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Exact charged black hole solutions in a Lorentz-violating Kalb-Ramond gravity with both nonminimal curvature couplings, together with their thermodynamic and topological phase structure.

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2026-08-04 11:38 UTC pith:FFGKREY2

load-bearing objection Useful new charged KR black-hole solutions with both curvature couplings, but the 'exact solution' status rests on a fixed-background approximation that needs explicit validation. the 3 major comments →

arxiv 2608.02196 v1 pith:FFGKREY2 submitted 2026-08-03 gr-qc

Charged Black Holes with a Lorentz--Violating Kalb--Ramond Background

Jia-Hui Yang , Xin-Yu Guo , Jia-Zhou Liu , Yu-Xiao Liu This is my paper
classification gr-qc
keywords blackchargedbackgroundholekalb--ramondthermodynamictopologicalcurvature
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Lorentz symmetry is the rule that the laws of physics look the same in every direction. Some quantum-gravity ideas suggest it may break slightly, and a useful way to model that is with a background Kalb-Ramond field, an antisymmetric tensor that picks out directions in spacetime. In this paper the authors look for static, spherically symmetric black holes with electric charge in such a theory, with two different nonminimal couplings between the Kalb-Ramond field and spacetime curvature. They find three exact metric solutions: one with no cosmological constant and quadratic potential, one with cosmological constant under a special relation between the two couplings, and one with a linear potential tuned to the cosmological constant.

Technically, the Kalb-Ramond field is not evolved; it is fixed as an external environment. The coupling constants eta and lambda are chosen so the field equations close. With these assumptions, the black hole metric is Reissner-Nordstrom-like, with Lorentz-violating parameters l1 and l2 shifting the horizon, the charge term, and the asymptotic solid angle. The authors then use the Iyer-Wald formalism to find entropy and energy, identify the cosmological constant with pressure, and plot Joule-Thomson inversion curves. They also use Duan's topological current to assign winding numbers to the black hole branches.

The main quantitative result is that, after a rescaling, the equation of state is the same as the charged AdS black hole, so the famous ratio of minimum inversion temperature to critical temperature stays 1/2. The Lorentz-violating parameters change the individual temperatures and pressures, but not this dimensionless ratio. The paper therefore extends the catalog of exact black holes in Lorentz-violating gravity and gives a thermodyn

Core claim

The paper claims to obtain exact static, spherically symmetric electrically charged black hole solutions in a Kalb-Ramond modified gravity with both nonminimal curvature couplings, Eqs. (3.14), (3.23), (3.27), and to derive their Iyer-Wald thermodynamics, Joule-Thomson inversion curves, and topological phase structure. If correct, the metrics satisfy the gravitational and modified-Maxwell field equations (3.4)-(3.6) with B_mu_nu fixed to (2.6), and the critical temperature and pressure from the defect curve, Eqs. (5.28)-(5.30), match the equation-of-state values, Eq. (4.30).

Load-bearing premise

The Kalb-Ramond field is treated as a fixed external background and its independent dynamical equation is never imposed. The paper states this explicitly after Eq. (2.10): 'the Kalb-Ramond field is treated as a fixed external tensor background, its independent dynamical equation is not analyzed.' If the full B_mu_nu equation of motion is enforced, the ansatz (2.6) with H=0 and, especially, the Case C solution with V' not equal to 0 and tuned lambda need not survive; hence the 'exact solution' claim is load-bearing on this non-dynamical approximation.

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Editorial analysis

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Referee Report

3 major / 4 minor

Summary. The paper studies static, spherically symmetric electrically charged black holes in a gravitational theory with a Kalb–Ramond two-form background and two nonminimal curvature couplings. It presents three families of exact solutions (Cases A, B, C in Sec. 3), with and without a cosmological constant, and then applies the Iyer–Wald covariant phase-space formalism to obtain corrected thermodynamic quantities. The paper further analyzes the Joule–Thomson expansion for the AdS Case B, finding the inversion curve and the universal ratio T_i^{min}/T_c = 1/2, and uses a topological thermodynamic approach to characterize van der Waals-like phase transitions and to recover the critical pressure and temperature from defect curves. The central claim is that Eqs. (3.14), (3.23), and (3.27) are exact solutions of the modified-gravity system and that the derived thermodynamic and topological properties follow from them.

Significance. If the solutions are genuine stationary points of the full action, the paper provides a charged extension of recent KR-gravity black hole solutions and a useful systematic account of their thermodynamics and phase structure. The paper is transparent in declaring after Eq. (2.10) that the Kalb–Ramond field is treated as a fixed external background and its independent dynamical equation is not analyzed. The thermodynamic derivation and the invariance of T_i^{min}/T_c are internally consistent within that declared approximation. The main value of the paper is therefore conditional: it is a complete thermodynamic analysis of a family of background-approximation solutions, but the claim that these are exact solutions of the action (2.1) requires an additional check of the B-field equation. The topological recovery of the critical point in Sec. 5.3 is best viewed as a consistency check rather than an independent derivation.

major comments (3)
  1. [Sec. 2 after Eq. (2.10); Sec. 3, Eqs. (3.14), (3.23), (3.27)] The paper does not impose the independent Kalb–Ramond field equation. The field equations (3.4)–(3.6) are obtained by varying only g and A with B inserted by hand, as explicitly stated after Eq. (2.10). For Cases A and B, V=V'=0 while the curvature is r-dependent (e.g., R in Eq. (3.21) is O(r^{-2})); the algebraic B_{mn} equation of motion then requires a nontrivial cancellation of curvature and electromagnetic terms for all r. No such cancellation is demonstrated. Thus Eqs. (3.14), (3.23), (3.27) are not yet established as solutions of the full action (2.1). This also affects the Iyer–Wald first law (4.14)–(4.15), which varies only g and A and may omit a B-sector charge. The manuscript should either solve the B-field equation, or consistently rephrase all claims as solutions of the truncated background system.
  2. [Sec. 5.3, Eqs. (5.19)–(5.30)] The claimed 'recovery' of the critical temperature and pressure from the defect curve is not an independent result. The defect curve (5.19) is constructed from the same mass formula and equation of state used in Sec. 4 (Eqs. (4.25)–(4.28)), so the agreement between Eqs. (5.28)–(5.30) and Eq. (4.30) is an internal algebraic consistency check, not a test of the thermodynamic model. This should be stated explicitly in the text; as written, Sec. 5.3 may be read as deriving new information from the topological construction.
  3. [Sec. 3, Eqs. (3.4)–(3.6), (3.15), (3.24), (3.26)] The exactness and scope of the solutions are difficult to audit. The field equations are stated and the final metrics are reported after 'substitution' without intermediate algebra, and the parameter restrictions (3.15), (3.24), and (3.26) are introduced as conditions without derivation. In particular, Case B restricts to l2 = -4l1, which is a codimension-one slice of the two-coupling parameter space. I request either an appendix or a supplementary notebook that verifies the vanishing of (3.4)–(3.6) for at least one case and explains whether the restrictions are necessary or are adopted only for simplicity.
minor comments (4)
  1. [Eq. (3.2) and Eq. (3.14)] The displayed formula for the pseudo-electric component tilde{E}(r) is garbled in the typesetting; the passage from the normalization condition to the constant value sqrt(2)/2 |b| in Eq. (3.14) should be written explicitly.
  2. [Sec. 4.1] The phrase 'In this chapter' should read 'In this section'. Also, the symbol lambda is used both as the potential coupling in Cases A/B and as the Lagrange multiplier in Case C; a brief clarification would avoid confusion.
  3. [Sec. 3, Eq. (3.7)] The bound (3.7) is quoted for a 'generic dimensionless Lorentz-violating parameter l', but the paper later varies l1 and l2 independently with different ranges. Please state the assumed sign range separately for l1 and l2.
  4. [Figs. 4 and 5] The captions of Figs. 4 and 5 mention relative deviation curves and continuous evolution of horizon radii, but the described quantitative deviation curves are not clearly visible in the text. Please check that the figures match the captions or clarify the plotted quantities.

Axiom & Free-Parameter Ledger

4 free parameters · 6 axioms · 0 invented entities

The central results rest on the effective-background approximation (B fixed), vacuum conditions for the potential, and a set of fine-tuned coupling relations. Beyond standard Einstein-Maxwell calculus and the cited Iyer-Wald/topological frameworks, the main burden is the un-imposed KR field equation and the l2 = -4l1 / eta / lambda constraints.

free parameters (4)
  • l1 = b^2 xi1 = not fitted; input coupling
    Dimensionless Lorentz-violating coupling entering all metrics and thermodynamic quantities; assumed small with bound (3.7).
  • l2 = b^2 xi2 = not fitted; input coupling
    Dimensionless Lorentz-violating coupling; independent in Cases A and C, but fixed to l2 = -4l1 in Case B.
  • eta = l2/(4 b^2 (1-l1)) in Cases A/C; -l1/(b^2(1-l1)) in Case B
    KR-Maxwell coupling chosen to satisfy the field equations; no independent measurement; central to the charged solution.
  • lambda (Case C) = (4 xi1 + xi2) Lambda/(1 - l1 - l2/2)
    Potential slope / Lagrange multiplier tuned in Eq. (3.26) to admit the closed-form AdS solution.
axioms (6)
  • domain assumption The KR field B_mu_nu is frozen to the fixed background (2.6) while its own Euler-Lagrange equation is ignored.
    Stated after Eq. (2.10): 'the Kalb-Ramond field is treated as a fixed external tensor background, its independent dynamical equation is not analyzed.' This converts 'exact solution of the full action' into 'solution in an effective-background approximation'.
  • domain assumption The vacuum conditions V=0 and V'=0 hold for Cases A and B (quadratic potential evaluated at X=0).
    Used to eliminate V and V' from field equations (3.4)-(3.6); for Case C, V' is nonzero and lambda is instead tuned to (3.26).
  • domain assumption The KR ansatz (2.6) with H_mu_nu_rho = 0 and A_mu = -Phi(r) delta_mu^t is a consistent truncation.
    Sections 2-3; the KR field strength vanishes and only the algebraic eta B^2 F^2 matter coupling remains.
  • standard math The Iyer-Wald covariant phase-space formalism applies with B_mu_nu not varied.
    Section 4.1; standard differential-geometric construction, but treating B as non-dynamical may affect the Noether charge and entropy.
  • standard math Duan's Phi-mapping topological current theory as reviewed in Sec. 5.1 is applicable.
    Prior framework used to classify black-hole phase transitions; cited as Refs. [98,99].
  • ad hoc to paper In Case B, the two nonminimal couplings are restricted to l2 = -4l1 (xi2 = -4 xi1), and eta is fixed by (3.24).
    Eq. (3.23); this fine-tuning reduces the two-coupling model to a one-parameter slice and is required for the stated closed-form solution.

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Pith. "Pith review of Charged Black Holes with a Lorentz--Violating Kalb--Ramond Background." pith.science (2026). https://pith.science/paper/FFGKREY2

@misc{pith2026260802196,
  author       = {Pith},
  title        = {Pith review of: Charged Black Holes with a Lorentz--Violating Kalb--Ramond Background},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FFGKREY2}},
  note         = {Machine review of arXiv:2608.02196}
}
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We investigate exact static, spherically symmetric electrically charged black hole solutions in a gravitational theory with spontaneous Lorentz-symmetry breaking induced by a background Kalb--Ramond field. In contrast to previous analyses that retained only one nonminimal curvature coupling, we include the combined effects of the two independent nonminimal curvature couplings and obtain charged black hole solutions both with and without a cosmological constant. Using the Iyer--Wald covariant phase-space formalism, we derive the corrected thermodynamic quantities and analyze the Joule--Thomson expansion, including the inversion curve and the cooling/heating regions. We further apply the topological approach to black hole thermodynamics to characterize the van der Waals-like phase transition and show how the thermodynamic critical temperature and pressure are encoded in the corresponding topological defect curve. These results clarify the thermodynamic and topological signatures of electrically charged black holes in gravity with a Lorentz-violating Kalb--Ramond background.

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Works this paper leans on

99 extracted references · 11 linked inside Pith

  1. [1]

    Kostelecký and S

    V.A. Kostelecký and S. Samuel, Spontaneous breaking of lorentz symmetry in string theory , Phys. Rev. D 39 (1989) 683

    1989

  2. [2]

    Alfaro, H.A

    J. Alfaro, H.A. Morales-Técotl and L.F. Urrutia, Loop quantum gravity and light propagation , Phys. Rev. D 65 (2002) 103509

    2002

  3. [3]

    Hoava, General covariance in gravity at a Lifshitz point , Classical and Quantum Gravity 28 (2011) 114012

    P. Hoava, General covariance in gravity at a Lifshitz point , Classical and Quantum Gravity 28 (2011) 114012

    2011

  4. [4]

    Carroll, J.A

    S.M. Carroll, J.A. Harvey, V.A. Kostelecký, C.D. Lane and T. Okamoto, Noncommutative Field Theory and Lorentz Violation , Phys. Rev. Lett. 87 (2001) 141601

    2001

  5. [5]

    Kostelecký, Gravity, Lorentz violation, and the standard model , Phys

    V.A. Kostelecký, Gravity, Lorentz violation, and the standard model , Phys. Rev. D 69 (2004) 105009

    2004

  6. [6]

    J. Liu, S. Wu, S. Wei et al., Exact black hole solutions in bumblebee gravity with lightlike or spacelike VEVs, Sci. China Phys. Mech. Astron. 69 (2026) 270411

    2026

  7. [7]

    Lai, Y.-Q

    X.-B. Lai, Y.-Q. Dong, Y.-Z. Fan and Y.-X. Liu, Stability analysis of cosmological perturbations in the bumblebee model: Parameter constraints and gravitational waves , Phys. Rev. D 113 (2026) 044003

    2026

  8. [8]

    Li, J.-Z

    B.-R. Li, J.-Z. Liu, W.-D. Guo and Y.-X. Liu, Quasinormal modes of a charged spherically symmetric black hole in bumblebee gravity , 2510.20503

    arXiv

  9. [9]

    Priyobarta Singh, I

    Y. Priyobarta Singh, I. Roshila Devi and T. Ibungochouba Singh, Quasinormal modes of spherically symmetric black hole with cosmological constant and global monopole in bumblebee gravity, Nuclear Physics B 1018 (2025) 117006

    2025

  10. [10]

    H.-F. Liu, W. Liu, Y.-X. Liu, Q. Su and D.-f. Zeng, Gravitational–Bumblebee perturbations: Exact decoupling and isospectrality , 2605.02820

    Pith/arXiv arXiv

  11. [11]

    Wentao, F

    L. Wentao, F. Xiongjun, J. Jiliang and W. Jieci, Lorentz violation induces isospectrality breaking in Einstein–bumblebee gravity theory , Sci. China Phys. Mech. Astron. 67 (2024) 280413

    2024

  12. [12]

    W. Deng, W. Liu, F. Long, K. Xiao and J. Jing, Quasinormal modes of a massive scalar field in slowly rotating Einstein–Bumblebee black holes , Journal of Cosmology and Astroparticle Physics 2025 (2025) 028

    2025

  13. [13]

    X. Liu, W. Liu, Z. Liu et al., Harvesting correlations from BTZ black hole coupled to a Lorentz-violating vector field , J. High Energy Phys. 2025 (2025) 94 . – 22 –

    2025

  14. [14]

    Y. Tang, W. Liu and J. Wang, Observational signature of Lorentz violation in acceleration radiation, Eur. Phys. J. C 85 (2025) 1108

    2025

  15. [15]

    W. Liu, X. Fang, J. Jing et al., QNMs of slowly rotating Einstein–Bumblebee black hole , Eur. Phys. J. C 83 (2023) 83

    2023

  16. [16]

    J. Liu, W. Guo, S. Wei et al., Charged spherically symmetric and slowly rotating charged black hole solutions in bumblebee gravity , Eur. Phys. J. C 85 (2025) 145

    2025

  17. [18]

    Atamurotov, D

    F. Atamurotov, D. Ortiqboev, A. Abdujabbarov and G. Mustafa, Particle dynamics and gravitational weak lensing around black hole in the Kalb-Ramond gravity , Eur. Phys. J. C 82 (2022) 659

    2022

  18. [19]

    Duan, J.Y

    Z.Q. Duan, J.Y. Zhao and K. Yang, Electrically charged black holes in gravity with a background Kalb–Ramond field , Eur. Phys. J. C 84 (2024) 798

    2024

  19. [20]

    Lin, J.-Z

    Y.-X. Lin, J.-Z. Liu and Y.-X. Liu, Dyonic Black Holes in Lorentz–Violating Gravity with a Background Kalb–Ramond Field , 2605.18371

    Pith/arXiv arXiv

  20. [21]

    W. Liu, D. Wu and J. Wang, Static neutral black holes in Kalb–Ramond gravity , Journal of Cosmology and Astroparticle Physics 2024 (2024) 017

    2024

  21. [22]

    Lessa, R

    L. Lessa, R. Oliveira, J. Silva and C. Almeida, Traversable wormhole solution with a background KalbRamond field , Annals of Physics 433 (2021) 168604

    2021

  22. [23]

    Maluf and C

    R. Maluf and C. Muniz, Exact solution for a traversable wormhole in a curvature-coupled antisymmetric background field , Eur. Phys. J. C 82 (2022) 445 [ 2110.12202]

    Pith/arXiv arXiv 2022

  23. [24]

    Araújo Filho, J

    A. Araújo Filho, J. Reis and H. Hassanabadi, Exploring antisymmetric tensor effects on black hole shadows and quasinormal frequencies , Journal of Cosmology and Astroparticle Physics 2024 (2024) 029

    2024

  24. [25]

    Gu, W.-D

    Y.-T. Gu, W.-D. Guo and Y.-X. LIU, Quasinormal modes of an electrically charged Kalb–Ramond black hole , Chinese Physics C (2026)

    2026

  25. [26]

    Zhong-Wu, L

    X. Zhong-Wu, L. Sheng, G. Huajie, P. Qiyuan and J. Jiliang, Scalar perturbation around a rotating Kalb–Ramond BTZ black hole , Sci. China Phys. Mech. Astron. 69 (2026) 260411

    2026

  26. [27]

    W. Deng, W. Liu, K. Xiao et al., Quasinormal modes of scalar, electromagnetic, and gravitational perturbations in slowly rotating Kalb–Ramond black holes , Eur. Phys. J. C 86 (2026) 232

    2026

  27. [28]

    W. Liu, D. Wu and J. Wang, Shadow of slowly rotating Kalb–Ramond black holes , Journal of Cosmology and Astroparticle Physics 2025 (2025) 017

    2025

  28. [29]

    Liu, S.-P

    J.-Z. Liu, S.-P. Wu, S.-W. Wei and Y.-X. Liu, Exact black hole solutions in gravity with a background Kalb-Ramond field , Journal of Cosmology and Astroparticle Physics 2025 (2025) 056

    2025

  29. [30]

    Bekenstein, Nonexistence of Baryon Number for Static Black Holes , Phys

    J.D. Bekenstein, Nonexistence of Baryon Number for Static Black Holes , Phys. Rev. D 5 (1972) 1239

    1972

  30. [31]

    Bekenstein, Black holes and entropy , Phys

    J.D. Bekenstein, Black holes and entropy , Phys. Rev. D 7 (1973) 2333. – 23 –

    1973

  31. [32]

    Bekenstein, Generalized second law of thermodynamics , Phys

    J.D. Bekenstein, Generalized second law of thermodynamics , Phys. Rev. D 9 (1974) 3292

    1974

  32. [33]

    Hawking, Black Hole Explosions? , Nature 248 (1974) 30

    S.W. Hawking, Black Hole Explosions? , Nature 248 (1974) 30

    1974

  33. [34]

    Hawking, Particle Creation by Black Holes , Commun

    S.W. Hawking, Particle Creation by Black Holes , Commun. Math. Phys. 43 (1975) 199

    1975

  34. [35]

    Davies, Thermodynamic phase transitions of Kerr-Newman black holes in de Sitter space, Classical and Quantum Gravity 6 (1989) 1909

    P.C.W. Davies, Thermodynamic phase transitions of Kerr-Newman black holes in de Sitter space, Classical and Quantum Gravity 6 (1989) 1909

    1989

  35. [36]

    Hawking and D.N

    S.W. Hawking and D.N. Page, Thermodynamics of Black Holes in anti-De Sitter Space , Commun. Math. Phys. 87 (1983) 577

    1983

  36. [37]

    Curir, Rotating black holes as dissipative spin-thermodynamical systems , Gen

    A. Curir, Rotating black holes as dissipative spin-thermodynamical systems , Gen. Rel. Grav. 13 (1981) 417

    1981

  37. [38]

    Curir, Black hole emissions and phase transitions , Gen

    A. Curir, Black hole emissions and phase transitions , Gen. Rel. Grav. 13 (1981) 1177

    1981

  38. [39]

    Pavón and J.M

    D. Pavón and J.M. Rubí, Nonequilibrium thermodynamic fluctuations of black holes , Phys. Rev. D 37 (1988) 2052

    1988

  39. [40]

    Pavón, Phase transition in Reissner-Nordström black holes , Phys

    D. Pavón, Phase transition in Reissner-Nordström black holes , Phys. Rev. D 43 (1991) 2495

    1991

  40. [41]

    Kaburaki, Critical behavior of extremal Kerr-Newman black holes , Gen

    O. Kaburaki, Critical behavior of extremal Kerr-Newman black holes , Gen. Rel. Grav. 28 (1996) 843

    1996

  41. [42]

    Cai, Z.-J

    R.-G. Cai, Z.-J. Lu and Y.-Z. Zhang, Critical behavior in (2+1)-dimensional black holes , Phys. Rev. D 55 (1997) 853

    1997

  42. [43]

    Cai and J.-H

    R.-G. Cai and J.-H. Cho, Thermodynamic curvature of the BTZ black hole , Phys. Rev. D 60 (1999) 067502

    1999

  43. [44]

    Wei, Thermodynamic critical and geometrical properties of charged BTZ black hole , Phys

    Y.-H. Wei, Thermodynamic critical and geometrical properties of charged BTZ black hole , Phys. Rev. D 80 (2009) 024029

    2009

  44. [45]

    Bhattacharya, S

    K. Bhattacharya, S. Dey, B.R. Majhi and S. Samanta, General framework to study the extremal phase transition of black holes , Phys. Rev. D 99 (2019) 124047

    2019

  45. [46]

    Kastor, S

    D. Kastor, S. Ray and J. Traschen, Enthalpy and the mechanics of AdS black holes , Classical and Quantum Gravity 26 (2009) 195011

    2009

  46. [47]

    Dolan, The cosmological constant and the black hole equation of state , Class

    B.P. Dolan, The cosmological constant and the black hole equation of state , Class. Quant. Grav. 28 (2011) 125020 [1008.5023]

    Pith/arXiv arXiv 2011

  47. [48]

    Dolan, Pressure and volume in the first law of black hole thermodynamics , Classical and Quantum Gravity 28 (2011) 235017

    B.P. Dolan, Pressure and volume in the first law of black hole thermodynamics , Classical and Quantum Gravity 28 (2011) 235017

    2011

  48. [49]

    Dolan, Compressibility of rotating black holes , Phys

    B.P. Dolan, Compressibility of rotating black holes , Phys. Rev. D 84 (2011) 127503

    2011

  49. [50]

    Dolan, Where is the PdV in the First Law of Black Hole Thermodynamics? , in Open Questions in Cosmology , G.J

    B.P. Dolan, Where is the PdV in the First Law of Black Hole Thermodynamics? , in Open Questions in Cosmology , G.J. Olmo, ed., (London), IntechOpen (2012), DOI

    2012

  50. [51]

    Kubiznak and R.B

    D. Kubiznak and R.B. Mann, P-V criticality of charged AdS black holes , JHEP 07 (2012) 033 [1205.0559]

    Pith/arXiv arXiv 2012

  51. [52]

    Kubizák, R.B

    D. Kubizák, R.B. Mann and M. Teo, Black hole chemistry: thermodynamics with Lambda , Classical and Quantum Gravity 34 (2017) 063001

    2017

  52. [53]

    Bhattacharya, B.R

    K. Bhattacharya, B.R. Majhi and S. Samanta, van der Waals criticality in AdS black holes: A phenomenological study, Phys. Rev. D 96 (2017) 084037 . – 24 –

    2017

  53. [54]

    Wei, Y.-X

    S.-W. Wei, Y.-X. Liu and R.B. Mann, Black Hole Solutions as Topological Thermodynamic Defects, Phys. Rev. Lett. 129 (2022) 191101 [2208.01932]

    Pith/arXiv arXiv 2022

  54. [55]

    Wei, Y.-X

    S.-W. Wei, Y.-X. Liu and R.B. Mann, Universal topological classifications of black hole thermodynamics, Phys. Rev. D 110 (2024) L081501 [2409.09333]

    Pith/arXiv arXiv 2024

  55. [56]

    Wei and Y.-X

    S.-W. Wei and Y.-X. Liu, Topology of black hole thermodynamics: A brief review , Sci. China Phys. Mech. Astron. 69 (2026) 260401 [ 2605.00037]

    Pith/arXiv arXiv 2026

  56. [57]

    S. Yang, S. Wu, S. Wei et al., Deciphering black hole phase transitions through photon spheres , Sci. China Phys. Mech. Astron. 68 (2025) 120412

    2025

  57. [58]

    Chen and S

    Z. Chen and S. Wei, Thermodynamical topology with multiple defect curves for dyonic AdS black holes , Eur. Phys. J. C 84 (2024) 1294

    2024

  58. [59]

    X.-D. Zhu, D. Wu and D. Wen, Topological classes of thermodynamics of the rotating charged AdS black holes in gauged supergravities , Physics Letters B 856 (2024) 138919

    2024

  59. [60]

    D. Wu, W. Liu, S.-Q. Wu and R.B. Mann, Novel topological classes in black hole thermodynamics, Phys. Rev. D 111 (2025) L061501

    2025

  60. [61]

    Wang and Y.-Z

    H. Wang and Y.-Z. Du, Topology of charged AdS black hole in restricted phase space* , Chinese Physics C 48 (2024) 095109

    2024

  61. [62]

    Sadeghi, S.N

    J. Sadeghi, S.N. Gashti, M.R. Alipour and M.A.S. Afshar, Thermodynamic topology of quantum corrected AdS-Reissner-Nordstrom black holes in Kiselev spacetime , Chinese Physics C 48 (2024) 115115

    2024

  62. [63]

    Wu and S.-W

    S.-P. Wu and S.-W. Wei, Thermodynamical topology of quantum BTZ black hole , Phys. Rev. D 110 (2024) 024054

    2024

  63. [64]

    Ökcü and E

    O. Ökcü and E. Aydıner, Joule–Thomson expansion of the charged AdS black holes , Eur. Phys. J. C 77 (2017) 24

    2017

  64. [65]

    Ahmed and E

    F. Ahmed and E. Silva, Thermodynamic geometry of charged AdS black holes with a string cloud in Lorentz-violating Einstein–Kalb–Ramond gravity , Eur. Phys. J. Plus 141 (2026) 709

    2026

  65. [66]

    Lekbich, A

    H. Lekbich, A. El Boukili, N. Mansour and M. Sedra, 4D AdS Einstein–Gauss–Bonnet black hole endowed with Lorentzian noncommutativity: P–V criticality, Joule–Thomson expansion, and shadow , Annals of Physics 458 (2023) 169451

    2023

  66. [67]

    Mo, G.-Q

    J.-X. Mo, G.-Q. Li, S.-Q. Lan and X.-B. Xu, Joule–Thomson expansion of d-dimensional charged AdS black holes , Phys. Rev. D 98 (2018) 124032

    2018

  67. [68]

    Media and T.I

    N. Media and T.I. Singh, Joule–Thomson Expansion of Kerr-Newman-de Sitter Black Hole Under Lorentz Violation Theory , International Journal of Theoretical Physics 64 (2025) 82

    2025

  68. [69]

    Altschul, Q.G

    B. Altschul, Q.G. Bailey and V.A. Kostelecký, Lorentz violation with an antisymmetric tensor , Phys. Rev. D 81 (2010) 065028

    2010

  69. [70]

    Higashijima and N

    K. Higashijima and N. Yokoi, Spontaneous Lorentz symmetry breaking by an antisymmetric tensor field , Phys. Rev. D 64 (2001) 025004

    2001

  70. [71]

    Maluf, A.A

    R.V. Maluf, A.A. Araújo Filho, W.T. Cruz and C.A.S. Almeida, Antisymmetric tensor propagator with spontaneous Lorentz violation , Europhysics Letters 124 (2019) 61001 . – 25 –

    2019

  71. [72]

    Lessa, J

    L. Lessa, J. Silva, R. Maluf and C. Almeida, Modified black hole solution with a background Kalb-Ramond field, Eur. Phys. J. C 80 (2020) 335 [ 1911.10296]

    Pith/arXiv arXiv 2020

  72. [73]

    Majumdar and S

    P. Majumdar and S. SenGupta, Parity-violating gravitational coupling of electromagnetic fields, Classical and Quantum Gravity 16 (1999) L89

    1999

  73. [74]

    Yang, Y.-Z

    K. Yang, Y.-Z. Chen, Z.-Q. Duan and J.-Y. Zhao, Static and spherically symmetric black holes in gravity with a background Kalb-Ramond field , Phys. Rev. D 108 (2023) 124004

    2023

  74. [75]

    Kostelecký and S

    V.A. Kostelecký and S. Samuel, Gravitational phenomenology in higher-dimensional theories and strings , Phys. Rev. D 40 (1989) 1886

    1989

  75. [76]

    Xiao, Y.-X

    Y. Xiao, Y.-X. Liu, Y. Tian and H. Zhang, Explicit and covariant formula for thermodynamic volume in extended black hole thermodynamics , 2512.01916

    Pith/arXiv arXiv

  76. [77]

    Y. Xiao, Y. Tian and Y.-X. Liu, Extended Black Hole Thermodynamics from Extended Iyer-Wald Formalism, Phys. Rev. Lett. 132 (2024) 021401

    2024

  77. [78]

    Liu, S.-P

    J.-Z. Liu, S.-P. Wu, S.-W. Wei and Y.-X. Liu, Black Hole Entropy Beyond the Wald Term in Nonminimally Coupled Gravity: A Covariant Phase Space Decomposition , 2605.22429

    Pith/arXiv arXiv

  78. [79]

    An, Notes on thermodynamics of Schwarzschild–like bumblebee black hole , Physics of the Dark Universe 45 (2024) 101520

    Y.-S. An, Notes on thermodynamics of Schwarzschild–like bumblebee black hole , Physics of the Dark Universe 45 (2024) 101520

    2024

  79. [80]

    P. Hu, L. Ma, H. Lü et al., Improved Reall–Santos method for AdS black holes in general 4-derivative gravities , Sci. China Phys. Mech. Astron. 67 (2024) 280412

    2024

  80. [81]

    Wu, Y.-X

    S.-P. Wu, Y.-X. Liu and S.-W. Wei, Generalized free energy landscapes from Iyer–Wald formalism, Physics of the Dark Universe 51 (2026) 102210

    2026

Showing first 80 references.