematical object is the nonlocal operator E-hat, which is constructed from the existing fields and a background volume V. The ledger lists V and the domain assumptions that the constraint analysis depends on, including compactness, sign-definiteness of W, and exclusion of the GR branch.

free parameters (1)

Background spatial volume V = Not fitted; arbitrary normalization scale (V = integral 1)
Appears in the kernel of the nonlocal operator E-hat and in the definition of spatial averages. It is an arbitrary regularization constant, and the letter does not show explicitly that physical results are independent of it.

assumptions (4)

domain assumption Spatial sections are compact, or the noncompact case is handled by the extension in [13], so averages over t = const hypersurfaces are well-defined.
Definitions of f, V, and the average-free decomposition (15), as well as kernel (7), require a finite volume; the letter defers the noncompact extension to companion [13].
domain assumption F(sqrt(gamma)) is monotonic and W(sqrt(gamma)) is sign-definite; exceptional cases W = 0, W = -1, and Omega = 0 are excluded or treated separately.
Footnote 1 and the discussion after Eq. (19) state these assumptions are needed to avoid pathologies in the constraint structure; UMG corresponds to the special case W = -1.
domain assumption The system is outside the GR branch H_perp = 0, where all constraints become first-class and the classification changes.
Footnotes 5 and 10 state that the discussed properties apply to the physical space outside the GR branch; the new description is claimed to handle it, but the proof is deferred to [12,13].
domain assumption The original GUMG constraint structure from [11], including the secondary constraints (W F H_perp),m = 0, is correct.
The derivation begins from the canonical action (16) and relies on the Dirac consistency analysis of GUMG presented in the authors' prior work [11].

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Cite this review

Pith. "Pith review of Henneaux-Teitelboim Form of the Generalized Unimodular Gravity Action." pith.science (2026). https://pith.science/paper/3IP6IE7C

@misc{pith2026241216139,
  author       = {Pith},
  title        = {Pith review of: Henneaux-Teitelboim Form of the Generalized Unimodular Gravity Action},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3IP6IE7C}},
  note         = {Machine review of arXiv:2412.16139}
}

read the original abstract

We propose an alternative description of generalized unimodular gravity (GUMG), extending the Henneaux-Teitelboim approach to unimodular gravity (UMG). The central feature of this formulation is the consistent incorporation of time reparameterization, which enhances the gauge structure and reveals a spatial nonlocality hidden in the dynamics of the original formulation. We examine the resulting dynamics, emphasizing the effects of spatial nonlocality, and outline the constraint structure. We show that the gauge symmetry in the gravitational sector is extended by a functionally incomplete symmetry, as occurs in the unimodular gravity. However, in contrast to the latter, the resulting action is not fully diffeomorphism-invariant.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

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with respect to the λ⊥, N ⊥, then introduce conjugate momenta πmn only for the spatial metric γmn. This yields a canonical theory on the phase space ( γmn , πmn) with a Hamiltonian density F (√γ )H⊥(γ, π) and primary constraints Hn(γ, π) = 0, cor- responding to spatial diﬀeomorphisms. The structures H⊥ and Hn are identical to the Hamiltonian and momen- tu...

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