Regular fluid of strings black hole with non trivial core and asymptotic structure by gravitational decoupling

Leonardo G. Barbosa; Luis C. N. Santos; Milko Estrada

REVIEW 2 major objections 6 minor 35 references

Regular fluid of strings black hole with non trivial core and asymptotic structure by gravitational decoupling

T0 review · 2 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A screening function on the whole string term, not just the mass, yields a regular string-supported black hole.

desk verdict Solid regular black hole construction with a real factor-2 slip in the topology section; the main results survive, the section needs fixing. read the letter →

arxiv 2608.02819 v1 pith:OJIDMHG3 submitted 2026-08-03 gr-qc hep-th

Milko Estrada , Luis C. N. Santos , Leonardo G. Barbosa This is my paper · ORCID

classification gr-qchep-th MSC 83C5783C1583C20

keywords regularblackholescloudofstringsstringfluidgravitationaldecouplingminimalgeometricdeformationdeSittercorequasinormalmodesholethermodynamics

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper sets out to show that a cloud-of-strings black hole cannot be regularized by the usual trick of replacing the point mass with a mass function $m(r)\sim r^3$: the string sector adds an independent term that leaves a $\sim r^{-4}$ divergence in the Kretschmann scalar. Using gravitational decoupling, it constructs a lapse $f(r)=1-(2M/r+\epsilon)s(r)$ with screening function $s(r)=r^3/(L^4M+r^3)$, which stays finite at $r=0$ and recovers the cloud-of-strings geometry at large distances with an extra $\epsilon L^4 M/r^3$ correction. If correct, this provides a regular black hole supported by an anisotropic string fluid whose string sector changes the de Sitter core, shifts the Davies phase transition and remnant size, and leaves a systematic imprint on scalar quasinormal modes.

What carries the argument

The load-bearing object is the screening function $s(r)$ together with the gravitational-decoupling (minimal geometric deformation) split of the lapse into a seed part $f_0=1-2Ms(r)/r$ and a deformation part $f_\epsilon=-s(r)$. Imposing the anisotropic equation of state $p_{t,\epsilon}=\omega_\epsilon(r)\rho_\epsilon$ turns $s(r)$ into a solution of $r^2s''+2r(1+\omega_\epsilon)s'+2\omega_\epsilon s=0$; choosing $\omega_\epsilon(0)=-3/2$ forces $s(r)\sim r^3$ near the core, the standard regularity condition, while $\omega_\epsilon\to 0$ at infinity makes $s(r)$ tend to a constant and restores the cloud-of-strings asymptotics. This mechanism is what lets the string sector itself, rather than a mass function, govern the ultraviolet behavior.

What would settle it

Compute the exact Kretschmann invariant for $f(r)=1-(2M/r+\epsilon)r^3/(L^4M+r^3)$ at $r=0$; any divergent subleading term would refute the regularity claim. A sharper test is to solve the differential equation for $s(r)$ with the equation of state of the fundamental string worldsheet action instead of assuming the given profile: if the resulting $s(r)$ does not behave as $\sim r^3$ near the core and tend to a constant at infinity, the claim that the string sector can be consistently screened while keeping its interpretation fails.

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Extended reading notes

Core claim

The central claim is that the cloud-of-strings spacetime admits a regular extension of the form $f(r)=1-(2M/r+\epsilon)s(r)$ with $s(r)=r^3/(L^4M+r^3)$, in which the string contribution is screened together with the mass rather than by an effective mass alone. Near the origin $f\simeq 1-2r^2/L^4-\epsilon r^3/(L^4M)$, so the Ricci and Kretschmann scalars remain finite, while at large distances $f\simeq 1-\epsilon-2M/r+\epsilon L^4M/r^3+2L^4M^2/r^4$, recovering the string-cloud asymptotics plus a longer-range $\epsilon r^{-3}$ correction that dominates the usual de Sitter-core (LQG) $r^{-4}$ term. The paper further claims that the solution admits non-extremal and extremal black holes and a regular horizonless compact object, that the string parameter shifts the Davies phase transition and the remnant size, and that the scalar quasinormal-mode spectrum changes systematically with $\epsilon$.

Load-bearing premise

The whole construction rests on the assumed screening profile $s(r)=r^3/(L^4M+r^3)$, chosen together with $\omega_\epsilon(0)=-3/2$; the paper does not derive this profile from the string-fluid action, so if a different profile were forced by the string-fluid equations of state, the core deformation, thermodynamics, and quasinormal-mode shifts would all change.

Editorial extensions

If this is right

The standard regular-black-hole prescription of replacing $M$ by $m(r)\sim r^3$ leaves a curvature singularity in string-supported geometries; the effective-mass mechanism must be applied to the combined term $2M/r+\epsilon$.
The spacetime interpolates between a deformed de Sitter core (local $S^3$ topology with an $\epsilon$-dependent deformation) and cloud-of-strings asymptotics, with an $\epsilon r^{-3}$ correction that dominates the usual LQG $r^{-4}$ term at intermediate distances.
The horizon structure is richer than the de Sitter-core case: non-extremal and extremal black holes exist, and beyond the critical value $\lambda^2=16/27$ the solution is a regular horizonless compact object; the extremal radius grows as $(1-\epsilon)^{-1}$.
The string sector turns the thermodynamically unstable cloud-of-strings black hole into one with a Davies-type phase transition, a locally stable branch, and a remnant whose size depends on $\epsilon$ and therefore need not be Planckian.
Scalar quasinormal modes shift systematically with $\epsilon$: increasing the string parameter moves the complex frequencies along nearly straight lines, raising both oscillation frequency and damping rate for higher multipoles.

Reading between the lines

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

The same screening strategy generalizes: any seed geometry with an independent matter sector that defeats the mass-function prescription could be regularized by screening the full combination, provided a suitable $\omega_\epsilon(r)$ is found.
The longer-range $\epsilon r^{-3}$ correction is a concrete observational target: at distances where the Newtonian $2M/r$ term dominates but the LQG $r^{-4}$ term is still negligible, the string-induced correction could show up in lensing or ringdown data if $\epsilon$ is not extremely small.
The choice $\omega_\epsilon(0)=-3/2$ fixes the exponent of the near-core power law; other constant negative values would give $s(r)\sim r^{-2\omega_\epsilon}$ and hence different core geometry and remnant physics, so the paper's quantitative predictions are tied to that specific exponent.
If the deformed core topology is generic, the local $S^3$ structure of regular black holes is not protected by regularity alone; mapping the topology change for these deformed cores would connect the construction to existing singularity theorems.

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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

Here's my take on arXiv:2608.02819.

The paper does something new and correct: it regularizes the cloud-of-strings black hole by screening the whole 2M/r + epsilon combination, not just the mass function. The metric family f = 1 - (2M/r + epsilon) r^3/(L^4 M + r^3) is clean, reduces to Hayward at epsilon=0 and to the string cloud at L=0, and the long-range epsilon r^-3 correction is a genuinely distinguishing feature. The thermodynamics is internally consistent; the temperature, heat capacity, Davies point, and epsilon-dependent remnant all follow from the metric without algebraic errors I could find. The QNM section uses standard sixth-order WKB and shows convergence with WKB order.

Soft spots are real but manageable. First, the topology section II.C has a genuine factor-2 inconsistency. Your own Eq. (55) gives R(0)=24/L^4, since L_eff^2=L^4/2. Eq. (59) uses R(0)=12/L^4, and Eq. (60) has the wrong coefficient; the coordinate transformation should be sin^2(xi)=4r^2/L^4, not 2r^2/L^4, and the epsilon sin^3(xi) coefficient becomes 5/12 epsilon L^2/M instead of sqrt(2)/3. The qualitative deformed-S3 conclusion survives, but the abstract and Section II.C need correction. I checked the second-pass note; it holds up. Second, the paper never states the domain 0 <= epsilon < 1 that the horizon analysis assumes; please make it explicit. Third, the screening ansatz is posited, not derived from a string-fluid action. The equation-of-state interpretation is reasonable but it is an assumption and should be flagged as such. Fourth, QNM results are only plotted; a table of numerical values and a check against an independent method would make them convincing.

Who is this for? People working on regular black holes, gravitational decoupling, and string-fluid phenomenology will get value from it. It deserves serious peer review: the central construction is correct and the flaws are fixable. I would send it out for review and ask for revision.

Referee Report

2 major / 6 minor

Summary. The paper constructs a static, spherically symmetric, regular black hole sourced by an effective anisotropic string fluid using the gravitational decoupling (MGD) scheme. The lapse function is f(r)=1-(2M/r+epsilon)s(r) with the Hayward-like screening function s(r)=r^3/(L^4 M+r^3), so that the O(epsilon) quasi-Einstein sector is interpreted as a string fluid with an r-dependent equation-of-state parameter omega_epsilon(r). The authors analyze the short- and long-distance behavior, the local topology of the core, the horizon structure, the Hawking temperature and heat capacity, and the scalar quasinormal-mode spectrum via sixth-order WKB. The paper's central claims are that the string sector deforms the de Sitter core, modifies the local S^3 topology, introduces a longer-range r^{-3} asymptotic correction, shifts the Davies phase transition and remnant size, and systematically changes the QNM spectrum.

Significance. If taken at face value, the paper provides a useful explicit example showing that the usual mass-function regularization of Hayward-type black holes does not automatically cure the cloud-of-strings singularity, and that a screening factor multiplying the combined source can do so while preserving the string-cloud infrared behavior. The main algebraic chain from the metric (18) through the horizon equation (65) and the thermodynamic expressions (76)-(77) is internally consistent, and the limits epsilon to 0 (Hayward) and L to 0 (Schwarzschild) check out. The QNM computation is a standard application of the sixth-order WKB method, and the convergence with WKB order is shown. The main limitation is that the central screening profile is posited rather than derived from the string-fluid action, so the advertised physical interpretation is conditional on that ansatz; the quantitative topology claim also contains a factor-2 inconsistency that must be corrected.

major comments (2)

[II.C (Eqs. 55-61)] There is a factor-2 inconsistency between the curvature calculation and the topology transformation. Equation (55) gives R(r,0)=12/L_eff^2=24/L^4, but Eq. (59) uses R(r,0)=12/L^4. For f(r)=1-2r^2/L^4-epsilon r^3/(L^4 M), the standard Ricci formula R=-f''-4f'/r+2(1-f)/r^2 gives R(0)=24/L^4 and a linear coefficient 20 epsilon/(L^4 M). Consequently, the correct core coordinate is sin^2(xi)=4r^2/L^4 with r=(L^2/2)sin(xi), and Eq. (60) becomes r^2 R/6 = sin^2(xi) + (5/12) epsilon L^2/M sin^3(xi), not the quoted sin^2(xi)+sqrt(2) epsilon L^2/(3M) sin^3(xi). Equation (61) and the deformation coefficient quoted in Eq. (64) must be corrected accordingly. The qualitative conclusion of a slightly deformed S^3 core survives with the corrected coefficients, but the quantitative topology/core result advertised in the abstract is not supported as written.
[II (Eqs. 27-32)] The text states that s(r) is not prescribed by hand but is dynamically determined through Eq. (27), and then adopts Eq. (32) as a Hayward-inspired ansatz. This is an overstatement: Eq. (27) only fixes the near-origin power-law behavior through omega_epsilon(0)=-3/2 and the asymptotic constant through omega_epsilon(to infinity)=0; the full profile s(r)=r^3/(L^4 M+r^3) is one member of that family, not a solution implied by the string-fluid equation of state. Since the core deformation, Davies radius, remnant size, and QNM shifts all depend on this specific profile, the claims that the string sector 'governs' the ultraviolet structure and that the geometry is a regular fluid-of-strings black hole are conditional on this choice. The authors should either derive Eq. (32) from a concrete string-fluid action or EoS, or explicitly state the ansatz dependence and test the robustness of their results under other r^3 screening profiles.

minor comments (6)

[II.A] The phrase 'As noted in Appendix I' should refer to Appendix A, which is where the Hayward seed properties are summarized.
[II.B (Eq. 48)] The short-distance expansion of rho_epsilon shown in Eq. (48) does not appear to follow from Eq. (34) with x=r/L; the small-x behavior of Eq. (34) is linear in x with a coefficient that does not match the displayed expression. Please check the algebra and state the units/conventions used for M and L.
[II.C (Eq. 69)] The text below Eq. (69) refers to 'lambda > lambda_cri' while the displayed cases are written in terms of lambda^2; the notation should be made consistent.
[V] The sentence 'Using the same notation as in [29], let us consider the mass terms of L in the form M=(1+L^4/2)/2' is unclear; the physical motivation for this particular parameter choice should be explained.
[Fig. 3] The label '3/2' in the heat-capacity figure is unexplained and should be removed or clarified in the caption.
[References] Reference [9] is incomplete: the Letelier 1983 fluid-of-strings paper should include the journal, volume, and page information.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the construction is an explicitly stated Hayward-type screening ansatz whose consequences are derived, with no fitted parameter, no load-bearing self-citation, and no prediction equivalent to its input by definition.

full rationale

The central chain is transparent and non-circular. The authors posit the screening function s(r)=r^3/(L^4 M + r^3) in Eq. (32), motivated by the Hayward/Planck-star form, and combine it with the minimal geometric deformation split in Eq. (18). From that explicit ansatz they compute the matter sector (Eqs. (33)-(39)), the long- and short-distance limits (Eqs. (43)-(53)), the curvature invariants (Eqs. (55)-(56)), horizons (Eq. (65)), thermodynamics (Eqs. (76)-(79)), and QNM spectra (Eqs. (86)-(87)). The asymptotic string-cloud behavior (rho_epsilon ~ epsilon/(8 pi r^2), pt_epsilon -> 0, W -> 0) is a derived consequence of s(r)->1, not an independent input, and the r^-3 metric correction is the Taylor expansion of the stated s(r). No parameter is fitted to data and then renamed a prediction; no uniqueness theorem from the authors' prior work is invoked to force the ansatz. The self-citations (Refs. [10], [11], [18]) are motivational or methodological and are not load-bearing for the main result. The only flagged issue is an internal arithmetic inconsistency in Sec. II.C: Eq. (59) uses R(0)=12/L^4 while Eq. (55) implies 12/L_eff^2 = 24/L^4, so the quoted deformed-S3 coefficients are not self-consistent. That is a correctness defect, not circularity, and it does not affect the regularity conclusion or the circularity score.

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

The construction imports the Hayward and Planck-star screening function and the cloud-of-strings equation of state; the only genuinely new input is the multiplicative screening of the string term, which is an ansatz rather than a derived consequence of string dynamics.

free parameters (4)

L (regularization length scale)
Introduced as the Hayward and Planck-star screening length in Eq. (32); sets the scale of the regular core and enters the horizon, temperature, heat capacity, and quasinormal-mode results. It is a model parameter, not fitted to observations.
epsilon (decoupling and string-cloud parameter)
Controls the string-cloud sector in f = 1 - (2M/r + epsilon) s(r). The horizon analysis in Eq. (65) divides by 1 - epsilon, so epsilon is implicitly assumed to be less than 1, but the bound is never stated.
omega_epsilon(0) = -3/2 = -3/2
Chosen in Eq. (31) so that the screening function behaves as r^3 near r = 0, matching the standard regular black hole and Hayward scaling. This choice is what makes the core regular.
QNM fixed mass M = (1 + L^4/2)/2 = 0.515625 for L = 1/2
Specific parameter choice used for the complex-frequency plots in Figs. 5-8, following Ref. [29]. It is not derived and not central to the geometric construction.

assumptions (5)

domain assumption The two gravitational sectors in the gravitational decoupling split are separately conserved, Eqs. (12)-(13), and interact only through the metric deformation.
This is a standard assumption of the minimal geometric deformation method; it excludes energy and momentum exchange between the Hayward seed and the string sector.
ad hoc to paper The screening function takes the Hayward form s(r) = r^3/(L^4 M + r^3) and multiplies the entire combined source 2M/r + epsilon.
Posited in Eq. (32) to interpolate between s approximately r^3 near the origin and s approaching 1 at infinity. All regularity, horizon, thermodynamic, and quasinormal-mode results follow from this ansatz.
ad hoc to paper The near-origin equation-of-state parameter omega_epsilon(0) is fixed to -3/2.
This value, given in Eq. (31), is chosen by hand to reproduce the standard r^3 screening behavior. It is not derived from the string-fluid dynamics.
domain assumption The full anisotropic matter with radial pressure p_r = -rho and tangential pressure p_t = W(r) rho is interpreted as a fluid of strings.
The paper verifies only the asymptotic equation-of-state behavior and the algebraic form of Appendix C; it does not derive the full interior matter distribution from the Nambu-Goto action.
ad hoc to paper The decoupling parameter epsilon satisfies 0 less than or equal to epsilon less than 1.
The horizon equation (65), the effective mass m = M/(1 - epsilon), and the asymptotic value f approaching 1 - epsilon require this bound, but the paper never states it explicitly.

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Cite this review

Pith. "Pith review of Regular fluid of strings black hole with non trivial core and asymptotic structure by gravitational decoupling." pith.science (2026). https://pith.science/paper/OJIDMHG3

@misc{pith2026260802819,
  author       = {Pith},
  title        = {Pith review of: Regular fluid of strings black hole with non trivial core and asymptotic structure by gravitational decoupling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OJIDMHG3}},
  note         = {Machine review of arXiv:2608.02819}
}

read the original abstract

Cloud-of-strings (CS) geometries provide an effective description of one-dimensional string distributions. However, their central singularity cannot be removed through the standard regular black holes (RBH) mechanism based on an effective mass function, since the string sector contributes independently to the ultraviolet structure of the spacetime. Motivated by this observation, we investigate whether string-supported black holes can be consistently regularized while preserving the CS asymptotics and admitting a physically meaningful string-fluid interpretation. Using the gravitational decoupling method, we construct a RBH supported by an effective anisotropic string fluid. We show that the string sector deforms the de Sitter core, modifies the local topology of the spacelike slices, and introduces a longer-range correction dominating the usual Hayward/LQG term. The geometry admits non-extremal and extremal RBH, as well as a regular horizonless compact object. Moreover, the string parameter qualitatively modifies the thermodynamic evolution by shifting the Davies phase transition and the size of the black-hole remnant. Finally, the scalar quasinormal-mode spectrum exhibits systematic changes in both the oscillation frequencies and damping rates. These results show that regularizing string supported black holes is a physically distinct problem, with the matter sector governing the ultraviolet structure, thermodynamics, and dynamical response of the spacetime.

Figures

Figures reproduced from arXiv: 2608.02819 by the authors.

**Figure 2.** FIG. 2: Hawking temperature [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗

**Figure 3.** FIG. 3: Heat capacity [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: FIG. 4: Effective potential [PITH_FULL_IMAGE:figures/full_fig_p023_4.png]

**Figure 5.** Figure 5: FIG. 5: Dependence of the quasinormal frequencies on the WKB approximation order for the [PITH_FULL_IMAGE:figures/full_fig_p024_5.png]

**Figure 6.** Figure 6: FIG. 6: Quasinormal frequencies in the complex plane for [PITH_FULL_IMAGE:figures/full_fig_p025_6.png]

**Figure 7.** Figure 7: FIG. 7: Quasinormal frequencies in the complex plane for the first overtone ( [PITH_FULL_IMAGE:figures/full_fig_p026_7.png]

**Figure 8.** Figure 8: FIG. 8: Quasinormal frequencies in the complex plane for the second overtone ( [PITH_FULL_IMAGE:figures/full_fig_p026_8.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Determine the location of the maximum of the effective potential V (r) and evaluate the re- quired derivatives at that point, since the WKB expansion is centered around this maximum

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Insert these quantities into the sixth-order WKB formula, which connects the complex fre- quency ω with the value and derivatives of the potential at its maximum

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Having established the scalar perturbation equation and the sixth-order WKB formalism, we now present the QNM results

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Pith tools

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

[1] [1]

Determine the location of the maximum of the effective potential V (r) and evaluate the re- quired derivatives at that point, since the WKB expansion is centered around this maximum

work page

[2] [2]

Insert these quantities into the sixth-order WKB formula, which connects the complex fre- quency ω with the value and derivatives of the potential at its maximum

work page

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Solve the resulting algebraic equation for ω to obtain the quasinormal frequencies corre- sponding to a chosen overtone number n

work page

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Having established the scalar perturbation equation and the sixth-order WKB formalism, we now present the QNM results

Repeat the analysis for different model parameters to investigate how the quasinormal spec- trum changes. Having established the scalar perturbation equation and the sixth-order WKB formalism, we now present the QNM results. It is instructive to examine the behavior of the effective potential, since its shape largely determines the oscillation frequencies...

work page 2024

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Consequently, the energy density of the cloud decays as r−2, a characteristic feature that will be relevant for the discussion presented in the main text

Imposing the conservation law ∇µT µν = 0 leads to −T 0 0 = −T 1 1 = ρ = √ −h ρcs = a r2 , (B4) where ρcs is interpreted as the proper energy density of the string cloud. Consequently, the energy density of the cloud decays as r−2, a characteristic feature that will be relevant for the discussion presented in the main text. Appendix C: A Brief Overview of ...

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