REVIEW 3 major objections 5 minor 1 cited by
Canonical Analysis of Eddington Gravity
T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash
Pith's one-line read The paper shows Eddington gravity's canonical action is identical to the ADM action of vacuum General Relativity — same phase space, constraints, and Hamiltonian, with the metric emerging from the connection's conjugate momentum.
desk verdict A clean but local, nondegenerate Hamiltonian reduction showing Eddington gravity matches ADM; worth refereeing once the qualifications are in the abstract. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The reducing engine is the Faddeev–Jackiw elimination of the non-dynamical connection components, organised around the composite phase-space pair $q^{ij}=\Pi^{00}\Pi^{ij}-\Pi^{0i}\Pi^{0j}$ and $K^{ij}=-(1/\Pi^{00})\Gamma^0_{ij}$. Solving $\Sigma^{\mu\nu}_i=0$ — equations that read like the vanishing of a tensor-density covariant derivative of $\Pi^{\mu\nu}$ — expresses every spatial connection component in terms of the momenta. The decisive identity is the compatibility condition $\partial_k q^{ij}+\gamma^i_{km}q^{mj}+\gamma^j_{km}q^{mi}-2\gamma^m_{mk}q^{ij}=0$, which, together with the decomposition $q^{ij}=(\sqrt q)^p h^{ij}$ and the choice $p=-2/(d-2)$, forces $\gamma^i_{jk}$ to be the Christoffel symbols of $h_{ij}$. That is the step that manufactures a metric theory out of connection data; the leftover momentum components $(\Pi^{00},\Pi^{0i})$ supply the lapse and shift, and the Hamiltonian becomes the ADM Hamiltonian and momentum constraints of vacuum General Relativity.
What would settle it
Repeat the reduction of Eqs. (20)–(37) while keeping every boundary term produced by the integrations by parts. If any surface term survives into the final action (51)–(52), then on manifolds with boundary the canonical actions of Eddington gravity and General Relativity differ by boundary data, and the claimed equivalence holds only modulo those terms; the places to check are the steps following (20), (36), and (37).
Extended reading notes
Core claim
The paper's central claim is that Eddington gravity, with action $S=\int d^d x\sqrt{\det R_{(\mu\nu)}}$ and no metric field, has exactly the same canonical action as vacuum General Relativity. Starting from the traceless combination $G^\lambda_{\mu\nu}=\Gamma^\lambda_{\mu\nu}-\tfrac{1}{2}(\delta^\lambda_\mu\Gamma^\rho_{\rho\nu}+\delta^\lambda_\nu\Gamma^\rho_{\rho\mu})$, the author computes the momenta $\Pi^{\mu\nu}$ conjugate to $G^0_{\mu\nu}$; the components $\Pi^{\mu\nu}_i$ vanish, so the spatial connection components are non-dynamical. Instead of the Dirac constraint cascade, he follows Faddeev and Jackiw: the equations $\Sigma^{\mu\nu}_i=0$ for the non-dynamical components are solved and the solutions substituted back. This introduces the canonical pair $q^{ij}=\Pi^{00}\Pi^{ij}-\Pi^{0i}\Pi^{0j}$ and $K^{ij}=-(1/\Pi^{00})\Gamma^0_{ij}$, and the compatibility condition on $q^{ij}$ forces the spatial connection $\gamma^i_{jk}$ to be the Levi-Civita connection of a metric $h_{ij}$ obtained from $q^{ij}$ up to a Weyl rescaling. With $\Pi^{00}=\sqrt{h}/\lambda$ and $\Pi^{0i}=\sqrt{h}\,\lambda^i/\lambda$ read as lapse and shift, the action takes the ADM form $S=\int d^d x(\partial_t h_{ij}\pi^{ij}-\lambda\tilde C-\lambda^i\tilde C_i)$, with $\tilde C$ and $\tilde C_i$ the Hamiltonian and momentum constraints of General Relativity; the constant term in $\tilde C$ plays the role of a non-zero cosmological constant, matching the known classical equivalence. The author concludes that Eddington gravity and GR are equivalent in the Hamiltonian formalism and notes that the metric components emerge as conjugate momenta of the connection.
Load-bearing premise
The whole reduction assumes that one component of the conjugate momentum — the one that later becomes the densitized lapse, $\Pi^{00}$ — is non-zero everywhere, so the equations of motion can be solved for the non-dynamical connection components, and that the derivation's dropped boundary terms really contribute nothing; if either premise fails, the claimed canonical equivalence need not hold.
Editorial extensions
If this is right
- For general dimension $d$, vacuum Eddington gravity and General Relativity share the same Hamiltonian dynamics: identical phase space $(h_{ij},\pi^{ij})$, identical Hamiltonian and momentum constraints, and lapse and shift as Lagrange multipliers.
- The spatial metric is not an input of Eddington gravity: it is manufactured from the connection's conjugate momentum, so the metric description of gravity is a derived, composite object in this formulation.
- Canonical quantization programs for vacuum GR, at least at the level of the phase space and constraint algebra, apply unchanged to Eddington gravity, since the constraints and Poisson brackets coincide.
- Applying the same Faddeev–Jackiw reduction to Born–Infeld-style modifications of Eddington gravity — the natural route to coupling matter — is expected to produce a Hamiltonian that differs from GR's; the author identifies this as the immediate next step.
Reading between the lines
- Because the derivation discards boundary terms, the natural reading is that the equivalence holds in the bulk of closed spatial manifolds; on manifolds with boundary — including black-hole horizons, where surface terms carry the thermodynamical charges — Eddington gravity and GR may differ by exactly the terms dropped at (20) and (36)–(37).
- The metric's emergence from a momentum density rather than from a configuration variable points toward a momentum-space formulation of gravity in which the familiar metric variables are composite; whether that formulation survives matter couplings, where Eddington gravity must be modified, would indicate which variables are more fundamental.
- A concrete stress test of the load-bearing condition $\Pi^{00}\neq 0$: on a foliation where the densitized lapse vanishes (the lapse diverges, as near a null slice), the solved equations for the connection become singular and the map to $(h_{ij},\pi^{ij})$ degenerates; locating such slices in explicit vacuum solutions would delimit where the GR description of Eddington gravity breaks down.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper performs a canonical analysis of Eddington gravity in d spacetime dimensions using the Faddeev–Jackiw method. It rewrites the action in terms of the traceless connection variable G^λ_μν, computes the momentum Π^{μν} conjugate to G^0_{μν}, solves the equations of motion for the non-dynamical connection components Γ^i_{μν}, and identifies q_{ij} and K^{ij} as a canonical pair. After a series of algebraic reductions, it redefines the spatial metric h_{ij} and momentum π^{ij}, and obtains the canonical action (51)–(52) with Hamiltonian and momentum constraints of the same form as the ADM action of general relativity. The paper concludes that Eddington gravity and general relativity are equivalent in the Hamiltonian formalism.
Significance. If the computation is correct, the paper is a useful consistency check: the affine-only Eddington action gives the same phase space and constraint structure as vacuum general relativity. Its strengths are a clean Faddeev–Jackiw route that bypasses a full Dirac constraint analysis, a plausible identification of q_{ij} and K^{ij} as the canonical pair, and a final action that reproduces the ADM form including the spatial curvature term. The main limitation is that the equivalence is established only on an open, nondegenerate region of field space and modulo discarded boundary terms; the abstract states the equivalence without these qualifications. The result is therefore significant but narrower than claimed.
major comments (3)
- [Section 2, Eqs. (15)–(19) and (39)] The derivation divides by Π^{00} repeatedly to solve for Γ^0_{0i} and Γ^j_{0i}, and the construction of h_{ij} from q_{ij} via Eq. (39) requires that q_{ij} be non-degenerate and (for Lorentzian signature) positive definite. These conditions are not stated in the abstract or Section 1, and no argument is given that they hold for all configurations of interest or that the configurations excluded by them are pure gauge or physically irrelevant. The claimed equivalence should be qualified as holding on the open region with Π^{00}≠0 and det q_{ij}≠0.
- [Section 2, Eqs. (20), (22), (36)–(37), and (51)] Total derivatives are discarded repeatedly, with the phrase 'ignore boundary terms' used at each step. On a manifold with boundary, the reduced Eddington action and the ADM action generically differ by boundary terms, so the equivalence has been established only as a bulk equivalence on closed manifolds. The paper should state this limitation explicitly, or show that the boundary terms can be canceled by the same boundary counterterms in both theories.
- [Section 2, Eqs. (30)–(35)] The central algebraic step of the paper, the reduction from Eq. (30) to the simplified Hamiltonian in Eqs. (33)–(38), is reported as 'we find' for each of the K^2, K^1, and K^0 terms without showing the calculation. Since the final equivalence claim depends entirely on these identities, the paper should present the intermediate steps or provide a supplementary file with the computation; otherwise the main result is not independently verifiable from the text.
minor comments (5)
- [Section 2, Eq. (27)] The notation is ambiguous: Π^{ij} denotes spatial components of the momentum density in some equations, while in Eq. (27) the same symbol is used for components of the inverse matrix. Use a distinct symbol for the inverse-matrix components.
- [Section 2, Eq. (39)] The displayed relation q_{ij}=√q p h_{ij} is ambiguous; it should be typeset unambiguously as q_{ij}=(√q)^p h_{ij} or q_{ij}=p√q h_{ij}, since the exponent is central to the subsequent determination of p in Eq. (41).
- [Section 2, Eq. (50)] Equation (50) writes Π^{00}=√h/λ and Π^{0i}=√h λ^i/λ and calls λ and λ^i scalar functions, but Π^{00} and Π^{0i} are densities; the transformation properties of λ and λ^i under spatial diffeomorphisms should be stated explicitly.
- [Section 2, Eq. (1)] The paper does not specify the signature of the manifold or the convention for the square root in Eq. (1); in Lorentzian signature det R can change sign, so a convention such as an absolute value or a Euclidean continuation should be stated.
- [Section 2, Eqs. (51)–(52)] The comparison with the ADM action would be easier to verify if the reference GR canonical action from [9,10] were transcribed explicitly with the same conventions for signs, density weights, and the cosmological term; as written, 'same form' is asserted without an explicit side-by-side comparison.
Circularity Check
No circularity: the canonical equivalence is derived from the Eddington action by explicit Hamiltonian reduction, not assumed or imported from a self-citation.
full rationale
The paper's derivation is self-contained: it begins with the Eddington action (1), computes momenta (6), solves the non-dynamical connection equations (15)-(19), introduces the symplectic variables q^{ij} and K_{ij} in (21), and derives the reduced action (37)-(38). The metric h_{ij} is then introduced through (39), with the exponent p fixed by the internal consistency condition (40)-(41) to cancel the ∂ sqrt(q) terms; it is not fitted to reproduce the General Relativity action. The final action (51)-(52) is an algebraic output of this reduction, and the identification of pi_{ij} in (46) is dictated by the kinetic term (45). The only self-citation, reference [7], appears in an introductory remark about related modified-gravity models and is not load-bearing for the derivation. The paper's assumptions about nonzero Pi^{00}, nonsingular A_{\mu u}, and dropped boundary terms are domain restrictions of the claimed equivalence, not circular reasoning. No prediction is fitted, no input is renamed as output, and no load-bearing argument reduces to an unverified self-citation.
Assumptions & free parameters
assumptions (6)
- domain assumption Connection coefficients are symmetric, Gamma^lambda_{mu nu} = Gamma^lambda_{nu mu}.
- domain assumption A_{mu nu} = R_{(mu nu)} is non-singular (det A != 0).
- domain assumption Pi^00 and det q^ij are non-vanishing.
- domain assumption Boundary terms may be dropped after integration by parts.
- standard math Faddeev-Jackiw reduction is equivalent to Dirac-Bergmann analysis.
- domain assumption Spatial dimension d > 2.
Cite this review
Pith. "Pith review of Canonical Analysis of Eddington Gravity." pith.science (2026). https://pith.science/paper/D74YNIWT
@misc{pith2026250616279,
author = {Pith},
title = {Pith review of: Canonical Analysis of Eddington Gravity},
year = {2026},
howpublished = {\url{https://pith.science/paper/D74YNIWT}},
note = {Machine review of arXiv:2506.16279}
}
read the original abstract
In this short note we perform canonical analysis of Eddington gravity using L. Faddeev and Jackiw formalism. We demonstrate that resulting canonical action has the same form as General Relativity canonical action which proves an equivalence of Eddington gravity and General Relativity in Hamiltonian formalism.
Forward citations
Cited by 1 Pith paper
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Canonical Form of Born-Infeld Inspired Gravity Coupled to Scalar Fields
Born-Infeld inspired gravity with minimally coupled scalar fields is rewritten in canonical form whose gravitational piece is identical to general relativity and whose matter piece carries complicated corrections.
Reference graph
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Reviewed August 15, 2026 · model on record in the stance chip above.
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