How a minimal length scale modifies thermodynamics of RN AdS Black Holes?

Suman Kumar Panja; Yu Shi

REVIEW 3 major objections 5 minor 24 references

A minimal length scale from kappa-deformed spacetime keeps the Van der Waals phase structure of charged AdS black holes but lowers the critical ratio and enlarges the cooling region in the Joule-Thomson expansion.

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2026-07-14 08:00 UTC pith:7VCDJEJI

load-bearing objection Competent extension of κ-deformed AdS thermodynamics to the charged case; algebra is clean and recovers known limits, but a stability-sign error and an unvalidated metric substitution keep it routine rather than decisive. the 3 major comments →

arxiv 2607.10964 v1 pith:7VCDJEJI submitted 2026-07-13 gr-qc

How a minimal length scale modifies thermodynamics of RN AdS Black Holes?

Suman Kumar Panja , Yu Shi This is my paper

classification gr-qc

keywords kappa-deformed spacetimeRN AdS black holeminimal lengthP-V criticalityJoule-Thomson expansionnoncommutative geometryblack-hole thermodynamicsVan der Waals analogy

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.

This paper asks how a fundamental minimal length, realized through kappa-deformed spacetime, changes the thermodynamics of a charged anti-de Sitter black hole. The authors build a deformed metric from phase-space commutation relations, then recompute temperature, entropy, enthalpy, heat capacity, equation of state, Gibbs free energy, and the Joule-Thomson coefficient. They find that the familiar small-to-large black-hole phase transition survives and remains analogous to a Van der Waals fluid, yet the critical ratio is reduced by a factor involving the deformation parameter and the cooling region in the temperature-pressure plane is expanded. The work supplies concrete, quantitative deformations of coexistence curves and inversion curves that could, in principle, serve as observational signatures of quantum-gravity effects at macroscopic scales.

Core claim

The kappa-deformed RN-AdS black hole still exhibits a small-to-large phase transition completely analogous to a Van der Waals fluid; the critical ratio becomes (1-2a epsilon) times 3/8, and the non-commutative parameter systematically raises inversion temperatures, thereby enlarging the cooling region of the Joule-Thomson expansion while preserving the universal ratio T_min_i / T_c = 1/2.

What carries the argument

The kappa-deformed line element obtained from the phase-space map [x-hat, P-hat] = i g-hat with the realization phi(a epsilon) = exp(-a epsilon), which multiplies only the spatial metric components by exp(-4 a epsilon) and thereby rescales temperature, entropy, volume, and all derived thermodynamic quantities.

Load-bearing premise

The entire thermodynamic analysis rests on one particular realization of the kappa-algebra and on the assumption that this realization can be substituted directly into the curved black-hole metric; if that map is invalid, every subsequent quantity collapses.

What would settle it

Compute the critical ratio or the minimum inversion temperature for a concrete value of the deformation parameter and compare with high-precision measurements of charged AdS black-hole thermodynamics (or with shadow-radius and lensing data that can constrain a); any measured ratio equal to the classical 3/8 with no enlargement of the cooling region would falsify the claimed deformations.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

Competent extension of κ-deformed AdS thermodynamics to the charged case; algebra is clean and recovers known limits, but a stability-sign error and an unvalidated metric substitution keep it routine rather than decisive.

read the letter

This paper takes the existing κ-deformed Schwarzschild-AdS program and does the charged RN-AdS version, including P-V criticality and the full Joule-Thomson analysis. That is the actual new content: the deformed metric (2.8), the critical ratio (1-2aε)3/8, the Gibbs swallowtails, and the inversion curves that shift the cooling region upward. Everything is written out explicitly and the commutative limits are recovered, so the algebra itself is transparent and reproducible.

What it does well is the systematic bookkeeping. Mass-horizon relation stays classical, T picks up e^{2aε}, S and V pick up e^{-4aε} and e^{-2aε} respectively, first law and Smarr hold, and the JT coefficient and inversion temperature follow by direct differentiation. The figures are clear enough that a reader can see the quantitative shifts without re-deriving everything. The universal ratio T_i^min/T_c = 1/2 survives, which is a nice check.

Soft spots are real but limited. First, §3 and the Fig. 3 caption invert the stability criterion (they call C_P < 0 stable). That is a straightforward sign error; the formula for C_P is fine, the prose is not. Second, the metric construction multiplies only the spatial components by e^{-4aε} via a particular realization φ = e^{-aε}, ψ = 1. Surface gravity and area are then read off those components with different powers of the exponential. The paper never re-derives surface gravity from the deformed geodesic equation or a non-commutative Bogoliubov calculation for the RN-AdS background, so the first-order shift in the critical ratio and the enlarged cooling region rest on that unvalidated substitution. If the powers are wrong, the quantitative claims move. The "observational signatures" paragraph is pure speculation and can be ignored.

This is for people already working on non-commutative or minimal-length black-hole thermodynamics who want the charged AdS numbers and the JT curves in one place. It is not a conceptual advance, but it is solid enough that a serious editor should send it to referees rather than desk-reject. I would cite the critical-ratio and inversion-temperature formulae if I needed the charged κ-deformed numbers; I would not build a new project on the metric construction without checking the surface-gravity step myself.

Referee Report

3 major / 5 minor

Summary. The paper constructs a κ-deformed RN-AdS metric by substituting a specific realization (φ=e^{-aε}, ψ=1) of the κ-algebra into the phase-space map, yielding a line element in which only the spatial components are rescaled by e^{-4aε}. From this metric the authors derive the modified Hawking temperature, entropy, enthalpy, thermodynamic volume and heat capacity, then analyze P–V criticality, Gibbs free energy and the Joule–Thomson expansion. They report that the small-to-large black-hole transition survives with a critical ratio lowered to (1−2aε)3/8, that the swallowtail structure of G persists, and that the NC parameter enlarges the cooling region in the T–P plane while preserving the universal ratio T_i^min/T_c=1/2.

Significance. If the deformed metric is accepted as a valid phenomenological model, the work supplies a complete, internally consistent thermodynamic catalogue for charged AdS black holes in κ-space-time, including the first systematic treatment of Joule–Thomson expansion in this setting. The quantitative shifts (critical ratio, inversion curves, coexistence region) are parameter-free once aε is fixed and recover the commutative limits, offering concrete, in-principle falsifiable signatures. The calculation fills a clear gap relative to existing κ-Schwarzschild-AdS studies and is a solid, incremental contribution to the non-commutative black-hole thermodynamics literature.

major comments (3)

[§3, Eq. (3.9), Fig. 3] §3 and Fig. 3 caption invert the thermodynamic stability criterion. The text states that the region with C_P<0 (left of the vertical asymptote) is stable and the region with C_P>0 is unstable. Standard black-hole thermodynamics identifies C_P>0 with local stability and C_P<0 with instability; for the large-r_+ branch of RN-AdS one has C_P>0. The locations of the divergences themselves are correctly identified, but the stable/unstable assignment and the accompanying prose must be reversed.
[§3, paragraph containing Eq. (3.2)] The intermediate surface-gravity formula written in §3, κ=√(−g̃_rr g̃_tt (½∂_r g̃_tt)²), evaluates to e^{−2aε} f′/2 for the metric (2.8), yet is equated to e^{2aε} f′/2. The final temperature is consistent with the standard GR expression κ=½√(−g^{rr}/g_tt)|∂_r g_tt|, but the displayed intermediate step is algebraically incorrect and should be replaced by a consistent formula so that a reader can reproduce the power of the exponential.
[§2 (choice of realization) and §§3–5] The deformation parameter aε is introduced as “related to the black-hole mass” (via ε), yet is treated throughout as an external constant when computing partial derivatives that define C_P, critical points and µ_JT, and when plotting isotherms and inversion curves. If ε scales with M (or T), those derivatives change. The manuscript should either justify the constant-aε approximation or recompute the thermodynamic response functions with a mass-dependent deformation.

minor comments (5)

[Abstract, title] Abstract and elsewhere: “systematically deform” → “systematically deforms”; “Reissner-Nordstroem” should be “Reissner–Nordström”.
[§4, after Eq. (4.2)] The identification v=2r_+ is retained from the commutative literature while the thermodynamic volume V carries an extra e^{−2aε}. A brief remark clarifying that the specific-volume map is unchanged by convention would avoid confusion.
[Figures] Fig. 1–7 captions are dense; several omit units or the fixed values of G (set to 1). Adding a single sentence on units and conventions would improve readability.
[§3, Eq. (3.1)] The claim that the mass-horizon relation is unmodified is stated without derivation from a Komar integral or similar; a short justification or reference would strengthen §3.
[Introduction] References [17,18] already treat κ-deformed Schwarzschild-AdS thermodynamics; a clearer sentence distinguishing the new charged + JT results from those works would help the reader assess novelty.

Circularity Check

1 steps flagged

No significant circularity: thermodynamic quantities and critical/JT results follow by direct differentiation from a fixed metric ansatz; self-citations supply the input deformation, not the target claims.

specific steps

self citation load bearing [Sec. 2, Eqs. (2.1)–(2.6) and surrounding text]
"Using Eq.(2.3) in Eq.(1.1) we find a specific realization for φ^α_µ to be φ^0_0=1, φ^0_i=0, φ^i_0=0, φ^i_j=δ^i_j e^{-aε}. … one find the κ-deformed metric as [16] … the generic form of the κ-deformed line element for spherically symmetric space-time to be [16] dŝ^{2}=g_{00}dx^{0}dx^{0}+g_{ij}e^{-4aε}dx^i dx^j."

The entire subsequent thermodynamics rests on the deformed line element (2.8), which is imported by citation to the authors' prior flat-space construction [16] rather than re-derived for the curved RN-AdS background. This is a minor self-citation of the input ansatz; it does not make the critical-ratio or JT results circular, because those results are still obtained by ordinary differentiation once the metric is fixed.

full rationale

The paper constructs a κ-deformed RN-AdS line element by a specific realization (φ=e^{-aε}, ψ=1) of the phase-space map, then extracts T, S, H, V, C_P, the equation of state, critical points, and inversion curves by the ordinary thermodynamic definitions (surface gravity, area law, first law, ∂P/∂v=0, µ_JT=0). The critical ratio (1-2aε)3/8 and the universal ratio T_i^min/T_c=1/2 are algebraic consequences of those differentiations; no free parameter is fitted to data and then re-labeled a prediction. Self-citations ([16],[17],[18],[20]) are used only to import the realization and the flat-space metric map; they do not assert the RN-AdS critical ratio or the enlarged cooling region. The skeptic's concern about inconsistent powers of e^{aε} in κ and A is a correctness/consistency issue of the substitution, not a circular reduction of output to input. Hence the derivation chain is self-contained against its own stated premises and scores 1 (minor self-citation that is not load-bearing for the central claim).

Axiom & Free-Parameter Ledger

1 free parameters · 4 axioms · 0 invented entities

The central claims rest on one free deformation strength, the standard black-hole thermodynamic dictionary, and a specific (but previously published) realization of the κ-algebra applied to a curved metric. No new particles or forces are invented; the only ad-hoc element is the choice of ordering function φ.

free parameters (1)

aε (NC deformation parameter) = 0-0.2 (illustrative)
Dimensionless product of the fundamental length a and an energy scale ε related to black-hole mass; scanned by hand over values 0, 0.01, 0.1, 0.2 to produce all plots and the first-order shift in the critical ratio. Not fitted to data but freely chosen.

axioms (4)

domain assumption κ-Minkowski algebra [x̂0,x̂i]=ia x̂i together with the realization φ(aε)=e^{-aε}, ψ=1 that yields the deformed line element (2.8)
Taken from the NC-geometry literature [14,16,20] and inserted without further justification into the RN-AdS geometry; all subsequent thermo quantities inherit this choice.
domain assumption Bekenstein-Hawking area law S=A/4 remains valid for the deformed horizon area
Used in §3 to obtain the modified entropy (3.4); not re-derived from a microscopic counting in the κ-deformed theory.
domain assumption Cosmological constant identified with thermodynamic pressure P=-Λ/8π and the extended first law dH=TdS+VdP+ΦdQ
Standard extended-phase-space dictionary of Kubizňák-Mann; assumed without modification for the deformed quantities.
domain assumption Surface gravity computed from the deformed metric components still gives the physical Hawking temperature
Eq. (3.2); the factor e^{2aε} is accepted as the physical temperature shift.

pith-pipeline@v1.1.0-grok45 · 16812 in / 3080 out tokens · 42295 ms · 2026-07-14T08:00:19.979804+00:00 · methodology

0 comments

read the original abstract

We investigate the thermodynamic modifications of the Reissner-Nordstroem anti-de Sitter (RN AdS) black hole induced by a minimal length scale,which naturally emerges in $\kappa$-deformed space-time. By constructing the modified metric via phase-space commutation relations,we derive the deformed Hawking temperature, entropy, and enthalpy. We analyze the thermal stability through the heat capacity and study the P-V criticality, revealing that the black hole undergoes a small-to-large phase transition analogous to the Van der Waals system, albeit with a critical ratio slightly lowered by non-commutativity. Furthermore, we examine the Joule-Thomson expansion and find that the non-commutative (NC) parameter expands the cooling region in the temperature-pressure plane. Our results demonstrate that while the overall thermodynamic analogy with the Van der Waals fluid persists, the minimal length scale systematically deform the coexistence region and inversion curves, offering potential observational signatures for quantum gravity.

discussion (0)

Reference graph

Works this paper leans on

24 extracted references

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[1] [1]

J. D. Bekenstein, Black Holes and Entropy,Phys. Rev.D 7, 2333 (1973)

1973

[2] [2]

S. W. Hawking, Particle creation by black holes,Commun. Math. Phys.43, 199 (1975)

1975

[3] [3]

R. M. Wald, The Thermodynamics of Black Holes,Living Rev. Rel.4, 6 (2001)

2001

[4] [4]

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

1983

[5] [5]

Chamblin, R

A. Chamblin, R. Emparan, C. V. Johnson and R. C. Myers, Charged AdS black holes and catastrophic holography,Phys. Rev.D 60, 064018 (1999)

1999

[6] [6]

Kubizňák and R

D. Kubizňák and R. B. Mann, P-V criticality of charged AdS black holes,J. High Energ. Phys.2012, 33 (2012)

2012

[7] [7]

Gunasekaran, R

S. Gunasekaran, R. B. Mann and D. Kubizňák, Extended phase space thermodynamics for charged and rotating black holes and Born-Infeld vacuum polarization,J. High Energ. Phys.1211, 110 (2012). – 14 –

2012

[8] [8]

Ökcü, and E

Ö. Ökcü, and E. Aydıner, Joule-Thomson Expansion of Charged AdS Black Holes,Eur. Phys. J.C 77, 24 (2017)

2017

[9] [9]

ZM. Xu, B. Wu and WL. Yang, van der Waals fluid and charged AdS black hole in the Landau theory,Class. Quan. Grav.38, 2014 (2021)

2014

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Connes,Non-Commutative Geometry; Academic Press: London, UK, (1994)

A. Connes,Non-Commutative Geometry; Academic Press: London, UK, (1994)

1994

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Kowalski-Glikman, Introduction to Doubly Special Relativity,Lect

J. Kowalski-Glikman, Introduction to Doubly Special Relativity,Lect. Notes. Phys.669, 131 (2005)

2005

[12] [12]

Doplicher, K

S. Doplicher, K. Fredenhagen and J. E. Roberts, Space-time quantization induced by classical gravity,Phys. Lett.B 331, 29 (1994)

1994

[13] [13]

Lukierski, A

J. Lukierski, A. Nowicki, H. Ruegg and V. N. Tolstoy, q-deformation of Poincaré algebra, Phys. Lett.B 264, 331 (1991)

1991

[14] [14]

Meljanac and M

S. Meljanac and M. Stojic, New realizations of Lie algebra kappa-deformed Euclidean space. Eur. Phys. J.C 47, 531 (2006)

2006

[15] [15]

Banerjee, B

R. Banerjee, B. R. Majhi, and S. Samanta, Noncommutative black hole thermodynamics, Phys. Rev.D 77124035 (2008)

2008

[16] [16]

Harikumar and N

E. Harikumar and N. S. Zuhair, Hawking radiation inκ-spacetime,Int. J. Mod. Phys.A 32, 1750072 (2017)

2017

[17] [17]

Wang, SJ

RB. Wang, SJ. Ma, L. You, JB. Deng and XR. Hu, Thermodynamics of Schwarzschild-AdS black hole in non-commutative geometry,Chin. Phys.C 49, 065101 (2025)

2025

[18] [18]

A. N. Kumara, V. Rajagopal and P. Wu, Thermodynamics and phase transitions of κ-deformed Schwarzschild-AdS black holes,Phys. Lett.B 880, 140724 (2026)

2026

[19] [19]

Bhagya, D

R. Bhagya, D. Parai, H. Sreekumar and S. K. Panja, Influence of the cosmological constant onκ-deformed Neutron Star,Gen. Relativ Gravit56, 91 (2024)

2024

[20] [20]

Harikumar, T

E. Harikumar, T. Juric and S. Meljanac, Geodesic equation inκ-Minkowski spacetime, Phys. Rev.D 86, 045002 (2012)

2012

[21] [21]

Kastor, S

D. Kastor, S. Ray and J. Traschen, Enthalpy and the mechanics of AdS black holes,Class. Quant. Grav.26, 195011 (2009)

2009

[22] [22]

L. You, RB. Wang, YC. Tang et al. Thermal chaos of quantum-corrected-AdS black hole in the extended phase space,Eur. Phys. J.C 84, 1133 (2024)

2024

[23] [23]

D. E. Winterbone, Advanced Thermodynamics for Engineers, 1st edn. Butterworth-Heinemann, Oxford, (1997)

1997

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N. J. L. S. Lobos and V. C. Fernandez, Observational signatures of quantum-corrected RN blackhole,arXiv:2506.13091. – 15 –

arXiv