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Alternative Action for Generalized Unimodular Gravity

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Every generalized unimodular gravity model admits an equivalent action in which general relativity couples to an auxiliary cosmological-constant field through a spatially nonlocal operator, with unimodular gravity as the exceptional local…

desk verdict A genuinely new HT-type action for GUMG with a real proof gap: the first-class status of the central constraint is asserted rather than demonstrated. read the letter →

arxiv 2505.13548 v1 pith:4ZCWGSQQ submitted 2025-05-19 gr-qc hep-th

Dmitry Nesterov This is my paper · ORCID
classification gr-qchep-th
keywords generalizedunimodulargravityHenneaux–Teitelboimactioncanonicalconstraintstimereparametrizationspatialnonlocalitycosmologicalperfectfluidbarotropicparameter
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

Generalized unimodular gravity is a family of modified gravity theories that keep the Einstein–Hilbert dynamics but partially break coordinate covariance by restricting the lapse function to a prescribed function $F(\sqrt{\gamma})$ of the spatial volume. This paper proves that every model in the family can be rewritten, by a constructive canonical time reparametrization, as an equivalent action of the Henneaux–Teitelboim type: the Einstein–Hilbert term plus a dynamical cosmological-constant field, with the model's characteristic function $W=d\ln F/d\ln\sqrt{\gamma}$ encoded in a spatially nonlocal operator $E$. A reader should care because the equivalence turns a family of apparently different constrained theories into one unified description, exposes their common gauge structure, and shows that the extra degree of freedom behaves as a cosmological perfect fluid with equation of state $p=W\rho$. For constant $W$ the action becomes local, and at $W=-1$ it reduces to the standard covariant form of unimodular gravity.

What carries the argument

The load-bearing object is the delocalization operator $E=I+W^{-1}\tilde I W$, where $I$ is the identity, $I$ projects onto spatially homogeneous (average) functions, and $\tilde I=I-I$ projects onto average-free functions; $W=d\ln F/d\ln\sqrt{\gamma}$ is the model's barotropic parameter. The operator appears when the Hamiltonian constraint of general relativity, which lives only in its spatially averaged part after consistent parameterization, must be merged with GUMG's secondary constraint into one functionally complete constraint. Because $E$ is invertible, the constraint basis can be rearranged as $\pi_0+E F H_\perp=0$ and $\pi_{0,m}=0$, which is what allows the metric momentum dependence to be reduced as in general relativity and the auxiliary fields to assemble into $\partial_\mu\mathcal{V}^\mu$. The consistency of the whole construction rests on the first-class constraint $P_I=\pi_0+F H_\perp+U_0^n H_n$ in the parameterized action. The same operator controls the nonlocal on-shell relation for $\Lambda$ and the quantum measure, whose determinant is $\mathrm{Det}\,E=W\,\overline{W^{-1}}\ge 1$.

What would settle it

Choose a GUMG model with spatially varying $W(\sqrt{\gamma})$ on a compact spatial slice and test the parameterized action's new constraint $P_I=\pi_0+F H_\perp+U_0^n H_n$ by the standard Dirac consistency procedure: if preserving that constraint in time generates new constraints, or if the rank of the constraint matrix changes, the two actions are not equivalent. The exceptional characteristic functions with $\Omega(\sqrt{\gamma})=0$, such as $F\propto|b+\sqrt{\gamma}^{-1}|$, are the natural place to look, since the paper itself identifies them as potentially pathological.

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Extended reading notes

Core claim

The central claim is a constructive proof that the generic GUMG action $S[g,\lambda_\perp]=\int dt\,dx\,\sqrt{|g|}\,R-\int dt\,dx\,\lambda_\perp(N^\perp-F(\sqrt{\gamma}))$ is classically equivalent to $S_{\rm alt}[g,\Lambda,\mathcal{V}]=\int dt\,dx\,\sqrt{|g|}(R-\Lambda)+\int dt\,dx\,\partial_\mu\mathcal{V}^\mu\,E F\sqrt{\gamma}\,\Lambda$, where $\Lambda(t,x)$ is an auxiliary cosmological-constant field, $\mathcal{V}^\mu$ is an auxiliary vector field entering only through $\partial_\mu\mathcal{V}^\mu$, and $E$ is an invertible, local-in-time but spatially nonlocal operator built from average and average-free projectors. The equivalence is established by introducing time parametrization into the canonical action, adding an auxiliary canonical pair, and rearranging the constraint basis so that all metric-momentum dependence sits in the usual general-relativity structures. The resulting action reproduces the original GUMG dynamics and gauge structure on both dynamical branches: the GR branch has the same degrees of freedom as general relativity, while the non-GR branch carries one extra degree of freedom that on shell is a cosmological perfect fluid with barotropic parameter $W(\sqrt{\gamma})$. When $W$ is constant the operator $E$ becomes the identity and the action is manifestly local; when $W=-1$ the whole action becomes the standard Henneaux–Teitelboim covariant action. On shell the effective cosmological constant feels the spatial average of $W^{-1}$, namely $\Lambda\sim\sqrt{\gamma}^{-1}F^{-1}\,\overline{W^{-1}}/W^{-1}\,c_0$.

Load-bearing premise

The whole equivalence chain rests on the claim that the reparametrization constraint $P_I=\pi_0+F H_\perp+U_0^n H_n$ is exactly first-class in the parameterized action, and that the nonlocal operator $E$ is invertible everywhere; if $W$ or $\Omega$ vanish on some configuration, the constraint algebra branches and the two actions describe different physics.

Editorial extensions

If this is right

  • Every GUMG model can be analyzed in one common representation, with all model dependence isolated in $F$, $W$, and $E$; this should simplify comparisons between different restriction functions and with unimodular gravity.
  • The on-shell value of the effective cosmological constant becomes a spatially nonlocal functional of the metric, so cosmological solutions in the same GUMG model can differ purely from the global spatial profile of $W$.
  • The gauge symmetry on the non-GR branch is exhausted by homogeneous time reparametrizations and transverse (volume-preserving) spatial diffeomorphisms; local time reparametrizations and longitudinal spatial diffeomorphisms are broken.
  • In the w-GUMG subfamily with $W=\mathrm{const}$, the alternative action is fully local while keeping the same mixed-class gauge structure, giving a simpler setting for studying GUMG dynamics.
  • The path-integral measure in the $\Lambda$ representation acquires a non-ultralocal factor $\mathrm{Det}\,E=W\,\overline{W^{-1}}\ge 1$, so spatially inhomogeneous $W$ configurations are weighted differently in the quantum theory.

Reading between the lines

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

  • A general criterion suggested by this construction is that a restricted gravity theory admits a Henneaux–Teitelboim-like covariantization exactly when its secondary constraint is, up to a local weight, the average-free part of the Hamiltonian; theories with additional structure would require genuinely new techniques.
  • A testable extension is to insert the nonlocal on-shell relation for $\Lambda$ into a homogeneous cosmological model, where $W(\sqrt{\gamma})$ depends on the scale factor, and compare the resulting effective dark-energy equation of state with reconstructions from supernovae and cosmic microwave background data.
  • Because the surviving spatial gauge symmetry is exactly volume-preserving diffeomorphisms, the whole family could be reformulated as general relativity on a spacetime with a fixed spatial volume form; the paper does not develop this coordinate-free reading, but the constraint analysis points directly to it.
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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

3 major / 4 minor

Summary. The paper claims to construct, by a sequence of canonical transformations, an alternative action (7) for the whole family of generalized unimodular gravity (GUMG) theories introduced in Eq. (1). The construction starts from the canonical extended action (31), implements time reparametrization, rearranges the constraint basis, and introduces a spatially nonlocal operator E defined in Eqs. (95)-(96). For the constant-W subfamily (w-GUMG) the action is claimed to become local and to reproduce the Henneaux-Teitelboim action of unimodular gravity in the special case W = -1. The paper also analyzes the on-shell dynamics, the gauge structure, and the effect of the nonlocality on the quantum measure. The central claim is that the alternative action (7)/(106) is classically equivalent to the original GUMG action for all nonexceptional characteristic functions F(√γ).

Significance. If the central equivalence claim were correct, the paper would be a valuable contribution: it would provide a Henneaux-Teitelboim-like covariant formulation for the entire GUMG family, make explicit the spatial delocalization inherent in the model, and open the way to a systematic study of the gauge structure and quantum measure. The paper is constructive, contains many explicit computations, and the w-GUMG subfamily (Section 3) appears to be handled carefully; the local action (55) for constant W is a credible result that could be useful on its own. However, the general-GUMG equivalence is the main result, and it is not established: the proof contains a load-bearing error in the constraint-basis rearrangement of Section 4.2, and the first-class property of the key constraint P_I in Section 4.1 is only asserted. Because the primary claim of the paper is the equivalence for the general GUMG family, the significance of the paper as it stands is substantially reduced.

major comments (3)
  1. [Section 4.2, Eq. (98)] The claimed equivalence of the constraint bases in (98) is not correct for nonconstant W(√γ). Using the explicit kernel (95), one has E(FH⊥) = FH⊥ − W^{-1} avg(W F H⊥) + avg(F H⊥). On the secondary surface (W F H⊥),m = 0, i.e. W F H⊥ = A(t), this gives E(FH⊥) = A(t) avg(W^{-1}), whereas F H⊥ = A(t) W^{-1}. Consequently the new first constraint π0 + E F H⊥ = 0 forces π0 = −A(t) avg(W^{-1}), which is spatially constant, while the old first constraint π0 + F H⊥ = 0 forces π0 = −A(t) W^{-1}, which is spatially constant only when W is constant. The two sets in (98) therefore describe different constraint surfaces in the extended phase space. This invalidates the derivation of the alternative action (7)/(106) and the subsequent on-shell relation (108) for general GUMG theories. The subsequent Lagrange-multiplier redefinition (100) does not repair the discrepancy, since it cannot change the constraint surface itself.
  2. [Section 4.1, Eq. (92)] The first-class property of P_I = π0 + F H⊥ + U_0^n H_n is asserted rather than proved. The text says that the system of involution conditions 'now has a solution given by (92)', but no Poisson-bracket computation is shown for {P_I, ∫f(π0+FH⊥)}, {P_I, ∫ξ^n H_n}, or {P_I, ∫η^n (WFH⊥),n}. This matters because U_0^n is only the on-shell Lagrange-multiplier solution (34); the standard Dirac construction yields an integrated first-class Hamiltonian, and it is not automatic that the local smeared density P_I is in involution with all constraints. The rank and degree-of-freedom equivalence between (91) and (31) depends directly on this point. A concrete bracket computation, or a precise reference to where it is performed, is required.
  3. [Section 4.2, Eq. (99)] The assertion that (91) and (99) are equivalent because 'the equivalence of representations is guaranteed by the freedom of choosing any equivalent constraint basis' is too quick. A constraint-basis change is permissible only if the two sets have the same constraint surface and the same rank structure. As shown in the first major comment, the two sets in (98) do not define the same surface for nonconstant W, so the equivalence of (91) and (99) is not established. This is not a presentation issue but a load-bearing gap in the constructive proof of the main result.
minor comments (4)
  1. [Section 4.2 and Appendix A] The notation for the identity operator and the averaging projector is extremely confusing: the same symbol I is used for both in equations such as (96) and (135). This ambiguity appears to be directly connected to the error in the reformulation of the first constraint in (98). The authors should use distinct symbols, e.g. Id for the identity and P_avg or I_avg for the averaging projector, and recheck all equations involving E.
  2. [Section 4.1, after Eq. (91)] The claim that the consistently parameterized action (91) is 'physically equivalent' to the original GUMG extended action (31) is stated without a full proof of the rank and transversality conditions; please provide the explicit consistency conditions and show that they are satisfied.
  3. [Section 5, Conclusions] There is a typo in the first paragraph: 'various issues are still are still open' should read 'various issues are still open'.
  4. [Appendix D] The treatment of noncompact spatial sections is explicitly heuristic and relies on a finite-volume regulator; this is acknowledged by the author, but the main text refers to the results as if they were established. Please state more clearly which results in Sections 4.3-4.4 depend on the compactness assumption and which are expected to survive in the noncompact case.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the alternative action is derived by explicit equivalence-preserving canonical transformations from the original GUMG action, and the perfect-fluid equation of state is inherited as a consistency check, not fitted.

full rationale

This paper is a constructive reformulation, not an empirical prediction paper. Section 2 re-derives the canonical extended action (31) from the GUMG action (1) with explicit Poisson brackets (32), so the starting constraint structure is re-derived rather than imported solely from [13]. The alternative actions (52), (55), (104), and (106) are obtained by parameterization, invertible constraint-basis changes, and reduction; the text states that all intermediate transformations preserve classical equivalence, and the equations of motion of the alternative action are therefore equivalent to the original action by construction. That is the stated theorem, not hidden circularity. The recovered perfect-fluid equation of state p = W rho (117) is the same relation already present in the original model (5) and is used as a consistency check, not as a prediction obtained by fitting; the constant c0 is an on-shell integration constant, not a fitted parameter. Self-citations to [13] and [19] provide context and prior constraint analysis, but Section 2 re-derives the needed structures, so they are not load-bearing in a circular way. The point most open to challenge is the asserted involution of P_I in (92) and the basis equivalence (98), but those are unexplained or terse mathematical steps, not reductions of a prediction to its inputs. No circular step can be exhibited from the paper's own equations.

Assumptions & free parameters 0 free parameters · 6 assumptions · 3 invented entities

The central construction introduces no fitted numbers. The model input is the characteristic function F(√γ) of the GUMG family, which is carried through as an arbitrary positive function; it is not tuned to data here. The derivation relies on the prior constraint analysis of [13], on sign-definiteness assumptions for W and Ω, on compact spatial sections for the averaging projectors, and on standard constrained-system results. Auxiliary fields are introduced for parameterization and have no independent observable handles.

assumptions (6)
  • domain assumption The canonical extended action and constraint structure of GUMG from Section 2 (taken from [13]) is correct and complete.
    The derivation starts from equations (12), (15), (31) and the secondary constraints (20), (27), all inherited from the earlier analysis.
  • domain assumption The characteristic function F(√γ) is such that W(√γ)=d ln F/d ln √γ and Ω(√γ)=d ln W/d ln √γ + W + 1 are sign-definite and nonzero on the domain.
    Stated in Section 2 as needed to avoid branchings of the constraint structure; this restricts the GUMG family to monotonic F and excludes Ω=0 exceptional models (except UMG).
  • domain assumption Spatial sections are compact for the central proof; noncompact sections are handled by a finite-volume regularization whose asymptotic behavior is not fully specified.
    The average/free decomposition (23) and the delocalization operator determinant rely on finite volume; Appendix D provides a regularization but admits the treatment is incomplete.
  • standard math The delocalization operator E defined by (95) is invertible on the relevant function space with inverse given by Lemma A.1.
    Invertibility is proven constructively in Appendix A.1 for bounded sign-definite W, and is used to eliminate H⊥ and to build the local action.
  • standard math Standard results from the theory of constrained systems (Dirac consistency, existence of first-class Hamiltonian, correspondence gauge reduction) as presented in [22] are assumed.
    Used throughout for parameterization equivalence and gauge analysis.
  • domain assumption Reduction of gravitational momenta by their own variational equations proceeds as in GR even though the lapse multiplier is associated with a mixed-class constraint on the non-GR branch.
    The paper reduces π_mn via (51)/(103) without discussing possible subtleties from the second-class average-free part of the lapse on the non-GR branch.
invented entities (3)
  • Auxiliary vector field V^μ (Henneaux-Teitelboim field)
    purpose: Appears only through ∂_μ V^μ and encodes the time parametrization fields τ0 and ν^m; it makes the alternative action covariant-looking and enforces Λ0 = const on shell.
    It is an auxiliary field introduced during parameterization; it has no direct observational handle and carries the gauge redundancy V→V + X with ∂·X=0.
  • Cosmological-constant field Λ0 (or rescaled Λ)
    purpose: Selects the GR branch (c0=0) versus non-GR branch (c0≠0) and supplies the perfect-fluid energy-momentum.
    It is a Lagrange-multiplier-like field, gauge-invariant and constant on shell; no independent physical prediction beyond what the original GUMG fluid already provides.
  • Time-parametrization pair τ0, π0 (homogeneous and average-free components)
    purpose: Implements the time reparametrization gauge degree of freedom in the canonical action and can be removed in the correspondence gauge τ0=t.
    Pure gauge/parameterization sector; the paper shows it decouples or becomes trivial in the correspondence gauge.

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Pith. "Pith review of Alternative Action for Generalized Unimodular Gravity." pith.science (2026). https://pith.science/paper/4ZCWGSQQ

@misc{pith2026250513548,
  author       = {Pith},
  title        = {Pith review of: Alternative Action for Generalized Unimodular Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4ZCWGSQQ}},
  note         = {Machine review of arXiv:2505.13548}
}
read the original abstract

We present an alternative formulation of generalized unimodular gravity (GUMG), a class of modifications to general relativity characterized by a special partial breaking of general coordinate covariance. The action for this formulation is derived constructively through a sequence of equivalent representations, starting from the original GUMG setup and extending the configuration space and gauge structure by introducing time parametrization. Our approach, based on a canonical formalism, parallels the method used by Henneaux and Teitelboim to covariantize the action of unimodular gravity (UMG), which was generalized to consistently accommodate a parameterization via local fields for the entire GUMG family. For completeness, we explore the dynamical structure of the theory and provide a detailed account of its gauge properties. A notable consequence of the consistent parametrization is the emergence of explicit spatial delocalization in the action, manifestations of which we carefully examine. Within the GUMG family, we identify a subfamily of models described by a local action. While preserving the essential structural features, these models avoid the complications of spatial nonlocality allowing a clearer analysis. The dynamical and constraint structures of GUMG family models are different from those of UMG, however the latter appears as an exceptional special case within the family, providing valuable comparative insights.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Henneaux-Teitelboim Form of the Generalized Unimodular Gravity Action

    hep-th 2024-12 conditional novelty 6.0 of 10

    A generalized Henneaux-Teitelboim action for generalized unimodular gravity is constructed, introducing a spatially nonlocal operator and showing that the theory is not fully diffeomorphism invariant.

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