arxiv: 2605.02813 · v1 · submitted 2026-05-04 · 🌀 gr-qc · hep-th

Recognition: 3 theorem links

· Lean Theorem

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

Authors on Pith no claims yet

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

?

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

?

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.

Reference graph

Works this paper leans on

78 extracted references · 73 canonical work pages · 2 internal anchors

[1]

Dymnikova, Gen

I. Dymnikova, “Vacuum nonsingular black hole,” Gen. Rel. Grav.24(1992),235- 242DOI:10.1007/BF00760226

work page doi:10.1007/bf00760226 1992
[2]

Ayon-Beato and A

E. Ayón-Beato and A. García, “Regular black hole in general relativity cou- pled to nonlinear electrodynamics,” Phys. Rev. Lett.80(1998),5056-5059 DOI:10.1103/PhysRevLett.80.5056[gr-qc/9911046[gr-qc]]

work page doi:10.1103/physrevlett.80.5056 1998
[3]

New regular black hole solu- tion from nonlinear electrodynamics,

E. Ayón-Beato and A. García, “New regular black hole solu- tion from nonlinear electrodynamics,” Phys. Lett. B464(1999),25 DOI:10.1016/S0370-2693(99)01038-2[hep-th/9911174[hep-th]]

work page doi:10.1016/s0370-2693(99)01038-2 1999
[4]

Nonsingular charged black hole solution for non- linear source,

E. Ayón-Beato and A. García, “Nonsingular charged black hole solution for non- linear source,” Gen. Rel. Grav.31(1999),629-633DOI:10.1023/A:1026640911319 [gr-qc/9911084[gr-qc]]

work page doi:10.1023/a:1026640911319 1999
[5]

Ayon-Beato and A

E. Ayón-Beato and A. García, “The Bardeen model as a non- linear magnetic monopole,” Phys. Lett. B493(2000),149-152 DOI:10.1016/S0370-2693(00)01125-4[gr-qc/0009077[gr-qc]]

work page doi:10.1016/s0370-2693(00)01125-4 2000
[6]

Four parametric regular black hole solution,

E. Ayon-Beato and A. Garcia, “Four parametric regular black hole solution,” Gen. Rel. Grav.37(2005),635DOI:10.1007/s10714-005-0050-y[hep-th/0403229[hep- th]]

work page doi:10.1007/s10714-005-0050-y 2005
[7]

Regular magnetic black holes and monopoles from nonlinear electro- dynamics,

K. A. Bronnikov, “Regular magnetic black holes and monopoles from nonlinear electrodynamics,” Phys. Rev. D63(2001),044005 DOI:10.1103/PhysRevD.63.044005[gr-qc/0006014[gr-qc]]. 45

work page doi:10.1103/physrevd.63.044005 2001
[8]

Regular rotating electrically charged black holes and solitons in non-linear electrodynamics mini- mally coupled to gravity,

I. Dymnikova and E. Galaktionov, “Regular rotating electrically charged black holes and solitons in non-linear electrodynamics mini- mally coupled to gravity,” Class. Quant. Grav.32(2015) no.16,165015 DOI:10.1088/0264-9381/32/16/165015[arXiv:1510.01353[gr-qc]]

work page doi:10.1088/0264-9381/32/16/165015 2015
[9]

Bronnikov,Regular black holes sourced by nonlinear electrodynamics,2211.00743

K. A. Bronnikov, “Regular black holes sourced by nonlinear electrodynamics,” [arXiv:2211.00743[gr-qc]]

work page arXiv
[10]

Nonlinear electrodynamics, regular black holes and wormholes,

K. A. Bronnikov, “Nonlinear electrodynamics, regular black holes and wormholes,” Int. J. Mod. Phys. D27(2018) no.06,1841005 DOI:10.1142/S0218271818410055[arXiv:1711.00087[gr-qc]]

work page doi:10.1142/s0218271818410055 2018
[11]

Dymnikova, Class

I. Dymnikova, “Regular electrically charged structures in nonlinear electrody- namics coupled to general relativity,” Class. Quant. Grav.21(2004),4417-4429 DOI:10.1088/0264-9381/21/18/009[gr-qc/0407072[gr-qc]]

work page doi:10.1088/0264-9381/21/18/009 2004
[12]

Mass formula for Kerr black holes,

L. Smarr, “Mass formula for Kerr black holes,” Phys. Rev. Lett.30(1973),71-73 [erratum: Phys. Rev. Lett.30(1973),521-521]DOI:10.1103/PhysRevLett.30.71

work page doi:10.1103/physrevlett.30.71 1973
[13]

Non-linear electrodynamics: zeroth and first laws of black hole mechanics,

D. A. Rasheed, “Nonlinear electrodynamics: Zeroth and first laws of black hole mechanics,” [hep-th/9702087[hep-th]]

work page arXiv
[14]

Smarr’s formula for black holes with non-linear electrody- namics,

N. Breton, “Smarr’s formula for black holes with non-linear electrody- namics,” Gen. Rel. Grav.37(2005),643-650DOI:10.1007/s10714-005-0051-x [gr-qc/0405116[gr-qc]]

work page doi:10.1007/s10714-005-0051-x 2005
[15]

Thermodynamics of charged black holes with a nonlinear electrodynamics source,

H. A. González, M. Hassaine and C. Martínez, “Thermodynamics of charged black holes with a nonlinear electrodynamics source,” Phys. Rev. D80(2009),104008 DOI:10.1103/PhysRevD.80.104008[arXiv:0909.1365[hep-th]]

work page doi:10.1103/physrevd.80.104008 2009
[16]

Energy and first law of thermodynamics for Born- Infeld-anti-de-Sitter black hole,

W. Yi-Huan, “Energy and first law of thermodynamics for Born- Infeld-anti-de-Sitter black hole,” Chin. Phys. B19(2010),090404 DOI:10.1088/1674-1056/19/9/090404

work page doi:10.1088/1674-1056/19/9/090404 2010
[17]

Extended phase space thermo- dynamics for charged and rotating black holes and Born-Infeld vacuum polar- ization,

S. Gunasekaran, R. B. Mann and D. Kubiznak, “Extended phase space thermo- dynamics for charged and rotating black holes and Born-Infeld vacuum polar- ization,” JHEP11(2012),110DOI:10.1007/JHEP11(2012)110[arXiv:1208.6251 [hep-th]]

work page doi:10.1007/jhep11(2012)110 2012
[18]

Thermodynamic analysis of black hole solutions in gravitating nonlinear electrodynamics,

J. Díaz-Alonso and D. Rubiera-García, “Thermodynamic analysis of black hole solutions in gravitating nonlinear electrodynamics,” Gen. Rel. Grav.45(2013), 1901-1950DOI:10.1007/s10714-013-1567-0[arXiv:1204.2506[gr-qc]]

work page doi:10.1007/s10714-013-1567-0 2013
[19]

First law and Smarr formula of black hole mechan- ics in nonlinear gauge theories,

Y. Zhang and S. Gao, “First law and Smarr formula of black hole mechan- ics in nonlinear gauge theories,” Class. Quant. Grav.35(2018) no.14,145007 DOI:10.1088/1361-6382/aac9d4[arXiv:1610.01237[gr-qc]]. 46

work page doi:10.1088/1361-6382/aac9d4 2018
[20]

Fan and X

Z. Y. Fan and X. Wang, “Construction of Regular Black Holes in General Rela- tivity,” Phys. Rev. D94(2016) no.12,124027DOI:10.1103/PhysRevD.94.124027 [arXiv:1610.02636[gr-qc]]

work page doi:10.1103/physrevd.94.124027 2016
[21]

A Note on Smarr Relation and Coupling Constants,

S. Q. Hu, X. M. Kuang and Y. C. Ong, “A Note on Smarr Relation and Coupling Constants,” Gen. Rel. Grav.51(2019) no.5,55DOI:10.1007/s10714-019-2540-3 [arXiv:1810.06073[gr-qc]]

work page doi:10.1007/s10714-019-2540-3 2019
[22]

Enthalpy and the Mechanics of AdS Black Holes

D. Kastor, S. Ray and J. Traschen, “Enthalpy and the Mechan- ics of AdS Black Holes,” Class. Quant. Grav.26(2009),195011 DOI:10.1088/0264-9381/26/19/195011[arXiv:0904.2765[hep-th]]

work page Pith review doi:10.1088/0264-9381/26/19/195011 2009
[23]

Black hole chem- istry: thermodynamics with Lambda,

D. Kubiznak, R. B. Mann and M. Teo, “Black hole chemistry: ther- modynamics with Lambda,” Class. Quant. Grav.34(2017) no.6,063001 DOI:10.1088/1361-6382/aa5c69[arXiv:1608.06147[hep-th]]

work page doi:10.1088/1361-6382/aa5c69 2017
[24]

Generalizations of the Smarr formula for black holes with nonlinear electromagnetic fields,

L. Gulin and I. Smoli´ c, “Generalizations of the Smarr formula for black holes with nonlinear electromagnetic fields,” Class. Quant. Grav.35(2018) no.2,025015 DOI:10.1088/1361-6382/aa9dfd[arXiv:1710.04660[gr-qc]]

work page doi:10.1088/1361-6382/aa9dfd 2018
[25]

Black holes: Lecture notes,

P . K. Townsend, “Black holes: Lecture notes,” [gr-qc/9707012[gr-qc]]

work page arXiv
[26]

Black hole thermodynamics in the pres- ence of nonlinear electromagnetic fields,

A. Bokuli´ c, T. Juri´ c and I. Smoli´ c, “Black hole thermodynamics in the pres- ence of nonlinear electromagnetic fields,” Phys. Rev. D103(2021) no.12,124059 DOI:10.1103/PhysRevD.103.124059[arXiv:2102.06213[gr-qc]]

work page doi:10.1103/physrevd.103.124059 2021
[27]

A Smarr formula for charged black holes in nonlinear electrodynamics,

L. Balart and S. Fernando, “A Smarr formula for charged black holes in nonlinear electrodynamics,” Mod. Phys. Lett. A32(2017) no.39,1750219 DOI:10.1142/S0217732317502194[arXiv:1710.07751[gr-qc]]

work page doi:10.1142/s0217732317502194 2017
[28]

Communications in Mathematical Physics31(2), 161–170 (1973) https://doi

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

work page doi:10.1007/bf01645742 1973
[29]

Covariant conservation laws in general relativity,

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

work page doi:10.1103/physrev.113.934 1959
[30]

Higher-form symmetries in supergravity, scalar charges and black-hole thermo- dynamics,

G. Barbagallo, J. L. V . Cerdeira, C. Gómez-Fayrén, P . Meessen and T. Ortín, “Higher-form symmetries in supergravity, scalar charges and black-hole thermo- dynamics,” [arXiv:2512.22565[hep-th]]. To be published in JHEP

work page arXiv
[31]

Black holes equilibrium states,

B. Carter, “Black holes equilibrium states,” Contribution to: Les Houches Summer School of Theoretical Physics,57-214
[32]

On Komar integrals in asymptotically anti-de Sitter space-times,

A. Magnon, “On Komar integrals in asymptotically anti-de Sitter space-times,” J. Math. Phys.26(1985),3112-3117DOI:10.1063/1.526690 47

work page doi:10.1063/1.526690 1985
[33]

A Gauss type law for gravity with a cosmological constant,

S. L. Bazanski and P . Zyla, “A Gauss type law for gravity with a cosmological constant,” Gen. Rel. Grav.22(1990),379-387

1990
[34]

Komar Integrals in Higher (and Lower) Derivative Gravity,

D. Kastor, “Komar Integrals in Higher (and Lower) Derivative Gravity,” Class. Quant. Grav.25(2008),175007DOI:10.1088/0264-9381/25/17/175007 [arXiv:0804.1832[hep-th]]

work page doi:10.1088/0264-9381/25/17/175007 2008
[35]

Smarr Formula and an Extended First Law for Lovelock Gravity,

D. Kastor, S. Ray and J. Traschen, “Smarr Formula and an Extended First Law for Lovelock Gravity,” Class. Quant. Grav.27(2010),235014 DOI:10.1088/0264-9381/27/23/235014[arXiv:1005.5053[hep-th]]

work page doi:10.1088/0264-9381/27/23/235014 2010
[36]

Komar integrals for theories of higher order in the Riemann curvature and black-hole chemistry,

T. Ortín, “Komar integrals for theories of higher order in the Riemann curvature and black-hole chemistry,” JHEP08(2021),023DOI:10.1007/JHEP08(2021)023 [arXiv:2104.10717[gr-qc]]

work page doi:10.1007/jhep08(2021)023 2021
[37]

Black hole chemistry, the cos- mological constant and the embedding tensor,

P . Meessen, D. Mitsios and T. Ortín, “Black hole chemistry, the cos- mological constant and the embedding tensor,” JHEP12(2022),155 DOI:10.1007/JHEP12(2022)155[arXiv:2203.13588[hep-th]]

work page doi:10.1007/jhep12(2022)155 2022
[38]

Smarr Formula for Lovelock Black Holes: a Lagrangian approach,

S. Liberati and C. Pacilio, “Smarr Formula for Lovelock Black Holes: a Lagrangian approach,” Phys. Rev. D93(2016) no.8,084044DOI:10.1103/PhysRevD.93.084044 [arXiv:1511.05446[gr-qc]]

work page doi:10.1103/physrevd.93.084044 2016
[39]

Local symmetries and constraints,

J. Lee and R. M. Wald, “Local symmetries and constraints,” J. Math. Phys.31 (1990),725-743DOI:10.1063/1.528801

work page doi:10.1063/1.528801 1990
[40]

Black hole entropy is the Noether charge,

R. M. Wald, “Black hole entropy is the Noether charge,” Phys. Rev. D48(1993) no.8, R3427-R3431DOI:10.1103/PhysRevD.48.R3427[gr-qc/9307038[gr-qc]]

work page doi:10.1103/physrevd.48.r3427 1993
[41]

The first law of black hole me- chanics in the Einstein-Maxwell theory revisited,

Z. Elgood, P . Meessen and T. Ortín, “The first law of black hole me- chanics in the Einstein-Maxwell theory revisited,” JHEP09(2020),026 DOI:10.1007/JHEP09(2020)026[arXiv:2006.02792[hep-th]]

work page doi:10.1007/jhep09(2020)026 2020
[42]

The first law of het- erotic stringy black hole mechanics at zeroth order inα’,

Z. Elgood, D. Mitsios, T. Ortín and D. Pereníguez, “The first law of het- erotic stringy black hole mechanics at zeroth order inα’,” JHEP07(2021),007 DOI:10.1007/JHEP07(2021)007[arXiv:2012.13323[hep-th]]

work page doi:10.1007/jhep07(2021)007 2021
[43]

The first law and Wald entropy for- mula of heterotic stringy black holes at first order inα ′,

Z. Elgood, T. Ortín and D. Pereníguez, “The first law and Wald entropy for- mula of heterotic stringy black holes at first order inα ′,” JHEP05(2021),110 DOI:10.1007/JHEP05(2021)110[arXiv:2012.14892[hep-th]]

work page doi:10.1007/jhep05(2021)110 2021
[44]

Komar integral and Smarr for- mula for axion-dilaton black holes versus S duality,

D. Mitsios, T. Ortín and D. Pereñiguez, “Komar integral and Smarr for- mula for axion-dilaton black holes versus S duality,” JHEP08(2021),019 DOI:10.1007/JHEP08(2021)019[arXiv:2106.07495[hep-th]]

work page doi:10.1007/jhep08(2021)019 2021
[45]

Magnetic charges and Wald entropy,

T. Ortín and D. Pereñiguez, “Magnetic charges and Wald entropy,” JHEP11(2022), 081DOI:10.1007/JHEP11(2022)081[arXiv:2207.12008[hep-th]]. 48

work page doi:10.1007/jhep11(2022)081 2022
[46]

Classical physics as geometry: Gravitation, electromagnetism, unquantized charge, and mass as properties of curved empty space,

C. W. Misner and J. A. Wheeler, “Classical physics as geometry: Gravitation, electromagnetism, unquantized charge, and mass as properties of curved empty space,” Annals Phys.2(1957),525-603DOI:10.1016/0003-4916(57)90049-0

work page doi:10.1016/0003-4916(57)90049-0 1957
[47]

Hairy black holes, scalar charges and ex- tended thermodynamics,

R. Ballesteros and T. Ortín, “Hairy black holes, scalar charges and ex- tended thermodynamics,” Class. Quant. Grav.41(2024) no.5,055007 DOI:10.1088/1361-6382/ad210a[arXiv:2308.04994[gr-qc]]

work page doi:10.1088/1361-6382/ad210a 2024
[48]

Generalized Komar charges and Smarr formu- las for black holes and boson stars,

R. Ballesteros and T. Ortín, “Generalized Komar charges and Smarr formu- las for black holes and boson stars,” SciPost Phys. Core8(2025) no.038,038 DOI:10.21468/SciPostPhysCore.8.2.038[arXiv:2409.08268[gr-qc]]

work page doi:10.21468/scipostphyscore.8.2.038 2025
[49]

On identically closed forms locally constructed from a field,

R. M. Wald, “On identically closed forms locally constructed from a field,” J. Math. Phys.31(1990) no.10,2378DOI:10.1063/1.528839

work page doi:10.1063/1.528839 1990
[50]

Non-closed scalar charge in four-dimensional Einstein-scalar-Gauss-Bonnet black hole thermodynamics

R. Ballesteros, M. Cárdenas and E. Lescano, “Nonclosed scalar charge in four- dimensional Einstein-scalar-Gauss-Bonnet black hole thermodynamics,” Phys. Rev. D113(2026) no.8,084039DOI:10.1103/1hz8-zn5h[arXiv:2510.08686[hep- th]]

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/1hz8-zn5h 2026
[51]

Supergravities in5Dimensions,

E. Cremmer, “Supergravities in5Dimensions,” LPTENS-80-17
[52]

Foundations of the new field theory,

M. Born and L. Infeld, “Foundations of the new field theory,” Proc. Roy. Soc. Lond. A144(1934) no.852,425-451DOI:10.1098/rspa.1934.0059

work page doi:10.1098/rspa.1934.0059 1934
[53]

Global structure of five-dimensional fuzzballs,

G. W. Gibbons and N. P . Warner, “Global structure of five-dimensional fuzzballs,” Class. Quant. Grav.31(2014),025016DOI:10.1088/0264-9381/31/2/025016 [arXiv:1305.0957[hep-th]]

work page doi:10.1088/0264-9381/31/2/025016 2014
[54]

Regular Stringy Black Holes?,

P . A. Cano, S. Chimento, T. Ortín and A. Ruipérez, “Regular Stringy Black Holes?,” Phys. Rev. D99(2019) no.4,046014DOI:10.1103/PhysRevD.99.046014 [arXiv:1806.08377[hep-th]]

work page doi:10.1103/physrevd.99.046014 2019
[55]

Black hole no-hair theorem for self-gravitating time- dependent spherically symmetric multiple scalar fields,

S. Yazadjiev and D. Doneva, “Black hole no-hair theorem for self-gravitating time- dependent spherically symmetric multiple scalar fields,” Eur. Phys. J. C84(2024) no.5,492DOI:10.1140/epjc/s10052-024-12822-6[arXiv:2401.13288[gr-qc]]

work page doi:10.1140/epjc/s10052-024-12822-6 2024
[56]

Symmetry inheritance of scalar fields,

I. Smoli´ c, “Symmetry inheritance of scalar fields,” Class. Quant. Grav.32(2015) no.14,145010DOI:10.1088/0264-9381/32/14/145010[arXiv:1501.04967[gr-qc]]

work page doi:10.1088/0264-9381/32/14/145010 2015
[57]

Conundrum of regular black holes with nonlinear electromagnetic fields,

A. Bokuli´ c, T. Juri´ c and I. Smoli´ c, “Conundrum of regular black holes with nonlinear electromagnetic fields,” Phys. Rev. D113(2026) no.2,024044 DOI:10.1103/z7gd-96ms[arXiv:2510.23711[gr-qc]]

work page doi:10.1103/z7gd-96ms 2026
[58]

Gravity and Strings

T. Ortín, “Gravity and Strings”,2nd edition, Cambridge University Press,2015. 49

2015
[59]

Introductory Notes on Non-linear Electrodynamics and its Appli- cations,

D. P . Sorokin, “Introductory Notes on Non-linear Electrodynamics and its Appli- cations,” Fortsch. Phys.70(2022) no.7-8,2200092DOI:10.1002/prop.202200092 [arXiv:2112.12118[hep-th]]

work page doi:10.1002/prop.202200092 2022
[60]

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

J. L. V . Cerdeira and T. Ortín, “On-shell Lagrangians as total derivatives and the generalized Komar charge,” JHEP09(2025),068DOI:10.1007/JHEP09(2025)068 [arXiv:2506.14024[hep-th]]

work page doi:10.1007/jhep09(2025)068 2025
[61]

Reloading Black Hole Thermodynamics with Noether Charges,

M. Golshani, M. M. Sheikh-Jabbari, V . Taghiloo and M. H. Vahidinia, “Reloading Black Hole Thermodynamics with Noether Charges,” [arXiv:2407.15994[hep- th]]

work page arXiv
[62]

Covariant phase space formalism for fluctuating boundaries,

H. Adami, M. Golshani, M. M. Sheikh-Jabbari, V . Taghiloo and M. H. Vahidinia, “Covariant phase space formalism for fluctuating boundaries,” JHEP09(2024), 157DOI:10.1007/JHEP09(2024)157[arXiv:2407.03259[hep-th]]

work page doi:10.1007/jhep09(2024)157 2024
[63]

A note on the Noether-Wald and gen- eralized Komar charges,

T. Ortín and M. Zatti, “A note on the Noether-Wald and gen- eralized Komar charges,” SciPost Phys. Core8(2025),094 DOI:10.21468/SciPostPhysCore.8.4.094[arXiv:2411.10420[gr-qc]]

work page doi:10.21468/scipostphyscore.8.4.094 2025
[64]

Exploring String Theory Solutions: Black hole thermodynamics withα ′ corrections and type II compactifications

M. Zatti, “Exploring String Theory Solutions: Black hole thermodynamics withα ′ corrections and type II compactifications”, PhD thesis, U. Autónoma de Madrid,
[65]

[arXiv:2505.01191[hep-th]]

work page arXiv
[66]

Comment on “Construction of reg- ular black holes in general relativity

B. Toshmatov, Z. Stuchlík and B. Ahmedov, “Comment on “Construction of reg- ular black holes in general relativity”,” Phys. Rev. D98(2018) no.2,028501 DOI:10.1103/PhysRevD.98.028501[arXiv:1807.09502[gr-qc]]

work page doi:10.1103/physrevd.98.028501 2018
[67]

A non-linear duality- invariant conformal extension of Maxwell’s equations,

I. Bandos, K. Lechner, D. Sorokin and P . K. Townsend, “A non-linear duality- invariant conformal extension of Maxwell’s equations,” Phys. Rev. D102(2020), 121703DOI:10.1103/PhysRevD.102.121703[arXiv:2007.09092[hep-th]]

work page doi:10.1103/physrevd.102.121703 2020
[68]

Nonlinear electrodynamics with the maximum allowable sym- metries,

B. P . Kosyakov, “Nonlinear electrodynamics with the maximum allowable sym- metries,” Phys. Lett. B810(2020),135840DOI:10.1016/j.physletb.2020.135840 [arXiv:2007.13878[hep-th]]

work page doi:10.1016/j.physletb.2020.135840 2020
[69]

The calculation of the thermodynamic quan- tities of the Bardeen black hole,

J. Man and H. Cheng, “The calculation of the thermodynamic quan- tities of the Bardeen black hole,” Gen. Rel. Grav.46(2014),1660 DOI:10.1007/s10714-013-1660-4[arXiv:1304.5686[hep-th]]

work page doi:10.1007/s10714-013-1660-4 2014
[70]

Bardeen black hole chemistry,

A. G. Tzikas, “Bardeen black hole chemistry,” Phys. Lett. B788(2019),219-224 DOI:10.1016/j.physletb.2018.11.036[arXiv:1811.01104[gr-qc]]

work page doi:10.1016/j.physletb.2018.11.036 2019
[71]

Poisson,A Relativist’s Toolkit: The Mathematics of Black-Hole Mechanics, Cambridge University Press (12, 2009), 10.1017/CBO9780511606601

E. Poisson, “A Relativist’s Toolkit: The Mathematics of Black-Hole Mechanics,” Cambridge University Press,2009,DOI:10.1017/CBO9780511606601 50

work page doi:10.1017/cbo9780511606601 2009
[72]

Bardeen Regular Black Hole With an Electric Source,

M. E. Rodrigues and M. V . de Sousa Silva, “Bardeen Regular Black Hole With an Electric Source,” JCAP06(2018),025DOI:10.1088/1475-7516/2018/06/025 [arXiv:1802.05095[gr-qc]]

work page doi:10.1088/1475-7516/2018/06/025 2018
[73]

Penrose, Phys

R. Penrose, “Gravitational collapse and space-time singularities,” Phys. Rev. Lett. 14(1965),57-59DOI:10.1103/PhysRevLett.14.57

work page doi:10.1103/physrevlett.14.57 1965
[74]

The Singularities of gravitational col- lapse and cosmology,

S. W. Hawking and R. Penrose, “The Singularities of gravitational col- lapse and cosmology,” Proc. Roy. Soc. Lond. A314(1970),529-548 DOI:10.1098/rspa.1970.0021

work page doi:10.1098/rspa.1970.0021 1970
[75]

The1965Penrose sin- gularity theorem,

J. M. M. Senovilla and D. Garfinkle, “The1965Penrose sin- gularity theorem,” Class. Quant. Grav.32(2015) no.12,124008 DOI:10.1088/0264-9381/32/12/124008[arXiv:1410.5226[gr-qc]]

work page doi:10.1088/0264-9381/32/12/124008 2015
[76]

Causality and energy conditions in non- linear electrodynamics,

J. G. Russo and P . K. Townsend, “Causality and energy conditions in non- linear electrodynamics,” JHEP06(2024),191DOI:10.1007/JHEP06(2024)191 [arXiv:2404.09994[hep-th]]

work page doi:10.1007/jhep06(2024)191 2024
[77]

Excising Cauchy Horizons with Nonlinear Electrodynamics,

T. Hale, R. A. Hennigar and D. Kubiznak, “Excising Cauchy Horizons with Nonlinear Electrodynamics,” Phys. Rev. D113(2026) no.6, L061502 DOI:10.1103/x8c8-j85k[arXiv:2506.20802[gr-qc]]

work page doi:10.1103/x8c8-j85k 2026
[78]

Black holes and causal nonlinear electrodynamics

J. G. Russo and P . K. Townsend, “Black holes and causal nonlinear electrodynam- ics,” [arXiv:2601.07789[hep-th]]. 51

work page internal anchor Pith review Pith/arXiv arXiv