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Van der Waals Black Holes: Universality, Quantum Corrections, and Topological Classifications

T0 review · 4 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper claims that a generalized universality relation, whose correction factor is fixed by the entropy's deviation from the area law, governs extremality shifts for GUP-, EUP-, and Rainbow-corrected Van der Waals black holes.

desk verdict The universality claim is broken by a sign error in the GUP inversion, but the corrected metrics and topology are new and worth separating from it. read the letter →

arxiv 2507.21663 v1 pith:LJ5VFNLP submitted 2025-07-29 gr-qc hep-th

Ankit Anand , Saeed Noori Gashti This is my paper
classification gr-qchep-th PACS 04.70.Dy04.60.-m
keywords blackholethermodynamicsVanderWaalsholesuniversalextremalityrelationgeneralizeduncertaintyprincipleextendedrainbowgravitythermodynamictopologywindingnumber
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 tries to establish that Van der Waals black holes keep a universal thermodynamic law once quantum-gravity-inspired corrections are switched on. In the classical case the entropy is the Bekenstein-Hawking area, and the usual relation between entropy corrections and extremality shifts holds. After deriving GUP-, EUP-, and Rainbow-gravity-corrected versions of the VdW black hole, the paper verifies a generalized version of that relation in which the proportionality constant is the derivative of the Bekenstein-Hawking entropy with respect to the corrected entropy. It also classifies the corrected black holes topologically by winding numbers, finding that GUP and Rainbow versions form a stable class W=0 while the EUP version splits into two classes. A reader should care because the relation, if true, turns the corrected entropy into a direct predictor of extremal behavior.

What carries the argument

The load-bearing object is the generalized universality relation, Eq. (3.2), imported from [74]: it asserts ∂M_ext/∂ǫ is proportional to -T ∂S/∂ǫ at extremality, with proportionality factor dS_BH/dS computed from the corrected entropy. The paper's checks use the perturbed-action method of [68], in which the cosmological constant is treated as a small perturbation parameter ǫ and mass and temperature are expanded to first order. For topology, the machinery is Duan's φ-mapping: the vector field φ=(∂F/∂r_h, -cotΘcscΘ) built from the generalized free energy F=M-S/τ has zeros that are thermodynamic equilibrium points, and the winding numbers of those zeros sum to the total topological charge W. Both pieces work together: the universality relation concerns the extremal limit, while the winding-number classification concerns stability away from extremality.

What would settle it

Take a corrected entropy that satisfies the first law but differs from the three studied here, e.g. an entropy with a different logarithmic coefficient, and compute the ratio ∂M_ext/∂ǫ / (-T∂S/∂ǫ) at extremality to first order in the deformation parameter. If the ratio is not ∂S_BH/∂S for that entropy, the generalized relation (3.2) is false; the same calculation for a model where S and r_h(S) are known exactly would settle it without relying on [74].

Watch

Extended reading notes

Core claim

On its own terms, the paper's central claim is that the universal relation between corrections to entropy and corrections to extremality, originally formulated for Bekenstein-Hawking entropy, has a generalized form that survives for Van der Waals black holes with GUP, EUP, and Rainbow gravity corrections. The generalized relation reads ∂M_ext/∂ǫ ∝ lim_{M→M_ext}(-T ∂S/∂ǫ), with proportionality constant ∂S_BH/∂S, where S_BH = π $r_h^{2}$ is the Bekenstein-Hawking entropy evaluated at the horizon radius obtained from the corrected entropy. The paper computes perturbed mass and temperature for each corrected model, inverts the corrected entropy to find r_h(S), and shows that the ratio of the two sides equals ∂S_BH/∂S exactly. On the topological side it finds that GUP-corrected and Rainbow-corrected VdW black holes always carry total charge W=0, with two charges (+1,-1) that do not change under parameter variation, while EUP-corrected holes fall into two classes: one with total charge W=-1 and another with W=0 whose number of charges can be two or four. The classical VdW black hole in the same topological scheme has a photon sphere of total charge -1; for the corrected models the photon-sphere zeros lie inside the horizon and are observationally inaccessible.

Load-bearing premise

The generalized universality relation (3.2), with its proportionality constant ∂S_BH/∂S, is taken from an as-yet-unpublished preprint by the paper's first author and is not derived here; if that relation is not valid in general, the three model checks only show consistency with an assumed formula.

Editorial extensions

If this is right

  • If the generalized universality relation is correct, the extremal mass shift under any quantum correction is fixed by the corrected entropy alone, so one can compute ∂S/∂ǫ and immediately get ∂M_ext/∂ǫ at extremality.
  • The GUP- and Rainbow-corrected VdW black holes have a total topological charge W=0 that is unchanged when a, b, α, or γ vary, so their phase-stability classification is parameter-independent.
  • The EUP-corrected VdW black hole has two topological classes: a W=-1 configuration with three charges, and a W=0 family with either two or four charges, so the EUP parameter β can change the phase structure qualitatively.
  • For the classical VdW black hole the photon sphere carries total topological charge -1; for the three corrected models the photon-sphere zeros lie inside the event horizon, making them inaccessible to external observers.

Reading between the lines

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

  • My inference: if Eq. (3.2) survives in other entropy modifications, such as Rényi or Barrow entropy, it would turn any proposed entropy function into a quantitative prediction for extremal mass shifts, giving a cheap observational discriminator between quantum-gravity models.
  • My inference: the difference in topological behavior—GUP and Rainbow frozen at W=0, EUP splitting into two classes—may track the sign or functional form of the entropy correction (logarithmic decrease vs. logarithmic increase), a pattern the paper does not explicitly isolate.
  • My inference: the paper's remark that corrected-model photon spheres sit inside the horizon suggests that, if such corrections are physical, they would hide the photon ring from external observers and weaken lensing-based tests of quantum gravity; verifying this numerically for the full corrected metrics would be a natural next step.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The paper studies asymptotically AdS black holes whose thermodynamics mimics a Van der Waals fluid, with additional corrections from GUP, EUP, and Rainbow Gravity. It claims that a generalized universality relation, proportional to dS_BH/dS, holds for the extremality shift in all three corrected models (Section 3), and that the topological charge distributions of the corrected models are stable or fall into a small number of classes (Section 4). It also analyzes photon spheres for the uncorrected case (Section 4.4).

Significance. If correct, the paper would extend the Goon–Penco universality relation to quantum-gravity-inspired corrections of Van der Waals black holes and would provide a topological classification of these models. The paper contains many explicit expressions and several figures, and it correctly identifies the classical case as satisfying the standard relation. However, the central universality claim is not established: the proportionality constant in Eq. (3.2) is forced to be unity for any entropy consistent with the first law, and the concrete inversions used in Sections 3.1 and 3.3 contain sign and algebraic errors. These are load-bearing, not presentation issues.

major comments (4)
  1. [§3.1, Eq. (3.9)] Equation (3.9) is not the correct first-order inversion of the GUP entropy (2.18). Expanding S = π r_h^2 − (απ/4) ln((4r_h^2+α)/α) around r_h = √(S/π) gives r_h = √(S/π) + (α√π/(8√S)) ln(4S/(πα)) + O(α^2), i.e. a positive correction, whereas Eq. (3.9) has a negative correction with a different coefficient. This error propagates into Eqs. (3.13) and (3.14), so the claimed match between the ratio and d(πr_h^2)/dS is an artifact of an incorrect inversion rather than a verification of the generalized universality relation.
  2. [§3, Eqs. (3.1)–(3.2)] The proportionality constant in Eq. (3.2) is forced to be 1 for any entropy S that satisfies the first law. Since the entropies in Eqs. (2.18), (2.31), and (2.41) are declared to be derived from the first law, the relation T = (∂M/∂S)_ǫ must hold. Then, at extremality, dM_ext/dǫ = M_ǫ = −T (∂S/∂ǫ)_M, making Eq. (3.1) an identity. The non-unit ratios reported in Eqs. (3.19) and (3.29) therefore imply that the M, T, and S used in those sections are not mutually consistent. Because the same entropy and temperature functions are used to define the free energy and on-shell condition in Section 4, this inconsistency also casts doubt on the topological classifications for the corrected models.
  3. [§3.3, Eq. (3.22)] Equation (3.22) is not the correct inversion of the Rainbow entropy (2.41). From S ≃ π r_h^2 + (γ/E_P^2) ln r_h, the leading-order inversion is r_h = √(S/π) + (γ/(4E_P^2√(πS))) ln(π/S) + O(γ^2), not the expression in Eq. (3.22), which contains a non-analytic γ log γ term and a different logarithmic argument. Since Eqs. (3.23)–(3.29) are all built from Eq. (3.22), the claimed Rainbow-gravity universality result is unsupported.
  4. [§3, Eq. (3.2)] The generalized universality relation is imported from ref. [74], an unpublished preprint by the same first author, and is not derived in this manuscript. Because this relation is the central hypothesis that Sections 3.1–3.3 purport to test, the paper should either provide a self-contained derivation or clearly state the relation as an assumption. As it stands, the calculations in Section 3 check consistency with an assumed formula rather than establish the relation from first principles.
minor comments (6)
  1. [Title] The title contains a typo: “Correctio ns” should be “Corrections”.
  2. [§1, p. 2] There is a grammatical error: “Ther are several discussion and for different cases can be foun d” should be rewritten, e.g., “There are several discussions, and cases can be found in [23–59].”
  3. [§4, Fig. 1] The figure captions mention “λ = 1” and “c = 8π/3”, but λ is not defined anywhere in the text, and c is introduced only as a computational shorthand in Eq. (4.1). The meaning of these parameters should be clarified.
  4. [§4.1–4.3] The winding numbers and topological charges are asserted from vector-field plots, but no quantitative calculation, table, or code is provided. The paper should include a table of the winding numbers for each panel or an explicit numerical method so that the topological classifications can be independently verified.
  5. [§2.3, Eq. (2.40)] The temperature expression in Eq. (2.40) contains a mix of r, r_h, and r_H in the same formula (e.g., terms with “r” in denominators), which appears to be a typographical error and makes the expression ambiguous.
  6. [§4.4] The photon-sphere analysis is performed only for the uncorrected Van der Waals black hole, and the claim that corrected photon spheres are hidden inside the horizon is not demonstrated. A clearer statement of the domain of validity and the limiting procedure would be helpful.

Circularity Check

2 steps flagged · score 7.0 of 10

Central universality claim is taken from a same-author unpublished preprint and then 'verified' by re-differentiating the same entropy inversion; the non-unit proportionality constants are internal-consistency artifacts, not independent predictions.

  1. self citation load bearing [Section 3, Eq. (3.2) and footnote 3 (definition of S_BH)]
    "It has been argued recently in [ 74], if the entropy is not the Bekenstein-Hawking entropy, then this relation can be modified as ... The proportionality constant is ∂S BH H ∂S where SBH H = πr2 h denotes the Bekenstein–Hawking entropy evaluated at the horizon radius obtained from the generalized entropy."

    The generalized universality relation, the paper's central physics claim, is not derived here. It is imported from ref. [74], an unpublished preprint by the same first author (Ankit Anand), and its proportionality constant is defined using the same entropy inversion that the later sections test. The subsequent verification therefore checks consistency with a formula that is itself the load-bearing input, rather than providing independent evidence for it.

  2. self definitional [Section 3.1, Eq. (3.14); same pattern in Eqs. (3.19)-(3.20) and (3.24)-(3.29)]
    "The proportionality constant can be computed using Eq. ( 3.9) as ∂(πr2 h) ∂S = ... and this exactly matches the result of [ 74]."

    The proportionality constant is defined as d(πr_h^2)/dS with r_h(S) obtained by inverting the modified entropy, and the same r_h(S) is used to express the perturbed mass and temperature whose ratio is then compared to that derivative. The 'verification' is thus a self-consistency check on one inversion formula, not an independent prediction. Moreover, since each entropy was obtained from the first law, T=(∂M/∂S) and Eq. (3.1) is an exact identity: the constant in Eq. (3.2) must be 1 for any consistent M,T,S. The reported dS_BH/dS≠1 values (e.g., the wrong-sign log in Eq. (3.9) and the irregular Rainbow inversion in Eq. (3.22)) indicate algebraic inconsistencies rather than a confirmed generalized universality.

full rationale

The paper's central Section 3 claim is that a generalized universality relation, Eq. (3.2), holds for GUP-, EUP-, and Rainbow-corrected VdW black holes. The relation itself is not derived in the paper; it is imported from ref. [74], an unpublished preprint by the same first author. The proportionality constant dS_BH/dS is then defined by inverting the modified entropy S(r_h) to obtain r_h(S), and the same r_h(S) is substituted into the perturbed mass and temperature. The ratio (∂M_ext/∂ǫ)/(-T ∂S/∂ǫ) is subsequently compared with d(π r_h^2)/dS, so the comparison is an identity by construction: the 'predicted' constant is the derivative of the same inversion used to write M and T. This constitutes a self-definitional verification rather than an independent check. Additionally, for any entropy obtained from the first law, T=(∂M/∂S), Eq. (3.1) is an exact mathematical identity, so Eq. (3.2) can only have proportionality 1; the paper's non-unit results (3.19) and (3.29) therefore indicate internal inconsistencies, such as the sign of the logarithmic term in the GUP inversion (3.9) and the non-analytic γ log γ term in the Rainbow inversion (3.22). The topological classification sections are largely self-contained applications of standard φ-mapping methods and are not circular; the circularity is confined to the universality claim, which is the paper's strongest advertised result. Score 7 reflects that the central claim reduces to checking an assumption imported from the authors' own unpublished work and verified by a derivative of the same inversion formula.

Assumptions & free parameters 7 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new particles or forces. The quantum corrections are encoded in entropy functions (2.18), (2.31), and (2.41) that are quoted from heuristic uncertainty-principle arguments. The main external input is the generalized universality relation (3.2) from the authors' previous preprint [74].

free parameters (7)
  • alpha (GUP parameter) = 0.5, 1 in topology plots
    Deformation parameter of the GUP; treated as a small expansion parameter in Section 3 and varied by hand in Section 4.
  • beta (EUP parameter) = 0.1, 0.5 in topology plots
    EUP deformation parameter; treated as a small expansion parameter and varied by hand in Section 4.
  • gamma (Rainbow parameter) = 0.1, 0.5 in topology plots
    Rainbow gravity deformation parameter; treated as small and varied by hand in Section 4.
  • a (VdW attraction parameter) = 1/(2 pi) and 1 in plots
    Parameter of the VdW equation of state, chosen by hand for the figures.
  • b (VdW covolume parameter) = 0.1, 0.5, 1.2 in plots
    Parameter of the VdW equation of state, chosen by hand for the figures.
  • l (AdS radius) = 1
    AdS length scale; related to pressure P = 3/(8 pi l^2).
  • c = 8 pi / 3
    Introduced 'just for computational purposes' in Eq (4.1) to simplify the free energy; it cancels a term when P = 3/(8 pi l^2).
assumptions (4)
  • domain assumption The metric ansatz (2.1) is a solution of Einstein equations with some matter source.
    The corrected metrics are reverse-engineered from the temperature matching the VdW equation of state; the stress-energy tensor is not exhibited or checked.
  • domain assumption The corrected entropies (Eqs 2.18, 2.31, 2.41) satisfy the first law dM = T dS + V dP.
    Used to derive the entropy and temperature relations; not proven from a Lagrangian.
  • ad hoc to paper The generalized universality relation (Eq 3.2) from ref [74] is valid.
    The relation is taken from an unpublished preprint by the same first author; no independent derivation is given in this paper.
  • domain assumption The free energy F = M - S/tau and the vector-field topology construction of refs [21,22] apply to the corrected thermodynamics.
    Standard in the literature, but its validity for quantum-corrected entropies is assumed without proof.

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Cite this review

Pith. "Pith review of Van der Waals Black Holes: Universality, Quantum Corrections, and Topological Classifications." pith.science (2026). https://pith.science/paper/LJ5VFNLP

@misc{pith2026250721663,
  author       = {Pith},
  title        = {Pith review of: Van der Waals Black Holes: Universality, Quantum Corrections, and Topological Classifications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LJ5VFNLP}},
  note         = {Machine review of arXiv:2507.21663}
}
read the original abstract

In this paper, we investigate the universal extremality relation and thermodynamic topology of Van der Waals (VdW) black holes-solutions of Einstein's equations whose thermodynamic behavior closely resembles that of Van der Waals fluids. In the classical case, the black hole entropy obeys the Bekenstein-Hawking area law and satisfies the standard universality relation. We then incorporate quantum corrections using three distinct frameworks: the Generalized Uncertainty Principle (GUP), the Extended Uncertainty Principle (EUP), and Rainbow Gravity. While these corrections modify the entropy law, we find that a generalized form of the universal extremality relation still holds. Next, we explore the thermodynamic topology of VdW black holes, focusing on the distribution of topological charges. Our analysis reveals that variations in the black hole and model parameters lead to significant changes in topological classifications and stability, as quantified by winding numbers. In the GUP-corrected case, topological charge distributions exhibit robustness against parameter variations, suggesting classification stability. For EUP-corrected black holes, we identify two distinct topological classes, with some configurations displaying three non-zero topological charges and others maintaining a total charge of zero, despite changes in individual charge counts. The Rainbow Gravity-corrected scenario shows similar consistency in topological behavior.

Figures

Figures reproduced from arXiv: 2507.21663 by the authors.

Figure 1
Figure 1. The (τ vs. rh) diagram, illustrating variations in free parameters for van der Waals (vdW) black holes, is presented in Figs. (1(a)), (1(c)), and (1(e)). Additionally, the normal vector field n in the (rh − Θ) plane is depicted, with Zero Points (ZPs) located at specific coordinates (rh, Θ). These ZPs correspond to parameter values b = 0.1, 0.5, 1.2, along with fixed values of c = 8π/3, λ = 1, and a = 1/2π, as shown… view at source ↗
Figure 2
Figure 2. The (τ vs. rh) diagram, illustrating variations in free parameters for GUP corrected black hole, is presented in Figs. (2(a)), (2(c)), (2(e)) and (2(g)). Additionally, the normal vector field n in the (rh − Θ) plane is depicted, with Zero Points (ZPs) located at specific coordinates (rh, Θ). These ZPs correspond to parameter values (α = 1, b = 0.1, a = 1/2π),(α = 1, b = 0.5, a = 1/2π),(α = 1, b = 1.2, a = 1) and (α … view at source ↗
Figure 3
Figure 3. The (τ vs. rh) diagram, illustrating the influence of varying free parameters on EUP-corrected black holes, is presented in Figs. (3(a)), (3(c)), (3(e)), (3(g)), (3(j)), and (3(m)). Furthermore, the normal vector field n is depicted in the (rh − Θ) plane, where Zero Points (ZPs) appear at specific coordinates (rh, Θ). These ZPs correspond to parameter values β = 0.1, 0.5, b = 0.1, 0.5, 1.2, and a = 1/2π, 1, providin… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: The (τ vs. rh) diagram, illustrating variations in free parameters for Rainbow Gravity-corrected black hole, is presented in Figs. (4(a)), (4(c)), (4(e)), (4(g)) and (4(i)). Additionally, the normal vector field n in the (rh − Θ) plane is depicted, with Zero Points (ZP…
Figure 5
Figure 5. Figure 5: The plotted photon spheres (PSs) of van der Waals (vdW) black holes are presented in Fig. (5(a)) for b = 0.1, Fig. (5(b)) for b = 0.5, and Fig. (5(c)) for b = 1.2. These visualizations correspond to the parameter values l = 1, a = 1/2π, c = 8π/3, and M = 0.1 logical ch…

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