Pith. sign in

REVIEW 3 major objections 3 minor 98 references

Planck-scale modified kinematics do not stabilize Schwarzschild black holes: the cubic entropy correction yields only unstable equilibrium branches with winding number -1.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-02 07:12 UTC pith:YNHORZ6Q

load-bearing objection The winding-number result is correct inside the entropy-geometry correspondence, but the MDR derivation has errors, the τ-domain threshold is arithmetically wrong, and the no-go claim evaporates if you use the standard fixed-background mass. the 3 major comments →

arxiv 2607.10600 v2 pith:YNHORZ6Q submitted 2026-07-12 gr-qc hep-th

A winding number analysis of Schwarzschild black hole stability in light of Planck-scale modified kinematics

Mohsen Khodadi , Nosratolla Jafari , Shahin Mamedov This is my paper
classification gr-qc hep-th PACS 04.70.Dy04.60.-m
keywords modified dispersion relationblack hole thermodynamicswinding numbertopological chargeentropy-geometry correspondenceSchwarzschild black holePlanck-scale correctionsthermodynamic topology
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.

The paper asks whether Planck-scale modifications of particle kinematics, encoded in a modified dispersion relation, can make a Schwarzschild black hole thermodynamically stable. Starting from the MDR E²=p²+ηE³/E_P, it derives a cubic correction to the Bekenstein-Hawking entropy via the tunneling method, and then feeds this entropy through an entropy-geometry correspondence to get mass, temperature, and an off-shell free energy. Imposing the physical requirements of positive ADM mass and positive temperature, it finds exactly one equilibrium branch for each sign of η, and in both cases the winding number is w=-1, giving total topological charge W=-1. A second mathematical root with w=+1 is excluded because it has negative mass and negative temperature and lies outside the regime where the entropy derivation is perturbatively controlled. So the paper concludes that this class of Planck-scale corrections does not stabilize Schwarzschild black holes, and the search must move to MDRs with different leading-order corrections.

Core claim

Under the entropy-geometry correspondence, for S(r_h)=πr_h² - α r_h³ with α=2η/(3κ), the ADM mass M=S'/(4π) and Hawking temperature T=S''/(4πS'). Physical black holes require S'>0 and T>0. For α<0 these hold for all r_h>0; for α>0 they restrict r_h<π/(3α). The equilibrium condition from the off-shell free energy yields two roots for α>0, but only the smaller root lies in the physical domain; it carries winding number w=-1. The larger root, carrying w=+1, violates both positivity conditions (S'<0 means negative mass, S''<0 means negative temperature) and is also outside the perturbative regime ηr_h/κ<~1. Thus every physical equilibrium branch has w=-1 and W=-1, identical to the classical Schw

What carries the argument

The entropy-geometry correspondence, taken as given, which fixes the metric from the entropy functional: f(r)=1-4πM/S'(r), M=S'(r_h)/(4π), T=S''(r_h)/(4πS'(r_h)). Together with the thermodynamic topology construction—generalized free energy F=M-S/τ, vector field φ=(∂F/∂r_h, -cotθ cscθ)—it maps equilibrium states to winding numbers w_i=±1, which correspond to stable/unstable only when S'>0 and M'>0. The paper complements this with the physical domain constraint S'(r_h)>0 and T>0.

Load-bearing premise

The load-bearing premise is the entropy-geometry correspondence, which sets the black hole mass to M=S'(r_h)/(4π) and temperature to T=S''/(4πS'); if that identification is wrong, the exclusion of the w=+1 branch as negative-mass/negative-temperature loses its basis.

What would settle it

Compute the specific heat C=dM/dT directly from M=S'/(4π) and T=S''/(4πS') for the physical branch r_h<π/(3α) with α>0; if C turns out positive for any r_h in that domain, the claim that all physical branches are unstable is wrong. Alternatively, choose a different entropy-geometry correspondence that preserves M=r_h/2 and re-evaluate the winding numbers; if the w=+1 root then satisfies S'>0 and T>0, the conclusion is correspondence-dependent.

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

If this is right

  • If the entropy-geometry correspondence is the right way to incorporate modified entropy into the metric, the cubic MDR correction cannot produce stable Schwarzschild black holes; the total topological charge remains -1.
  • The w=+1 branch that appears in the mathematical classification is not a physical stable phase; any analysis that reports stability from winding numbers must first verify S'>0 and T>0.
  • The result distinguishes this cubic entropy from logarithmic and Kaniadakis corrections, which have W=0 with coexisting stable and unstable branches; the difference is that the would-be stable branch for the cubic correction violates positivity.
  • The perturbative derivation of the entropy correction breaks down near r_h~κ/η, so even ignoring positivity, the stable branch lies in a regime where the entropy formula itself is unreliable.

Where Pith is reading between the lines

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

  • The paper's method effectively turns the topological classification into a filter: only entropy functions that keep S' and S'' positive over a sufficiently wide domain can yield w=+1 physical branches; this suggests a systematic criterion for scanning entropies for Planck-scale stabilization.
  • Because the exclusion of the stable branch relies on identifying ADM mass with S'/(4π), a different entropy-geometry embedding (e.g., one that keeps M=r_h/2 while correcting only the entropy) could leave the w=+1 branch physical; the conclusion is therefore tied to the correspondence, not to the entropy correction alone.
  • A concrete extension: apply the same physical-constraint check to other proposed entropy corrections (Barrow, Tsallis, Rényi, logarithmic, Kaniadakis) and recompute winding numbers; the paper predicts only entropies maintaining S'>0 and S''>0 throughout the equilibrium domain can host stable branches.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper studies the thermodynamic topology of Schwarzschild black holes with a cubic entropy correction S(r_h) = πr_h² − αr_h³, which it claims to derive from the Planck-scale-modified dispersion relation E² = p² + ηE³/E_P via the Hamilton–Jacobi tunneling method. Using the entropy–geometry correspondence of Refs. [65,66], the authors impose physical constraints S′(r_h)>0 and T>0, solve the equilibrium condition ∂F/∂r_h=0, and compute winding numbers. They find that every physical equilibrium branch has w=−1 and total charge W=−1 for both signs of η; a second mathematical root with w=+1 for η>0 is discarded as unphysical because it has S′(r_h)<0 (negative ADM mass) and lies outside the perturbative regime. The paper concludes that this class of MDRs does not yield stable Schwarzschild black holes, while acknowledging that the entropy–geometry correspondence was taken as given and that alternative approaches could give different results.

Significance. If the derivation and framework were sound, the paper would provide a useful cautionary example for the thermodynamic-topology literature: winding-number classifications must be restricted to the physical domain, and a formal w=+1 root can be an artifact of a non-standard mass–entropy relation. The topological step itself is clean: for the cubic entropy, Q(r)=4π²−12παr+18α²r²>0, so every root with S′(r)>0 necessarily has w=−1. The paper is transparent in listing its limitations, including the perturbative validity of the entropy formula and the borrowed nature of the entropy–geometry correspondence. However, the input derivation contains an internal error (the energy E is dropped from the temperature), the integrated entropy coefficient is inconsistent with the preceding equation, and the central no-stabilization claim is contingent on a correspondence that is not defended. These issues significantly limit the generality of the advertised conclusion, although the core topological calculation remains instructive as a framework-specific consistency check.

major comments (3)
  1. [Section II, Eqs. (2.8)–(2.13)] The derivation of the cubic entropy is internally inconsistent. Equation (2.8) gives T_eff(E) ≈ T_H[1+ηE/(2κ)], but Eq. (2.9) removes the energy dependence without any stated identification of E, writing T ≈ 1/(8πM)+η/(16πκ). Different choices for E (e.g., E=T_H or E=M) yield different r_h-dependences after integration, so Eq. (2.13) is not established. Moreover, integrating Eq. (2.12) gives S ≈ A/4 − η/(48√π κ) A^{3/2}, not Eq. (2.13); the coefficient in Eq. (2.13) is larger by a factor 4/π. Consequently, the definition α ≡ 2η/(3κ) used in Eq. (4.1) is not the coefficient that follows from the preceding equations. This is load-bearing because S(r_h)=πr_h²−αr_h³ is the input for the entire topological analysis.
  2. [Section III.C, Section VI, and Eq. (4.2)] The central conclusion that no stable branch exists is entirely conditioned on the entropy–geometry correspondence, specifically M=S′(r_h)/(4π) and T=S″(r_h)/(4πS′(r_h)). The paper explicitly states in Section VI that this correspondence 'was taken as given.' This is not a harmless caveat: using the standard fixed-background Schwarzschild mass M=r_h/2 (the approach of Refs. [67,68]) with the same cubic entropy, the equilibrium condition τ=2S′(r_h) admits, for α>0, a root in the interval r_max<r_h<r_c where S′(r_h)>0 and S″(r_h)<0. In that interval the specific heat C=−S′²/S″ is positive and ∂²F/∂r_h²=−S″/τ>0, giving w=+1 and a thermodynamically stable branch. Thus the no-stabilization claim is not a property of the MDR-derived entropy alone; it is an artifact of the adopted dictionary. The abstract and Section V should either restrict the claim to the entropy–geometry correspondence or p
  3. [Section IV, Eqs. (4.24)–(4.27) and Table II] The evaluation of Q(r_max) contains an algebraic error. Substituting r_max=π/(3α) into Eq. (4.24) gives Q(r_max)=2π²/(3α)−τ−4π²/(3α)+τ = −2π²/(3α), which is independent of τ and always negative. Equation (4.26), Q(r_max)=τ/2−2π²/(3α), is therefore incorrect. As a consequence, the threshold τ_max=4π²/(3α) in Eq. (4.27) is spurious: the smaller root r₁*(τ) is physical for every τ>0, not only for τ≤τ_max, and the claim that no physical root exists for large τ is false. In addition, the larger root satisfies r₂*>r_c, so both S′(r₂*) and S″(r₂*) are negative; Eq. (4.3) then gives T>0, contradicting Table II, which lists T<0 for that branch. The branch is still excluded by its negative ADM mass and by the breakdown of the perturbative expansion, but the stated reasons need to be corrected.
minor comments (3)
  1. [Section III.B, Eq. (3.17)] Equation (3.17) is missing a factor of S′. From Eqs. (3.14) and (3.16), ∂²F/∂r_h² = (M″S′−M′S″)/S′ = M′S′/C, not M′S′²/C. The sign conclusion in Eq. (3.18) still holds in the physical domain S′>0, but the displayed formula is incorrect.
  2. [Throughout] There are numerous typographical and grammatical issues: 'By serving Hamilton-Jacobi tunneling method' (Section II), 'We analyzes the resulting phase portraits' (Section I), 'Now we are in a right position to extracting' (Section IV), and 'the cubic correction at hand' (Section V). A careful editing pass is needed.
  3. [Abstract and Section V] The abstract's concluding sentence, 'Thus, this class of MDRs does not yield stable Schwarzschild black holes,' is too strong given the explicit framework dependence acknowledged in Section VI. The wording should include the qualification 'within the entropy–geometry correspondence adopted here.'

Circularity Check

0 steps flagged

No significant circularity: the winding-number result is a direct analytic consequence of the stated MDR and entropy-geometry correspondence, with no fitted parameter or self-citation chain at the core.

full rationale

The claimed derivation is self-contained given its explicitly stated inputs. The MDR (2.1) is used to derive the cubic entropy S = π r_h^2 − α r_h^3 through the Hamilton–Jacobi tunneling calculation (Sec. II). The winding number is then computed analytically from the generalized free energy F = S'(r_h)/(4π) − S(r_h)/τ that follows from the entropy-geometry correspondence of Refs. [65,66] (Secs. III–IV); no parameter is fitted to the conclusion and no 'prediction' is a renamed fit. The exclusion of the w = +1 root is a direct consequence of imposing S'(r_h) > 0 and T > 0 within that correspondence (Eqs. (4.4)–(4.7) and (4.32)), and the paper explicitly flags in Sec. VI that the correspondence was 'taken as given' and that alternative correspondences could give different conclusions. The topological method itself is imported from Wei–Liu–Mann and Anand et al., not from the present authors, so there is no load-bearing self-citation. Internal algebraic/interpretive issues (e.g., the entropy coefficient in Eq. (2.13) and the temperature/mass status of r_2^* in Table II) are correctness concerns, not circularity. Hence no circular step can be exhibited.

Axiom & Free-Parameter Ledger

2 free parameters · 6 axioms · 0 invented entities

The paper introduces no new particles, forces, or conserved quantities. Its only free input is the phenomenological MDR parameter η, carried over from prior literature. The main assumption burden is the entropy-geometry correspondence and the perturbative derivation of the cubic entropy.

free parameters (2)
  • η = unconstrained; |η| of order 1–10 from GRB bounds cited
    Dimensionless MDR parameter in E²=p²+ηE³/E_P. Its sign sets the sign of α and hence the structure of physical roots. It is not fitted in this paper, but the entropy correction and all conclusions scale with it.
  • α = α=2η/(3κ)
    Combination defining the cubic entropy correction S=πr_h²−αr_h³. Because η is unconstrained, α is effectively a free strength parameter controlling the magnitude of the correction.
axioms (6)
  • domain assumption Modified dispersion relation E²=p²+ηE³/E_P (Eq. 2.1)
    Phenomenological parametrization of Planck-scale modified kinematics, taken from the LIV/DSR literature. The paper does not commit to a specific framework.
  • domain assumption Hamilton-Jacobi tunneling with corrected group velocity yields T_eff and the integrated entropy (Eqs. 2.6–2.13)
    The method is standard in the MDR-tunneling literature, but the paper's implementation contains an unflagged step where the emitted energy E disappears between Eq. (2.8) and Eq. (2.9).
  • domain assumption Entropy-geometry correspondence: M=S'(r_h)/(4π), T=S''(r_h)/(4πS'(r_h)) (Eqs. 3.21–3.24)
    Taken as given from Refs. [65,66]. This is the load-bearing premise that defines ADM mass and temperature from the entropy function and justifies discarding the w=+1 branch.
  • domain assumption Physical constraints S'(r_h)>0 and T>0 define the physical domain
    Standard positivity requirements for positive ADM mass and positive Hawking temperature, used to exclude the second root for α>0.
  • standard math Thermodynamic topology formalism: generalized free energy, vector field, winding number (Eqs. 3.1–3.12)
    The Wei-Liu-Mann framework is adopted without modification. The paper carefully states the sign conditions under which w=+1 means stability.
  • domain assumption Perturbative smallness of ηr_h/κ in deriving the entropy correction
    The expansion after Eq. (2.11) assumes η/(κ√A)≪1. The paper acknowledges that the physical regime approaches ηr_h/κ∼π/2 or π, near the boundary of perturbative control.

pith-pipeline@v1.3.0-alltime-deepseek · 20760 in / 23788 out tokens · 207821 ms · 2026-08-02T07:12:07.899448+00:00 · methodology

0 comments
read the original abstract

Determining whether Planck-scale effects can stabilize black holes addresses fundamental questions about black hole evaporation and quantum gravity consistency. Here, we analyze the thermodynamic topology of Schwarzschild black holes under Planck-scale modified kinematics, using a cubic entropy correction derived from a well-known phenomenological MDR with leading correction \(\eta E^3/E_P\). Enforcing physical constraints (\(S'(r_h) > 0\), \(T > 0\)) via the entropy-geometry correspondence, we find a single unstable branch with \(w = -1\) and \(W = -1\) for both signs of the correction parameter. A second root suggesting stability (\(w = +1\)) is excluded due to negative mass/temperature and lies outside the perturbative regime. Thus, this class of MDRs does not yield stable Schwarzschild black holes. However, MDRs with different leading-order corrections may behave otherwise, leaving the search for Planck-scale stabilization an open endeavor.

Figures

Figures reproduced from arXiv: 2607.10600 by Mohsen Khodadi, Nosratolla Jafari, Shahin Mamedov.

Figure 1
Figure 1. Figure 1: FIG. 1. The [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

Review history (2 revisions) →

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

98 extracted references · 71 linked inside Pith

  1. [1]

    At equilibrium,τ −1 =T(r h), the Hawking temperature

    Thermodynamic stability from winding number The generalized off-shell free energy is F(r h, τ) =M(rh)− S(rh) τ ,(A1) whereM(r h)is the black-hole mass expressed as a function of the horizon radiusr h. At equilibrium,τ −1 =T(r h), the Hawking temperature. The specific heat at constant volume is C=T ∂S ∂T = T dT /dS.(A2) As shown in Eqs. (3.16)–(3.19), with...

  2. [2]

    The unstable node atr ∗ 2 corresponds to a region whereS ′(rh)<0(negative mass) andT <0(negative temperature), rendering it unphysical

    However, only the saddle point atr ∗ 1 satisfies the physical constraintsS ′(rh)>0andT >0(positive ADM mass and positive temperature). The unstable node atr ∗ 2 corresponds to a region whereS ′(rh)<0(negative mass) andT <0(negative temperature), rendering it unphysical. The physical branch is therefore a single saddle with winding numberw=−1. The apparent...

  3. [3]

    Mukhi, Class

    S. Mukhi, Class. Quant. Grav.28(2011), 153001 [arXiv:1110.2569 [physics.pop-ph]]

    Pith/arXiv arXiv 2011

  4. [4]

    The Jacobian at such a point is diagonal: J= λr 0 0 1 ! , λ r ≡ ∂ϕrh ∂rh r∗ h = ∂2F ∂r 2 h r∗ h .(A8) The eigenvalues areλ 1 =λ r andλ 2 = 1

    Phase-space dynamics The auxiliary vector field ϕ(rh, θ) = ϕrh , ϕθ , ϕ rh = ∂F ∂rh , ϕ θ =−cotθcscθ, (A6) defines a fictitious dynamical system on the(r h, θ)plane: drh dt =ϕ rh , dθ dt =ϕ θ.(A7) Equilibrium points(r ∗ h, π/2)satisfyϕ rh = 0andϕ θ = 0. The Jacobian at such a point is diagonal: J= λr 0 0 1 ! , λ r ≡ ∂ϕrh ∂rh r∗ h = ∂2F ∂r 2 h r∗ h .(A8) T...

  5. [5]

    D. J. Gross and P. F. Mende, Nucl. Phys. B303(1988), 407-454

    1988

  6. [6]

    Amati, M

    D. Amati, M. Ciafaloni and G. Veneziano, Phys. Lett. B216 (1989), 41-47

    1989

  7. [7]

    L. J. Garay, Int. J. Mod. Phys. A10(1995), 145-166 [arXiv:gr- qc/9403008 [gr-qc]]

    arXiv 1995

  8. [8]

    Rovelli and L

    C. Rovelli and L. Smolin, Nucl. Phys. B442(1995), 593- 622 [erratum: Nucl. Phys. B456(1995), 753-754] [arXiv:gr- qc/9411005 [gr-qc]]

    arXiv 1995

  9. [9]

    Ashtekar and J

    A. Ashtekar and J. Lewandowski, Class. Quant. Grav.14 (1997), A55-A82 [arXiv:gr-qc/9602046 [gr-qc]]

    Pith/arXiv arXiv 1997

  10. [10]

    Wallden, J

    P. Wallden, J. Phys. Conf. Ser.453(2013), 012023

    2013

  11. [11]

    Khodadi, K

    M. Khodadi, K. Nozari and A. Hajizadeh, Phys. Lett. B770 (2017), 556-563 [arXiv:1702.06357 [gr-qc]]

    Pith/arXiv arXiv 2017

  12. [12]

    Calmet, M

    X. Calmet, M. Graesser and S. D. H. Hsu, Phys. Rev. Lett.93 (2004), 211101 [arXiv:hep-th/0405033 [hep-th]]

    Pith/arXiv arXiv 2004

  13. [13]

    A. F. Ali, Class. Quant. Grav.28(2011), 065013 [arXiv:1101.4181 [hep-th]]

    Pith/arXiv arXiv 2011

  14. [14]

    Farag Ali, M

    A. Farag Ali, M. M. Khalil and E. C. Vagenas, EPL112(2015) no.2, 20005 [arXiv:1510.06365 [gr-qc]]

    Pith/arXiv arXiv 2015

  15. [15]

    Tee and N

    P. Tee and N. Jafari, Eur. Phys. J. C82(2022) no.6, 571 [arXiv:2207.05063 [gr-qc]]

    Pith/arXiv arXiv 2022

  16. [16]

    Nozari, M

    K. Nozari, M. Khodadi and M. A. Gorji, EPL112(2015) no.6, 60003 [arXiv:1512.07779 [gr-qc]]

    Pith/arXiv arXiv 2015

  17. [17]

    Khodadi, K

    M. Khodadi, K. Nozari and E. N. Saridakis, Class. Quant. Grav. 35(2018) no.1, 015010 [arXiv:1612.09254 [gr-qc]]

    Pith/arXiv arXiv 2018

  18. [18]

    Khodadi, K

    M. Khodadi, K. Nozari, H. Abedi and S. Capozziello, Phys. Lett. B783(2018), 326-333 [arXiv:1805.11310 [gr-qc]]

    Pith/arXiv arXiv 2018

  19. [19]

    Nojiri and S

    S. Nojiri and S. D. Odintsov, Phys. Rept.505(2011), 59-144 [arXiv:1011.0544 [gr-qc]]

    Pith/arXiv arXiv 2011

  20. [20]

    Casadio, W

    R. Casadio, W. Feng, I. Kuntz and F. Scardigli, Phys. Lett. B 838(2023), 137722 [arXiv:2210.12801 [hep-th]]

    Pith/arXiv arXiv 2023

  21. [21]

    Y . J. Ng, [arXiv:1102.4109 [gr-qc]]

    Pith/arXiv arXiv

  22. [22]

    T. P. Sotiriou and V . Faraoni, Rev. Mod. Phys.82(2010), 451- 497 [arXiv:0805.1726 [gr-qc]]

    Pith/arXiv arXiv 2010

  23. [23]

    Amelino-Camelia and T

    G. Amelino-Camelia and T. Piran, Phys. Rev. D64(2001), 036005 [arXiv:astro-ph/0008107 [astro-ph]]

    Pith/arXiv arXiv 2001

  24. [24]

    Salvio, Front

    A. Salvio, Front. in Phys.6(2018), 77 [arXiv:1804.09944 [hep- th]]

    Pith/arXiv arXiv 2018

  25. [25]

    Biswas, E

    T. Biswas, E. Gerwick, T. Koivisto and A. Mazumdar, Phys. Rev. Lett.108(2012), 031101 [arXiv:1110.5249 [gr-qc]]

    Pith/arXiv arXiv 2012

  26. [26]

    Colladay and V

    D. Colladay and V . A. Kostelecky, Phys. Rev. D58(1998), 116002 [arXiv:hep-ph/9809521 [hep-ph]]

    Pith/arXiv arXiv 1998

  27. [27]

    Kowalski-Glikman, Phys

    J. Kowalski-Glikman, Phys. Lett. A286(2001), 391-394 [arXiv:hep-th/0102098 [hep-th]]

    Pith/arXiv arXiv 2001

  28. [28]

    Amelino-Camelia, Phys

    G. Amelino-Camelia, Phys. Lett. B510(2001), 255-263 [arXiv:hep-th/0012238 [hep-th]]

    Pith/arXiv arXiv 2001

  29. [29]

    Amelino-Camelia, Int

    G. Amelino-Camelia, Int. J. Mod. Phys. D11(2002), 35-60 [arXiv:gr-qc/0012051 [gr-qc]]

    Pith/arXiv arXiv 2002

  30. [30]

    Amelino-Camelia, [arXiv:gr-qc/0309054 [gr-qc]]

    G. Amelino-Camelia, [arXiv:gr-qc/0309054 [gr-qc]]

    Pith/arXiv arXiv

  31. [31]

    Jafari and M

    N. Jafari and M. R. R. Good, Phys. Lett.B(2020), 135735 [arXiv:2009.06096 [gr-qc]]

    Pith/arXiv arXiv 2020

  32. [32]

    Amelino-Camelia, L

    G. Amelino-Camelia, L. Freidel, J. Kowalski-Glikman and L. Smolin, Phys. Rev. D84(2011), 084010 [arXiv:1101.0931 [hep-th]]

    Pith/arXiv arXiv 2011

  33. [33]

    Magueijo and L

    J. Magueijo and L. Smolin, Phys. Rev. Lett.88(2002), 190403 [arXiv:hep-th/0112090 [hep-th]]

    Pith/arXiv arXiv 2002

  34. [34]

    Amelino-Camelia, J

    G. Amelino-Camelia, J. R. Ellis, N. E. Mavromatos, D. V . Nanopoulos and S. Sarkar, Nature393(1998), 763-765 [arXiv:astro-ph/9712103 [astro-ph]]

    Pith/arXiv arXiv 1998

  35. [35]

    Magueijo and L

    J. Magueijo and L. Smolin, Phys. Rev. D67(2003), 044017 [arXiv:gr-qc/0207085 [gr-qc]]

    Pith/arXiv arXiv 2003

  36. [36]

    R. C. Myers and M. Pospelov, Phys. Rev. Lett.90(2003), 211601 [arXiv:hep-ph/0301124 [hep-ph]]

    Pith/arXiv arXiv 2003

  37. [37]

    Kowalski-Glikman and S

    J. Kowalski-Glikman and S. Nowak, Int. J. Mod. Phys. D12 (2003), 299-316 [arXiv:hep-th/0204245 [hep-th]]

    Pith/arXiv arXiv 2003

  38. [38]

    Amelino-Camelia, Nature418(2002), 34-35 [arXiv:gr- qc/0207049 [gr-qc]]

    G. Amelino-Camelia, Nature418(2002), 34-35 [arXiv:gr- qc/0207049 [gr-qc]]

    arXiv 2002

  39. [39]

    Guvendi, O

    A. Guvendi, O. Mustafa and N. Jafari, Phys. Lett. B866(2025), 139547 [arXiv:2504.10537 [physics.gen-ph]]

    arXiv 2025

  40. [40]

    Majid and H

    S. Majid and H. Ruegg, Phys. Lett. B334(1994), 348-354 [arXiv:hep-th/9405107 [hep-th]]

    Pith/arXiv arXiv 1994

  41. [41]

    J. M. Carmona, J. L. Cortes and J. J. Relancio, Phys. Rev. D94 (2016) no.8, 084008 [arXiv:1609.01347 [hep-th]]

    Pith/arXiv arXiv 2016

  42. [42]

    Majid, Lect

    S. Majid, Lect. Notes Phys.541(2000), 227-276 [arXiv:hep- th/0006166 [hep-th]]

    arXiv 2000

  43. [43]

    Khodadi and K

    M. Khodadi and K. Nozari, Commun. Theor. Phys.71(2019) no.6, 677-684

    2019

  44. [44]

    Jafari and A

    N. Jafari and A. Guvendi, Phys. Lett. B866(2025), 139515 [arXiv:2511.05641 [gr-qc]]

    arXiv 2025

  45. [45]

    Jafari and A

    N. Jafari and A. Boumali, Phys. Lett. B871(2025), 140019 [arXiv:2510.05117 [physics.gen-ph]]

    arXiv 2025

  46. [46]

    Boumali, N

    A. Boumali, N. Jafari, B. Shukirgaliyev and F. Serdouk, Eur. Phys. J. C85(2025) no.10, 1147 [arXiv:2511.11709 [gr-qc]]

    arXiv 2025

  47. [47]

    E. P. Verlinde, JHEP04(2011), 029 [arXiv:1001.0785 [hep- th]]

    Pith/arXiv arXiv 2011

  48. [48]

    J. D. Bekenstein, Phys. Rev. D7(1973), 2333-2346

    1973

  49. [49]

    S. W. Hawking, Commun. Math. Phys.43(1975), 199-220 [er- ratum: Commun. Math. Phys.46(1976), 206]

    1975

  50. [50]

    Padmanabhan, Rept

    T. Padmanabhan, Rept. Prog. Phys.73(2010), 046901 [arXiv:0911.5004 [gr-qc]]

    Pith/arXiv arXiv 2010

  51. [51]

    C. Q. Geng, Y . T. Hsu, J. R. Lu and L. Yin, Eur. Phys. J. C80 (2020) no.1, 21 [arXiv:1911.06046 [astro-ph.CO]]

    Pith/arXiv arXiv 2020

  52. [52]

    A. J. M. Medved, Class. Quant. Grav.22(2005), 133-142 [arXiv:gr-qc/0406044 [gr-qc]]

    Pith/arXiv arXiv 2005

  53. [53]

    Pourhassan and M

    B. Pourhassan and M. Faizal, Eur. Phys. J. C77(2017) no.2, 96 [arXiv:1703.00517 [quant-ph]]

    Pith/arXiv arXiv 2017

  54. [54]

    Nojiri, S

    S. Nojiri, S. D. Odintsov and E. N. Saridakis, Eur. Phys. J. C79 (2019) no.3, 242 [arXiv:1903.03098 [gr-qc]]

    Pith/arXiv arXiv 2019

  55. [55]

    Nojiri, S

    S. Nojiri, S. D. Odintsov and V . Faraoni, Phys. Rev. D105 (2022) no.4, 044042 [arXiv:2201.02424 [gr-qc]]

    Pith/arXiv arXiv 2022

  56. [56]

    Chatterjee and A

    A. Chatterjee and A. Ghosh, Phys. Rev. Lett.125(2020) no.4, 041302 [arXiv:2007.15401 [gr-qc]]

    Pith/arXiv arXiv 2020

  57. [57]

    Jusufi and A

    K. Jusufi and A. Sheykhi, Phys. Lett. B836(2023), 137621 [arXiv:2210.01584 [gr-qc]]

    Pith/arXiv arXiv 2023

  58. [58]

    D. J. Zamora and C. Tsallis, Eur. Phys. J. C82(2022) no.8, 689 [arXiv:2201.03385 [gr-qc]]

    Pith/arXiv arXiv 2022

  59. [59]

    Jalalzadeh, H

    S. Jalalzadeh, H. Moradpour and P. V . Moniz, Phys. Dark Univ. 42(2023), 101320 [arXiv:2308.12089 [gr-qc]]

    Pith/arXiv arXiv 2023

  60. [60]

    Nojiri, S

    S. Nojiri, S. D. Odintsov and T. Paul, Phys. Lett. B831(2022), 137189 [arXiv:2205.08876 [gr-qc]]

    Pith/arXiv arXiv 2022

  61. [61]

    G. G. Luciano and A. Sheykhi, Phys. Dark Univ.42(2023), 101319 [arXiv:2304.11006 [hep-th]]

    Pith/arXiv arXiv 2023

  62. [62]

    Basilakos, A

    S. Basilakos, A. Lymperis, M. Petronikolou and E. N. Saridakis, Eur. Phys. J. C84(2024) no.3, 297 [arXiv:2308.01200 [gr-qc]]

    Pith/arXiv arXiv 2024

  63. [63]

    Jacobson, Phys

    T. Jacobson, Phys. Rev. Lett.75(1995), 1260-1263 [arXiv:gr- 14 qc/9504004 [gr-qc]]

    arXiv 1995

  64. [64]

    Ebrahimi and A

    E. Ebrahimi and A. Sheykhi, Phys. Dark Univ.45(2024), 101518 [arXiv:2405.13096 [gr-qc]]

    Pith/arXiv arXiv 2024

  65. [65]

    Capozziello and M

    S. Capozziello and M. Shokri, Eur. Phys. J. C85(2025) no.2, 200 [arXiv:2501.12987 [gr-qc]]

    Pith/arXiv arXiv 2025

  66. [66]

    Nojiri and S

    S. Nojiri and S. D. Odintsov, [arXiv:2607.06582 [gr-qc]]

    Pith/arXiv arXiv

  67. [67]

    Nojiri, S

    S. Nojiri, S. D. Odintsov and V . Faraoni, Astrophysics65 (2022) no.4, 534-551 [arXiv:2208.10235 [gr-qc]]

    Pith/arXiv arXiv 2022

  68. [68]

    H. B. Callen, Thermodynamics and an Introduction to Thermo- statistics. John Wiley & Sons, 2nd ed., 1985

    1985

  69. [69]

    Anand, S

    A. Anand, S. Devdutt, K. Jusufi and E. N. Saridakis, Eur. Phys. J. C86(2026) no.5, 534 [arXiv:2511.04613 [gr-qc]]

    Pith/arXiv arXiv 2026

  70. [70]

    Anand, K

    A. Anand, K. Jusufi, S. Basilakos and E. N. Saridakis, Eur. Phys. J. C86(2026) no.2, 126 [arXiv:2512.13769 [gr-qc]]

    arXiv 2026

  71. [71]

    S. W. Hawking and D. N. Page, Commun. Math. Phys.87 (1983), 577

    1983

  72. [72]

    Elizalde, S

    E. Elizalde, S. Nojiri and S. D. Odintsov, Universe11(2025) no.2, 60 [arXiv:2502.05801 [gr-qc]]

    Pith/arXiv arXiv 2025

  73. [73]

    P. C. W. Davies, Rept. Prog. Phys.41(1978), 1313-1355

    1978

  74. [74]

    R. M. Wald, Living Rev. Rel.4(2001), 6 [arXiv:gr-qc/9912119 [gr-qc]]

    Pith/arXiv arXiv 2001

  75. [75]

    Vasileiou, J

    V . Vasileiou, J. Granot, T. Piran and G. Amelino-Camelia, Na- ture Phys.11(2015) no.4, 344-346

    2015

  76. [76]

    Chamblin, R

    A. Chamblin, R. Emparan, C. V . Johnson and R. C. Myers, Phys. Rev. D60(1999), 064018 [arXiv:hep-th/9902170 [hep- th]]

    Pith/arXiv arXiv 1999

  77. [77]

    S. W. Wei, Y . X. Liu and R. B. Mann, Phys. Rev. Lett.129 (2022) no.19, 191101 [arXiv:2208.01932 [gr-qc]]

    Pith/arXiv arXiv 2022

  78. [78]

    S. W. Wei, Y . X. Liu and R. B. Mann, Phys. Rev. D110(2024) no.8, L081501 [arXiv:2409.09333 [gr-qc]]

    Pith/arXiv arXiv 2024

  79. [79]

    F. W. Stecker, Astropart. Phys.35(2011), 95-97 [arXiv:1102.2784 [astro-ph.HE]]

    Pith/arXiv arXiv 2011

  80. [80]

    Jacob and T

    U. Jacob and T. Piran, Nature Phys.3(2007), 87-90 [arXiv:hep- ph/0607145 [hep-ph]]

    arXiv 2007

Showing first 80 references.