arxiv: 2604.08629 · v1 · submitted 2026-04-09 · 🌀 gr-qc

Recognition: unknown

Thermodynamics and phase transitions of kappa-deformed Schwarzschild-AdS black holes

A. Naveena Kumara , Vishnu Rajagopal , Puxun Wu

Authors on Pith no claims yet

Pith reviewed 2026-05-10 16:42 UTC · model grok-4.3

classification 🌀 gr-qc

keywords black hole thermodynamicskappa deformationphase transitionsSchwarzschild-AdSextended phase spacecritical phenomenanon-commutative geometry

0 comments

The pith

κ-deformation induces critical behavior and phase transitions in uncharged Schwarzschild-AdS black holes

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

The paper constructs an effective metric for the Schwarzschild-AdS black hole by starting from a κ-deformed Newtonian potential in non-commutative geometry. Treating the deformation parameter κ itself as a thermodynamic variable with an identifiable conjugate potential yields a modified first law and Smarr relation. The resulting thermodynamics shows that this deformation alone produces critical points and phase transitions in an otherwise uncharged black hole, with a universal critical ratio Pc vc/Tc approximately 0.37 that matches the van der Waals value closely. The Gibbs free energy versus temperature plot displays a distinctive double-loop structure rather than the usual swallow-tail shape.

Core claim

κ-deformation induces critical behaviour and phase transitions in an uncharged Schwarzschild-AdS black hole, with a critical ratio Pc vc/Tc ≃ 0.370 that is independent of the deformation parameter and close to the Van der Waals value. The G-T curve exhibits a peculiar double-loop structure, deviating from the standard swallow-tail behaviour associated with a first-order phase transition.

What carries the argument

The effective κ-deformed Schwarzschild-AdS metric constructed from the κ-deformed Newtonian potential, with κ promoted to an additional thermodynamic variable whose conjugate potential is fixed by consistency with the first law.

If this is right

Uncharged black holes acquire a van der Waals-like phase structure solely through the deformation.
The critical ratio remains fixed near 0.37 for any value of the deformation parameter.
The Gibbs free energy develops a double-loop form instead of the conventional swallow-tail shape.
Phase transitions appear in the extended phase space even though the black hole carries no electric charge.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Non-commutative deformations may act as a geometric substitute for charge in generating black-hole phase transitions.
The same construction could be applied to other AdS black-hole families to check whether similar critical ratios emerge.
Analog condensed-matter or fluid systems with controlled non-commutativity might be engineered to reproduce the double-loop feature.

Load-bearing premise

The effective metric obtained from the κ-deformed Newtonian potential is a faithful description of the black-hole geometry that preserves the symmetries needed for a consistent thermodynamic identification of the conjugate to κ.

What would settle it

Explicit computation of the equation of state showing that the critical ratio Pc vc/Tc changes when the value of κ is varied, or numerical plots of the Gibbs free energy lacking the reported double-loop structure.

Figures

Figures reproduced from arXiv: 2604.08629 by A. Naveena Kumara, Puxun Wu, Vishnu Rajagopal.

**Figure 2.** Figure 2: FIG. 2: Plot of [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3: Plot of [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5: Plot of [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗

read the original abstract

We investigate the thermodynamics of the Schwarzschild-AdS black hole in the framework of \(\kappa\)-deformed non-commutative geometry by constructing an effective \(\kappa\)-deformed Schwarzschild-AdS metric from the \(\kappa\)-deformed Newtonian potential. In the extended phase space, we derive a modified first law and the corresponding Smarr relation by treating the \(\kappa\)-deformation parameter as an additional thermodynamic variable and identifying its conjugate potential. Our analysis shows that \(\kappa\)-deformation induces critical behaviour and phase transitions in an uncharged Schwarzschild-AdS black hole, with a critical ratio \(P_c v_c/T_c \simeq 0.370\) that is independent of the deformation parameter and close to the Van der Waals value. Interestingly, the \(G - T\) curve exhibits a peculiar double-loop structure, deviating from the standard swallow-tail behaviour associated with a first-order phase transition.

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 claims κ-deformation forces van der Waals-like transitions into the uncharged Schwarzschild-AdS case with a κ-independent critical ratio near 0.37 and a double-loop G-T curve, but the effective metric from the Newtonian potential is the weakest link.

read the letter

The main takeaway is that a κ-deformation appears to induce critical behavior and phase transitions in the uncharged Schwarzschild-AdS black hole. They report a critical ratio Pc vc/Tc around 0.37 that stays fixed regardless of the deformation strength, plus a double-loop structure in the Gibbs free energy versus temperature plot instead of the usual swallowtail. This extends the fluid analogy into non-commutative gravity models in a concrete way. The paper does a solid job of treating κ as an extra thermodynamic variable, writing down the modified first law and Smarr relation, and then mapping out the critical points and phase structure from the effective metric. Those steps are laid out clearly enough to follow. The soft spot is the starting metric. It comes from tweaking the Newtonian potential rather than solving the deformed field equations, so it is not obvious that the geometry keeps the right AdS asymptotics or satisfies the curvature conditions needed for consistent thermodynamics. If that construction introduces artifacts, the κ-independence of the ratio and the double-loop feature could be tied to the specific choice rather than a genuine feature of the deformation. The stress-test concern about boundary conditions lands on the paper and should be checked directly. This work is aimed at people already working on black hole thermodynamics in deformed or non-commutative settings. A reader who wants to see how the extended phase space behaves under κ-deformation will get something concrete to test. It deserves a serious referee because the claims are specific and the calculations can be verified or falsified, even if the metric foundation needs extra scrutiny. I would send it to peer review with a note to focus on the validity of the effective geometry and the thermodynamic consistency.

Referee Report

3 major / 1 minor

Summary. The manuscript constructs an effective κ-deformed Schwarzschild-AdS black hole metric by modifying the standard solution with a κ-deformed Newtonian potential. In the extended phase space, it treats the deformation parameter κ as an additional thermodynamic variable with an identified conjugate potential, derives a modified first law and Smarr relation, and analyzes the resulting thermodynamics, finding that the deformation induces critical behavior and phase transitions with a critical ratio P_c v_c / T_c ≈ 0.370 independent of κ, and a double-loop structure in the Gibbs free energy versus temperature diagram.

Significance. Should the construction of the effective metric prove consistent with the requirements of black hole thermodynamics in AdS space, the results would indicate that non-commutative κ-deformations can trigger van der Waals-like phase transitions even in uncharged black holes. The reported critical ratio being close to the van der Waals value of 0.375 and independent of the deformation parameter would highlight a robust feature potentially arising from the deformation framework.

major comments (3)

[Metric construction (as per abstract)] The effective κ-deformed Schwarzschild-AdS metric is built from the κ-deformed Newtonian potential, which is valid only in the weak-field limit. This raises a concern for the strong-field geometry required for black hole thermodynamics: the paper must demonstrate that the resulting metric preserves AdS asymptotics and satisfies the necessary curvature conditions, as no derivation from κ-deformed Einstein equations is mentioned.
[Thermodynamic derivations] The identification of the conjugate potential to κ and the derivation of the modified first law and Smarr relation need to be shown explicitly, including all steps, to confirm they are consistent and not circular with the metric construction choices.
[Critical behavior analysis] The claim that P_c v_c / T_c ≃ 0.370 is independent of the deformation parameter κ is central; explicit calculations showing how this ratio emerges from the equations of state and remains constant across κ values are required to rule out it being an artifact of the effective metric.

minor comments (1)

[Abstract] The abstract mentions derivations but provides no key equations or error estimates; including at least the form of the modified first law would improve clarity.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We appreciate the opportunity to clarify the effective nature of our construction and to provide more explicit details on the derivations and critical point analysis. We address each major comment below and have revised the manuscript accordingly.

read point-by-point responses

Referee: The effective κ-deformed Schwarzschild-AdS metric is built from the κ-deformed Newtonian potential, which is valid only in the weak-field limit. This raises a concern for the strong-field geometry required for black hole thermodynamics: the paper must demonstrate that the resulting metric preserves AdS asymptotics and satisfies the necessary curvature conditions, as no derivation from κ-deformed Einstein equations is mentioned.

Authors: We acknowledge that the metric is an effective construction based on the κ-deformed Newtonian potential in the weak-field regime, rather than a solution derived from κ-deformed Einstein equations. This phenomenological approach is standard for incorporating non-commutative deformations into black hole geometries. In the revised manuscript, we have added explicit verification in Section 2: the metric is expanded at large r to confirm it recovers the standard Schwarzschild-AdS asymptotics (g_tt → −(1 + r²/l²)). We have also computed the leading curvature invariants (Ricci scalar and Kretschmann scalar) to verify consistency with the AdS background. A full derivation from modified field equations lies outside the scope of this effective model and is noted as a limitation for future work. revision: partial
Referee: The identification of the conjugate potential to κ and the derivation of the modified first law and Smarr relation need to be shown explicitly, including all steps, to confirm they are consistent and not circular with the metric construction choices.

Authors: We have expanded Section 3 with the complete step-by-step derivations. Starting from the horizon condition and the expression for the mass M(r_+, P, κ), we compute the conjugate potential Φ_κ = (∂M/∂κ)_{S,P} explicitly. The modified first law dM = T dS + V dP + Φ_κ dκ follows directly from the differential of M, and the Smarr relation is obtained by applying Euler's theorem to the scaling properties of the thermodynamic variables. These steps are independent of the metric construction details beyond the horizon thermodynamics and are presented without circular reasoning. revision: yes
Referee: The claim that P_c v_c / T_c ≃ 0.370 is independent of the deformation parameter κ is central; explicit calculations showing how this ratio emerges from the equations of state and remains constant across κ values are required to rule out it being an artifact of the effective metric.

Authors: We agree that explicit demonstration is essential. The revised manuscript includes a new Appendix A containing the full analytical derivation. The equation of state is obtained from the metric, and the critical point conditions (∂P/∂v)_T = 0 and (∂²P/∂v²)_T = 0 are solved. The κ-dependent terms cancel algebraically in the combination P_c v_c / T_c, yielding the constant value ≈ 0.370. This is confirmed both symbolically and numerically for multiple κ values (0.1, 0.5, 1.0), ruling out an artifact. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained given the effective-metric ansatz

full rationale

The paper constructs an effective κ-deformed Schwarzschild-AdS metric from the κ-deformed Newtonian potential, augments the first law by treating κ as an additional thermodynamic variable with an identified conjugate, and derives the equation of state and critical ratio from that geometry. The reported Pc vc/Tc ≃ 0.370 (independent of κ) is obtained by explicit computation on the modified metric rather than by definitional reduction or by fitting a parameter to the target quantity. No self-definitional loop, fitted-input-renamed-as-prediction, or load-bearing self-citation chain appears in the central steps; the independence of the ratio from κ is a derived feature, not an input. The framework is therefore self-contained against external benchmarks once the initial effective-metric construction is granted.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on the validity of constructing an effective metric from the κ-deformed Newtonian potential and on the domain assumption that extended phase space thermodynamics applies directly to the deformed geometry with κ as a thermodynamic variable.

axioms (2)

domain assumption Extended phase space thermodynamics applies to the κ-deformed black hole with κ treated as an additional variable having a conjugate potential
Invoked to derive the modified first law and Smarr relation.
ad hoc to paper The effective κ-deformed Schwarzschild-AdS metric obtained from the κ-deformed Newtonian potential is a consistent starting point for thermodynamic analysis
Central construction step stated in the abstract.

invented entities (1)

κ-deformed Schwarzschild-AdS metric no independent evidence
purpose: To incorporate non-commutative geometry effects into the black hole spacetime for thermodynamic study
Effective construction introduced in the paper; no independent evidence provided in abstract.

pith-pipeline@v0.9.0 · 5465 in / 1519 out tokens · 80567 ms · 2026-05-10T16:42:33.992883+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

75 extracted references · 65 canonical work pages · 1 internal anchor

[1]

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

1973
[2]

S. W. Hawking, Particle Creation by Black Holes, Com- mun. Math. Phys.43, 199 (1975), [Erratum: Com- mun.Math.Phys. 46, 206 (1976)]

1975
[3]

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

1983
[4]

Anti-de Sitter Space, Thermal Phase Transition, And Confinement In Gauge Theories

E. Witten, Anti-de Sitter space, thermal phase transition, and confinement in gauge theories, Adv. Theor. Math. Phys.2, 505 (1998), arXiv:hep-th/9803131

work page Pith review arXiv 1998
[5]

Enthalpy and the Mechanics of AdS Black Holes

D. Kastor, S. Ray, and J. Traschen, Enthalpy and the Mechanics of AdS Black Holes, Class. Quant. Grav.26, 195011 (2009), arXiv:0904.2765 [hep-th]

work page Pith review arXiv 2009
[6]

Black hole chem- istry: thermodynamics with Lambda,

D. Kubiznak, R. B. Mann, and M. Teo, Black hole chemistry: thermodynamics with Lambda, Class. Quant. Grav.34, 063001 (2017), arXiv:1608.06147 [hep-th]

work page arXiv 2017
[7]

P-V criticality of charged AdS black holes

D. Kubiznak and R. B. Mann, P-V criticality of charged AdS black holes, JHEP07, 033, arXiv:1205.0559 [hep- th]

work page Pith review arXiv
[8]

P-V criticality in the extended phase space of Gauss-Bonnet black holes in AdS space

R.-G. Cai, L.-M. Cao, L. Li, and R.-Q. Yang, P-V criti- cality in the extended phase space of Gauss-Bonnet black holes in AdS space, JHEP09, 005, arXiv:1306.6233 [gr- qc]

work page Pith review arXiv
[9]

Gunasekaran, R

S. Gunasekaran, R. B. Mann, and D. Kubiznak, Ex- tended phase space thermodynamics for charged and ro- tating black holes and Born-Infeld vacuum polarization, JHEP11, 110, arXiv:1208.6251 [hep-th]. 9

work page arXiv
[10]

M. H. Dehghani, S. Kamrani, and A. Sheykhi,P−V criticality of charged dilatonic black holes, Phys. Rev. D 90, 104020 (2014), arXiv:1505.02386 [hep-th]

work page arXiv 2014
[11]

R. A. Hennigar, W. G. Brenna, and R. B. Mann,P− vcriticality in quasitopological gravity, JHEP07, 077, arXiv:1505.05517 [hep-th]

work page arXiv
[12]

Ballon-Bayona, H

A. Ballon-Bayona, H. Boschi-Filho, E. F. Capossoli, and D. M. Rodrigues, Criticality from Einstein-Maxwell- dilaton holography at finite temperature and density, Phys. Rev. D102, 126003 (2020), arXiv:2006.08810 [hep- th]

work page arXiv 2020
[13]

D. V. Singh and S. Siwach, Thermodynamics and P-v criticality of Bardeen-AdS Black Hole in 4DEinstein- Gauss-Bonnet Gravity, Phys. Lett. B808, 135658 (2020), arXiv:2003.11754 [gr-qc]

work page arXiv 2020
[14]

Zhang, C.-M

M. Zhang, C.-M. Zhang, D.-C. Zou, and R.-H. Yue, P- V criticality and Joule-Thomson expansion of Hayward- AdS black holes in 4D Einstein-Gauss-Bonnet gravity, Nucl. Phys. B973, 115608 (2021), arXiv:2102.04308 [hep-th]

work page arXiv 2021
[15]

Li and J

R. Li and J. Wang, Hawking radiation andP−v criticality of charged dynamical (Vaidya) black hole in anti-de Sitter space, Phys. Lett. B813, 136035 (2021), arXiv:2009.09319 [gr-qc]

work page arXiv 2021
[16]

W. Cong, D. Kubiznak, R. B. Mann, and M. R. Visser, Holographic CFT phase transitions and crit- icality for charged AdS black holes, JHEP08, 174, arXiv:2112.14848 [hep-th]

work page arXiv
[17]

A. Sood, A. Kumar, J. K. Singh, and S. G. Ghosh, Ther- modynamic stability andP−Vcriticality of nonsingular- AdS black holes endowed with clouds of strings, Eur. Phys. J. C82, 227 (2022), arXiv:2204.05996 [gr-qc]

work page arXiv 2022
[18]

Li, H.-P

X.-Q. Li, H.-P. Yan, L.-L. Xing, and S.-W. Zhou, Critical behavior of AdS black holes surrounded by dark fluid with Chaplygin-like equation of state, Phys. Rev. D107, 104055 (2023), arXiv:2305.03028 [gr-qc]

work page arXiv 2023
[19]

Astefanesei, P

D. Astefanesei, P. Cabrera, R. B. Mann, and R. Ro- jas, Extended phase space thermodynamics for hairy black holes, Phys. Rev. D108, 104047 (2023), arXiv:2304.09203 [hep-th]

work page arXiv 2023
[20]

S.-J. Yang, M. S. Ali, S.-W. Wei, and Y.-X. Liu, Holo- graphic thermodynamics of a five-dimensional neutral Gauss–Bonnet AdS black hole, Eur. Phys. J. C85, 420 (2025), arXiv:2408.00318 [hep-th]

work page arXiv 2025
[21]

H. S. Snyder, Quantized space-time, Phys. Rev.71, 38 (1947)

1947
[22]

Doplicher, K

S. Doplicher, K. Fredenhagen, and J. E. Roberts, Space- time quantization induced by classical gravity, Phys. Lett. B331, 39 (1994)

1994
[23]

M. R. Douglas and N. A. Nekrasov, Noncommutative field theory, Rev. Mod. Phys.73, 977 (2001), arXiv:hep- th/0106048

work page arXiv 2001
[24]

Arzano and J

M. Arzano and J. Kowalski-Glikman, An Introduction to κ-Deformed Symmetries, Phase Spaces and Field Theory, Symmetry13, 946 (2021)

2021
[25]

String Theory and Noncommutative Geometry

N. Seiberg and E. Witten, String theory and noncommu- tative geometry, JHEP09, 032, arXiv:hep-th/9908142

work page Pith review arXiv
[26]

J. W. Moffat, Noncommutative quantum gravity, Phys. Lett. B491, 345 (2000), arXiv:hep-th/0007181

work page arXiv 2000
[27]

Aschieri, C

P. Aschieri, C. Blohmann, M. Dimitrijevic, F. Meyer, P. Schupp, and J. Wess, A Gravity theory on noncom- mutative spaces, Class. Quant. Grav.22, 3511 (2005), arXiv:hep-th/0504183

work page internal anchor Pith review arXiv 2005
[28]

Harikumar and V

E. Harikumar and V. O. Rivelles, Noncommutative Grav- ity, Class. Quant. Grav.23, 7551 (2006), arXiv:hep- th/0607115

work page arXiv 2006
[29]

A. H. Chamseddine, Deforming Einstein’s gravity, Phys. Lett. B504, 33 (2001), arXiv:hep-th/0009153

work page arXiv 2001
[30]

Chaichian, M

M. Chaichian, M. R. Setare, A. Tureanu, and G. Zet, On Black Holes and Cosmological Constant in Non- commutative Gauge Theory of Gravity, JHEP04, 064, arXiv:0711.4546 [hep-th]

work page arXiv
[31]

Juri´ c, A

T. Juri´ c, A. N. Kumara, and F. Poˇ zar, Constructing non- commutative black holes, Nucl. Phys. B1017, 116950 (2025), arXiv:2503.08560 [hep-th]

work page arXiv 2025
[32]

Nicolini,Noncommutative Black Holes, The Final Appeal To Quantum Gravity: A Review,Int

P. Nicolini, Noncommutative black holes, the final appeal to quantum gravity: A review, Int. of Mod. Phys. A24, 1229 (2009), arXiv:0807.1939 [hep-th]

work page arXiv 2009
[33]

Nicolini, A

P. Nicolini, A. Smailagic, and E. Spallucci, Noncommu- tative geometry inspired Schwarzschild black hole, Phys. Lett. B632, 547 (2006), arXiv:gr-qc/0510112

work page arXiv 2006
[34]

Noncommutative geometry inspired charged black holes

S. Ansoldi, P. Nicolini, A. Smailagic, and E. Spallucci, Noncommutative geometry inspired charged black holes, Phys. Lett. B645, 261 (2007), arXiv:gr-qc/0612035

work page Pith review arXiv 2007
[35]

Spallucci, A

E. Spallucci, A. Smailagic, and P. Nicolini, Non- commutative geometry inspired higher-dimensional charged black holes, Phys. Lett. B670, 449 (2009), arXiv:0801.3519 [hep-th]

work page arXiv 2009
[36]

Modesto and P

L. Modesto and P. Nicolini, Charged rotating noncom- mutative black holes, Phys. Rev. D82, 104035 (2010), arXiv:1005.5605 [gr-qc]

work page arXiv 2010
[37]

Y. S. Myung, Y.-W. Kim, and Y.-J. Park, Thermody- namics and evaporation of the noncommutative black hole, JHEP02, 012, arXiv:gr-qc/0611130

work page arXiv
[38]

Banerjee, B

R. Banerjee, B. R. Majhi, and S. Samanta, Noncommu- tative Black Hole Thermodynamics, Phys. Rev. D77, 124035 (2008), arXiv:0801.3583 [hep-th]

work page arXiv 2008
[39]

Nozari and S

K. Nozari and S. H. Mehdipour, Hawking Radia- tion as Quantum Tunneling from Noncommutative Schwarzschild Black Hole, Class. Quant. Grav.25, 175015 (2008), arXiv:0801.4074 [gr-qc]

work page arXiv 2008
[40]

Banerjee, B

R. Banerjee, B. R. Majhi, and S. K. Modak, Noncom- mutative Schwarzschild Black Hole and Area Law, Class. Quant. Grav.26, 085010 (2009), arXiv:0802.2176 [hep- th]

work page arXiv 2009
[41]

Banerjee, S

R. Banerjee, S. Gangopadhyay, and S. K. Modak, Voros product, Noncommutative Schwarzschild Black Hole and Corrected Area Law, Phys. Lett. B686, 181 (2010), arXiv:0911.2123 [hep-th]

work page arXiv 2010
[42]

Ma and R

M.-S. Ma and R. Zhao, Noncommutative geometry in- spired black holes in Rastall gravity, Eur. Phys. J. C77, 629 (2017), arXiv:1706.08054 [hep-th]

work page arXiv 2017
[43]

Liang, Z.-H

J. Liang, Z.-H. Guan, Y.-C. Liu, and B. Liu, P–v criti- cality in the extended phase space of a noncommutative geometry inspired Reissner–Nordstr¨ om black hole in AdS space-time, Gen. Rel. Grav.49, 29 (2017)

2017
[44]

S. G. Ghosh, Noncommutative geometry inspired Ein- stein–Gauss–Bonnet black holes, Class. Quant. Grav.35, 085008 (2018), arXiv:1707.08174 [gr-qc]

work page arXiv 2018
[45]

S. R. Chowdhury, D. Deb, F. Rahaman, S. Ray, and B. K. Guha, Noncommutative black hole in the Fins- lerian spacetime, Class. Quant. Grav.38, 145019 (2021), arXiv:1907.03558 [physics.gen-ph]

work page arXiv 2021
[46]

Lekbich, A

H. Lekbich, A. El Boukili, N. Mansour, and M. B. Sedra, 4D AdS Einstein–Gauss–Bonnet black hole en- dowed with Lorentzian noncommutativity: P−V criti- cality, Joule–Thomson expansion, and shadow, Annals 10 Phys.458, 169451 (2023)

2023
[47]

A. A. A. Filho, J. R. Nascimento, A. Y. Petrov, P. J. Porf´ ırio, and A.¨Ovg¨ un, Effects of non-commutative ge- ometry on black hole properties, Phys. Dark Univ.46, 101630 (2024), arXiv:2406.12015 [gr-qc]

work page arXiv 2024
[48]

Hamil and B

B. Hamil and B. C. L¨ utf¨ uo˘ glu, Noncommutative Schwarzschild black hole surrounded by quintessence: Thermodynamics, Shadows and Quasinormal modes, Phys. Dark Univ.44, 101484 (2024), arXiv:2401.09295 [gr-qc]

work page arXiv 2024
[49]

Ahmed, A

F. Ahmed, A. R. P. Moreira, and A. Bouzenada, Non- commutative geometry inspired AdS black hole with a cloud of strings surrounded by quintessence-like fluid, Eur. Phys. J. C86, 152 (2026), arXiv:2508.00740 [gr-qc]

work page arXiv 2026
[50]

Miao and Z.-M

Y.-G. Miao and Z.-M. Xu, Thermodynamics of noncom- mutative high-dimensional AdS black holes with non- Gaussian smeared matter distributions, Eur. Phys. J. C 76, 217 (2016), arXiv:1511.00853 [hep-th]

work page arXiv 2016
[51]

Miao and Z.-M

Y.-G. Miao and Z.-M. Xu, Phase transition and entropy inequality of noncommutative black holes in a new ex- tended phase space, JCAP03, 046, arXiv:1604.03229 [hep-th]

work page arXiv
[52]

J. P. M. Gra¸ ca, E. F. Capossoli, and H. Boschi- Filho, Joule-Thomson expansion for noncommuta- tive uncharged black holes, EPL135, 41002 (2021), arXiv:2107.05781 [hep-th]

work page arXiv 2021
[53]

Wang, S.-J

R.-B. Wang, S.-J. Ma, L. You, J.-B. Deng, and X.-R. Hu, Thermodynamics of Schwarzschild-AdS black hole in non-commutative geometry, Chin. Phys. C49, 065101 (2025), arXiv:2410.03650 [gr-qc]

work page arXiv 2025
[54]

Tan, Thermodynamics of high order correction for Schwarzschild-AdS black hole in non-commutative geometry, Nucl

B. Tan, Thermodynamics of high order correction for Schwarzschild-AdS black hole in non-commutative geometry, Nucl. Phys. B1014, 116868 (2025), arXiv:2410.16799 [gr-qc]

work page arXiv 2025
[55]

R.-B. Wang, L. You, S.-J. Ma, J.-B. Deng, and X.-R. Hu, Thermodynamic properties and Joule-Thomson ex- pansion of AdS black hole with Gaussian distribution in non-commutative geometry (2025), arXiv:2503.12363 [gr- qc]

work page arXiv 2025
[56]

Sadeghi, B

J. Sadeghi, B. Pourhassan, and F. Khosravani, The im- pact of non-expansive entropies on thermodynamics of high order correction of Schwarzschild-AdS black hole in non-commutative geometry, Phys. Lett. B872, 140111 (2026)

2026
[57]

Kowalski-Glikman and S

J. Kowalski-Glikman and S. Nowak, Doubly special rel- ativity theories as different bases of kappa Poincare algebra, Phys. Lett. B539, 126 (2002), arXiv:hep- th/0203040

work page arXiv 2002
[58]

Lukierski, A

J. Lukierski, A. Nowicki, and H. Ruegg, New quantum Poincare algebra and k deformed field theory, Phys. Lett. B293, 344 (1992)

1992
[59]

Cianfrani, J

F. Cianfrani, J. Kowalski-Glikman, D. Pranzetti, and G. Rosati, Symmetries of quantum spacetime in three dimensions, Phys. Rev. D94, 084044 (2016), arXiv:1606.03085 [hep-th]

work page arXiv 2016
[60]

Lukierski, H

J. Lukierski, H. Ruegg, and W. J. Zakrzewski, Classi- cal quantum mechanics of free kappa relativistic systems, Annals Phys.243, 90 (1995), arXiv:hep-th/9312153

work page arXiv 1995
[61]

B. P. Dolan, K. S. Gupta, and A. Stern, Noncommutative BTZ black hole and discrete time, Class. Quant. Grav. 24, 1647 (2007), arXiv:hep-th/0611233

work page arXiv 2007
[62]

Schupp and S

P. Schupp and S. Solodukhin, Exact Black Hole Solutions in Noncommutative Gravity (2009), arXiv:0906.2724 [hep-th]

work page arXiv 2009
[63]

Rozental and O

D. Rozental and O. Birnholtz,κ-general-relativity I: a non-commutative GR theory with theκ-Minkowski spacetime as its flat limit, Class. Quant. Grav.42, 185006 (2025), arXiv:2408.11955 [gr-qc]

work page arXiv 2025
[64]

K. S. Gupta, E. Harikumar, T. Juric, S. Meljanac, and A. Samsarov, Effects of Noncommutativity on the Black Hole Entropy, Adv. High Energy Phys.2014, 139172 (2014), arXiv:1312.5100 [hep-th]

work page arXiv 2014
[65]

Harikumar and N

E. Harikumar and N. S. Zuhair, Hawking radiation in κ-spacetime, Int. J. Mod. Phys. A32, 1750072 (2017), arXiv:1609.05288 [hep-th]

work page arXiv 2017
[66]

K. S. Gupta, E. Harikumar, T. Juri´ c, S. Meljanac, and A. Samsarov, Noncommutative scalar quasinormal modes and quantization of entropy of a BTZ black hole, JHEP09, 025, arXiv:1505.04068 [hep-th]

work page arXiv
[67]

Juri´ c and A

T. Juri´ c and A. Samsarov, Entanglement entropy renor- malization for the noncommutative scalar field coupled to classical BTZ geometry, Phys. Rev. D93, 104033 (2016), arXiv:1602.01488 [hep-th]

work page arXiv 2016
[68]

K. S. Gupta, T. Juri´ c, A. Samsarov, and I. Smoli´ c, Non- commutativity and logarithmic correction to the black hole entropy, JHEP02, 060, arXiv:2209.07168 [hep-th]

work page arXiv
[69]

Scardigli and R

F. Scardigli and R. Casadio, Gravitational tests of the Generalized Uncertainty Principle, Eur. Phys. J. C75, 425 (2015), arXiv:1407.0113 [hep-th]

work page arXiv 2015
[70]

Scardigli, G

F. Scardigli, G. Lambiase, and E. Vagenas, GUP parame- ter from quantum corrections to the Newtonian potential, Phys. Lett. B767, 242 (2017), arXiv:1611.01469 [hep-th]

work page arXiv 2017
[71]

Rajagopal and P

V. Rajagopal and P. Wu, Entropic force and bouncing behaviour inκ-Minkowski space-time, Phys. Lett. B868, 139670 (2025), arXiv:2502.15831 [gr-qc]

work page arXiv 2025
[72]

Regular Black Hole in General Relativity Coupled to Nonlinear Electrodynamics

E. Ayon-Beato and A. Garcia, Regular black hole in general relativity coupled to nonlinear electrodynamics, Phys. Rev. Lett.80, 5056 (1998), arXiv:gr-qc/9911046

work page Pith review arXiv 1998
[73]

Ma and R

M.-S. Ma and R. Zhao, Corrected form of the first law of thermodynamics for regular black holes, Class. Quant. Grav.31, 245014 (2014), arXiv:1411.0833 [gr-qc]

work page arXiv 2014
[74]

Wang, S.-J

R.-B. Wang, S.-J. Ma, L. You, Y.-C. Tang, Y.-H. Feng, X.-R. Hu, and J.-B. Deng, Thermodynamics of AdS- Schwarzschild-like black hole in loop quantum gravity, Eur. Phys. J. C84, 1161 (2024), arXiv:2405.08241 [gr- qc]

work page arXiv 2024
[75]

M.-S. Ma, Y. He, X.-M. Wang, and H.-F. Li, From sin- gular to regular: Revisiting thermodynamics of Bardeen- Ads black holes, Phys. Lett. B870, 139961 (2025), arXiv:2510.06576 [hep-th]

work page arXiv 2025