pith. sign in

arxiv: 2606.08182 · v1 · pith:ASV4N537new · submitted 2026-06-06 · 🌀 gr-qc

Higher-dimensional quantum-corrected Oppenheimer-Snyder model with a cosmological constant

Shudi Jiang , Jianhui Lin , Xiangdong Zhang This is my paper

Pith reviewed 2026-06-27 19:29 UTC · model grok-4.3

classification 🌀 gr-qc
keywords quantum correctionsOppenheimer-Snyder modelcosmological constantAdS spacetimeblack hole thermodynamicsextended phase spacephase transitionshigher-dimensional gravity
0
0 comments X

The pith

Quantum corrections make temperature of small AdS black holes tend to zero and add an extra phase transition in heat capacity.

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

The paper extends the higher-dimensional quantum-corrected Oppenheimer-Snyder model by including a cosmological constant. It examines the anti-de Sitter case using the extended phase space formalism for thermodynamics and compares results to classical black holes. For quantum-corrected small black holes the temperature no longer diverges but approaches zero. The heat capacity instead displays behavior that signals an additional phase transition caused by the quantum corrections. These changes illustrate how quantum effects alter black hole thermodynamics relative to the classical limit.

Core claim

Incorporating a cosmological constant into the higher-dimensional quantum-corrected Oppenheimer-Snyder model yields modified thermodynamics in AdS spacetime: the temperature of small quantum-corrected black holes tends to zero rather than diverging, while the heat capacity exhibits characteristic behavior that indicates an extra phase transition induced by the quantum corrections.

What carries the argument

The higher-dimensional quantum-corrected Oppenheimer-Snyder model extended by a cosmological constant, with thermodynamic quantities evaluated in the extended phase space formalism.

If this is right

  • Temperature of small quantum-corrected black holes in AdS tends to zero instead of diverging.
  • Heat capacity displays behavior consistent with an extra phase transition from quantum corrections.
  • Thermodynamic quantities differ from those of classical black holes under the same extended phase space treatment.
  • Quantum effects produce a measurable impact on black hole thermodynamics in higher dimensions with nonzero cosmological constant.

Where Pith is reading between the lines

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

  • The temperature stabilization at small scales may alter the expected endpoint of black hole evaporation in AdS.
  • The same extension procedure could be applied to the de Sitter case to check whether the extra phase transition persists.
  • If the model holds, thermodynamic stability criteria for small black holes would need revision in the presence of quantum corrections.

Load-bearing premise

The quantum correction from the original higher-dimensional Oppenheimer-Snyder model applies unchanged once a cosmological constant is added and thermodynamics are computed in extended phase space.

What would settle it

A direct computation showing that temperature still diverges for small quantum-corrected black holes in AdS or that heat capacity lacks any signature of the additional phase transition would disprove the reported thermodynamic changes.

Figures

Figures reproduced from arXiv: 2606.08182 by Jianhui Lin, Shudi Jiang, Xiangdong Zhang.

Figure 1
Figure 1. Figure 1: FIG. 1. Hawking temperature for the quantum-corrected and Schwarzschild black hole in 5-dimensional spacetime with Λ=- [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Hawking temperature for the quantum-corrected and Schwarzschild black hole in 6-dimensional spacetime with Λ=- [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Entropy of quantum-corrected and Schwarzschild (classical) black hole in 5-dimensional space. [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Entropy of quantum-corrected and Schwarzschild black hole in 6-dimensional space. [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Heat capacity of the 5-dimensional quantum-corrected and Schwarzschild black hole. [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Heat capacity of the 5-dimensional quantum-corrected black hole with different values of [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Gibbs free energy against temperature in 4-dimensional space [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
read the original abstract

We extended the higher-dimensional quantum Oppenheimer-Snyder model to the case with a cosmological constant. For AdS case, we discuss its thermodynamic properties in extend phase space formalism and make comparison with classical black holes. For quantum-corrected small black holes in AdS spacetime, the temperature no longer diverges but tends to zero. Additionally, the heat capacity exhibits characteristic behavior indicative of an extra phase transition induced by quantum corrections, highlighting the profound impact of quantum effects on black hole thermodynamics.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript extends a prior higher-dimensional quantum-corrected Oppenheimer-Snyder collapse model to include a nonzero cosmological constant. For the AdS case it computes thermodynamic quantities in the extended phase-space formalism and compares them to the classical (uncorrected) black-hole case, claiming that the quantum correction causes the Hawking temperature of small black holes to approach zero instead of diverging and produces an additional phase transition in the heat capacity.

Significance. If the central construction is valid, the work would demonstrate that quantum corrections can qualitatively change the thermodynamic stability and phase structure of higher-dimensional AdS black holes, removing the usual small-black-hole temperature divergence and adding a new transition. Such results would be of interest for quantum-gravity-inspired modifications of black-hole thermodynamics.

major comments (2)
  1. [Construction of the quantum-corrected metric (likely §3 or §4)] The headline claims (T → 0 for small AdS black holes; extra phase transition in C) rest on the assumption that the quantum correction term previously derived for the Λ = 0 higher-D OS model remains form-invariant when a cosmological constant is introduced. The junction conditions, interior dust solution, and effective mass function all acquire additional curvature terms once Λ ≠ 0; the manuscript does not demonstrate that the correction is unmodified under this change. This assumption is load-bearing for both thermodynamic results.
  2. [Thermodynamic analysis in extended phase space (likely §5)] The temperature and heat-capacity expressions in the extended phase space must be derived explicitly from the modified metric; without the intermediate steps showing how the quantum term enters the surface gravity and the first law, it is impossible to confirm that the claimed T → 0 behavior and the additional C discontinuity are not artifacts of an inconsistent insertion of the correction.
minor comments (2)
  1. [Abstract] The abstract contains the typographical error 'extend phase space' instead of 'extended phase space'.
  2. [Results section] A side-by-side table or plot comparing the classical and quantum-corrected temperature and heat-capacity curves for several values of the quantum parameter and Λ would make the claimed differences easier to assess.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need for explicit derivations. We address both major comments below and will revise the manuscript to supply the missing steps.

read point-by-point responses

  1. Referee: [Construction of the quantum-corrected metric (likely §3 or §4)] The headline claims (T → 0 for small AdS black holes; extra phase transition in C) rest on the assumption that the quantum correction term previously derived for the Λ = 0 higher-D OS model remains form-invariant when a cosmological constant is introduced. The junction conditions, interior dust solution, and effective mass function all acquire additional curvature terms once Λ ≠ 0; the manuscript does not demonstrate that the correction is unmodified under this change. This assumption is load-bearing for both thermodynamic results.

    Authors: We agree that the manuscript does not explicitly re-derive the junction conditions with nonzero Λ to confirm form-invariance of the quantum correction. The correction originates from a quantum modification to the interior dust energy density in the higher-dimensional OS collapse; because this modification is introduced at the level of the interior stress-energy tensor before matching, it remains additive to the classical mass function even after the exterior is changed to include Λ. In the revision we will add the explicit junction-condition calculation for the Λ ≠ 0 case, showing that the quantum term enters the effective mass function in the same functional form as in the Λ = 0 paper. revision: yes

  2. Referee: [Thermodynamic analysis in extended phase space (likely §5)] The temperature and heat-capacity expressions in the extended phase space must be derived explicitly from the modified metric; without the intermediate steps showing how the quantum term enters the surface gravity and the first law, it is impossible to confirm that the claimed T → 0 behavior and the additional C discontinuity are not artifacts of an inconsistent insertion of the correction.

    Authors: We acknowledge that the intermediate steps relating the quantum-corrected metric to surface gravity and the extended first law were omitted. The temperature is obtained from the surface gravity of the modified metric function f(r) = 1 - 2M/r^{D-3} + (quantum term) - (Λ r^2)/(D-1)(D-2); the quantum term prevents the usual 1/r divergence as r → 0, yielding T → 0. The heat capacity follows by differentiating the enthalpy with respect to T at fixed pressure. In the revision we will insert these explicit derivations, including the modified first-law verification, to substantiate the reported T → 0 limit and the additional discontinuity in C. revision: yes

Circularity Check

0 steps flagged

No significant circularity; extension uses prior correction as starting point but computes new thermodynamics independently

full rationale

The paper takes the quantum correction term from earlier Lambda=0 work and inserts it into a metric now containing a cosmological constant, then derives thermodynamic quantities in extended phase space. This is an explicit modeling assumption rather than a self-definitional loop, fitted-input prediction, or reduction by construction. No equations are shown that force the new temperature or heat-capacity results to equal the input correction term; the claimed T->0 behavior and extra phase transition emerge from the new junction conditions and extended-phase-space analysis. Self-citations to the prior model are present but supply an independent ansatz that is then applied to a distinct geometry, satisfying the criteria for non-circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the ledger is therefore empty.

pith-pipeline@v0.9.1-grok · 5597 in / 1071 out tokens · 18567 ms · 2026-06-27T19:29:36.062933+00:00 · methodology

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

48 extracted references · 17 linked inside Pith

  1. [1]

    After expandingk T in powers of ˜v, we find that the expression is proportional to 1 ˜t , i.e.λ= 1

    To findk T , we need to change the parameters in (37) into ˜vand ˜p ∂P ∂V = ∂P ∂˜v ∂˜v ∂V = Pc Vc ∂˜p ∂˜v,(44) thenk T becomes kT =− 1 ˜v+ 1· 1 Pc ∂˜p ∂˜v .(45) We can also ignore the 1 Pc term in expression since it is a constant. After expandingk T in powers of ˜v, we find that the expression is proportional to 1 ˜t , i.e.λ= 1. Eq. (38) can also be writ...

  2. [2]

    Carballo-Rubio, F

    R. Carballo-Rubio, F. Di Filippo, S. Liberati, and M. Visser, Phenomenological aspects of black holes beyond general relativity, Phys. Rev. D98, 124009 (2018), arXiv:1809.08238 [gr-qc]

    Pith/arXiv arXiv 2018

  3. [3]

    C. A. Z. Vasconcellos, D. Hadjimichef, M. Razeira, G. Volkmer, and B. Bodmann, Pushing the limits of General Relativity beyond the Big Bang singularity, Astron. Nachr.340, 857 (2019)

    2019

  4. [4]

    Thiemann, Introduction to Modern Canonical Quantum General Relativity, (2001), arXiv:gr-qc/0110034

    T. Thiemann, Introduction to Modern Canonical Quantum General Relativity, (2001), arXiv:gr-qc/0110034

    Pith/arXiv arXiv 2001

  5. [5]

    Ashtekar and J

    A. Ashtekar and J. Lewandowski, Background independent quantum gravity: A Status report, Class. Quant. Grav.21, R53 (2004), arXiv:gr-qc/0404018

    Pith/arXiv arXiv 2004

  6. [6]

    M. Han, W. Huang, and Y. Ma, Fundamental structure of loop quantum gravity, Int. J. Mod. Phys. D16, 1397 (2007), arXiv:gr-qc/0509064

    Pith/arXiv arXiv 2007

  7. [7]

    Ashtekar, J

    A. Ashtekar, J. Baez, A. Corichi, and K. Krasnov, Quantum geometry and black hole entropy, Phys. Rev. Lett.80, 904 (1998), arXiv:gr-qc/9710007

    Pith/arXiv arXiv 1998

  8. [8]

    Ashtekar and M

    A. Ashtekar and M. Bojowald, Quantum geometry and the Schwarzschild singularity, Class. Quant. Grav.23, 391 (2006), arXiv:gr-qc/0509075

    Pith/arXiv arXiv 2006

  9. [9]

    Rovelli, Black hole entropy from loop quantum gravity, Phys

    C. Rovelli, Black hole entropy from loop quantum gravity, Phys. Rev. Lett.77, 3288 (1996), arXiv:gr-qc/9603063

    Pith/arXiv arXiv 1996

  10. [10]

    Zhang, Loop Quantum Black Hole, Universe9, 313 (2023), arXiv:2308.10184 [gr-qc]

    X. Zhang, Loop Quantum Black Hole, Universe9, 313 (2023), arXiv:2308.10184 [gr-qc]

    arXiv 2023

  11. [11]

    J. Lin, X. Zhang, and M. Bravo-Gaete, Mass inflation and strong cosmic censorship conjecture in the covariant quantum black hole, Phys. Rev. D111, 106025 (2025), arXiv:2412.01448 [gr-qc]

    arXiv 2025

  12. [12]

    Y. Du, Y. Liu, and X. Zhang, Spinning particle dynamics and the innermost stable circular orbit in covariant loop quantum gravity, JCAP05, 045, arXiv:2411.13316 [gr-qc]

    arXiv

  13. [13]

    Du, J.-R

    Y. Du, J.-R. Sun, and X. Zhang, Information paradox and island of covariant black holes in LQG, Phys. Rev. D113, 046017 (2026), arXiv:2510.11921 [gr-qc]

    arXiv 2026

  14. [14]

    J. R. Oppenheimer and H. Snyder, On Continued gravitational contraction, Phys. Rev.56, 455 (1939)

    1939

  15. [15]

    R. C. Tolman, Static solutions of Einstein’s field equations for spheres of fluid, Phys. Rev.55, 364 (1939). 12

    1939

  16. [16]

    Lewandowski, Y

    J. Lewandowski, Y. Ma, J. Yang, and C. Zhang, Quantum Oppenheimer-Snyder and Swiss Cheese Models, Phys. Rev. Lett.130, 101501 (2023), arXiv:2210.02253 [gr-qc]

    arXiv 2023

  17. [17]

    M. Han, C. Rovelli, and F. Soltani, Geometry of the black-to-white hole transition within a single asymptotic region, Phys. Rev. D107, 064011 (2023), arXiv:2302.03872 [gr-qc]

    arXiv 2023

  18. [18]

    Z. Shi, X. Zhang, and Y. Ma, Higher-dimensional quantum Oppenheimer-Snyder model, Phys. Rev. D110, 104074 (2024), arXiv:2408.15821 [gr-qc]

    arXiv 2024

  19. [19]

    S. W. Hawking, Black hole explosions, Nature248, 30 (1974)

    1974

  20. [20]

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

    1975

  21. [21]

    J. M. Bardeen, B. Carter, and S. W. Hawking, The Four laws of black hole mechanics, Commun. Math. Phys.31, 161 (1973)

    1973

  22. [22]

    J. D. Bekenstein, Black holes and entropy, Phys. Rev. D7, 2333 (1973)

    1973

  23. [23]

    Strominger and C

    A. Strominger and C. Vafa, Microscopic origin of the Bekenstein-Hawking entropy, Phys. Lett. B379, 99 (1996), arXiv:hep- th/9601029

    arXiv 1996

  24. [24]

    A. G. Riesset al.(Supernova Search Team), Observational evidence from supernovae for an accelerating universe and a cosmological constant, Astron. J.116, 1009 (1998), arXiv:astro-ph/9805201

    Pith/arXiv arXiv 1998

  25. [25]

    Weinberg, The Cosmological Constant Problem, Rev

    S. Weinberg, The Cosmological Constant Problem, Rev. Mod. Phys.61, 1 (1989)

    1989

  26. [26]

    P. J. E. Peebles and B. Ratra, The Cosmological Constant and Dark Energy, Rev. Mod. Phys.75, 559 (2003), arXiv:astro- ph/0207347

    arXiv 2003

  27. [27]

    J. M. Maldacena, The LargeNlimit of superconformal field theories and supergravity, Adv. Theor. Math. Phys.2, 231 (1998), arXiv:hep-th/9711200

    Pith/arXiv arXiv 1998

  28. [28]

    Witten, Anti de Sitter space and holography, Adv

    E. Witten, Anti de Sitter space and holography, Adv. Theor. Math. Phys.2, 253 (1998), arXiv:hep-th/9802150

    Pith/arXiv arXiv 1998

  29. [29]

    Kubiznak and R

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

    Pith/arXiv arXiv

  30. [30]

    Cai, L.-M

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

    Pith/arXiv arXiv

  31. [31]

    Wei and Y.-X

    S.-W. Wei and Y.-X. Liu, Triple points and phase diagrams in the extended phase space of charged Gauss-Bonnet black holes in AdS space, Phys. Rev. D90, 044057 (2014), arXiv:1402.2837 [hep-th]

    Pith/arXiv arXiv 2014

  32. [32]

    S.-W. Wei, B. Liang, and Y.-X. Liu, Critical phenomena and chemical potential of a charged AdS black hole, Phys. Rev. D96, 124018 (2017), arXiv:1705.08596 [gr-qc]

    Pith/arXiv arXiv 2017

  33. [33]

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

    1983

  34. [34]

    Li and J

    R. Li and J. Wang, Thermodynamics and kinetics of Hawking-Page phase transition, Phys. Rev. D102, 024085 (2020)

    2020

  35. [35]

    Israel, Singular hypersurfaces and thin shells in general relativity, Il Nuovo Cimento B (1965-1970)44, 1 (1966)

    W. Israel, Singular hypersurfaces and thin shells in general relativity, Il Nuovo Cimento B (1965-1970)44, 1 (1966)

    1965

  36. [36]

    Ou and X

    M. Ou and X. Zhang, Quantum Oppenheimer-Snyder models in loop quantum cosmology with Lorentz term, Phys. Rev. D112, 126016 (2025), arXiv:2508.01183 [gr-qc]

    arXiv 2025

  37. [37]

    Zhang, Higher dimensional Loop Quantum Cosmology, Eur

    X. Zhang, Higher dimensional Loop Quantum Cosmology, Eur. Phys. J. C76, 395 (2016), arXiv:1506.05597 [gr-qc]

    Pith/arXiv arXiv 2016

  38. [38]

    Ashtekar, Loop Quantum Cosmology: An Overview, Gen

    A. Ashtekar, Loop Quantum Cosmology: An Overview, Gen. Rel. Grav.41, 707 (2009), arXiv:0812.0177 [gr-qc]

    Pith/arXiv arXiv 2009

  39. [39]

    S. F. Mirekhtiary and I. Sakalli, Hawking Radiation of Grumiller Black Hole, Commun. Theor. Phys.61, 558 (2014), arXiv:1402.5514 [gr-qc]

    Pith/arXiv arXiv 2014

  40. [40]

    B. P. Dolan, The cosmological constant and black-hole thermodynamic potentials, Class. Quant. Grav.28, 125020 (2011)

    2011

  41. [41]

    Eslam Panah, Super-entropy bumblebee AdS black holes, Phys

    B. Eslam Panah, Super-entropy bumblebee AdS black holes, Phys. Lett. B861, 139273 (2025), arXiv:2501.09317 [gr-qc]

    arXiv 2025

  42. [42]

    T. V. Fernandes and J. P. S. Lemos, Grand canonical ensemble of a d-dimensional Reissner-Nordstr¨ om black hole in a cavity, Phys. Rev. D108, 084053 (2023), arXiv:2309.12388 [hep-th]

    arXiv 2023

  43. [43]

    Witten, Introduction to black hole thermodynamics, Eur

    E. Witten, Introduction to black hole thermodynamics, Eur. Phys. J. Plus140, 430 (2025), arXiv:2412.16795 [hep-th]

    arXiv 2025

  44. [44]

    Chatzifotis, P

    N. Chatzifotis, P. Dorlis, N. E. Mavromatos, and E. Papantonopoulos, Thermal stability of hairy black holes, Phys. Rev. D107, 084053 (2023), arXiv:2302.03980 [gr-qc]

    arXiv 2023

  45. [45]

    H. E. Stanley and V. K. Wong, Introduction to phase transitions and critical phenomena, American Journal of Physics 40, 927 (1972)

    1972

  46. [46]

    Lin and X

    J. Lin and X. Zhang, Effective four-dimensional loop quantum black hole with a cosmological constant, Phys. Rev. D110, 026002 (2024), arXiv:2402.13638 [gr-qc]

    arXiv 2024

  47. [47]

    S. Li, Y. Ma, F. Yuan, and X. Zhang, Effective dynamics of loop quantum Kaluza-Klein cosmology, Phys. Rev. D113, L041503 (2026), arXiv:2508.07962 [gr-qc]

    arXiv 2026

  48. [48]

    I. H. Belfaqih, M. Bojowald, S. Brahma, and E. I. Duque, Hawking Evaporation and the Fate of Black Holes in Loop Quantum Gravity, Phys. Rev. Lett.135, 161501 (2025), arXiv:2504.11998 [gr-qc]

    arXiv 2025