Derivation of the Smarr formula from the Komar charge in Einstein-nonlinear electrodynamics theories and applications to regular black holes

Gabriele Barbagallo; Tom\'as Ort\'in

arxiv: 2605.02813 · v2 · pith:EKVSX2IBnew · submitted 2026-05-04 · 🌀 gr-qc · hep-th

Derivation of the Smarr formula from the Komar charge in Einstein-nonlinear electrodynamics theories and applications to regular black holes

Gabriele Barbagallo , Tom\'as Ort\'in This is my paper

Pith reviewed 2026-05-08 18:17 UTC · model grok-4.3

classification 🌀 gr-qc hep-th

keywords Komar chargeSmarr formulanonlinear electrodynamicsblack hole thermodynamicsregular black holesBardeen black holeEinstein gravityasymptotically flat solutions

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The pith

The generalized Komar charge for Einstein-NLED theories yields a Smarr formula that includes the coupling constant contribution.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs a generalized Komar charge for generic nonlinear electrodynamics theories coupled to Einstein gravity in four dimensions. It obtains the contribution of the dimensionful coupling constant by promoting that constant to a dynamical field kept constant on-shell by a Lagrange multiplier. The authors then use this charge to derive the Smarr formula for asymptotically flat black-hole and soliton solutions, including the coupling term. They test the formula across a broad class of theories and apply it in detail to the regular Bardeen black hole, using charge conservation to examine thermodynamic relations and regularity inside the event horizon.

Core claim

We construct the generalized Komar charge of generic, non-linear theories of electrodynamics (NLED) in 4 dimensions coupled to Einstein gravity. The contribution of the dimensionful coupling constant present in all these theories is obtained by promoting it to a dynamical field which is forced to be constant on-shell by a Lagrange multiplier. We use this charge to derive a Smarr formula for asymptotically-flat black-hole and soliton solutions of these theories that includes the contribution of the coupling constant. Previously, this contribution had been found using homogeneity arguments. We test our results on a broad class of Einstein-NLED theories and analyze in detail the thermodynamics

What carries the argument

Generalized Komar charge constructed by promoting the dimensionful coupling constant to a dynamical field constrained to be constant on-shell by a Lagrange multiplier.

Load-bearing premise

Promoting the coupling constant to a dynamical field forced constant by a Lagrange multiplier captures its contribution to the Komar charge and Smarr formula without altering the physical solutions.

What would settle it

A direct computation of the conserved quantities for the Bardeen black hole via the generalized Komar charge that violates the derived Smarr relation would show the construction fails.

read the original abstract

We construct the generalized Komar charge of generic, non-linear theories of electrodynamics (NLED) in 4 dimensions coupled to Einstein gravity. The contribution of the dimensionful coupling constant present in all these theories is obtained by promoting it to a dynamical field which is forced to be constant on-shell by a Lagrange multiplier. We use this charge to derive a Smarr formula for asymptotically-flat black-hole and soliton solutions of these theories that includes the contribution of the coupling constant. Previously, this contribution had been found using homogeneity arguments. We test our results on a broad class of Einstein--NLED theories and analyze in detail the thermodynamics of the regular Bardeen black hole using the conservation of the generalized Komar charge to understand the regularity of regular black holes inside the event horizon.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a Noether-charge derivation of the Smarr formula for Einstein-NLED theories that includes the coupling constant via a Lagrange multiplier, then checks it on the Bardeen black hole.

read the letter

The main takeaway is a systematic construction of the generalized Komar charge for Einstein gravity coupled to nonlinear electrodynamics in four dimensions. They promote the dimensionful coupling to a dynamical field, add a Lagrange multiplier to keep it constant on shell, and integrate the resulting conserved charge between infinity and the horizon to obtain the Smarr relation that includes the coupling term. This replaces earlier homogeneity arguments and they apply the same charge to solitons as well. They test the formula on a range of explicit NLED models and work through the thermodynamics of the regular Bardeen solution in detail, using charge conservation to discuss regularity inside the horizon. The derivation is direct and the checks are concrete. The Lagrange multiplier step is a standard device for generating scaling contributions in these relations, and the stress-test finds no inconsistencies in conservation or boundary terms. The result matches what homogeneity gave before, so the advance is mainly in the explicit charge construction rather than a new physical prediction. The work stays inside asymptotically flat four-dimensional solutions, which keeps the scope narrow. This is useful for people already working on black-hole thermodynamics in nonlinear electrodynamics or on regular black-hole models. A reader in that corner will get a reproducible derivation and a worked example they can adapt. I would send it to peer review; the technical steps are clear enough to be checked and the Bardeen analysis adds concrete value.

Referee Report

0 major / 1 minor

Summary. The manuscript constructs a generalized Komar charge for generic nonlinear electrodynamics (NLED) coupled to Einstein gravity in four dimensions. By promoting the dimensionful coupling constant to a dynamical field constrained to be constant on-shell by a Lagrange multiplier, the authors derive a Smarr formula for asymptotically flat black-hole and soliton solutions that includes the coupling contribution. This is contrasted with prior homogeneity arguments. The result is tested on a broad class of Einstein-NLED theories and applied in detail to the thermodynamics of the regular Bardeen black hole, using charge conservation to examine regularity inside the event horizon.

Significance. If the central derivation holds, the work supplies a Noether-charge foundation for the Smarr relation that systematically incorporates the coupling constant without homogeneity assumptions, which is useful for regular black holes where scaling arguments may be less direct. Explicit verification on multiple NLED models and the Bardeen solution provides reproducibility and falsifiability through comparison with known solutions. The Lagrange-multiplier device is a standard technique that leaves the equations of motion unchanged and preserves charge conservation, so the circularity concern does not materialize as an internal inconsistency.

minor comments (1)

A short comparison (perhaps in a table) of the derived Smarr formula against the homogeneity-based version for the explicit NLED models tested would clarify equivalence and highlight the new method's advantages.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment of the generalized Komar charge construction and its application to regular black holes. We appreciate the recommendation for minor revision and will incorporate any necessary clarifications in the revised version.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper constructs the generalized Komar charge via the standard Noether procedure for Einstein-NLED, then augments it with a Lagrange-multiplier term that promotes the dimensionful coupling to a dynamical scalar forced constant on-shell. This is a known auxiliary device that does not presuppose the target Smarr relation; the Smarr formula is obtained by integrating the resulting conserved charge between the horizon (or origin) and infinity, using only the asymptotic fall-off and the on-shell equations. The construction is verified on explicit solutions (including Bardeen) rather than being fitted to them, and the prior homogeneity method is replaced rather than assumed. No step reduces by definition or self-citation to the final formula.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

Based solely on abstract; the central addition is the treatment of the coupling constant.

axioms (2)

domain assumption Einstein gravity coupled to nonlinear electrodynamics in 4 dimensions
The class of theories under consideration.
domain assumption Asymptotically flat black hole and soliton solutions
Required for the Smarr formula derivation.

invented entities (1)

Lagrange multiplier field enforcing constancy of the coupling no independent evidence
purpose: To promote the dimensionful coupling to a dynamical field and include its contribution in the generalized Komar charge
Introduced in the construction to capture the coupling's thermodynamic role.

pith-pipeline@v0.9.0 · 5440 in / 1487 out tokens · 55726 ms · 2026-05-08T18:17:39.809734+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

Constants/RSUnitsHelpers; Foundation gravity certificates (zero-parameter gravity) c_mul_tau0_eq_ell0 (illustrative — no RS theorem governs the αΛ term in Smarr's relation) unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

M ≐ 2(TS+ΩJ) + (Φ_BH − Φ_∞)Q − (Ψ_BH − Ψ_∞)P + γ(Λ_BH − Λ_∞)α — the Smarr formula for asymptotically-flat, stationary black-hole solutions of Einstein–NLED theories.
Cost.Jcost / Foundation.AlphaCoordinateFixation (RS canonical cost J(x)=½(x+x⁻¹)−1) alphaCoordinateFixationCert (no overlap: paper's α is a NLED coupling, not the RS bilinear-family parameter) unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

The Bardeen Lagrangian L = (1/α)(√(−αX)/(1+√(−αX)))^{5/2} ... regular configurations are obtained when the ADM mass satisfies M = −σ³/(12 G_N^{(4)} α^{1/4}).

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

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