On-shell Lagrangians as total derivatives and the generalized Komar charge

Jos\'e Luis V. Cerdeira; Tom\'as Ort\'in

On-shell Lagrangians as total derivatives and the generalized Komar charge

T0 review · 0 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A homogeneous Lagrangian is on-shell an exact form, and this identity builds generalized Komar charges.

desk verdict A clean field-theoretic Euler identity that powers a worked Komar-charge construction; minor presentational gaps only, so it deserves a serious referee. read the letter →

arxiv 2506.14024 v2 pith:4ODDPTRZ submitted 2025-06-16 hep-th gr-qc

José Luis V. Cerdeira , Tomás Ortín This is my paper · ORCID

classification hep-thgr-qc

keywords KomarchargeNoether–Waldon-shellLagrangianEulertheoremforhomogeneousfunctionsSmarrformulablackholethermodynamicsmomentummapsquasi-homogeneousLagrangians

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 proves a functional version of Euler's theorem for homogeneous functions: if the fields rescale with weights $\omega_i$ and the Lagrangian rescales with weight $\omega_L$, then $L=\frac{\omega_i}{\omega_L}E_i\wedge\phi^i+d\Theta(\phi,\frac{\omega}{\omega_L}\phi)$ holds identically, with $E_i$ the Euler–Lagrange forms and $\Theta$ the pre-symplectic potential, the boundary term generated by varying the action. On solutions, $E_i=0$, so the Lagrangian is the total derivative of a universal expression that depends only on the fields and the fixed weights, not on the particular solution. Because the generalized Komar construction needs the on-shell pullback of the Lagrangian by a Killing vector to be exact, this identity supplies exactly the boundary term needed to close the charge. The authors demonstrate the method on Einstein–Maxwell–Dilaton gravity, minimal five-dimensional supergravity, and a self-interacting scalar, where nonlinearity is handled by promoting the coupling constant to a constrained field. The result is a systematic route to conserved surface charges of the form needed for Smarr formulas in black-hole thermodynamics.

What carries the argument

The load-bearing identity is the functional Euler theorem (Eq. (2.7)): $L=\frac{\omega_i}{\omega_L}E_i\wedge\phi^i+d\Theta(\phi,\frac{\omega}{\omega_L}\phi)$, where $E_i$ are the Euler–Lagrange forms and $\Theta$ is the pre-symplectic potential, the boundary term generated by an arbitrary field variation. On solutions it becomes $L\doteq dJ_0$ with $J_0=\Theta(\phi,\frac{\omega}{\omega_L}\phi)$. This identity is the engine of the paper: it turns the on-shell Lagrangian into a total derivative without solving the field equations, and when combined with the generalized-Komar requirement $i_k L+B(\Lambda_k,\phi)\doteq d\omega(k,\phi)$, it yields the exact $(d-2)$-form $\omega(k,\phi)$ and hence the closed charge $K[k]$.

What would settle it

Take the proposed generalized Komar charge for Einstein–Maxwell–Dilaton theory (Eq. (4.23)) and evaluate its exterior derivative on a known non-extremal black-hole solution; if $dK[k]$ does not vanish on-shell, or if the horizon integral differs from the integral at infinity, the exactness assumption is false. A simpler algebraic check is to substitute arbitrary field configurations into Eq. (2.7) and verify the identity term by term; any counterexample would falsify the Euler-theorem claim.

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Extended reading notes

Core claim

The central claim is that for any Lagrangian $d$-form that transforms homogeneously under a global transformation of the fields, the Lagrangian can be written identically as a sum of the equations of motion contracted with the fields plus an exact piece: $L=\frac{\omega_i}{\omega_L}E_i\wedge\phi^i+d\Theta(\phi,\frac{\omega}{\omega_L}\phi)$. On-shell this reduces to $L\doteq dJ_0$, where $J_0=\Theta(\phi,\frac{\omega}{\omega_L}\phi)$ is solution-independent. The paper shows that $J_0$ is unique only up to conserved currents $J_n$ associated with scaling symmetries of the theory, and that the same Euler-theorem logic works for any global transformation with nonzero weight, not just rescalings. It then plugs this identity into the generalized Komar algorithm: given a Killing vector $k$, the on-shell interior product $i_k L$ becomes an exact $(d-2)$-form, and the resulting closed charge $K[k]$ reproduces and organises the known charges of Einstein–Maxwell–Dilaton theory and minimal $d=5$ supergravity, and gives a potential-times-charge form for the self-interacting scalar.

Load-bearing premise

The load-bearing assumption is that a symmetry that leaves all fields fixed also makes the on-shell Lagrangian pullback exact: there must always exist some $(d-2)$-form $\omega(k,\phi)$ with $i_k L+B(\Lambda_k,\phi)\doteq d\omega(k,\phi)$. The paper takes this exactness from the earlier generalized-Komar framework rather than proving it; if it fails, the constructed charge is not conserved on-shell and the Gauss law behind Smarr formulas is lost.

Editorial extensions

If this is right

In any theory with a homogeneous rescaling of the fields, the action evaluated on solutions is a pure boundary term, $S_{\mathrm{on-shell}}=\int dJ_0$.
For every Killing vector $k$, the on-shell interior product $i_k L$ is exact, so the generalized Komar charge $K[k]$ is closed on-shell and satisfies a Gauss law.
The ambiguity in $J_0$ by conserved currents is physical: in Einstein–Maxwell–Dilaton theory, adding $\alpha J_1$ shifts the Komar charge by a term proportional to the conserved scalar charge $Q_k$, while leaving Smarr formulas unchanged when the scalar-charge Gauss law is used.
Promoting a dimensionful coupling constant to a constrained scalar field extends the Euler-theorem identity to nonlinear Lagrangians, as demonstrated for a self-interacting scalar; the authors state that the same trick applies to nonlinear electrodynamics coupled to gravity.

Reading between the lines

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

The paper does not pursue it, but Eq. (2.7) also gives a shortcut for Euclidean on-shell actions: the boundary term is read off from the pre-symplectic potential before solving equations, which could simplify free-energy computations for stationary black holes.
Because the identity is algebraic in the fields, it should extend to higher-curvature or Chern–Simons theories; testing it on Lovelock gravity or generalized Proca models would be a direct next step.
Applying the coupling-promotion trick to a cosmological constant rather than a scalar coupling should yield a Komar charge whose boundary integral is the pressure–volume term of extended black-hole thermodynamics.

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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 thing to know about 2506.14024: the engine is a two-line variational identity that is correct and genuinely useful, and the paper is worth referee time. The central result, Eq. (2.7), is a field-theoretic Euler theorem: for a Lagrangian that scales homogeneously under global field rescalings, the on-shell Lagrangian is the exterior derivative of the pre-symplectic potential evaluated at the scaling variation, L ≑ dJ0. No fitting, no hidden assumptions; it is a parameter-free consequence of the standard variational identity. The genuinely new part is not the identity itself — it is morally a scale-symmetry Noether current — but the systematic use of it inside the Ortín program for constructing generalized Komar charges, the non-uniqueness analysis (the α-family), and the coupling-promotion trick that extends the method to a non-linear scalar theory. The three worked examples all reproduce known charges from earlier work, which is the right kind of grounding: the method gives familiar answers before it gives new ones.

The soft spots are minor. The reader flagged Eq. (3.17) as an inherited assumption; on reading Section 3, that is too harsh. The exactness of ı_k L + B(Λ_k, φ) on-shell follows from the Noether second identity (3.3) together with the Killing condition δ_k φ ≑ 0, with the usual tacit Poincaré lemma for local exactness. What is genuinely under-discussed is the global side: the construction is local, and topological obstructions are never mentioned. The Lagrange-multiplier equivalence in Section 6 is asserted rather than checked in detail; it is a standard trick from their earlier work and it is plausible, but a referee should ask for the equations spelled out. Presentationally, Eq. (2.10) leaves the α_0 = 1 and ω^n_i conventions implicit, and Eq. (3.24b) carries a stray '= 0'. The prose has small typos ('a a matter of fact', 'the its definition'). None of this touches the main derivation.

The citation pattern is self-heavy, which is normal for a group mining its own program; the external anchors (Komar, Magnon, Pacilio) are real and the examples are independently reproducible.

Who this is for: specialists in black hole thermodynamics and conserved-charge technology. It deserves a serious referee and, I think, acceptance after minor revisions. I would not desk-reject it.

Referee Report

0 major / 6 minor

Summary. The paper proves a field-theoretic analogue of Euler's homogeneous-function theorem: if a Lagrangian d-form L is homogeneous under a global rescaling of the fields with weights ω_i and transforms with weight ω_L, then L = (ω_i/ω_L) E_i∧φ_i + dΘ(...), so that on-shell L is an exact form with a solution-independent representative (Eq. 2.7). The paper then combines this identity with the generalized Komar charge algorithm of Refs. [12–16] to construct Komar charges in three examples: Einstein–Maxwell–Dilaton theory, the bosonic sector of minimal 5-dimensional supergravity, and a self-interacting scalar field coupled to gravity (using a Lagrange-multiplier extension for the non-linear potential). It also discusses the non-uniqueness of the total-derivative representative coming from additional scaling symmetries.

Significance. The central identity in Section 2 is clean, self-contained, and parameter-free; it follows directly from the variational identity and does not depend on the earlier Komar-charge algorithm. The application sections show that this identity supplies the needed on-shell total-derivative form in three non-trivial theories and reproduce known generalized Komar charges, which is a valuable consistency check. The Lagrange-multiplier trick in Section 6 extends the method to non-linear potentials. If correct, the paper provides a systematic and practical tool for constructing Smarr formulas and generalized Komar charges.

minor comments (6)

[Section 2, Eq. (2.8)] The sign in Eq. (2.8) appears incorrect: from Eq. (2.6) with ω_L = 0 one obtains dJ_n = -ω^n_i E_i∧φ_i, as used in Eq. (4.11), not dJ_n = +ω^n_i E_i∧φ_i.
[Section 3, Eq. (3.24b)] The trailing '= 0' in Eq. (3.24b) is inconsistent with the definition of the generalized Komar charge and with the non-trivial results in the subsequent examples; it should be removed, or replaced by a statement that dK[k] ≑ 0.
[Section 4, Eq. (4.22)] The placement of the parentheses in Eq. (4.22) gives the αQ_k term a factor of 1/2; the derivation via Eq. (4.21a) gives the term +αQ_k outside the factor 1/2, matching Eq. (4.23).
[Section 5, Eqs. (5.9)–(5.10)] The sign of τ(χ,φ) in Eq. (5.10) appears inconsistent with a direct explicit computation of δχJ0 from J0 = (1/3)⋆G∧V using Eq. (5.3b); please check and correct the sign.
[Section 6, Eq. (6.3)] The on-shell equivalence of the extended action (6.3) to the original theory is asserted but not proven; a short proof (integrating out C and g, or a brief citation plus explanation) would make the example self-contained.
[Throughout] There are several small typos: 'generalized generalized Komar charges' before Eq. (4.23), 'a a matter of fact' after Eq. (2.10), 'Let as assume' at the start of Section 3, and 'equtions' in Section 3.1.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Euler-theorem identity is derived from first principles; earlier Komar-charge work is cited contextually, not as load-bearing input.

full rationale

The central claim, Eq. (2.7), is obtained directly from the variational identity (2.2) by setting the variation to the assumed rescaling δλφi = ωi λ φi and δλL = ωL λ L. No quantity in this derivation is fitted, nor is the conclusion L = dΘ(φ, (ω/ωL)φ) imported from the references; it is an algebraic consequence of the homogeneity assumption. Eq. (2.10) merely parameterizes the freedom to add on-shell-closed currents Jn, and the coefficients αn are explicitly left as free choices, not tuned to match a precomputed answer. The generalized-Komar discussion in Section 3 is a review of the authors' earlier algorithm (Refs. [12-16]) and supplies the standard Noether-Wald construction; the exactness condition (3.17) is posed as the condition defining the desired ω(k, φ), not as a result secretly supplied by the cited papers. The examples reproduce known charges and thus serve as independent consistency checks rather than circular confirmations. Although self-citations are numerous, they are not load-bearing for the paper's main derivation, so there is no circular step.

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

The central Euler-theorem identity requires only the standard action variation formula and the homogeneity assumption. The Komar charge construction adds the inherited exactness assumption Eq. (3.17). The scalar-field example introduces an auxiliary Lagrange-multiplier field and an ad hoc equivalence assumption. No numerical parameters are fitted to data; the only free parameters are the arbitrary coefficients alpha_n in the non-uniqueness analysis.

free parameters (1)

alpha_n (and alpha in Section 4) = arbitrary, set to 0 in EMD example
Coefficients of on-shell closed currents that can be added to the total derivative expression, Eq. (2.10) and Eq. (4.13). They parametrize a family of Komar charges without affecting the Smarr formula when the scalar-charge relation is used.

assumptions (4)

standard math The action variation formula (2.2) with Euler-Lagrange forms and pre-symplectic potential
Standard result in Lagrangian field theory, used to derive Eq. (2.7).
domain assumption The Lagrangian is homogeneous under a global rescaling with weights omega_i and weight omega_L (Eq. 2.4)
Assumed; for non-homogeneous theories such as the scalar example, the coupling constant is promoted to a scalar field to recover homogeneity.
domain assumption On-shell exactness of the pulled-back Lagrangian for Killing vectors, Eq. (3.17)
Inherited from Refs. [12-16]; not proved in this paper, and it is the basis of the generalized Komar charge closure.
ad hoc to paper The Lagrange-multiplier extended action (6.3) is on-shell equivalent to the original theory, preserving charges
Introduced in Section 6; the paper asserts equivalence but does not check boundary terms or charge preservation.

invented entities (1)

Auxiliary (d-1)-form Lagrange-multiplier field C
purpose: Enforces constancy of the promoted coupling constant g in the self-interacting scalar example
Introduced in Eq. (6.3); a non-dynamical auxiliary field with no independent observable prediction.

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Pith. "Pith review of On-shell Lagrangians as total derivatives and the generalized Komar charge." pith.science (2026). https://pith.science/paper/4ODDPTRZ

@misc{pith2026250614024,
  author       = {Pith},
  title        = {Pith review of: On-shell Lagrangians as total derivatives and the generalized Komar charge},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4ODDPTRZ}},
  note         = {Machine review of arXiv:2506.14024}
}

read the original abstract

Lagrangians which transform homogeneously under a global transformation of the fields (a global rescaling, for instance) can be written on-shell as a total derivative which has a universal, solution-independent expression, using a functional version of the Euler theorem for homogeneous functions. We study the uniqueness of this expression and how this result can be used in the construction of generalized Komar charges.

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

GGI lectures on boundary and asymptotic symmetries
hep-th 2025-12 conditional novelty 4.0 of 10

A lecture-note review of boundary and asymptotic symmetries that re-derives the BMS group as the asymptotic symmetry group of Minkowski spacetime alone and constructs an integral Hamiltonian generator for scalar-field...

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Works this paper leans on

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

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L. F. Abbott and S. Deser, “Stability of Gravity with a Cosmological Constant,” Nucl. Phys. B 195 (1982), 76-96 DOI:10.1016/0550-3213(82)90049-9

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G. Barnich and F. Brandt, “Covariant theory of asymptotic symmetries, conservation laws and central charges,” Nucl. Phys. B 633 (2002), 3-82 DOI:10.1016/S0550-3213(02)00251-1 [hep-th/0111246 [hep-th]]

arXiv 2002

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Energy in generic higher curvature gravity theories,

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Conserved Charges in Extended Theories of Gravity,

H. Adami, M. R. Setare, T. C. Sisman and B. Tekin, “Conserved Charges in Extended Theories of Gravity,” Phys. Rept. 834 (2019), 1 DOI:10.1016/j.physrep.2019.08.003 [arXiv:1710.07252 [hep-th]]

arXiv 2019

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Covariant conservation laws in general relativity,

A. Komar, “Covariant conservation laws in general relativity,” Phys. Rev. 113 (1959), 934-936 DOI:10.1103/PhysRev.113.934

doi:10.1103/physrev.113.934 1959

[6] [6]

Boundary charges in gauge theories: Using Stokes theorem in the bulk,

G. Barnich, “Boundary charges in gauge theories: Using Stokes theorem in the bulk,” Class. Quant. Grav. 20 (2003), 3685-3698 DOI:10.1088/0264-9381/20/16/310 [hep-th/0301039 [hep-th]]

arXiv 2003

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J. M. Bardeen, B. Carter and S. W. Hawking, “The Four laws of black hole me- chanics,” Commun. Math. Phys. 31 (1973), 161-170 DOI:10.1007/BF01645742

doi:10.1007/bf01645742 1973

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B. Carter, “Black holes equilibrium states,” Contribution to: Les Houches Summer School of Theoretical Physics, 57-214