Off-diagonal deformations of regular Schwarzschild black holes and general relativistic G. Perelman thermodynamics

El\c{s}en Veli Veliev; Sergiu I. Vacaru

REVIEW 5 major objections 5 minor 1 cited by

Off-diagonal deformations of regular Schwarzschild black holes and general relativistic G. Perelman thermodynamics

T0 review · 5 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper claims that regular Schwarzschild black holes can be deformed into generic off-diagonal Einstein solutions whose thermodynamics is Perelman's geometric-flow entropy rather than Bekenstein-Hawking entropy.

desk verdict New off-diagonal deformations of regular Schwarzschild metrics, but the GR claim rests on an unverified Levi-Civita extraction and the thermodynamics is formal. read the letter →

arxiv 2505.18208 v1 pith:ROXLQWPM submitted 2025-05-22 physics.gen-ph

Sergiu I. Vacaru , Elşen Veli Veliev This is my paper · ORCID

classification physics.gen-ph

keywords off-diagonalmetricdeformationsregularblackholesSchwarzschildholeanholonomicframemethodPerelmanentropygeometricflowthermodynamicsBekenstein-Hawkinggravitationalpolarizations

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 regular Schwarzschild black holes admit a broad class of generic off-diagonal deformations that remain solutions of the Einstein equations. Using a nonholonomic 2+2 splitting with an auxiliary connection, the field equations decouple into integrable systems, and the resulting target metrics encode de Sitter condensates, solitonic vacuum structure, deformed horizons, and gravitational polarizations of constants. Because these generic off-diagonal solutions do not have hypersurface or holographic descriptions, the paper argues that the Bekenstein-Hawking thermodynamic paradigm is not applicable to them, and that G. Perelman's W-entropy from geometric flows provides the appropriate thermodynamics. If correct, this gives new exact and parametric regular black-hole models in general relativity and a concrete set of thermodynamic variables computed from volume forms and effective cosmological constants rather than horizon area.

What carries the argument

The carrying object is the anholonomic frame and connection deformation method (AFCDM): a nonholonomic 2+2 splitting of spacetime with a nonlinear connection $N$ and a canonical d-connection $\hat D$, adapted to the quasi-stationary ansatz (27). The decoupling formulas (29)-(30) express the metric coefficients in terms of a generating function $\Psi$ and h/v generating sources, turning the field equations into integrable systems of nonlinear PDEs. Zero-torsion constraints (36) are then imposed to extract Levi-Civita configurations. The nonlinear symmetries (33)-(34) re-encode generating data through an effective cosmological constant $\Lambda$, and these symmetries reduce the Perelman W-entropy calculation to volume forms (65) and the explicit thermodynamic variables (66)-(68).

What would settle it

Take the target d-metrics (47) or (51) for the regular mass functions (55)/(56)/(57), impose the Levi-Civita constraints (36), and compute the Einstein tensor of the resulting metric. If the constraints admit no smooth nonzero solution for these generating data, or the resulting Ricci tensor is not of the claimed source form, the central claim that these are GR solutions fails.

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Extended reading notes

Core claim

The central claim is that the quasi-stationary d-metrics (47) and (51), generated from a prime regular Schwarzschild metric by gravitational polarization functions $\eta^4(r,\theta,\phi)$ or $\chi^4(r,\theta,\phi)$, are exact or parametric solutions of the Einstein equations written with the canonical d-connection, with Levi-Civita configurations extracted whenever the zero-torsion constraints (36) are imposed. For the regular mass functions (55), (56), and (57), these solutions describe deformed regular black holes with Schwarzschild exterior, an off-diagonal de Sitter condensate, and possible solitonic or void structures in the interior and exterior. The paper further claims that such generic off-diagonal solutions lack well-defined horizons and holographic boundaries, so Bekenstein-Hawking entropy and temperature are not valid for them; instead the paper defines G. Perelman thermodynamic variables $\hat E(\tau)$, $\hat S(\tau)$, and $\hat Z(\tau)$ using nonlinear symmetries that introduce an effective cosmological constant $\Lambda(\tau)$, and computes them explicitly in formulas (66) and (68).

Load-bearing premise

The load-bearing premise is that the deformed metrics (47)/(51) satisfy the standard Einstein equations with the Levi-Civita connection once the zero-torsion constraints are imposed, an extraction the paper asserts is always possible but does not demonstrate for the specific regular-black-hole deformations it constructs.

Editorial extensions

If this is right

The deformed regular Schwarzschild metrics give new exact or parametric solutions in general relativity with six degrees of freedom, so unmodified GR can host nonsingular interiors with solitonic vacuum condensates.
For generic off-diagonal deformations, Bekenstein-Hawking entropy does not apply; thermodynamic properties must be read from the Perelman variables $\hat E(\tau)$ and $\hat S(\tau)$.
The volume-form formula (65) makes the entropy ratio of two deformed solutions equal to the ratio of their volume functionals, giving a concrete comparison criterion independent of horizon areas.
For rotoid deformations (52), the Bekenstein-Hawking and Perelman descriptions coexist at fixed $\tau_0$ but describe different physics, so the two entropies can be checked against each other.

Reading between the lines

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

Beyond the paper, the no-holography argument would apply to any generic off-diagonal stationary configuration, not only these regular black holes, which would make area-based entropy bounds non-universal in GR.
Beyond the paper, a direct numerical check of the Levi-Civita constraints (36) on the $n,l$ mass families would settle whether the claimed solutions actually exist as Levi-Civita Einstein metrics.
Beyond the paper, the entropy ratio (67) suggests a thermodynamic preference ordering among deformed solutions; comparing that ordering with linearized stability would be a concrete testable extension.

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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

Quick take: the paper does construct genuinely new off-diagonal deformations of the regular Schwarzschild metrics from Casadio et al., but the central claim that these are GR solutions is not supported because the Levi-Civita extraction is asserted, not checked for the specific metrics. The Perelman thermodynamics part is formal.

What's actually new: the combination of the regular mass functions (56)/(57) with the η4 and χ4 polarizations in the target d-metrics (47)/(51) is not in the cited literature. The paper gives a clear recipe: start from a prime diagonal metric, apply the anholonomic frame method, and produce parametric families with solitonic or ellipsoidal features. The authors are upfront that generic off-diagonal solutions don't have horizons, which is why they invoke Perelman entropy. That is an honest and interesting move.

The soft spot is load-bearing. Equations (25) are solved with the auxiliary hat connection, not with ∇. To get Einstein equations in GR you need the canonical d-torsion to vanish, i.e. constraints (36). The paper says LC configurations can always be extracted (Sec. 3.1.1) and refers to the earlier method, but never verifies that the specific functions in (47) or (51) satisfy w*_i = e_i ln √|h3|, n*_i = 0, etc. Without that verification, (47)/(51) are only solutions of the modified system. The thermodynamic variables in (66)-(68) inherit this gap, and they are also formal: the volume functionals J are not evaluated for any concrete metric, so there are no numbers to compare. A referee would need to see at least one explicit check that a target metric is torsion-free (or a concrete counterexample where it isn't).

The paper relies heavily on earlier works by the same group for the decoupling machinery; that in itself isn't a flaw, but here the missing step is exactly in the part they've outsourced. The citation pattern is mostly fair, with the regular BH literature properly acknowledged.

Who's this for? People working on off-diagonal exact solutions and geometric flow thermodynamics. It's a niche but real community. The paper deserves peer review: a serious referee could push the authors to either prove the LC conditions for the explicit families or restrict the claims to the hat-connection theory. Right now the GR claim is unproven, so I would not cite it as an exact GR solution.

Referee Report

5 major / 5 minor

Summary. The paper constructs quasi-stationary off-diagonal d-metric families intended to describe deformations of regular Schwarzschild black holes, using the authors' anholonomic frame and connection deformation method (AFCDM). The construction first solves Einstein-type equations for an auxiliary canonical d-connection with nontrivial d-torsion, and then claims that Levi-Civita (LC) configurations can be extracted by imposing the constraints (36). Sections 3 and 4 present explicit target d-metrics (47) and (51) built from gravitational polarization functions, and Section 5 proposes G. Perelman thermodynamic variables for these families, asserting that Bekenstein–Hawking thermodynamics is inapplicable to generic off-diagonal solutions. The paper does not verify that the specific target d-metrics satisfy the LC constraints (36), and the thermodynamic variables are expressed through volume functionals that are never evaluated for the displayed metrics.

Significance. If the LC-extraction step were rigorously checked and the volume functionals evaluated, the paper would contribute a large class of explicit off-diagonal regular black hole metrics and a thermodynamic framework alternative to Bekenstein–Hawking entropy. The manuscript is transparent about its auxiliary-connection strategy and lists the constraints needed for GR solutions, which is a strength. However, the central physical claim that (47) and (51) are solutions of the standard Einstein equations is not established, and the thermodynamic results remain formal. The significance is therefore conditional on closing these gaps.

major comments (5)

[Sec. 2.4 and Sec. 3.1.2, Eqs. (36) and (47)] The central claim that the target d-metrics (47) are solutions of the Einstein equations is unsupported: the construction solves the hat-connection equations (25), and to obtain LC configurations the coefficients must satisfy the zero-d-torsion constraints (36). For the d-metric (47), the functions w_i and n_i are fixed by the formulas (30) in terms of the polarization function η4(r,θ,ϕ) and the sources; no computation is shown that these functions obey w*_i = e_i ln sqrt|h3|, e_i ln sqrt|h4| = 0, ∂_i w_j = ∂_j w_i, n*_i = 0, and ∂_i n_j = ∂_j n_i. Since η4 depends on r, θ, and ϕ, the condition e_i ln sqrt|h4| = 0 is generically violated. Unless explicit restrictions on η4, the sources, and the integration functions are stated and verified, (47) is only a solution of the modified hat-D system, not of the standard Einstein equations (1).
[Sec. 3.1.3, Eq. (51)] The same LC-extraction gap affects the κ-parametric d-metrics (51). The n-coefficients in (51) involve terms proportional to 2n_k times integrals over ϕ; their ϕ-derivatives are generically nonzero, which contradicts n*_i = 0 in the LC constraints (36) unless 2n_k is set to zero with appropriate choices of the remaining integration data. No such restrictions or verification are provided, so the claim that (51) describes small deformations of a prime Schwarzschild metric in GR is not established.
[Sec. 5.2.1 and Sec. 5.2.2, Eqs. (63)–(68)] The thermodynamic variables are formal rather than computed. Equation (65) defines the volume functional J_ηV as an unspecified integral over the quasi-stationary d-metric, and the formulas (66) and (68) merely multiply this functional by factors involving Λ(τ) and τ. No explicit evaluation is given for any of the d-metrics (47), (51), or the n/l-labeled regular BH families of Section 4. The paper therefore does not deliver concrete thermodynamic predictions such as explicit entropy or energy functions of τ and physical parameters; it only provides a template whose numerical content depends on an unevaluated functional.
[Sec. 5.2.2, Eq. (67)] The entropy ratio in Eq. (67) is printed as q_1Ŝ(τ)/q_2Ŝ(τ) = J_ηV[q_1g(τ)]/J_ηV[q_1g(τ)], with the same metric q_1 in both numerator and denominator. As written, the ratio is identically 1 and cannot compare two different quasi-stationary solutions. The denominator should presumably be J_ηV[q_2g(τ)]. This typo affects the interpretational claim about thermodynamic priorities and must be corrected.
[Sec. 4.2 and Sec. 4.3, Eqs. (56)–(57)] The regular BH families are introduced through mass functions Λ_em(r) and l_em(r) inherited from Refs. [1,26], but the off-diagonal deformations of these primaries are not shown to preserve the stated regularity and horizon properties, nor are they checked against the LC constraints (36). The assertion that these constructions define 'regular oﬀ-diagonal modiﬁcations of the Schwarzschild exterior' therefore requires an explicit verification for the deformed metrics, not only for the diagonal primaries.

minor comments (5)

[Sec. 3.1.1, Eq. (40)] The mass-function definition in Eq. (40) contains an incomplete formulation: '˚m for 0<r, we obtain the Schwarzschild solution' is grammatically and logically garbled and should be rewritten as a clean piecewise definition.
[Throughout] There are numerous typos and misspellings, including 'quass-stationary', 'beomg', 'Ffor', 'symple', and 'Thorston-Poincaré' in Section 6; these should be corrected in a revised version.
[Sec. 5.3.2] The phrase '9 diﬀerent type τ -running quasi-stationary conﬁgurations' does not match the preceding enumeration, which lists nine notational variants; the count and the list should be aligned.
[Sec. 2.2, Eqs. (29)–(31)] The notation for the integration function oscillates between h[0]_4 and g[0]_4 in Eqs. (29)–(31) and in the quadratic element; a single consistent symbol should be used throughout.
[References] Several reference entries contain errors: the Riess et al. arXiv identifier should be astro-ph/9805201 rather than astro-ph/0905201, and the Ovalle arXiv identifier in Ref. [26] appears as '2405/06731' instead of '2405.06731'.

Circularity Check

1 steps flagged · score 4.0 of 10

The central solution-generation is an application of a self-cited but checkable decoupling method; the LC-extraction step is asserted rather than verified, which is a correctness gap rather than circularity. The one by-construction reduction is the entropy ratio (67), which is a tautology of definition (66).

self definitional [Section 5.2.2, Eq. (67)]
"q1 ˆS(τ )/ q2 ˆS(τ ) = Jη V[ q1g(τ )]/ Jη V[ q1g(τ )]. (67) ... Such relations can be always derived using the nonlinea symm etries (A.5)."

Equation (66) defines qηS(τ) = [1−Λ(τ)]/[4π²τ²] JηV[qg(τ)]. Substituting this definition into the left side of (67) gives JηV[q1g]/JηV[q2g] identically, because the common factor [1−Λ]/[4π²τ²] cancels. The reference to 'nonlinear symmetries (A.5)' plays no role in this identity. Thus the claimed relation is not a derived physical prediction but a restatement of the definition of the entropy variable, and the subsequent comparison of 'thermodynamic and informational priorities' carries no independent content beyond the chosen volume functional.

full rationale

The paper's construction pipeline is: solve the hat-connection equations (25) using the quasi-stationary ansatz (27), then extract Levi-Civita configurations by imposing constraints (36). The decoupling formulas (29)-(30) and the assertion that LC configurations can always be extracted are imported from the authors' previous papers [2-5], but these are mathematical statements that can in principle be checked by substitution, so the self-citation is not itself circular. The main weakness is that for the target regular-BH d-metrics (47) and (51), the paper never verifies the specific LC constraints (36); it only says, after Eq. (46), that 'LC configurations can be always extracted as for d-metrics (37)'. This is an omitted proof and a correctness risk, not a circular step, because no equation identity is exhibited that makes the GR claim equal to its inputs. The thermodynamic variables (66) are defined as functionals of the already-constructed metric, which is a definition rather than a prediction; the statement that Bekenstein-Hawking thermodynamics is inapplicable follows from the absence of horizons/holographic structure and is likewise not circular. The one genuinely circular element is Eq. (67): the entropy ratio is identically the ratio of volume functionals by virtue of definition (66), so it reduces by construction to the input volume functional. Overall, the central solution-generating claim retains independent content, but the paper contains a load-bearing unverified LC-extraction step and a minor self-definitional thermodynamic relation, giving a partial circularity score of 4.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The paper rests on a self-cited decoupling method and on arbitrary generating, polarization, and integration functions. The resulting solutions and thermodynamic variables are therefore parameterized by choices rather than derived uniquely from physical principles.

free parameters (5)

Generating function Psi(xi,y3) and source vUpsilon(xi,y3)
Prescribed by hand; every solution in (31)/(35) is parameterized by these functions, and changing them changes the physical configuration without any selection principle.
Gravitational polarization functions eta4(r,theta,phi) and chi4(r,theta,phi) = e.g., chi4 = chi sin(omega0 phi + phi0) in Eq. (52)
Chosen ad hoc to produce ellipsoidal or solitonic deformations; not derived from equations of motion.
Integration functions 1n_k, 2n_k, h[0]_4
Determined only by unspecified boundary or asymptotic conditions; no explicit values are given for physical examples.
Mass-function coefficients c_n and integer labels n,l = e.g., Lambda e m(r) in Eq. (56), l e m(r) in Eq. (57)
Imported from [1] and prescribed to ensure regularity; not fitted to data in this paper but essential to the prime metrics.
Effective cosmological constant Lambda(tau)
Introduced via nonlinear symmetries (50) and allowed to run with tau; no observational anchoring.

assumptions (4)

domain assumption The canonical d-connection hat D allows decoupling and integration of the Einstein equations in the form (29)-(30).
This is the core of AFCDM, cited to the authors' previous works [2-5]; no proof is included in this paper.
ad hoc to paper LC configurations can always be extracted from hat D solutions by imposing constraints (36).
Stated in Sec. 2.4 and assumed in Sec. 3.1.1; not demonstrated for the target regular BH deformations.
ad hoc to paper Perelman's W-functional and thermodynamic variables can be generalized to relativistic nonholonomic Lorentz manifolds by postulating (B.1)-(B.2) and (62).
The relativistic generalization is postulated and references the authors' prior work; no derivation from GR or Ricci flow theory is given.
ad hoc to paper For quasi-stationary solutions with hat R_sc = 4Lambda(tau), the thermodynamic variables reduce to (63).
This simplification is used to compute entropy and energy from volume functionals; it depends on the unproved W-functional monotonicity in the Lorentzian setting.

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Pith. "Pith review of Off-diagonal deformations of regular Schwarzschild black holes and general relativistic G. Perelman thermodynamics." pith.science (2026). https://pith.science/paper/ROXLQWPM

@misc{pith2026250518208,
  author       = {Pith},
  title        = {Pith review of: Off-diagonal deformations of regular Schwarzschild black holes and general relativistic G. Perelman thermodynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ROXLQWPM}},
  note         = {Machine review of arXiv:2505.18208}
}

read the original abstract

We construct new classes of solutions describing generic off-diagonal deformations of regular Schwarzschild black holes (BHs) in general relativity (GR). Examples of such (primary) diagonal metrics reducing the Einstein equations to integrable systems of nonlinear ordinary differential equations were studied in a recent work by R. Casadio, A. Kamenshchik and J. Ovalle in Phys. Rev. D 111 (2025) 064036. We develop and apply our anholonomic frame and connection deformations method, which allows us to generate new classes of target off-diagonal solutions. Ansatz that reduces the gravitational field equations to systems of (exactly or parametric) integrable systems of nonlinear partial differential equations are used. We find and analyze certain families of deformed regular BHs containing an off-diagonal de Sitter condensate encoding solitonic vacuum configurations, with possible deformations of horizons and/or gravitational polarizations of constants. We emphasize that general off-diagonal solutions do not involve certain hypersurface or holographic configurations. This means that the Bekenstein-Hawking thermodynamic paradigm is not applicable for characterizing the physical properties of such target regular solutions. We argue that the concept of G. Perelman's entropy and relativistic geometric flow thermodynamics is more appropriate. Using nonlinear symmetries involving effective cosmological constants, we show how to compute thermodynamic variables for various classes of physically essential solutions in GR.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Metric--Measure Geometry and Geometric Analogues of Holographic Extremal Surfaces
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The paper re-derives the standard weighted-area condition H=(1/2)∂_n f and labels it a holographic analogue; the example scales are inserted through hand-chosen f.

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Bubuianu, J

L. Bubuianu, J. O. Seti, S. Vacaru, and E. V. Veliev, Nonas sociative cosmological solitonic R-ﬂux defor- mations in gauge gravity and G. Perelman geometric ﬂow therm odynamics, Annals of Physics 465 (2024) 169689; arXiv: 2410.03698

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Vacaru, Nonassociative gauge gravity theories with R-ﬂux star products and Batalin-Vilkovisky quanti- zation in algebraic quantum ﬁeld theory, Class

S. Vacaru, Nonassociative gauge gravity theories with R-ﬂux star products and Batalin-Vilkovisky quanti- zation in algebraic quantum ﬁeld theory, Class. Quant. Grav . 42 (2025) 015006; arXiv: 2411.12158 35

2025 arXiv

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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Casadio, A

R. Casadio, A. Kamenshchik and J. Ovalle, Regular Schwar zschild black holes and cosmological models, Phys. Rev. D 111 (2025) 064036, arXiv: 2502.13627

arXiv 2025

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S. Vacaru, Decoupling of ﬁeld equations in Einstein and m odiﬁed gravity, J. Phys.: Conf. Ser. 543 (2013) 012021; arXiv: 1108.2022v3

arXiv 2013

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Vacaru, Oﬀ-diagonal ekpyrotic scenarios and equival ence of modiﬁed, massive and/or Einstein gravity, Phys

S. Vacaru, Oﬀ-diagonal ekpyrotic scenarios and equival ence of modiﬁed, massive and/or Einstein gravity, Phys. Lett. B 752 (2016) 27-33; arXiv: 1304.1080

arXiv 2016

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Deforming Black Hole and Cosmological Solutions by Quasiperiodic and/or Pattern Forming Structures in Modified and Einstein Gravity

L. Bubuianu and S. Vacaru, Deforming black hole and cosmol ogical solutions by quasiperiodic and/or pattern forming structures in modiﬁed and Einstein gravity , Eur. Phys. J. C 78 (2018) 33; arXiv: 1706.02584

work page Pith review arXiv 2018

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Decoupling and integrability of nonassociative vacuum phase space gravitational equations with star and R-flux parametric deformations

Elsen Veli Veliev, Laurentiu Bubuianu, and Sergiu I. Vaca ru, Decoupling and integrability of nonassociative vacuum phase space gravitational equations with star and R- ﬂux parametric deformations, Fortschr. Phys. 69 (2021) 2100030; arXiv: 2106.01869

work page Pith review arXiv 2021

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Vacaru, Nonholonomic Ricci ﬂows: II

S. Vacaru, Nonholonomic Ricci ﬂows: II. Evolution equat ions and dynamics, J. Math. Phys. 49 (2008) 043504; arXiv: math.DG/0702598

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T. Gheorghiu, V. Ruchin, O. Vacaru and S. Vacaru, Geometr ic ﬂows and Perelman’s thermodynamics for black ellipsoids in R2 and Einstein gravity theories; An nals of Physics, NY, 369 (2016) 1-35; arXiv: 1602.08512

arXiv 2016

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Bubuianu, D

L. Bubuianu, D. Singleton, S. Vacaru, and E, V. Veliev, Non associative geometric and quantum information ﬂows and R-ﬂux deformations of wormhole solutions in string gravity, Fortschr. Phys. 72 (2024) 202300212; arXiv: 2402.10993

arXiv 2024