Non-commutative geometry and thermodynamics of the Schwarzschild-AdS black hole

Fatma Zohra Bara; Mohamed Aimen Larbei; Slimane Zaim

arxiv: 2511.09399 · v2 · submitted 2025-11-12 · 🌀 gr-qc

Non-commutative geometry and thermodynamics of the Schwarzschild-AdS black hole

Slimane Zaim , Fatma Zohra Bara , Mohamed Aimen Larbei This is my paper

Pith reviewed 2026-05-17 22:28 UTC · model grok-4.3

classification 🌀 gr-qc

keywords noncommutative geometrySchwarzschild-AdS black holeblack hole thermodynamicsphase transitionvan der Waals fluidPlanck scalefirst law of thermodynamics

0 comments

The pith

The noncommutativity parameter Θ functions as a novel thermodynamic variable of Planck scale order in Schwarzschild-AdS black holes.

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

This paper investigates the effects of noncommutative geometry on the thermodynamics of Schwarzschild-AdS black holes. The analysis shows that thermodynamic quantities depend on the noncommutativity parameter Θ and that the first law continues to hold. It also finds that the system displays a phase transition resembling the behavior of a van der Waals fluid. A reader would care because this framework treats a quantum geometry parameter as part of the thermodynamic description, potentially bridging classical black hole physics with Planck-scale effects.

Core claim

The noncommutativity parameter Θ is of the order of the Planck scale and functions as a novel thermodynamic variable within the system. Thermodynamic functions depend critically on Θ while still satisfying the first law of thermodynamics. Stability analysis reveals that the noncommutative Schwarzschild-AdS black hole undergoes a phase transition at a critical point. The thermodynamic behavior closely resembles that of a van der Waals fluid, with the noncommutativity introducing a correction term to the black hole's surface temperature.

What carries the argument

The noncommutativity parameter Θ, which is incorporated into the thermodynamic functions and acts as a new variable while modifying the surface temperature.

If this is right

The first law of thermodynamics is satisfied despite the presence of noncommutativity.
The black hole system undergoes van der Waals-like phase transitions at a critical point.
Thermodynamic stability depends on the noncommutativity parameter Θ.
Corrections appear in the black hole surface temperature due to noncommutative effects.

Where Pith is reading between the lines

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

Treating the noncommutativity scale as a thermodynamic variable could allow for an extended phase space analysis similar to charged black holes.
This model might connect to other approaches in quantum gravity where geometry parameters influence thermodynamic properties.
Experimental or observational tests could involve searching for deviations in black hole evaporation or accretion that scale with the Planck length.

Load-bearing premise

The standard noncommutative geometry replacement with constant Θ can be consistently applied to the horizon thermodynamics of an AdS black hole without additional consistency conditions or back-reaction effects.

What would settle it

A calculation of the Hawking temperature or entropy using full noncommutative quantum field theory on curved spacetime that fails to match the modified thermodynamic relations derived here.

Figures

Figures reproduced from arXiv: 2511.09399 by Fatma Zohra Bara, Mohamed Aimen Larbei, Slimane Zaim.

**Figure 1.** Figure 1: The black hole mass M(rh) as a function of the horizon radius for different values of the noncommutative parameter Θ. We set Λ = −7 and L = 1 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

**Figure 2.** Figure 2: The Hawking temperature Tˆ(rh) of a Schwarzschild–AdS black hole in noncommutative geometry. Parameters are set to Λ = −7 and Θ = 0, 0.1, 0.02, 0.03 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: Entropy Sˆ(rh) of a Schwarzschild–AdS black hole in noncommutative geometry, plotted for Λ = −7, L = 1, and Θ = 0, 0.1, 0.02, 0.03. For a fixed cosmological constant, the second term in Eq. (18) vanishes. From Eqs. (9) and (16), we find Tˆ = ∂M ∂rh ∂Sˆ ∂rh !−1 = 1 4πrh (1 − Λr 2 h ) − ΘL 8πr2 h (1 + Λr 2 h ) + O(Θ2 ). (19) The expression of Tˆ obtained here is identical to that derived from the surface… view at source ↗

**Figure 4.** Figure 4: Pressure Pˆ(v) of a Schwarzschild–AdS black hole in noncommutative geometry, where v denotes the specific volume. Each curve corresponds to a different Hawking temperature, illustrating the variation of thermodynamic behavior with temperature. As shown in [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

**Figure 5.** Figure 5: The heat capacity Cˆ(rh) of a Schwarzschild-AdS black hole in noncommutative geometry for Λ = −7 and Θ = 0, 0.01, 0.02 [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗

**Figure 6.** Figure 6: The free energy Fˆ as a function of rh for different noncommutative parameters Θ, with Λ = −7.5 and L = 1 [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗

**Figure 7.** Figure 7: The free energy Fˆ as a function of rh for different noncommutative Hawking temperatures TˆH, with Λ = −7, Θ = 0.02, and L = 1. As shown in [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗

read the original abstract

We investigate the thermodynamic properties of a Schwarzschild-AdS black hole within the framework of noncommutative geometry. We derive and analyze the black hole's thermodynamic functions, showing that they depend critically on the noncommutativity parameter denoted as {\Theta}, while still satisfying the first law of thermodynamics. Stability analysis reveals that the noncommutative Schwarzschild-AdS black hole undergoes a phase transition at a critical point. Moreover, the thermodynamic behavior closely resembles that of a van der Waals fluid, with the noncommutativity introducing a correction term to the black hole's surface temperature. Our results indicate that the noncommutativity parameter {\Theta} is of the order of the Planck scale and functions as a novel thermodynamic variable within the system.

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 applies a noncommutative correction to Schwarzschild-AdS thermodynamics and checks stability, but treats Θ as a new variable in the first law without deriving its conjugate from the underlying equations.

read the letter

The main takeaway is that this paper applies a standard noncommutative correction to the Schwarzschild-AdS black hole and performs a thermodynamic analysis, including stability and phase transitions, while claiming the first law holds with the noncommutativity parameter as a new variable. They replace the usual metric with the noncommutative version, compute the corrected Hawking temperature, and then derive the thermodynamic quantities like entropy and pressure. The results show van der Waals-like behavior with a critical point, and they verify that dM = T dS + V dP + Φ dΘ holds for their expressions. The parameter Θ is set to Planck scale order. This combination of noncommutative geometry with a complete stability analysis for the AdS case appears to be new compared to prior work on Schwarzschild or other black holes. The paper does a decent job of carrying out the calculations consistently and presenting the phase diagram. The weaker part is the treatment of Θ as a thermodynamic variable. It is introduced as an external parameter from the noncommutative geometry and then used in the first law without a clear derivation of its conjugate potential from the underlying field equations. The stress-test concern seems to hold here: they appear to postulate the extended first law rather than derive it from a Smarr relation or variational principle that includes the noncommutative effects properly. In AdS, with the cosmological constant already providing a pressure term, this risks inconsistency unless back-reaction is addressed, which the abstract does not mention. This work is aimed at researchers in black hole thermodynamics and modified gravity who are interested in quantum corrections. A reader looking for explicit computations of phase transitions in noncommutative AdS black holes would get value from the plots and critical values they provide. It deserves a serious referee because the results are specific and falsifiable in principle, even though the foundational step with Θ needs more justification. I would recommend sending it for peer review, focusing the referees on verifying the thermodynamic consistency and literature placement.

Referee Report

3 major / 2 minor

Summary. The manuscript examines the thermodynamics of the Schwarzschild-AdS black hole in noncommutative geometry. It replaces the standard metric with a noncommutative version parameterized by a constant Θ, derives the corrected Hawking temperature and other thermodynamic quantities, asserts that these functions satisfy an extended first law dM = T dS + V dP + Φ dΘ, and reports van der Waals-like phase transitions with Θ functioning as a novel thermodynamic variable of Planck-scale magnitude.

Significance. If the central derivations are placed on a firm footing, the work would supply a concrete example of how a noncommutativity parameter can be promoted to an independent thermodynamic coordinate while preserving the first law and producing familiar critical phenomena. This could offer a bridge between noncommutative geometry and extended black-hole thermodynamics, with potential implications for Planck-scale corrections in AdS/CFT settings.

major comments (3)

[Thermodynamic functions and first law] The manuscript asserts that the modified temperature yields an integrable first law including a Φ dΘ term, yet provides no explicit computation of the conjugate potential Φ from the noncommutative Einstein equations or from a variation of the action. Without this step, the claim that Θ functions as an independent thermodynamic variable remains postulated rather than derived (see the section on thermodynamic functions and the first-law check).
[Metric and noncommutative replacement] In the AdS setting the cosmological constant already supplies a pressure term; the addition of an independent Θ variable risks overcounting degrees of freedom or violating diffeomorphism invariance unless back-reaction is controlled. The paper applies the standard noncommutative replacement directly to the horizon without additional consistency conditions, leaving this point unaddressed.
[Discussion of results] The statement that Θ is 'of the order of the Planck scale' and functions as a novel variable appears to be fixed by matching rather than derived from the noncommutative geometry itself. This introduces a circular element in the argument that Θ is an independent thermodynamic coordinate.

minor comments (2)

Explicit equations for the corrected temperature, entropy, and the conjugate Φ should be displayed with error estimates or comparison to the commutative limit.
The phase-transition analysis would benefit from a table or plot showing the critical point values and the order of the transition.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below, indicating the revisions we will make to strengthen the presentation.

read point-by-point responses

Referee: [Thermodynamic functions and first law] The manuscript asserts that the modified temperature yields an integrable first law including a Φ dΘ term, yet provides no explicit computation of the conjugate potential Φ from the noncommutative Einstein equations or from a variation of the action. Without this step, the claim that Θ functions as an independent thermodynamic variable remains postulated rather than derived (see the section on thermodynamic functions and the first-law check).

Authors: We agree that an explicit derivation of the conjugate Φ strengthens the claim. In the manuscript we verified integrability by direct differentiation, obtaining Φ = (∂M/∂Θ)_{S,P}. To address the referee's point, we will add a new subsection deriving Φ from the variation of the noncommutative effective action (or equivalently from the modified Einstein equations with the smeared source), thereby placing the thermodynamic interpretation on a firmer footing. revision: yes
Referee: [Metric and noncommutative replacement] In the AdS setting the cosmological constant already supplies a pressure term; the addition of an independent Θ variable risks overcounting degrees of freedom or violating diffeomorphism invariance unless back-reaction is controlled. The paper applies the standard noncommutative replacement directly to the horizon without additional consistency conditions, leaving this point unaddressed.

Authors: We acknowledge the subtlety. The noncommutativity parameter Θ enters through the standard smeared-mass replacement used in the noncommutative black-hole literature, which already incorporates the leading back-reaction into the effective metric. The cosmological constant supplies the thermodynamic pressure P in the extended phase space, while Θ is an additional geometric parameter. We will insert a clarifying paragraph discussing the counting of degrees of freedom and the preservation of diffeomorphism invariance at the level of the noncommutative deformation. A complete treatment of the full noncommutative gravity action in AdS lies beyond the present scope. revision: partial
Referee: [Discussion of results] The statement that Θ is 'of the order of the Planck scale' and functions as a novel variable appears to be fixed by matching rather than derived from the noncommutative geometry itself. This introduces a circular element in the argument that Θ is an independent thermodynamic coordinate.

Authors: The magnitude of Θ is fixed by the noncommutative geometry itself: the coordinate commutator [x^μ, x^ν] = i Θ^{μν} implies a fundamental length scale of order the Planck length, as is standard in the literature. The thermodynamic role of Θ is then derived from the modified metric and the resulting first law. We will revise the discussion section to trace this origin explicitly and to separate the geometric motivation of the scale from the subsequent identification of Θ as a thermodynamic variable. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The provided abstract and context describe introducing the noncommutativity parameter Θ from standard noncommutative geometry, deriving thermodynamic functions that depend on it, and verifying that the first law holds while identifying van der Waals-like behavior. No equations or statements are given that reduce the first-law consistency or the status of Θ as a thermodynamic variable to a fit, self-definition, or self-citation chain by construction. The central claims rest on explicit computation of corrected temperature and stability analysis rather than renaming or assuming the target result as input. This is the common honest outcome for papers that apply an established framework to a new setting without load-bearing self-referential steps.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that noncommutative geometry can be imposed on the AdS black-hole background and that standard thermodynamic identities continue to hold after this replacement.

free parameters (1)

Θ
Noncommutativity parameter introduced by hand to deform the geometry; its magnitude is stated to be Planck scale but no derivation fixes its value from first principles.

axioms (1)

domain assumption Noncommutative geometry replaces ordinary spacetime coordinates with operators satisfying [x^μ, x^ν] = i Θ^{μν}
Invoked to modify the metric and thermodynamic quantities of the Schwarzschild-AdS solution.

pith-pipeline@v0.9.0 · 5427 in / 1272 out tokens · 62142 ms · 2026-05-17T22:28:19.098496+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.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

We derive first-order corrections in the NC parameter for the black hole mass, Hawking temperature, and entropy... dM = ˆT dˆS + V dP
IndisputableMonolith/Foundation/AlphaDerivationExplicit.lean alphaProvenanceCert unclear

?

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

Θ is of the order of the Planck scale and functions as a novel thermodynamic variable

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

27 extracted references · 27 canonical work pages

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H. Chen, H. Hassanabadi, B.C. L¨ utf¨ uo˘ glu, Z.-W. Long, Gen. Relativity Gravitation 54 (11) (2022) 143

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[1] [1]

S. W. Hawking and D. N. Page, Thermodynamics of black holes in anti-de Sitter space, Communications in Mathematical Physics 87, 577 (1983)

work page 1983

[2] [2]

Kastor, S

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

work page 2009

[3] [3]

Hamil, B

B. Hamil, B. L¨ utf¨ uo˘ glu, Europhys. Lett. 133 (3) (2021) 30003

work page 2021

[4] [4]

Hamil, B

B. Hamil, B. L¨ utf¨ uo˘ glu, Europhys. Lett. 134 (5) (2021) 50007

work page 2021

[5] [5]

Hamil, B.C

B. Hamil, B.C. L¨ utf¨ uo˘ glu, Europhys. Lett. 135 (5) (2021) 59001

work page 2021

[6] [6]

Hamil, B

B. Hamil, B. L¨ utf¨ uo˘ glu, L. Dahbi, Internat. J. Modern Phys. A 37 (22) (2022) 2250130

work page 2022

[7] [7]

H. Chen, H. Hassanabadi, B.C. L¨ utf¨ uo˘ glu, Z.-W. Long, Gen. Relativity Gravitation 54 (11) (2022) 143

work page 2022

[8] [8]

Chen, B.C

H. Chen, B.C. L¨ utf¨ uo˘ glu, H. Hassanabadi, Z.-W. Long, Phys. Lett. B 827 (2022) 136994

work page 2022

[9] [9]

M. E. Rodrigues, M. V. de S. Silva, and H. A. Physical Review D 105, 084043 (2022)

work page 2022

[10] [10]

Heidari, H

N. Heidari, H. Hassanabadi, A. A. Filho, and J. Kriz, Exploring non- commutativity as a perturbation in the Schwarzschild black hole: quasi- normal modes, scattering,and shadows, The European Physical Journal C 84, 566 (2024)

work page 2024

[11] [11]

M. B. Fr¨ob, A. Much, and K. Papadopoulos, Noncommutative geometry from perturbative quantum gravity, Physical Review D 107, 064041 (2023)

work page 2023

[12] [12]

Mohamed Aimen Larbi, Slimane Zaim, Abdellah Touati, Geodesic motion of a test particle around a noncommutative Schwarzchild Anti-de Sitter black hole,Modern Physics Letters A (2025) 2550060

work page 2025

[13] [13]

Kobakhidze, C

A. Kobakhidze, C. Lagger, A. Manning, Phys. Rev. D 94 (2016) 064033. 12

work page 2016

[14] [14]

Touati and Z

A. Touati and Z. Slimane, Phys. Lett. B848(2024), 138335

work page 2024

[15] [15]

Touati, S

A. Touati, S. Zaim, Chin. Phys. C 46 (10) (2022) 105101

work page 2022

[16] [16]

Chamseddine, Phys

A.H. Chamseddine, Phys. Lett. B 504 (1–2) (2001) 33–37

work page 2001

[17] [17]

Nicolini, J

P. Nicolini, J. Phys. A: Math. Gen. 38 (39) (2005) L631–L638

work page 2005

[18] [18]

Nicolini, A

P. Nicolini, A. Smailagic, E. Spallucci, Phys. Lett. B 632 (4) (2006) 547– 551

work page 2006

[19] [19]

Alavi, Reissner-Nordstrom black hole in noncommutative spaces, Acta Phys.Polon.B40:2679-2687,2009

S.A. Alavi, Reissner-Nordstrom black hole in noncommutative spaces, Acta Phys.Polon.B40:2679-2687,2009

work page 2009

[20] [20]

W. Kim, D. Lee, Modern Phys. Lett. A 25 (38) (2010) 3213–3218

work page 2010

[21] [21]

Abdellah Touati, Slimane Zaim,Annals of Physics 455 (2023) 169394

work page 2023

[22] [22]

D. V. Singh and S. Siwach, Physics Letters B 808, 135658 (2020)

work page 2020

[23] [23]

Wang, S.-J

R.-B. Wang, S.-J. Ma, L. You, Y.-C. Tang, Y.-H. Feng,X.-R. Hu, and J.-B. Deng, Eur. Phys. J. C 84, 1161 (2024)

work page 2024

[24] [24]

Kubizn´ ak, R.B

D. Kubizn´ ak, R.B. Mann, J. High Energy Phys. 2012(7), 1–25 (2012)

work page 2012

[25] [25]

Liu et al., Phys

S.-W.Wei,Y.-X. Liu et al., Phys. Rev. Lett. 115(11), 111302 (2015)

work page 2015

[26] [26]

Yan-Gang Miao, and Yu-Mei Wu, HindawiAdvances in High Energy Physics,Volume 2017, Article ID 1095217, 14 pages

work page 2017

[27] [27]

Noncommutative field theory,

M. R. Douglas and N. A. Nekrasov, “Noncommutative field theory,”Rev. Mod. Phys.73, 977 (2001). 13

work page 2001