Covariant Canonical Formalism For Born-Infeld Inspired Gravity in Palatini Formulation

J. Kluson

REVIEW 3 major objections 5 minor 19 references

Covariant Canonical Formalism For Born-Infeld Inspired Gravity in Palatini Formulation

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The covariant canonical formalism reproduces the equations of motion of Palatini Born-Infeld gravity, reducing to general relativity with a cosmological constant in vacuum.

desk verdict A useful, mostly correct covariant canonical treatment of Palatini BI gravity; the advertised equivalence checks out in the key limits, but the proof has typos and one unproved uniqueness step that need referee attention. read the letter →

arxiv 2412.17294 v1 pith:GDIBZ3GU submitted 2024-12-23 gr-qc

J. Kluson This is my paper · ORCID

classification gr-qc PACS 04.20.Fy04.50.Kd

keywords Born-InfeldinspiredgravityPalatiniformalismcovariantcanonicalmultisymplecticfieldtheoryWeyl-DeDonderHamiltonianmodifiedcosmologicalconstant

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 show that Born-Infeld inspired gravity in the Palatini formulation admits a manifestly covariant Hamiltonian description, and that the equations of motion derived from that Hamiltonian agree with the usual Lagrangian ones. The motivation is practical: standard 3+1 canonical analysis of modified gravity is complicated and breaks manifest covariance, while a covariant canonical treatment keeps all derivatives on an equal footing. In vacuum the canonical equations are claimed to reduce exactly to $R_{ab} + M_{\mathrm{BI}}^2(1-\lambda)g_{ab}=0$, the field equations of general relativity with a cosmological constant $\Lambda = M_{\mathrm{BI}}^2(1-\lambda)$. With matter, the same equations reduce to general relativity in the weak-field limit $|T_{ab}| \ll M_p^2 M_{\mathrm{BI}}^2$, while differing from general relativity when the curvature scale approaches $M_{\mathrm{BI}}^2$. If the equivalence holds, the covariant Hamiltonian is a valid starting point for a manifestly covariant quantization or thermodynamic analysis of this theory.

What carries the argument

The load-bearing object is the covariant canonical Hamiltonian density (26), built from the momenta $\Pi^{abd}{}_c$ conjugate to $\partial_d\Gamma^c{}_{ab}$. The key identity is $\Pi^{abc}{}_c = \frac{3}{2}M_p^2\sqrt{-\det A}\,A^{ab}$, which lets the action's square-root determinant be written as $\sqrt{-\det \Pi^{abc}{}_c}$; this makes the Hamiltonian a function of momenta only. The argument then rests on a momentum ansatz (37), $\Pi^{abc}{}_d = \sqrt{-g}(K g^{ab}\delta^c_d + L(g^{ac}\delta^b_d + g^{bc}\delta^a_d))$, with constants $K,L$ fixed by consistency, which turns the connection equations of motion into the metric-compatibility condition and ultimately into the field equations of general relativity.

What would settle it

Solve the canonical equations (33)-(34) without imposing the ansatz (37) in a simple vacuum geometry, such as a homogeneous but anisotropic metric, and check whether any momentum configuration with non-constant $K,L$ or with terms not of the form $g^{ab}\delta^d_c$ satisfies them; the existence of even one such solution would break the claimed equivalence with the Lagrangian equations of motion.

Watch

Extended reading notes

Core claim

The central claim is that applying the covariant canonical formalism to Palatini Born-Infeld gravity yields a Hamiltonian whose equations of motion are equivalent to the Lagrangian equations of motion. The canonical momenta conjugate to connection derivatives are $\Pi^{abd}{}_c = \frac{1}{2}M_p^2 \sqrt{-\det A}(A^{ab}\delta^d_c - \frac{1}{2}(A^{ad}\delta^b_c+A^{bd}\delta^a_c))$; through the identity $\Pi^{abc}{}_c = \frac{3}{2} M_p^2 \sqrt{-\det A}\,A^{ab}$, the determinant of $A$ is re-expressed in terms of momenta, so the Hamiltonian density becomes a function of momenta rather than velocities. The equations of motion are then solved with a two-constant momentum ansatz, enforcing metric compatibility; in vacuum this yields $R_{ab}+M_{\mathrm{BI}}^2(1-\lambda)g_{ab}=0$, and with matter the weak-field limit reproduces the field equations of general relativity. The paper reads this as a consistency check of the covariant canonical formalism for this theory.

Load-bearing premise

The argument assumes, rather than proves, that the canonical momentum has the two-constant form used in the ansatz; if other momentum configurations solve the equations, the Hamiltonian covers only part of the theory.

Editorial extensions

If this is right

In vacuum, the canonical equations of the Born-Infeld gravity Hamiltonian reduce to $R_{ab}+M_{\mathrm{BI}}^2(1-\lambda)g_{ab}=0$, i.e. the field equations of general relativity with cosmological constant $\Lambda=M_{\mathrm{BI}}^2(1-\lambda)$; this is the paper's explicit consistency check.
When matter is included, the canonical equations reproduce general relativity in the weak-field regime $|T_{ab}|\ll M_p^2 M_{\mathrm{BI}}^2$, so ordinary gravity is recovered at low curvatures.
The Hamiltonian density (26) is manifestly covariant and depends on momenta, in contrast to the 3+1 canonical treatment where covariance is lost; this offers a covariant starting point for quantization or thermodynamics.
Because the connection is determined algebraically, no new propagating degrees of freedom appear, preserving the ghost-free character of Palatini Born-Infeld gravity.
Deviations from general relativity become significant when curvature is of order $M_{\mathrm{BI}}^2$, which the paper identifies as the regime where Born-Infeld gravity is most interesting.

Reading between the lines

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

The paper leaves open whether the momentum ansatz (37) is the unique solution of the canonical equations; a uniqueness proof would turn the equivalence result into a full equivalence, while a counterexample would confine the covariant Hamiltonian to a constrained sector.
The auxiliary metric $\hat{g}_{ab}$ introduced via $\Pi^{abc}{}_c=\sqrt{-\det\hat{g}}\,\hat{g}^{ab}$ suggests the canonical dynamics is governed by an effective metric related to $g$ and the matter stress tensor; one could test whether geodesics of $\hat{g}$, rather than $g$, carry physical meaning.
Because the variational principle of Born-Infeld gravity has no boundary term, the covariant Hamiltonian may provide the missing tool for a thermodynamic treatment of horizons in this theory.
A natural testable extension is to repeat the analysis with torsion included; the paper restricts to a torsion-free connection, and the momentum ansatz would likely need modification.

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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

Short version: this is a legitimate extension of the covariant canonical (multisymplectic) formalism to Palatini Born-Infeld gravity, following the author's metric-formulation paper [16]. The new piece is the Hamiltonian (26) and the demonstration that its equations reproduce the known vacuum BI equations and the weak-field Einstein equations with matter. If you work in this subfield, it is worth a careful read.

What I liked: the construction is straightforward and mostly explicit. The momentum (19) depends on the connection only through R_(ab), so the Legendre transform is much simpler than in the metric formulation, and the Hamiltonian genuinely depends on momenta. The final checks against known BI/Einstein limits are the right sanity checks, and they pass once you allow for typos.

The soft spots are real but mostly cosmetic. The determinant identity (25) is unproved—it is easy to verify, so this is a minor presentation issue. Eq. (38) drops the λ from Eq. (36); as printed it is inconsistent with the preceding equation, and (47) quietly sets λ=1. Eq. (58) has a prefactor problem: working from (55) and (56) puts √(-det Ω) in the numerator, not the denominator. The weak-field limit (61) seems to be using the correct version, so I read this as a typo, but it must be fixed.

The concern I take seriously is the one raised about the ansatz (37)/(68). The momentum has 40 components, while Π^abc_c is the only combination that appears in the Hamiltonian. The paper solves for a two-parameter family and shows it satisfies the canonical equations, but it never proves this family is the general solution or that the primary constraints coming from the singular Legendre map are preserved by the dynamics. So the claimed equivalence with the Lagrangian equations is conditional on that uniqueness. I don't think this sinks the paper—the endpoint equations are strong evidence the result is right—but the proof should be upgraded.

Citation pattern is clean; [16] is a natural predecessor. Who is this for: people doing modified-gravity canonical methods or BI gravity thermodynamics. It deserves a serious referee, with requests to fix the typos and justify the ansatz.

Referee Report

3 major / 5 minor

Summary. The paper develops a covariant canonical (Weyl-De Donder / multisymplectic) formulation of Born-Infeld inspired gravity in the Palatini formalism. After reviewing the action and the Lagrangian equations of motion, it defines the covariant conjugate momentum (19), constructs the Hamiltonian density (26), and writes the canonical action (30). The remainder of the paper aims to show that the canonical equations reproduce the Lagrangian field equations: in vacuum it obtains Rab + M_BI^2(1-lambda)gab = 0 (Eq. 51), and with a scalar-field matter source it obtains the Einstein equations in the weak-field limit (Eq. 61). The paper concludes that the covariant canonical formalism provides a valid manifestly covariant Hamiltonian description of Palatini BI gravity and emphasizes that the theory deviates from general relativity at finite M_BI.

Significance. The claimed result is of genuine interest because a manifestly covariant Hamiltonian formulation of Born-Infeld type gravity has been missing; such a formulation could be a starting point for thermodynamic or quantization studies. The final equations (51) and (61) agree with known results for BI gravity, which strongly suggests that the central claim is correct. The derivation is explicit and the free parameters (lambda, M_BI) are clearly identified. However, the equivalence proof is conditional on an unproven momentum ansatz and on the omission of primary constraints; the paper does not supply a complete proof that the canonical equations are equivalent to the Lagrangian equations on the full phase space. If these gaps are repaired, the paper would be a useful contribution to the canonical-analysis literature on BI gravity.

major comments (3)

[Sec. 3.1, Eq. (37); Sec. 3.2, Eq. (68)] The reduction from the canonical equations to the Lagrangian equations of motion is obtained by imposing the constant-coefficient ansatz (37) and its analogue (68). This establishes sufficiency only: it shows that if the momentum takes this special form and the connection is metric-compatible, then the BI field equations follow. It does not show that every solution of the canonical equations (32)-(34) lies in this family. This matters because the Legendre map (19) is singular: the BI Lagrangian depends on the forty derivatives partial_d Gamma^c_ab only through the ten components R_(ab), so the phase space contains roughly thirty primary constraints. The canonical action (30) includes only the constraints M^abc = 0; the constraints that enforce the image of the Legendre transform (19) are absent. Unless the author proves that (37)/(68) is the general solution of (34), or shows that the omitted primary constraints are preserved by the Hamiltonian flow generated by (26), the Hamiltonian (26) may admit spurious sectors that do not correspond to Palatini BI gravity. This is the load-bearing gap in the claimed equivalence and should be addressed with a constraint analysis or a uniqueness argument.
[Eq. (58)] Equation (58) is printed with sqrt(-det Omega) in the denominator: Rab + M_BI^2 gab - (sqrt(-g) M_BI^2 / sqrt(-det Omega)) hatOmega_ab = 0. Direct substitution of (56) into (55) gives Rab + M_BI^2 gab - M_BI^2 sqrt(-g) sqrt(-det Omega) hatOmega_ab = 0, i.e. the factor sqrt(-det Omega) belongs in the numerator. With the numerator form, the weak-field expansion (60) correctly reduces to the Einstein equation (61); with the denominator form the term has the wrong dimension and cannot reduce to (61). Since (58) is the central matter equation of the paper, this typo must be corrected.
[Eqs. (38), (47)] Substituting the ansatz (37) into the trace condition (36) gives 2K + L = (3/4) lambda M_p^2, not (3/4) M_p^2. Consequently the momentum solution should be Pi^abc_d = (lambda M_p^2/2) sqrt(-g) (g^ab delta^c_d - (1/2)(g^ac delta^b_d + g^bc delta^a_d)); that is, Eq. (47) is missing a factor lambda. As printed, (47) does not satisfy Eq. (36) unless lambda = 1. The final vacuum equation (51) is unaffected because the later derivation uses (36) directly rather than (47), but the displayed solution for the momentum is internally inconsistent and should be corrected.

minor comments (5)

[Eq. (25)] The determinant identity det A = (2/(3M_p^2))^4 det(Pi^abc_c) is asserted without proof. It is true and deserves a one-line derivation from (24), since Pi^ab = (3/2) M_p^2 sqrt(-det A) A^ab implies det(Pi) = (3/2 M_p^2)^4 det A.
[Notation] The momentum is written Pi^abd_c in (19) and Pi^abc_d in (26)-(37), with the derivative index and the free lower index interchanged. Please define the index ordering once and use it consistently.
[Eq. (37)] Equation (37) has an unmatched parenthesis and the delta-index placement is inconsistent with the surrounding formulas; for example, the last term should be g^bc delta^a_d with the closing parenthesis.
[Eq. (56)] In (56) the expression uses -delta H/delta g^ab, but the equation it derives from, (33), contains delta H_M/delta g^ab. Since H in (26) includes gravitational contributions, the notation should be H_M (or the gravitational part must be shown to vanish) to avoid an apparent sign error.
[Throughout] There are several typos: "imediatelly" (Introduction), "thermodymics" and "od" (Introduction), and "Facul ty" (author affiliation). These should be corrected in the final version.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the covariant Hamiltonian is derived by direct Legendre transform and the reduction to the Lagrangian equations is an explicit consistency check, with the main caveat being an unproven ansatz rather than a circular dependency.

full rationale

The paper's central derivation is self-contained. The covariant Hamiltonian (26) is obtained directly from the Born-Infeld Palatini action (17) by Legendre transformation: the conjugate momenta are defined in (19), and the determinant of A is expressed in terms of the canonical momenta through (24)-(25). No parameter is fitted to data and no result is imported from prior work as a load-bearing premise; reference [16] is only contextual. The canonical equations of motion (32)-(34) are derived by varying the canonical action (30). The vacuum reduction reproduces the Lagrangian equation Rab + M_BI^2(1-lambda)gab = 0 by solving the algebraic relation (36) with an ansatz (37), fixing K and L through consistency conditions (38) and (44), and then combining (49) and (50). The final equation is not assumed in advance. Similarly, the matter case uses the definitions (56)-(57) and expansion (60) to recover the weak-field Einstein equations (61). The paper explicitly calls this a 'nice consistency check of covariant canonical formalism,' which is an honest description rather than a disguised prediction. The only substantive caveat is that the ansatz (37)/(68) is not proved to be the general solution of the canonical equations, so the equivalence is demonstrated for a particular sector of momentum configurations. That is a completeness and correctness concern, not circularity: the ansatz is not equivalent to the target equations by construction, and no claim reduces to its own input. Therefore the circularity score is 0.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The derivation rests on a small set of standard Palatini assumptions (torsion-free connection, metric-only matter coupling), plus invertibility of A and Pi. The only model parameters are lambda and M_BI, both present in the starting action and not fitted to data. The auxiliary metric ghat introduced in Eq. (64) is a bookkeeping device, not a newly postulated physical entity.

free parameters (2)

lambda = not fitted (theory parameter)
Coefficient of the sqrt(-g) term in the action (1); determines the cosmological constant via Lambda = (lambda - 1) M_BI^2. It is an input to the model, not fitted to data.
M_BI = not fitted (theory parameter)
Born-Infeld mass scale in the action (1); sets the scale at which BI corrections become important. It is a model input, not fitted in this paper.

assumptions (4)

domain assumption The connection is torsion-free, Gamma^c_ab = Gamma^c_ba (Eq. 4).
Restricts the BI gravity action to symmetric connections; the most general case with torsion is excluded.
domain assumption The matter action depends on the metric only, so delta S_M / delta Gamma = 0 (Section 2).
Matter is assumed not to couple to the connection; this is standard in Palatini BI gravity with minimally coupled matter.
standard math The matrix A^a_b and the momentum matrix Pi^ab are non-singular, so their inverses exist (Eqs. 16, 35).
Invertibility is required to define the auxiliary connection and the Hamiltonian; it is assumed throughout.
domain assumption The weak-field expansion in Section 3.2 assumes T_ab << M_p^2 M_BI^2 so that Omega^ab = lambda g^ab + corrections, giving the GR limit.
The reduction to Einstein equations is shown only in this limit, which is the regime where BI corrections are small.

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Cite this review

Pith. "Pith review of Covariant Canonical Formalism For Born-Infeld Inspired Gravity in Palatini Formulation." pith.science (2026). https://pith.science/paper/GDIBZ3GU

@misc{pith2026241217294,
  author       = {Pith},
  title        = {Pith review of: Covariant Canonical Formalism For Born-Infeld Inspired Gravity in Palatini Formulation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GDIBZ3GU}},
  note         = {Machine review of arXiv:2412.17294}
}

read the original abstract

We analyze Born-Infeld inspired gravity in Palatini formulation in the framework of covariant canonical formalism. We determine covariant Hamiltonian and corresponding equations of motion.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

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

[1] [16]

Born-Infeld Inspired Gravity in Covariant Canonical Formalism

J. Kluson, “Born-Infeld Inspired Gravity in Covariant Canonical For- malism,” [arXiv:2410.22921 [hep-th]]

work page Pith review arXiv

[2] [1]

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S. Chakraborty and T. Padmanabhan, “Boundary term in the grav- itational action is the heat content of the null surfaces,” Phys. Rev. D 101 (2020) no.6, 064023 doi:10.1103/PhysRevD.101.064023 [arXiv:1909.00096 [gr-qc]]

work page Pith review arXiv 2020

[3] [2]

General Relativity from a Thermodynamic Perspective

T. Padmanabhan, “General Relativity from a Thermodynamic Perspec- tive,” Gen. Rel. Grav. 46 (2014), 1673 doi:10.1007/s10714-014-1673-7 [arXiv:1312.3253 [gr-qc]]

work page Pith review arXiv 2014

[4] [3]

The Mathematical Theory of Relativity,

A. Eddington, "The Mathematical Theory of Relativity," Cambridge University Press, Cambridge, UK, 2 ed., 1924. 14

work page 1924

[5] [4]

Eddington ’s theory of gravity and its progeny,

M. Banados and P. G. Ferreira, “Eddington ’s theory of gravity and its progeny,” Phys. Rev. Lett. 105 (2010), 011101 [erratum: Phys. Rev. Lett. 113 (2014) no.11, 119901] doi:10.1103/PhysRevLett.105.0111 01 [arXiv:1006.1769 [astro-ph.CO]]

arXiv 2010

[6] [5]

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M. Born and L. Infeld, “Foundations of the new ﬁeld theory,” Proc. Roy. Soc. Lond. A 144 (1934) no.852, 425-451 doi:10.1098/rspa.1934.0059

arXiv 1934

[7] [6]

Born-Infeld-Einstein actions?,

S. Deser and G. W. Gibbons, “Born-Infeld-Einstein actions?,” Class. Quant. Grav. 15 (1998), L35-L39 doi:10.1088/0264-9381/15/5/001 [arXiv:hep-th/9803049 [hep-th]]

arXiv 1998

[8] [7]

Palatini approach to Born-Infeld-Einstein theory and a geometric description of electrodynamics,

D. N. Vollick, “Palatini approach to Born-Infeld-Einstein theory and a geometric description of electrodynamics,” Phys. Rev. D 69 (2004), 064030 doi:10.1103/PhysRevD.69.064030 [arXiv:gr-qc/03 09101 [gr-qc]]

doi:10.1103/physrevd.69.064030 2004