Thermodynamics and P-v criticality of RN-AdS black hole surrounded by PFDM on the EGUP framework

Bo-Li Liu; Qi-Quan Li; Yu Zhang

REVIEW 3 major objections 5 minor 61 references

Thermodynamics and P-v criticality of RN-AdS black hole surrounded by PFDM on the EGUP framework

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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

Pith's one-line read The paper claims that extended generalized uncertainty principle (EGUP) corrections to the thermodynamics of a charged, dark-matter-surrounded anti-de Sitter black hole shrink the unstable intermediate branch, lower the critical pressure…

desk verdict Routine EGUP extension to PFDM; temperature and P-v parts are plausible, but the entropy fails the first law, so the swallowtail phase-transition claim is unsupported. read the letter →

arxiv 2509.08005 v1 pith:QSTBTO3B submitted 2025-09-08 gr-qc

Bo-Li Liu , Yu Zhang , Qi-Quan Li This is my paper · ORCID

classification gr-qc

keywords extendedgeneralizeduncertaintyprincipleperfectfluiddarkmatterReissner-Nordstromanti-deSitterblackholethermodynamicsP-vcriticalityfirst-orderphasetransitionHawkingtemperatureGibbsfreeenergy

open problems Dark Matter

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper sets out to graft the extended generalized uncertainty principle (EGUP) onto the thermodynamics of a Reissner-Nordström anti-de Sitter black hole surrounded by perfect fluid dark matter (PFDM), producing explicit corrected formulas for Hawking temperature, heat capacity, entropy, and Gibbs free energy. It argues that the corrections favor thermodynamic stability by shrinking the negative-heat-capacity region of the intermediate black hole branch, and that the familiar small-to-large black hole phase transition survives with a slightly moved equilibrium point. A sympathetic reader should care because EGUP is a proposed bridge between quantum gravity and black hole physics, and this work tests whether that bridge preserves the van der Waals-like phase structure of AdS black holes. The paper also derives remnant mass and temperature, giving a concrete prediction for where black hole evaporation stops.

What carries the argument

The load-bearing device is the EGUP uncertainty relation, $\Delta x_i \Delta p_j \geq (\hbar/2)\delta_{ij}[1 + \beta l_p^2 (\Delta p_j)^2/\hbar^2 + \alpha (\Delta x_i)^2/L^2]$, combined with the information-theoretic substitution $dA/dS \simeq (\gamma/\ln 2)\Delta X\Delta P$ used to convert surface gravity into a corrected Hawking temperature. With $\Delta X \simeq 2r_+$ and $\gamma = 4\ln 2$ fixed by demanding the HUP limit reproduce $T = 1/(4\pi r_+)$, this machinery injects the EGUP parameters into every subsequent quantity: the heat capacity follows from $C_P = (\partial M/\partial T)_P$, the entropy from integrating $C/T$, the equation of state from inverting the corrected temperature for $P$, and the Gibbs free energy from $G = M - TS$. The same substitution also produces the minimum horizon radii $r_{\rm rem(EGUP)}$ and $r_{\rm rem(GUP)}$ that define black hole remnants.

What would settle it

Recompute the EGUP-corrected Hawking temperature using the full position-uncertainty bounds of Eq. (6) instead of the simplifying assumption $\Delta X \simeq 2r_+$, then re-derive the heat capacity; if the negative-heat-capacity region does not shrink with $\alpha$ and $\beta$, the stability conclusion is an artifact of the simplification. A direct semiclassical calculation of $dA/dS$ for the PFDM metric would also either confirm or overturn the substitution at the heart of the paper.

Watch

Extended reading notes

Core claim

The paper claims that applying the EGUP to the RN-AdS black hole surrounded by PFDM yields a consistent set of corrected thermodynamic quantities: the Hawking temperature in Eq. (18), heat capacity in Eq. (33), entropy in Eq. (44), and Gibbs free energy in Eq. (53). Using these, the paper argues that the negative-heat-capacity region corresponding to the intermediate black hole shrinks as the EGUP parameters $\alpha$ and $\beta$ increase, so the correction favors thermodynamic stability. It further claims that the first-order phase transition signaled by a swallowtail in the $G$-$T$ diagram survives for $P < P_c$, with the equilibrium point shifting only slightly, and that for $P = P_c$ and $P > P_c$ the phase-transition behavior mirrors the uncorrected case, even though the critical pressure, critical temperature, and the ratio $P_c v_c / T_c$ all decrease with the EGUP parameters.

Load-bearing premise

The entire chain rests on treating the uncertainty-product substitution $dA/dS \simeq (\gamma/\ln 2)\Delta X\Delta P$ with $\Delta X \simeq 2r_+$ as valid for a charged, dark-matter background; if that heuristic fails, every corrected quantity—temperature, heat capacity, entropy, and free energy—changes.

Editorial extensions

If this is right

For fixed charge, dark matter parameter, and pressure, increasing $\alpha$ and $\beta$ shrinks the interval of horizon radii with negative heat capacity, so the small and large black hole branches become stable over a wider range.
The critical pressure $P_c$, critical specific volume $v_c$, and the ratio $P_c v_c / T_c$ all decrease as $\alpha$ and $\beta$ grow, shifting the phase diagram toward lower pressure and temperature.
The swallowtail in the Gibbs free energy versus temperature plot persists for $P < P_c$, meaning the first-order small-black-hole/large-black-hole transition survives EGUP corrections.
For $P = P_c$, a second-order phase transition remains, signaled by a single heat-capacity divergence and the disappearance of the intermediate branch, while for $P > P_c$ no phase transition occurs.
The EGUP-corrected entropy is smaller than the area-law entropy for the same horizon radius, and the deficit grows with the EGUP parameters.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial extension: the same $dA/dS$ substitution with $\Delta X \simeq 2r_+$ could be applied to other AdS black holes, such as rotating or higher-dimensional ones; the paper's logic suggests the qualitative stabilization and preservation of the phase transition would persist, but the size of the shifts would depend on the metric details.
Editorial extension: because $\gamma$ is calibrated to recover the HUP limit, the correction is forced to vanish at large radii; a sharper test of the framework would probe intermediate radii where the correction peaks, for instance by computing quasinormal mode frequencies or lensing signatures that depend on the corrected horizon temperature.
Editorial extension: the monotonic decrease of $P_c$ with $\alpha$ and $\beta$ in Table 1 hints at a systematic relation that could be extracted by scanning more parameter values; such a relation would let future observations or analogue experiments bound the EGUP parameters.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

What you should know: this is a routine extension of the EGUP-based black hole thermodynamics program to the PFDM background. The temperature and P-v criticality sections are plausible; the entropy, however, fails the first law with the stated temperature, so the Gibbs free energy swallowtail plots and the first-order phase transition conclusion are not supported.

The genuinely new bit is the specific background: no one has done EGUP corrections for this RN-AdS+PFDM line element before. The temperature in Eq. (18), the EOS in Eq. (50), and the critical-point table are consistent with the uncorrected limits and follow the same path as Tan's quintessence paper. The heat capacity analysis, Figs. 2 and 6, is based on C = (∂M/∂T) and does not depend on the entropy, so the claim that EGUP shrinks the unstable intermediate branch may be right.

The problem is the entropy. Take the alpha=0 limit. From Eq. (19), T_GUP can be written as F/(2π r_+ (1+sqrt(...))), with F the metric function combination. The first law dM = T dS then requires dS/dr_+ = π r_+ (1+sqrt(...)) = 2π r_+ - π β l_p^2/(8 r_+) to leading order. Differentiating the paper's Eq. (45) gives 2π r_+ - π β l_p^2/(4 r_+), off by a factor of two. So dM = T dS fails already at O(β). That error carries into the full EGUP entropy Eq. (44) and into G = M - T S in Eq. (53). The swallowtail curves in Fig. 5 are therefore not Legendre transforms of a consistent thermodynamic system. This is a load-bearing flaw, not a typo.

The heuristic input—Delta X ≈ 2r_+ and γ fitted to the HUP limit—is imported from prior work. That is a model assumption, not a derivation, and makes the whole framework phenomenological. I would not call it fatal, but it is worth noting.

Who gets value: readers working on GUP/EUP-corrected black hole thermodynamics will find the temperature and P-v parts useful, and the entropy error is instructive. It deserves a serious referee, because the mistake is fixable and the rest is standard. Recommendation: send to review, but the current version should not be accepted; the entropy and G must be redone before any reliable phase-transition statement.

Referee Report

3 major / 5 minor

Summary. This paper studies extended generalized uncertainty principle (EGUP) corrections to the thermodynamics and P-v criticality of the Reissner-Nordström-AdS black hole surrounded by perfect fluid dark matter. The authors start from the first law in the extended phase space, use a heuristic relation between entropy-area change and position-momentum uncertainties to derive an EGUP-corrected Hawking temperature (Eq. (18)), and then compute the heat capacity, entropy, remnant quantities, equation of state, and Gibbs free energy. They analyze heat-capacity divergence and G-T 'swallowtail' diagrams. The central claims are that larger EGUP parameters shrink the unstable intermediate black hole branch, that the EGUP correction is beneficial for thermodynamic stability, and that a first-order phase transition persists for P<P_c with a slightly shifted equilibrium point and reduced critical pressure and temperature as alpha and beta increase.

Significance. Should the derivations hold, the paper would provide a self-contained extension of GUP-corrected black-hole thermodynamics to a PFDM background, with explicit analytic formulas and a parameter-free calibration of gamma from the HUP limit. The temperature and equation-of-state parts reduce correctly to the HUP case, and the graphical analysis is systematic. However, the central thermodynamic consistency condition dM = T dS fails for the derived entropy, and because the Gibbs free energy (and hence the swallowtail phase-transition curves) is built from that entropy, the paper's main phase-transition conclusions are not currently supported by the manuscript. The first-law failure is a mathematical error localized in the entropy integration; the temperature and P-v criticality results may survive a correction, but all S-dependent results need to be rederived before the phase-transition claims can be assessed.

major comments (3)

[Section 3, Eqs. (19), (45), and (53)] The GUP entropy is not thermodynamically conjugate to the GUP temperature, so the Gibbs free energy and the swallowtail analysis are not supported. With F = 1 - Q^2/r_+^2 + 3r_+^2/l^2 + lambda/r_+, Eq. (11) gives dM/dr_+ = F/2, and Eq. (19) is equivalent to T_GUP = F/[2 pi r_+ (1 + sqrt(1 - beta l_p^2/(4r_+^2)))]. The first law (13) at fixed Q, P, lambda then requires dS/dr_+ = pi r_+ (1 + sqrt(1 - beta l_p^2/(4r_+^2))) = 2 pi r_+ - pi beta l_p^2/(8r_+) + O(beta^2). Differentiating the paper's Eq. (45) gives dS_GUP/dr_+ = 2 pi r_+ - pi beta l_p^2/(4r_+), disagreeing already at first order in beta. This inconsistency propagates into the full EGUP entropy Eq. (44) and into G = M - T S in Eq. (53), which is the quantity whose swallowtail curves are shown in Fig. 5. The phase-transition conclusions drawn from G are therefore not established; the temperature and equation-of-state results, which do not use S, are not affected by this particular error.
[Section 3, Eq. (43)] The entropy integral is not written correctly, and the stated derivation cannot produce a consistent S. The formula S = Integral (partial M / T)_{Q,P,lambda} should read S = Integral (1/T)(partial M/partial r_+) d r_+ at fixed Q, P, lambda, and the replacement by Integral C dT/T is only meaningful if C is computed as (partial M/partial T)_P. As written, the display is formally ambiguous. More importantly, the failure of T and S to satisfy dM = T dS shows that Eq. (44) is not the result of a correct integration; the authors should recompute S directly from dS = dM/T and then rederive all S-dependent quantities, including Eq. (53) and Fig. 5.
[Section 2 and Eq. (15)] The entire EGUP correction is fixed by the heuristic dA/dS ~ (gamma/ln2) Delta X Delta P with Delta X ~ 2 r_+, and gamma = 4 ln2 is chosen to reproduce the HUP limit. This is an imported assumption rather than a derivation, and the HUP limit is enforced by construction rather than obtained from the model. The functional form of every corrected quantity, including the critical pressure and temperature in Table 1, depends on this choice. The authors should test the robustness of their qualitative conclusions by, for example, repeating the analysis for Delta X = k r_+ with k a free parameter of order one, and discuss whether the stability and phase-transition conclusions are independent of k.

minor comments (5)

[Eqs. (44), (45), and (53)] Logarithms of dimensionful quantities appear (L^4, r_+, and combinations with l_p^2). Since the figures set L = l_p = 1, this is hidden, but in a dimensionful formulation the arguments of the logarithms should be made dimensionless.
[Eq. (12) and surrounding text] The paper defines kappa as f'(r_+), whereas the standard surface gravity is f'(r_+)/2. The subsequent conventions are internally consistent after fixing gamma, but the nonstandard definition should be stated explicitly to avoid confusion.
[Eq. (37)] The remnant-mass expression appears to contain typos, including 'alpha beta 12 p' in the denominator and an inconsistent power of l_p. Please check and correct the formula.
[Eqs. (23)-(26)] The roots r_+1 and r_+2 are displayed with plus-minus signs, but only the physical positive root is retained for the constraint analysis. Please specify which branches are used and why.
[Section 4 and Table 1] Table 1 shows that the ratio P_c v_c/T_c decreases from 0.389 to 0.284 as alpha and beta increase, a change of about 27 percent. The text says the phase-equilibrium point changes only slightly; this quantitative shift in the critical parameters should be mentioned and interpreted.

Circularity Check

0 steps flagged · score 2.0 of 10

No material circularity: the EGUP corrections are explicit model assumptions, and the phase-transition results are computed outputs rather than fitted or redefined inputs.

full rationale

The paper's central chain is: adopt EGUP (Eq. 4), import the standard heuristic dA/dS ~ (gamma/ln2) Delta X Delta P (Eq. 15) and Delta X ~ 2 r_+ from Refs. [50,56,57,58], then compute T_EGUP (Eq. 18), heat capacity (Eq. 33), entropy (Eq. 44), equation of state (Eq. 50), and Gibbs free energy (Eq. 53). These later quantities are genuine outputs of the stated assumptions: no parameter is fitted to the phase-transition data, alpha and beta are varied by hand, and gamma = 4 ln 2 is fixed by reproducing the HUP limit rather than by the target results. The critical points in Table 1 and the swallowtail behavior in Fig. 5 are obtained by solving the standard critical-point condition and from G = M - T S, so they are not inserted by construction. The only mild concern is that the temperature correction is not derived from first principles but is inherited from prior work, including one author self-citation [57] supporting Eq. (15); however, the same relation is also attributed to non-self references [50,56], so the self-citation is not load-bearing. A possible failure of the first law dM = T dS associated with Eq. (44) is an algebraic consistency issue, not a circularity, because the entropy is not used as an input to force the claimed phase transition. Accordingly, no specific circular step is identified, and the score reflects only the minor reliance on imported heuristics rather than any reduction of the central claim to its inputs.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The model depends on two phenomenological parameters, alpha and beta, and on three imported heuristics: the EGUP relation, the identification Delta X approximately 2 r_plus, and the dA/dS versus Delta X Delta P relation. No data constrain these ingredients; the paper is an illustrative parameter scan over hand-chosen values.

free parameters (3)

alpha (EUP parameter)
Dimensionless large-scale correction parameter; varied by hand as 0.01, 0.03, 0.05, and 0.06 in figures, not fitted to data.
beta (GUP parameter)
Dimensionless minimum-length correction parameter; same hand-chosen values as alpha.
gamma (correction factor) = 4 ln 2
Chosen so that the EGUP temperature reduces to the standard Hawking temperature in the HUP limit; a calibration constant, not data-fitted.

assumptions (5)

domain assumption The EGUP relation of Eq. (4) governs position-momentum uncertainty near black-hole horizons.
Borrowed from quantum-gravity phenomenology (Refs. [45,46]); not derived in this paper.
ad hoc to paper A particle absorbed by the black hole satisfies Delta X approximately 2 r_plus, so the uncertainty term depends on the horizon radius.
Introduced just before Eq. (17); this is the main route by which EGUP enters the temperature.
ad hoc to paper dA/dS approximately gamma over ln2 times Delta X Delta P, with gamma fixed to 4 ln2, relates entropy change to the uncertainty product.
Equation (15), imported from Ref. [56]; it normalizes the corrected temperature to the HUP result.
domain assumption The PFDM metric is f(r)=1-2M/r+Q^2/r^2+r^2/l^2+lambda over r times ln(r over |lambda|) with lambda greater than 0.
Taken from Kiselev and Xu et al. (Refs. [18,19,28]) as the dark-matter background.
standard math The extended first law dM=TdS+Phi dQ+VdP+Pi dlambda holds, with P equal to negative Lambda over 8pi.
Basis of extended-phase-space black hole thermodynamics; used to define T, S, V, and G.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Thermodynamics and P-v criticality of RN-AdS black hole surrounded by PFDM on the EGUP framework." pith.science (2026). https://pith.science/paper/QSTBTO3B

@misc{pith2026250908005,
  author       = {Pith},
  title        = {Pith review of: Thermodynamics and P-v criticality of RN-AdS black hole surrounded by PFDM on the EGUP framework},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QSTBTO3B}},
  note         = {Machine review of arXiv:2509.08005}
}

abstract

In this paper, we systematically study the thermodynamic properties and P-v criticality of the RN-AdS black hole surrounded by PFDM in the framework of the extended generalized uncertainty principle (EGUP). In the extended phase space, starting from the first law of black hole thermodynamics, the EGUP-corrected Hawking temperature, heat capacity, and entropy are derived. Then, we use graphical methods to analyze the effects of the EGUP parameters on these thermodynamic quantities. Analysis of the heat capacity indicates that the EGUP correction is beneficial for maintaining the thermodynamic stability of black holes. Finally, based on the modified thermodynamic quantities, we obtain the EGUP-corrected black hole equation of state and Gibbs free energy. The G-T diagram with the EGUP correction effect is analyzed, and the results show that when $P<P_c$, a ``swallowtail'' behavior emerges, indicating that the black hole undergoes a first order phase transition. The position of the phase equilibrium point changes slightly as the EGUP parameters increase. For the cases of $P=P_c$ and $P>P_c$, we combine the analysis of the relationship between heat capacity and event horizon radius, finding that the phase transition characteristics depicted in the G-T diagram are similar to those without the correction effect.

Figures

Figures reproduced from arXiv: 2509.08005 by the authors.

**Figure 2.** shows that there are divergent points in heat capacity, indicating the occurrence of phase transition. We take the case of α = β = 0.03 as an example for discussion. When r+ < 0.483 and r+ > 0.811, the black hole heat capacity is positive, indicating that the black hole exists in a stable state, namely the small black hole (SBH) and the large black hole (LBH). In the range 0.483 < r+ < 0.811, the black hole heat ca… view at source ↗

**Figure 3.** The behavior of S EGUP vs. r+ for different parameters α and β (lp = L = 1). 4. The P-v criticality in the EGUP framework From Eq. (18), we can deduce the relationship between P and r+, which is expressed as P = L 2Q 2 − r+ L 2λ + r+ L 2 − 4Q 2α + r+ (4α (λ + r+) − ω) 8πr 4 + L 2 + 4αr 2 + , (47) where ω = 2L 2 πT   1 + s 1 + β [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: The behavior of P vs. v for different parameters α and β (lp = L = 1, Q = 0.3, λ = 0.1, Tc (see [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: c and Fig. 5d, we plot the corresponding relationship between CEGUP and r+ ( [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: Using l = q 3 8πP , we plot two distinct behaviors of CEGUP vs. r+ for different parameters α and β (lp = L = 1, Q = 0.3, λ = 0.1, Pc (see [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

61 extracted references · 60 canonical work pages

[1]

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

work page 1973
[2]

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

work page 1975
[3]

J. M. Bardeen, B. Carter and S. W. Hawking, Commun. Math. Phys.31(1973), 161-170

work page 1973
[4]

J. D. E. Creighton and R. B. Mann, Phys. Rev. D52 (1995), 4569-4587

work page 1995
[5]

V . E. Hubeny, Class. Quantum Gravity32(2015), 124010

work page 2015
[6]

Padmanabhan, Class

T. Padmanabhan, Class. Quantum Gravity19(2002), 5387-5408

work page 2002
[7]

Teitelboim, Phys

C. Teitelboim, Phys. Lett. B158(1985), 293-297

work page 1985
[8]

J. D. Brown and C. Teitelboim, Phys. Lett. B195(1987), 177-182

work page 1987

Show all 61 references

[9]

Sadeghi, B

J. Sadeghi, B. Pourhassan, S. Noori Gashti, S. Upadhyay and E. N. Mezerji, Phys. Scripta98(2023), 025305

2023
[10]

S. W. Wei and Y . X. Liu, Phys. Rev. D90(2014), 044057

2014
[11]

D. V . Singh and S. Siwach, Phys. Lett. B808(2020), 135658

2020
[12]

Hamil, B

B. Hamil, B. C. Lütfüo ˘glu and L. Dahbi, Mod. Phys. Lett. A39(2024), 2450161

2024
[13]

G. G. Luciano and E. Saridakis, JHEP12(2023), 114. 8

2023
[14]

X. Y . Guo, H. F. Li, L. C. Zhang and R. Zhao, Phys. Rev. D100(2019), 064036

2019
[15]

Zhang, X

J. Zhang, X. Chen and J. Tao, Phys. Dark Univ.46(2024), 101594

2024
[16]

R. B. Wang, S. J. Ma, L. You, Y . C. Tang, Y . H. Feng, X. R. Hu and J. B. Deng, Eur. Phys. J. C84(2024), 1161

2024
[17]

P. A. R. Adeet al.[Planck], Astron. Astrophys.594 (2016), A13

2016
[18]

V . V . Kiselev, Class. Quantum Gravity20(2003), 1187- 1198

2003
[19]

V . V . Kiselev, [arXiv:gr-qc/0303031 [gr-qc]]

arXiv
[20]

Liang, Y

X. Liang, Y . P. Hu, C. H. Wu and Y . S. An, Eur. Phys. J. C 83(2023), 1009

2023
[21]

A. Sood, M. S. Ali, J. K. Singh and S. G. Ghosh, Chin. Phys. C48(2024), 065109

2024
[22]

Hamil and B

B. Hamil and B. C. Lütfüo ˘glu, Annals Phys.472(2025), 169861

2025
[23]

D. V . Singh, S. Upadhyay, Y . Myrzakulov, K. Myrzakulov, B. Singh and M. Kumar, Nucl. Phys. B1016(2025), 116915

2025
[24]

Q. Q. Li, Y . Zhang, Q. Sun, C. H. Xie and Y . L. Lou, Chin. J. Phys.92(2024), 1-9

2024
[25]

Abdusattar, Phys

H. Abdusattar, Phys. Dark Univ.40(2023), 101228

2023
[26]

Sekhmani, S

Y . Sekhmani, S. K. Maurya, M. K. Jasim, ˙I. Sakallı, J. Rayimbaev and I. Ibragimov, Eur. Phys. J. C85(2025), 229

2025
[27]

M. H. Li and K. C. Yang, Phys. Rev. D86(2012), 123015

2012
[28]

Z. Xu, X. Hou, J. Wang and Y . Liao, Adv. High Energy Phys.2019(2019), 2434390

2019
[29]

L. J. Garay, Int. J. Mod. Phys. A10(1995), 145-166

1995
[30]

Capozziello, G

S. Capozziello, G. Lambiase and G. Scarpetta, Int. J. Theor. Phys.39(2000), 15-22

2000
[31]

Konishi, G

K. Konishi, G. Paffuti and P. Provero, Phys. Lett. B234 (1990), 276-284

1990
[32]

Nozari and A

K. Nozari and A. Etemadi, Phys. Rev. D85(2012), 104029

2012
[33]

Maggiore, Phys

M. Maggiore, Phys. Lett. B319(1993), 83-86

1993
[34]

A. F. Ali, S. Das and E. C. Vagenas, Phys. Lett. B678 (2009), 497-499

2009
[35]

Menculini, O

L. Menculini, O. Panella and P. Roy, Phys. Rev. D87 (2013), 065017

2013
[36]

Majumder, Phys

B. Majumder, Phys. Rev. D84(2011), 064031

2011
[37]

Battista, S

E. Battista, S. Capozziello and A. Errehymy, Eur. Phys. J. C84(2024), 1314

2024
[38]

D. J. Gross and P. F. Mende, Phys. Lett. B197(1987), 129-134

1987
[39]

Amati, M

D. Amati, M. Ciafaloni and G. Veneziano, Phys. Lett. B 216(1989), 41-47

1989
[40]

Maggiore, Phys

M. Maggiore, Phys. Lett. B304(1993), 65-69

1993
[41]

Bojowald and A

M. Bojowald and A. Kempf, Phys. Rev. D86(2012), 085017

2012
[42]

Scardigli, Phys

F. Scardigli, Phys. Lett. B452(1999), 39-44

1999
[43]

Scardigli and R

F. Scardigli and R. Casadio, Class. Quantum Gravity20 (2003), 3915-3926

2003
[44]

Banerjee and S

R. Banerjee and S. Ghosh, Phys. Lett. B688(2010), 224- 229

2010
[45]

M. I. Park, Phys. Lett. B659(2008), 698-702

2008
[46]

Bolen and M

B. Bolen and M. Cavaglia, Gen. Relativ. Gravit.37 (2005), 1255-1262

2005
[47]

Tan, Phys

B. Tan, Phys. Lett. B861(2025), 139289

2025
[48]

S. Wu, B. Q. Wang, H. Chen and Z. W. Long, Eur. Phys. J. Plus137(2022), 970

2022
[49]

Chaudhary, M

S. Chaudhary, M. D. Sultan, A. Malik, A. ur Rehman, A. Övgün and A. A. Ghfar, Nucl. Phys. B1006(2024), 116635

2024
[50]

H. Chen, B. C. Lütfüo ˘glu, H. Hassanabadi and Z. W. Long, Phys. Lett. B827(2022), 136994

2022
[51]

Farahani, H

N. Farahani, H. Hassanabadi, W. S. Chung, B. C. Lüt- füo˘glu and S. Zarrinkamar, EPL132(2020), 50009

2020
[52]

Hamil, B

B. Hamil, B. C. Lütfüo ˘glu and L. Dahbi, Int. J. Mod. Phys. A38(2023), 2350052

2023
[53]

Hassanabadi, N

H. Hassanabadi, N. Farahani, W. S. Chung and B. C. Lüt- füo˘glu, EPL130(2020), 40001

2020
[54]

Cavaglia and S

M. Cavaglia and S. Das, Class. Quantum Gravity21 (2004), 4511-4522

2004
[55]

Shaymatov, B

S. Shaymatov, B. Ahmedov and M. Jamil, Eur. Phys. J. C 81(2021), 588 [erratum: Eur. Phys. J. C81(2021), 724]

2021
[56]

Xiang and X

L. Xiang and X. Q. Wen, JHEP10(2009), 046

2009
[57]

B. L. Liu, Y . Zhang, Q. Q. Li and C. H. Xie, Eur. Phys. J. Plus140(2025), 295

2025
[58]

Hassanabadi, E

H. Hassanabadi, E. Maghsoodi and W. S. Chung, Eur. Phys. J. C79(2019), 358

2019
[59]

Z. W. Lin, C. Ma, Y . Zhang, Q. Li and P. F. Duan, Int. J. Mod. Phys. A37(2022), 2250124. 9

2022
[60]

Z. W. Lin, Y . Zhang, Q. Li, C. Ma and P. F. Duan, Int. J. Theor. Phys.61(2022), 199

2022
[61]

Kubiznak and R

D. Kubiznak and R. B. Mann, JHEP07(2012), 033. 10

2012

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

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

work page 1973

[2] [2]

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

work page 1975

[3] [3]

J. M. Bardeen, B. Carter and S. W. Hawking, Commun. Math. Phys.31(1973), 161-170

work page 1973

[4] [4]

J. D. E. Creighton and R. B. Mann, Phys. Rev. D52 (1995), 4569-4587

work page 1995

[5] [5]

V . E. Hubeny, Class. Quantum Gravity32(2015), 124010

work page 2015

[6] [6]

Padmanabhan, Class

T. Padmanabhan, Class. Quantum Gravity19(2002), 5387-5408

work page 2002

[7] [7]

Teitelboim, Phys

C. Teitelboim, Phys. Lett. B158(1985), 293-297

work page 1985

[8] [8]

J. D. Brown and C. Teitelboim, Phys. Lett. B195(1987), 177-182

work page 1987