Thermodynamic Curvature and Topological Insights of Hayward Black Holes with String Fluids

Aditya Singh; Ankit Anand; Saeed Noori Gashti

REVIEW 5 major objections 6 minor 1 cited by

Thermodynamic Curvature and Topological Insights of Hayward Black Holes with String Fluids

T0 review · 5 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read For Hayward-AdS black holes with string fluids, the sign of the string-fluid energy density sets the thermodynamic topological charge: W=0 for one sign and W=+1 for the other.

desk verdict Load-bearing unstated l^2 = -3/Λ identification breaks the enthalpy-volume consistency; the Ruppeiner and topology results are not yet supported, though the review and regularity analysis are useful. read the letter →

arxiv 2506.23736 v1 pith:GDZPTEOV submitted 2025-06-30 gr-qc hep-th

Ankit Anand , Saeed Noori Gashti , Aditya Singh This is my paper

classification gr-qchep-th MSC 83C5783C2053Z05

keywords HaywardblackholesstringfluidsRuppeinercurvaturethermodynamictopologyextendedphasespacevanderWaals-liketransitionholemicrostructuretopologicalcharge

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

This paper tries to establish that adding a surrounding fluid of strings to a Hayward-AdS black hole produces an extended thermodynamics whose phase structure is controlled by a handful of parameters, most importantly the sign $\epsilon$ of the string-fluid energy density. It claims that the normalized Ruppeiner curvature $R_N$ in the $(T,V)$ plane is negative over most of parameter space, indicating attractive microstructures, turns slightly positive at small volumes for high charge, and diverges at the same places as heat-capacity singularities. It further claims that the thermodynamic topological charge is $W=0$ when $\epsilon=-1$ and $W=+1$ when $\epsilon=+1$, with the parameter $b$ changing the multiplicity of winding numbers but not the total. If correct, the string fluid is a switch between Reissner–Nordström-like and AdS-Reissner–Nordström-like topological classes, and regular black holes become a concrete laboratory for black hole chemistry and holographic duality.

What carries the argument

The load-bearing objects are the normalized Ruppeiner curvature $R_N$ and the topological winding number $W$. $R_N$ is the scalar curvature of the line element $ds^2 = (1/T)(\partial P/\partial V)_T\,dV^2 + (C_V/T^2)\,dT^2$ on the $(T,V)$ plane, normalized to remove the $C_V=0$ singularity; it carries the microscopic-interaction signal through its sign and the criticality signal through its divergences. $W$ is computed from the zero points of the vector field $\phi = (\partial F/\partial r_+,\;-\cot\Theta\,\csc\Theta)$ with $F = M - S/\tau$, each zero contributing a Hopf index times the sign of a Jacobian determinant; it classifies the global phase structure. The entire argument chains the metric function Eq. (14) to the mass Eq. (20), then to the equation of state, temperature, $R_N$, and $W$.

What would settle it

Evaluate the mass by solving $f(r_+)=0$ in Eq. (14) with $P=-\Lambda/(8\pi)$, and compare the result to Eq. (20); if they disagree for any allowed charge $q$, the claimed enthalpy, temperature, equation of state, $R_N$, and $W$ do not follow from the stated metric. This is a direct numerical check.

Watch

Extended reading notes

Core claim

The paper's central claim, stated on its own terms, is that Hayward-AdS black holes surrounded by string fluids have a van der Waals-like equation of state, a microstructure probed by the normalized Ruppeiner curvature $R_N$, and a topological class fixed by $\epsilon$. The curvature $R_N$, built from the $(T,V)$ metric, is negative in the dominant volume regime, indicating attractive interactions among the hypothesized black hole molecules, with a small positive pocket at small volume and large charge; its divergences coincide with the small-to-large black hole phase transition. In the phi-mapping topological current formalism, the generalized free energy $F = M - S/\tau$ defines a vector field whose zero points carry winding numbers: two zero points ($+1$ and $-1$) for $\epsilon=-1$, giving total $W=0$, and one zero point ($+1$) for $\epsilon=+1$, giving $W=+1$; increasing $b$ can add charges without changing the total. The paper reads this as identifying Hayward-string black holes with the Reissner–Nordström class ($W=0$) or the AdS-Reissner–Nordström class ($W=+1$), with $\epsilon$ as the controlling switch.

Load-bearing premise

The entire thermodynamic chain assumes, without stating it, that the Hayward regularization length squared equals $-3/\Lambda$, so that the AdS radius and the Hayward parameter are the same length; if that identification is wrong, the mass formula and every downstream quantity change.

Editorial extensions

If this is right

For $\epsilon=-1$ the net topological charge is $W=0$, placing these Hayward-string black holes in the Reissner–Nordström topological class; for $\epsilon=+1$ it is $W=+1$, the AdS-Reissner–Nordström class.
Increasing $b$ (when $\epsilon=+1$) can create additional zero points and winding numbers, but the total charge stays $W=+1$, so $b$ tunes the complexity of the phase space without changing its class.
$R_N$ is negative across most of parameter space, becomes positive at small volume for high charge, and diverges exactly at phase transitions, giving a one-to-one correspondence between curvature singularities and heat-capacity divergences.
The $P$–$v$ diagram shows van der Waals-like oscillating isotherms and first-order transitions, with critical pressure decreasing as charge increases, so string fluids leave a clear thermodynamic signature.

Reading between the lines

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

The paper leaves implicit that if the identification $l^2 = -3/\Lambda$ is intended, the Hayward regularization scale and the AdS radius are being silently identified; in that reading the results describe a specific coincidence limit, and repeating the analysis with $l$ and $\Lambda$ independent would test how much of the $W=0$/$W=+1$ dichotomy survives.
The claim that $W$ depends only on the sign of $\epsilon$ suggests a structural stability that could be checked by adding other matter couplings: one would predict $W$ stays fixed as long as the leading string-fluid term keeps the same sign.
The same $(T,V)$ Ruppeiner construction could be run on other regular black hole families with string fluids; the paper's mechanism predicts the same $\epsilon$-controlled switch if the string fluid enters the metric in the same way.
Since $W$ is not directly observable, the testable content for astrophysics is the metric parameters themselves; the paper's proposed shadow, quasi-normal mode, and ringdown comparisons are the natural way to look for a string-fluid component.

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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

Two things to know before you spend time on arXiv:2506.23736. The combination is new: normalized Ruppeiner curvature plus Duan phi-mapping applied to Hayward-AdS black holes with a string fluid. The paper also gives a clean review of the metric and a credible Kretschmann regularity analysis, with the −1 ≤ β < 0 range for regularity. That part is worth having.

The problem is in the thermodynamics chain. Eq. (20) follows from Eq. (14) only if you silently set l^2 = −3/Λ, identifying the Hayward regularization scale with the AdS radius. That identification is never stated. If it is not intended, every formula after Eq. (20) changes. If it is intended, it produces an internal inconsistency: differentiating Eq. (20) at fixed q gives ∂M/∂P = 4π(r_+^3 + q^3)/3, not the claimed V = 4πr_+^3/3. Either way, the extended first law is not satisfied as written.

The entropy S = πr_+^2 − πq^3/r_+ in Eq. (22) is asserted, not derived from M. The Ruppeiner expression Eq. (29) is stated without derivation, so the sign changes in R_N that drive the microstructure claims cannot be checked. The conclusion says R_N diverges exactly at phase transitions, but no direct comparison with heat-capacity divergences is shown. The topology figures also omit some parameter values in the captions, making the W = 0 vs W = +1 claims hard to verify.

I would not send this to a referee as submitted. The central results are not independently verifiable and one relation is internally inconsistent. If the authors state the l-Λ identification, correct the volume, and derive R_N, it could be a routine incremental paper in black hole chemistry. The review material is fine; the new calculations are not.

Referee Report

5 major / 6 minor

Summary. The paper studies the extended thermodynamics, thermodynamic geometry, and thermodynamic topology of Hayward-AdS black holes surrounded by string fluids. Starting from the Hayward-AdS metric with a string-fluid source, the authors write down the mass, temperature, entropy, and volume in the extended phase space, then use them to construct the normalized Ruppeiner curvature in the (T,V) plane and the Duan φ-mapping topological charge. The main claims are that the normalized curvature R_N changes sign with charge and volume, signaling a crossover between attractive and repulsive microstructures, and that the total topological charge is W=0 for ε=−1 and W=+1 for ε=+1, so the sign of the string-fluid parameter acts as a switch between RN-like and AdS-RN-like topological classes.

Significance. If the results were correct, the paper would extend the standard thermodynamic-geometry and topology program to a regular Hayward-AdS black hole surrounded by a string fluid, providing a concrete example in which the string-fluid sign ε controls the topological class and in which R_N offers a microstructure diagnostic. The manuscript contains explicit analytical expressions for R_N and for the topological free energy, which is useful for reproducibility if the underlying thermodynamic relations are valid. However, the central derivations rest on an unstated identification between the Hayward scale and the AdS radius, an incorrect thermodynamic volume, and an entropy that does not satisfy the first law; until these are corrected, the claimed phase-transition detection and topological classification are not supported.

major comments (5)

[II, Eq. (20)] The mass formula (20) does not follow from the metric function (14) unless one makes the unstated identification l^2 = −3/Λ. Solving f(r_+)=0 from Eq. (14) gives M = (r_+^3+q^3)/(2r_+^2)[1 + εβ/(β−2)(b/r_+)^{2/β} − Λr_+^2/3], whereas Eq. (20) expands to M = (r_+^3+q^3)/(2r_+^2)[1 + r_+^2/l^2 + εβ/(β−2)(b/r_+)^{2/β}]. These coincide only if l^2 = −3/Λ, an identification that is never stated. The symbol l was introduced in Eq. (4) as the Hayward Planck-scale length, while Λ is an independent cosmological constant; the substitution therefore merges two distinct physical scales. Since Eqs. (21), (27), (28), (29)–(31), and the topological analysis all inherit this substitution, the entire thermodynamic chain is unsupported until the identification is stated and justified.
[II, Eq. (22)] The thermodynamic volume is claimed to be V = ∂M/∂P = 4πr_+^3/3. However, differentiating Eq. (20) with l^2 = 3/(8πP) gives ∂M/∂P = 4π(q^3+r_+^3)/3, not 4πr_+^3/3. The volume used in the equation of state (28) and in the normalized Ruppeiner curvature (29)–(31) is therefore inconsistent with the enthalpy from which it is supposed to be derived. This is a load-bearing error: the P–v diagram, the critical-point analysis, and the sign changes of R_N all depend on the correct relation between V and r_+.
[II, Eq. (22)] The entropy S = πr_+^2 − πq^3/r_+ is simply asserted as ∫ dM/T, with no derivation from the first law or from a Lagrangian/Wald analysis. With M from Eq. (20), T from Eq. (21), and this S, the first law dM = T dS + V dP is not satisfied. For example, in the limit of vanishing string fluid (ε=0) with l=1, q=0.5, r_+=1, one finds ∂M/∂r_+ ≈ 1.875 whereas T ∂S/∂r_+ ≈ 1.77. The entropy must be derived consistently from the same M, T, and V, otherwise the free energy used in the topological section is also invalid.
[III A, Eqs. (29)–(31)] The normalized Ruppeiner curvature R_N is presented without showing the intermediate computation that turns the line element (26) and the equation of state into the closed forms (29)–(31). The expression is extremely opaque, and because it is built from the inconsistent volume and entropy identified above, the curvature results in Fig. 4 cannot be verified. The authors should either provide the full derivation or a reproducibility package, and they should recompute R_N using a thermodynamically consistent set of variables.
[IV A, Eq. (36) and Fig. 5] The topological analysis uses the generalized free energy F = M − S/τ with the unproven entropy (22). Moreover, the plots in Fig. 5 do not specify the values of q and l (or, equivalently, P) used for each panel, even though the location and number of zero points depend on these parameters. The reported results W=0 for ε=−1 and W=+1 for ε=+1 are therefore not reproducible and may change once the first law and the volume relation are corrected.

minor comments (6)

[II, Eq. (4)] The relation q_m = 3√4m^2l/2 is garbled; please define q_m, m, and l explicitly and write the equation in unambiguous notation.
[II, Eq. (14) and Eq. (20)] The symbol l is used for the Hayward regularization length in Eq. (4) and then for the object that replaces the cosmological constant in Eq. (20). Rename one of these to avoid the conflation that underlies the major concern in Eq. (20).
[II, Eq. (15)] The Kretschmann scalar expression uses m, but the metric function (14) uses M; please define the relation between m and M, or use a single symbol.
[III, Eq. (26)] The statement that C_V=0 holds for 'all static AdS black holes' is imprecise; C_V vanishes because volume is not an independent fluctuation coordinate in the chosen ensemble, not because of AdS asymptotics alone.
[IV A, Fig. 5] The figure captions contain notation such as 'ε 1' that should read 'ε = 1'; also, each panel should list the values of q, l, and b used in the plot.
[V, Conclusion] There are a few typographical errors, including 'Hyward' in the vicinity of Eq. (36), and the reference list appears to contain duplicates (e.g., refs. [13] and [14] both refer to the Soleng article).

Circularity Check

1 steps flagged · score 2.0 of 10

No circular reduction: the Ruppeiner curvature RN and topological charges W are computed in-paper from the stated enthalpy and equation of state, with no fitted inputs; the only self-citation is the RN normalization-convention cluster (refs. 74-77, 99), which is minor because the construction is externally grounded, while the unstated l^2 = -3/Lambda identification behind Eq.

self citation load bearing [Section III (Thermodynamic Curvature), paragraph following Eq. (26); also Section V conclusions invoking the RN diagnostic.]
"The given metric is diagonal but becomes singular when the specific heat at constant volume, CV, vanishes-a condition satisfied by all static AdS black holes. By rescaling the metric or its scalar curvature by CV, this singularity can be resolved, resulting in a normalized thermodynamic scalar curvature that is frequently written as RN in the literature[74-77]."

Refs. [74]-[77] are prior works of co-author A. Singh, and ref. [99] is by co-authors Anand and Gashti; they are cited as the literature in which the normalized (T,V)-plane curvature 'is frequently written as RN.' The normalization-by-CV construction and its microstructure interpretation carry the paper's central diagnostic claims, and the citation frames the authors' own prior applications as established external notation. This is minor, however: the same passage explicitly credits the construction to the external works of Wei, Liu and Mann (refs. [52,53]), and the Hayward-specific RN (Eqs. 29-31) and the topological charges are computed in-paper rather than imported.

full rationale

The paper's two central outputs — the normalized Ruppeiner curvature RN in Eqs. (29)-(31) and the topological charges W = 0 for epsilon = -1 and W = +1 for epsilon = +1 — are computed in-paper from the stated mass formula, temperature, and equation of state; no parameter is fitted to any target and no conclusion is read off from an external result. The (T,V) line element, Eq. (26), is derived in Appendix B from the first law, and the normalized-curvature construction is explicitly traced to the external Wei-Liu-Mann works (refs. [52,53]), so the self-citation cluster (refs. [74-77], [99], all by the present co-authors) functions only as a notation/convention citation for 'RN' and is not load-bearing. The flagged l^2 = -3/Lambda identification is a genuine derivation gap: Eq. (20) coincides with the horizon condition f(r_+) = 0 of Eq. (14) only under that unstated identification, and the same identification is needed to pass from Eq. (21) to Eq. (27). Also, Eq. (22) asserts V = 4*pi*r_+^3/3 while differentiating Eq. (20) at fixed q yields an additional q^3 term. These are missing-support and correctness defects in the claimed derivation chain, not circular reductions: the predicted RN and W are functions of the model parameters computed from the paper's own algebra, and they are not equal to any input by construction. Overall the derivation is self-contained against external benchmarks with no significant circularity; the score of 2 reflects only the minor self-citation for the RN convention.

Assumptions & free parameters 5 free parameters · 4 assumptions · 1 invented entities

The central claims depend on the manually chosen model parameters epsilon, beta, b, q and on the implicit identification l^2=-3/Lambda. No parameters are fitted to external data, so this is not data fitting, but the parameter choices are not derived and the entropy is assumed rather than integrated from M.

free parameters (5)

epsilon = ±1
Sign of the string-fluid energy density; selected by hand. It is the main control parameter in the topological result (W=0 versus W=+1).
beta = -0.1, -0.5, -0.9
Anisotropy parameter for the string fluid; chosen by hand inside the regularity window -1 <= beta < 0; affects the curvature and winding structure.
b = 1, 2, 2.5, 5.5
Integration constant of the string fluid; varied by hand and claimed to change the number of topological charges.
q = 0.06 to 0.25 in plots
Charge parameter of the Hayward solution; varied by hand. The sign change of Ruppeiner curvature is attributed to q and volume.
l = not specified; implicitly l^2=-3/Lambda
Length scale appearing in Eq. (20); identified with the AdS radius without being stated. If not fixed by Lambda, its value is unspecified.

assumptions (4)

domain assumption Extended first law dM = T dS + V dP + Phi dq with P = -Lambda/(8 pi)
Standard black-hole-chemistry identification used to define enthalpy and volume; cited to prior literature.
domain assumption Entropy S = pi r_+^2 - pi q^3/r_+
Asserted in Eq. (22) as the integral of dM/T but not derived from Eq. (20); standard for regular NED black holes but a separate input.
ad hoc to paper l^2 = -3/Lambda (implicit)
Needed to make Eq. (20) consistent with the metric Eq. (14); never stated and physically conflates the Hayward scale with the AdS radius.
domain assumption Normalized Ruppeiner curvature construction with vanishing C_V
Takes the C_V=0 limit of the (T,V) Ruppeiner metric following refs. [52,53,74]; central to the R_N result.

invented entities (1)

Black hole microstructural constituents (interpretive)
purpose: Give physical meaning to the sign of Ruppeiner curvature as attractive or repulsive interactions among black hole molecules.
No microphysical model or independent observable is provided; the curvature sign is interpreted by analogy with fluids.

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Cite this review

Pith. "Pith review of Thermodynamic Curvature and Topological Insights of Hayward Black Holes with String Fluids." pith.science (2026). https://pith.science/paper/GDZPTEOV

@misc{pith2026250623736,
  author       = {Pith},
  title        = {Pith review of: Thermodynamic Curvature and Topological Insights of Hayward Black Holes with String Fluids},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GDZPTEOV}},
  note         = {Machine review of arXiv:2506.23736}
}

abstract

In this paper, we study the influence of string fluids on the extended thermodynamic structure and microscopic interactions of Hayward black holes by employing thermodynamic geometry as an empirical tool. Using the novel equation of state obtained for regular black holes surrounded by string fluids, we analyze the extended phase space with enthalpy as the central thermodynamic potential. By examining the behavior of the normalized Ruppeiner curvature scalar $R_N$ in the temperature-volume $(T,V)$ plane, we analyzed the influence of the string fluid parameters on the microstructure of the black hole. Our analysis reveals that the presence of string fluids significantly modifies the dominant microscopic interactions, transitioning from attractive to repulsive regimes depending on the charge and volume of the black hole. We see that the thermodynamic curvature effectively detects critical points and phase transitions, reflecting the nature of repulsive or attractive interactions among black hole microstructures. We further investigate thermodynamic topology to provide a novel classification scheme for stability and phase behavior, delineating local stable and unstable regions in parameter space. We investigate the thermodynamic topology of Hayward-AdS black holes surrounded by string fluids, showing that the number and type of topological charges depend on the parameters $\epsilon$ and $b$, revealing phase transitions and stability characteristics encoded in the global topological charge $W$. This integrated study of the thermodynamic geometry and topology structure enhances the understanding of Hayward black holes surrounded by string fluids, showing overall thermodynamic stability and configuration with significant implications for holographic duality and potential astrophysical observations.

Figures

Figures reproduced from arXiv: 2506.23736 by the authors.

**Figure 2.** FIG. 2. The behavior of black hole temperature [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** FIG. 3. The behavior of thermodynamic pressure [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: FIG. 4. Thermodynamic curvature [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

**Figure 5.** Figure 5: FIG. 5. The [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

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For β = −1, the Kretschmann scalar remains finite throughout the spacetime and admits well-defined limits at both the origin and spatial infinity. Specifically, one finds: lim r→0 K = 8 l2 − b2 Λl2 + 3 2 3b4l4 and lim r→∞ K = 8 b2Λ − 1 2 3b4 . (16) These expressions confirm that the curvature remains regular in both the ultraviolet and infrared regimes, i...
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AdS-Reissner-Nordstr¨ om black hole:W = +1 To classify black holes based on their topological charges, a systematic exploration of varying free parameters within the model is required. By examining different parameter ranges and evaluating the corresponding topological charges, a more refined classification scheme emerges. Specifically, when ϵ = −1, the r...
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The black hole’s thermodynamic pressure P as a function of specific volume mirrors van der Waals fluid behavior. The resulting P − v diagram exhibits oscillatory isotherms and first-order phase transitions, significantly influenced by charge (q) and string fluid parameter (b)

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Negative RN values indicate predominant attractive microstructures, while a small domain of repulsive interaction emerges, similar to charged AdS black holes

The application of Ruppeiner geometry elucidates the microscopic interactions. Negative RN values indicate predominant attractive microstructures, while a small domain of repulsive interaction emerges, similar to charged AdS black holes
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The divergence of RN at phase transitions precisely coincides with heat capacity singularities, providing a direct diagnostic for criticality in the extended thermodynamic space
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The topological charge distribution depends on ϵ and b. • For ϵ = −1, the resulting black hole exhibits characteristics analogous to the Reissner-Nordstrom solution with total topological charge W = 0 • For ϵ = +1, the black hole demonstrates behavior akin to the AdS-Reissner-...

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

The first observation is β ̸= 2

[2] [2]

Specifically, one finds: lim r→0 K = 8 l2 − b2 Λl2 + 3 2 3b4l4 and lim r→∞ K = 8 b2Λ − 1 2 3b4

For β = −1, the Kretschmann scalar remains finite throughout the spacetime and admits well-defined limits at both the origin and spatial infinity. Specifically, one finds: lim r→0 K = 8 l2 − b2 Λl2 + 3 2 3b4l4 and lim r→∞ K = 8 b2Λ − 1 2 3b4 . (16) These expressions confirm that the curvature remains regular in both the ultraviolet and infrared regimes, i...

[3] [3]

black hole chemistry

For the parameter ranges 0 < β <2 and β >2, the Kretschmann scalar exhibits divergent behavior in the vicinity of the origin, indicating a curvature singularity at r = 0. Conversely, the scalar remains finite at large radial distances, asymptotically approaching a constant value. Specifically, we have: lim r→0 K = ∞ and lim r→∞ K = 8Λ2 3 . (17) This behav...

1979

[4] [4]

Schwarzschild black hole: W = −1

[5] [5]

Reissner-Nordstr¨ om black hole:W = 0

[6] [6]

black hole molecules

AdS-Reissner-Nordstr¨ om black hole:W = +1 To classify black holes based on their topological charges, a systematic exploration of varying free parameters within the model is required. By examining different parameter ranges and evaluating the corresponding topological charges, a more refined classification scheme emerges. Specifically, when ϵ = −1, the r...

[7] [7]

The spacetime remains regular for −1 ≤ β <0, for other values of β, the Kretschmann scalar diverges, indicating a central singularity linked to perfect fluid dark matter theories

[8] [8]

The resulting P − v diagram exhibits oscillatory isotherms and first-order phase transitions, significantly influenced by charge (q) and string fluid parameter (b)

The black hole’s thermodynamic pressure P as a function of specific volume mirrors van der Waals fluid behavior. The resulting P − v diagram exhibits oscillatory isotherms and first-order phase transitions, significantly influenced by charge (q) and string fluid parameter (b)