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arxiv: 1907.10876 · v1 · pith:L4WPU4GYnew · submitted 2019-07-25 · ✦ hep-th · gr-qc

Quasi-topological Electromagnetism: Dark Energy, Dyonic Black Holes, Stable Photon Spheres and Hidden Electromagnetic Duality

Hai-Shan Liu , Zhan-Feng Mai , Yue-Zhou Li , H. Lu This is my paper

Pith reviewed 2026-05-24 16:25 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords quasi-topological electromagnetismdark energydyonic black holesphoton sphereselectromagnetic dualitycosmological constantFLRW solutionsWheeler-DeWitt patch
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The pith

Adding the square of the topological 4-form to electromagnetism produces a dark-energy fluid with pressure exactly opposite its density.

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

The paper defines quasi-topological electromagnetism as the squared norm of F wedge F. Its energy-momentum tensor is an isotropic perfect fluid with pressure equal in magnitude but opposite in sign to the energy density, enabling it to serve as a dark energy model. This term does not affect purely electric or magnetic Reissner-Nordstrom black holes but completely modifies dyonic solutions, allowing up to four horizons and three photon spheres with one stable. In cosmology the term alone replicates a cosmological constant, yet when coupled to a scalar field it supports standard FLRW solutions. Although the term breaks electromagnetic duality, the symmetry is restored in the on-shell action evaluated in the Wheeler-DeWitt patch.

Core claim

The quasi-topological term ((F ∧ F)^2) added to the Einstein-Maxwell action has an energy-momentum tensor that is precisely that of a cosmological constant fluid, p = -ρ. This leads to unmodified electric or magnetic black holes but new dyonic black hole solutions with four horizons and multiple photon spheres including a stable one. The term is equivalent to a cosmological constant in pure cosmology but allows coupling to matter, and restores electromagnetic duality in the on-shell action within the Wheeler-DeWitt patch.

What carries the argument

The quasi-topological term defined as the square of the topological 4-form F∧F, which supplies an energy-momentum tensor of isotropic perfect fluid with p = −ρ.

Load-bearing premise

The quasi-topological term can be added to the Einstein-Maxwell theory in four dimensions without introducing ghosts or other inconsistencies.

What would settle it

Detection of a stable photon sphere around a dyonic black hole or confirmation that dark energy has exactly w = -1 in a coupled scalar model would support the claim, while absence of such features in observations could falsify it.

read the original abstract

We introduce the quasi-topological electromagnetism which is defined to be the squared norm of the topological 4-form $F\wedge F$. A salient property is that its energy-momentum tensor is of the isotropic perfect fluid with the pressure being precisely the opposite to its energy density. It can thus provide a model for dark energy. We study its application in both black hole physics and cosmology. The quasi-topological term has no effect on the purely electric or magnetic Reissner-Nordstr\"om black holes, the dyonic solution is however completely modified. We find that the dyonic black holes can have four real horizons. For suitable parameters, the black hole can admit as many as three photon spheres, with one being stable. Another intriguing property is that although the quasi-topological term breaks the electromagnetic duality, the symmetry emerges in the on-shell action in the Wheeler-DeWitt patch. In cosmology, we demonstrate that the quasi-topological term alone is equivalent to a cosmological constant, but the model provides a mechanism for the dark energy to couple with other types of matter. We present a concrete example of the quasi-topological electromagnetism coupled to a scalar field that admits the standard FLRW cosmological solutions.

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.

Referee Report

2 major / 2 minor

Summary. The paper introduces 'quasi-topological electromagnetism' as the squared norm of the topological 4-form (F ∧ F). Its key property is that the associated energy-momentum tensor is that of an isotropic perfect fluid with p = −ρ exactly, providing a model for dark energy. The work examines consequences for dyonic Reissner-Nordström black holes (modified solutions admitting up to four horizons and three photon spheres, one stable), restoration of electromagnetic duality in the on-shell action within the Wheeler-DeWitt patch, and cosmological applications where the term alone mimics a cosmological constant but permits coupling to other matter (explicit FLRW example with a scalar field).

Significance. If the construction holds, the explicit result that the EMT of the (F ∧ F)^2 term is T_μν = −φ² g_μν (with φ = (1/2) F ∧ *F) supplies a parameter-free realization of dark-energy behavior from a higher-order electromagnetic term whose equations remain second-order. The black-hole phenomenology (multiple horizons, stable photon sphere) and the on-shell duality restoration are concrete, falsifiable predictions. The cosmological coupling mechanism extends the model beyond a pure cosmological constant. These features are strengths of the manuscript.

major comments (2)
  1. [§2] The central EMT derivation (action variation yielding T_μν = −φ² g_μν) is load-bearing for the dark-energy claim; the manuscript should display the explicit steps of this variation (including the precise definition of the 4-form norm and the factor of 1/2) rather than stating the result, to allow verification that no metric-dependent factors remain.
  2. [§4] §4 (dyonic black holes): the assertion that solutions can possess four real horizons and three photon spheres (one stable) is central to the black-hole phenomenology; the manuscript must supply the explicit metric function, the quartic equation for horizons, and the effective potential for null geodesics together with the parameter ranges that realize these features, rather than only numerical examples.
minor comments (2)
  1. [Eq. (1)] The action is written with measure (−g)^{−1/2} d⁴x; confirm this is the intended convention or correct to the standard √(−g) d⁴x, and ensure consistency with the stress-tensor computation.
  2. [§2] Notation for the dual *F and the 4-form F ∧ F should be introduced once with a clear definition before repeated use in the black-hole and cosmological sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and recommendation of minor revision. We address each major comment below and will incorporate the requested explicit material into the revised manuscript.

read point-by-point responses

  1. Referee: [§2] The central EMT derivation (action variation yielding T_μν = −φ² g_μν) is load-bearing for the dark-energy claim; the manuscript should display the explicit steps of this variation (including the precise definition of the 4-form norm and the factor of 1/2) rather than stating the result, to allow verification that no metric-dependent factors remain.

    Authors: We agree that the explicit variation steps are essential for verification. In the revised manuscript we will add a dedicated subsection deriving the energy-momentum tensor from the quasi-topological term. This will include the precise definition of the 4-form norm (F ∧ F)^2, the conventional factor of 1/2, the full variation with respect to the metric, and an explicit check that the result is exactly T_μν = −φ² g_μν with no leftover metric-dependent prefactors. revision: yes

  2. Referee: [§4] §4 (dyonic black holes): the assertion that solutions can possess four real horizons and three photon spheres (one stable) is central to the black-hole phenomenology; the manuscript must supply the explicit metric function, the quartic equation for horizons, and the effective potential for null geodesics together with the parameter ranges that realize these features, rather than only numerical examples.

    Authors: We accept that the analytic expressions are required to substantiate the black-hole claims. The revised §4 will present the explicit dyonic metric function, the resulting quartic equation for the horizons, the effective potential for null geodesics, and the ranges of the electric/magnetic charges and quasi-topological coupling constant that permit four real horizons and three photon spheres (one of which is stable). revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper defines the quasi-topological term explicitly as the square of the topological 4-form F∧F and computes its energy-momentum tensor by direct variation of the corresponding action, producing T_μν = −φ² g_μν (with φ = (1/2)F∧*F) as an output. This property is not presupposed in the definition nor obtained by fitting parameters to data; the equations of motion remain second-order, and the cosmological and black-hole solutions follow from the derived EMT without self-referential steps. No self-citations, uniqueness theorems, or ansatze imported from prior work are invoked to establish the central dark-energy claim, rendering the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The construction rests on the standard Einstein-Maxwell action plus one new term whose only free parameter is an overall coupling constant; no additional fitted quantities or new particles are introduced.

free parameters (1)
  • overall coupling constant of the quasi-topological term
    An arbitrary coefficient multiplying (F∧F)^2 that sets the strength of the new contribution.
axioms (1)
  • standard math Einstein-Maxwell theory provides the base gravitational and electromagnetic dynamics in four dimensions
    The paper begins from the standard Einstein-Maxwell action and augments it with the new term.
invented entities (1)
  • quasi-topological electromagnetism term (F∧F)^2 no independent evidence
    purpose: To generate a perfect-fluid stress tensor with p = -ρ
    New term postulated in the action; no independent experimental signature is supplied in the abstract.

pith-pipeline@v0.9.0 · 5762 in / 1400 out tokens · 24896 ms · 2026-05-24T16:25:37.143909+00:00 · methodology

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supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
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Forward citations

Cited by 3 Pith papers

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

  1. Dyonic black holes supporting nearly-black self-gravitating thin shells

    gr-qc 2026-05 unverdicted novelty 5.0

    Dyonic black holes support self-gravitating nearly-black thin shells at discrete universal radii independent of central mass.

  2. Topologically equivalent yet radiatively distinct orbits in EMRI system

    gr-qc 2026-04 unverdicted novelty 5.0

    In dyonic black holes, periodic orbits with identical rotation numbers but spanning different curvature regions generate radiatively distinct gravitational waveforms in EMRIs.

  3. Topology of black hole thermodynamics: A brief review

    gr-qc 2026-04 unverdicted novelty 2.0

    Topological numbers categorize black hole systems into universality classes based on thermodynamic behavior, with calculations for critical points and phase transitions.

Reference graph

Works this paper leans on

31 extracted references · 31 canonical work pages · cited by 3 Pith papers · 15 internal anchors

  1. [1]

    Foundations of the new field theory ,

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

    work page doi:10.1098/rspa.1934.0059 1934

  2. [2]

    A new cubic theory of gravity in five dimensions: Black hole, Birkhoff's theorem and C-function

    J. Oliva and S. Ray, “A new cubic theory of gravity in five di mensions: Black hole, Birkhoff’s theorem and C-function,” Class. Quant. Grav. 27, 225002 (2010) doi:10. 1088/0264-9381/27/22/225002 [arXiv:1003.4773 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  3. [3]

    Black holes in quasi-topolo gical gravity,

    R.C. Myers and B. Robinson, “Black holes in quasi-topolo gical gravity,” JHEP 1008, 067 (2010) doi:10.1007/JHEP08(2010)067 [arXiv:1003.535 7 [gr-qc]]

    work page doi:10.1007/jhep08(2010)067 2010

  4. [4]

    Black holes in quartic quasitopological g ravity,

    M.H. Dehghani, A. Bazrafshan, R.B. Mann, M.R. Mehdizade h, M. Ghanaatian and M.H. Vahidinia, “Black holes in quartic quasitopological g ravity,” Phys. Rev. D 85, 104009 (2012) doi:10.1103/PhysRevD.85.104009 [arXiv:11 09.4708 [hep-th]]

    work page doi:10.1103/physrevd.85.104009 2012

  5. [5]

    Quasi-topological Ricci po lynomial gravities,

    Y.Z. Li, H.S. Liu and H. L¨ u, “Quasi-topological Ricci po lynomial gravities,” JHEP 1802, 166 (2018) doi:10.1007/JHEP02(2018)166 [arXiv:1708.07 198 [hep-th]]

    work page doi:10.1007/jhep02(2018)166 2018

  6. [6]

    Dark Matter and Dark Energy Interactions: Theoretical Challenges, Cosmological Implications and Observational Signatures

    B. Wang, E. Abdalla, F. Atrio-Barandela and D. Pavon, “Da rk matter and dark energy interactions: theoretical challenges, cosmological impl ications and observational signa- 30 tures,” Rept. Prog. Phys. 79, no. 9, 096901 (2016) doi:10.1088/0034-4885/79/9/096901 [arXiv:1603.08299 [astro-ph.CO]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/0034-4885/79/9/096901 2016

  7. [7]

    Conservation of charge and the Einstei n-Maxwell field equations,

    G.W. Horndeski, “Conservation of charge and the Einstei n-Maxwell field equations,” J. Math. Phys. 17, 1980 (1976). doi:10.1063/1.522837

    work page doi:10.1063/1.522837 1980

  8. [8]

    Energy conditions for el ectromagnetic field in pres- ence of cosmological constant,

    O. Goldoni and M.F.A. da Silva, “Energy conditions for el ectromagnetic field in pres- ence of cosmological constant,” PoS ISFTG , 072 (2009). doi: 10.22323/1.081.0072

    work page doi:10.22323/1.081.0072 2009

  9. [9]

    Black hole and cosmos wi th multiple horizons and multiple singularities in vector-tensor theories,

    C. Gao, Y. Lu, S. Yu and Y.G. Shen, “Black hole and cosmos wi th multiple horizons and multiple singularities in vector-tensor theories,” Ph ys. Rev. D 97, no. 10, 104013 (2018) doi:10.1103/PhysRevD.97.104013 [arXiv:1711.009 96 [gr-qc]]

    work page doi:10.1103/physrevd.97.104013 2018

  10. [10]

    Black holes coupled to a m assive dilaton,

    J.H. Horne and G.T. Horowitz, “Black holes coupled to a m assive dilaton,” Nucl. Phys. B 399, 169 (1993) doi:10.1016/0550-3213(93)90621-U [hep-th/9 210012]

    work page doi:10.1016/0550-3213(93)90621-u 1993

  11. [11]

    Nontopological Magnetic Monopoles and New Magnetically Charged Black Holes

    K.M. Lee and E.J. Weinberg, “Nontopological magnetic m onopoles and new magneti- cally charged black holes,” Phys. Rev. Lett. 73, 1203 (1994) doi:10.1103/PhysRevLett. 73.1203 [hep-th/9406021]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett 1994

  12. [12]

    Topology, entropy and W itten index of dilaton black holes,

    G.W. Gibbons and R.E. Kallosh, “Topology, entropy and W itten index of dilaton black holes,” Phys. Rev. D 51, 2839 (1995) doi:10.1103/PhysRevD.51.2839 [hep-th/9407 118]

    work page doi:10.1103/physrevd.51.2839 1995

  13. [13]

    A Classical inst ability of Reissner-Nordstrom solutions and the fate of magnetically charged black holes,

    K.M. Lee, V.P. Nair and E.J. Weinberg, “A Classical inst ability of Reissner-Nordstrom solutions and the fate of magnetically charged black holes, ” Phys. Rev. Lett. 68, 1100 (1992) doi:10.1103/PhysRevLett.68.1100 [hep-th/911104 5]

    work page doi:10.1103/physrevlett.68.1100 1992

  14. [14]

    Static Equilibria of Charged Particles Around Charged Black Holes: Chaos Bound and Its Violations

    Q.Q. Zhao, Y.Z. Li and H. L¨ u, “Static equilibria of char ged particles around charged black holes: chaos bound and its violations,” Phys. Rev. D 98, no. 12, 124001 (2018) doi:10.1103/PhysRevD.98.124001 [arXiv:1809.04616 [gr- qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.98.124001 2018

  15. [15]

    Black holes that repel,

    H. L¨ u, Z.L. Wang and Q.Q. Zhao, “Black holes that repel, ” Phys. Rev. D 99, no. 10, 101502 (2019) doi:10.1103/PhysRevD.99.101502 [arXiv:19 01.02894 [hep-th]]

    work page doi:10.1103/physrevd.99.101502 2019

  16. [16]

    Enthalpy and the Mechanics of AdS Black Holes

    D. Kastor, S. Ray and J. Traschen, “Enthalpy and the mech anics of AdS black holes,” Class. Quant. Grav. 26, 195011 (2009) doi:10.1088/0264-9381/26/19/195011 [arXiv:0904.2765 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/0264-9381/26/19/195011 2009

  17. [17]

    Black Hole Enthalpy and an Entropy Inequality for the Thermodynamic Volume

    M. Cvetiˇ c, G.W. Gibbons, D. Kubiznak and C.N. Pope, “Bl ack hole enthalpy and an entropy inequality for the thermodynamic volume,” Phys. Re v. D 84, 024037 (2011) doi:10.1103/PhysRevD.84.024037 [arXiv:1012.2888 [hep- th]]. 31

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.84.024037 2011

  18. [18]

    Hairy Black Holes and Null Circular Geodesics

    S. Hod, “Hairy black holes and null circular geodesics, ” Phys. Rev. D 84, 124030 (2011) doi:10.1103/PhysRevD.84.124030 [arXiv:1112.3286 [gr-q c]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.84.124030 2011

  19. [19]

    The fastest way to circle a black hole

    S. Hod, “The fastest way to circle a black hole,” Phys. Re v. D 84, 104024 (2011) doi:10.1103/PhysRevD.84.104024 [arXiv:1201.0068 [gr-q c]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.84.104024 2011

  20. [20]

    Photon Spheres and Sonic Horizons in Black Holes from Supergravity and Other Theories

    M. Cvetiˇ c, G.W. Gibbons and C.N. Pope, “Photon spheres and sonic horizons in black holes from supergravity and other theories,” Phys. Rev. D 94, no. 10, 106005 (2016) doi:10.1103/PhysRevD.94.106005 [arXiv:1608.02202 [gr- qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.94.106005 2016

  21. [21]

    The correspondence between sonic points of ideal photon gas accretion and photon spheres

    Y. Koga and T. Harada, “Correspondence between sonic po ints of ideal photon gas ac- cretion and photon spheres,” Phys. Rev. D 94, no. 4, 044053 (2016) doi: 10.1103/Phys- RevD.94.044053 [arXiv:1601.07290 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/phys- 2016

  22. [22]

    Duality transformations o f abelian and nonabelian gauge fields,

    S. Deser and C. Teitelboim, “Duality transformations o f abelian and nonabelian gauge fields,” Phys. Rev. D 13, 1592 (1976). doi:10.1103/PhysRevD.13.1592

    work page doi:10.1103/physrevd.13.1592 1976

  23. [23]

    Off-shell electromagnetic duality invarianc e,

    S. Deser, “Off-shell electromagnetic duality invarianc e,” J. Phys. A 15, 1053 (1982). doi:10.1088/0305-4470/15/3/039

    work page doi:10.1088/0305-4470/15/3/039 1982

  24. [24]

    Electric-mag netic black hole duality,

    S. Deser, M. Henneaux and C. Teitelboim, “Electric-mag netic black hole duality,” Phys. Rev. D 55, 826 (1997) doi:10.1103/PhysRevD.55.826 [hep-th/960718 2]

    work page doi:10.1103/physrevd.55.826 1997

  25. [25]

    Dualisation of Dualities, II: Twisted self-duality of doubled fields and superdualities

    E. Cremmer, B. Julia, H. L¨ u and C.N. Pope, “Dualization of dualities. 2. Twisted self-duality of doubled fields, and superdualities,” Nucl. Phys. B 535, 242 (1998) doi:10.1016/S0550-3213(98)00552-5 [hep-th/9806106]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/s0550-3213(98)00552-5 1998

  26. [26]

    Dyonic (A)dS Black Holes in Einstein-Born-Infeld Theory in Diverse Dimensions

    S. Li, H. L¨ u and H. Wei, “Dyonic (A)dS black holes in Eins tein-Born-Infeld the- ory in diverse dimensions,” JHEP 1607, 004 (2016) doi:10.1007/JHEP07(2016)004 [arXiv:1606.02733 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep07(2016)004 2016

  27. [27]

    Holographic com- plexity equals bulk action?

    A.R. Brown, D.A. Roberts, L. Susskind, B. Swingle and Y. Zhao, “Holographic com- plexity equals bulk action?” Phys. Rev. Lett. 116, no. 19, 191301 (2016), arXiv:1509. 07876 [hep-th]

    work page 2016

  28. [28]

    Complexity, action, and black holes

    A.R. Brown, D.A. Roberts, L. Susskind, B. Swingle and Y. Zhao, “Complexity, action, and black holes,” Phys. Rev. D 93, no. 8, 086006 (2016), arXiv:1512.04993 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  29. [29]

    Holographic Complexity Equals Which Action?

    K. Goto, H. Marrochio, R.C. Myers, L. Queimada and B. Yos hida, “Holographic com- plexity equals which action?” JHEP 1902, 160 (2019), arXiv:1901.00014 [hep-th]. 32

    work page internal anchor Pith review Pith/arXiv arXiv 1902

  30. [30]

    Action growth of dyonic black holes and electromagnetic duality,

    H.S. Liu and H. L¨ u, “Action growth of dyonic black holes and electromagnetic duality,” arXiv:1905.06409 [hep-th]

    work page arXiv 1905

  31. [31]

    Gravitational action with null boundaries

    L. Lehner, R.C. Myers, E. Poisson and R.D. Sorkin, “Grav itational action with null boundaries,” Phys. Rev. D 94, no. 8, 084046 (2016), arXiv:1609.00207 [hep-th]. 33

    work page internal anchor Pith review Pith/arXiv arXiv 2016