Repetitive Penrose Process in Rastall Rotating Black Holes Immersed in Quintessence Dark Energy

Abdul Malik Sultan; Ali Ahmad Sabir; Hamood Ur Rehman; Ke Wang; Muhammad Israr Aslam; Yakup Yildirim

REVIEW 4 major objections 5 minor 53 references

Repetitive Penrose Process in Rastall Rotating Black Holes Immersed in Quintessence Dark Energy

T0 review · 4 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read The paper claims that, for incident particles with unit specific energy, the repetitive Penrose process in a Rastall rotating black hole with quintessence stops at a spin threshold set uniquely by Particle 0, and that the Rastall structure

desk verdict Incremental but solid extension of the repetitive Penrose formalism to the Rastall rotating metric; tables are consistent, but overclaims and an unverified turning-point assumption need referee attention. read the letter →

arxiv 2607.19541 v1 pith:KO6DZZZD submitted 2026-07-21 gr-qc

Ali Ahmad Sabir , Muhammad Israr Aslam , Abdul Malik Sultan , Ke Wang , Hamood Ur Rehman , Yakup Yildirim This is my paper · ORCID

classification gr-qc MSC 83C5783D0583C10

keywords PenroseprocessrotationalenergyextractionRastallgravityrotatingblackholequintessencedarkirreduciblemassergoregionminimumspinthreshold

topics Dark Energy

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

What the paper tries to establish: in a rotating black hole described by Rastall gravity and surrounded by quintessence, the Penrose process can be repeated—each splitting inside the ergoregion feeds the next—and the iteration does not continue until the black hole stops spinning. Instead, for incident particles with unit specific energy, the process is terminated by a spin threshold set uniquely by Particle 0, the fragment that falls into the hole; that particle's required minimum spin is always the largest among the three decay products. The paper also claims that the Rastall structure parameter N-hat_s and, to a much smaller degree, the Rastall coupling alpha control the energetics: lower N-hat_s improves energy utilization at smaller decay radii, shifts the peak extracted energy inward, and leaves more extractable energy behind, while larger values boost efficiency at larger radii. A sympathetic reader would care because this is a concrete prediction of how modified gravity and dark-energy surroundings leave a signature on the maximum amount of rotational energy that can be mined from a black hole.

What carries the argument

The machinery is the triple turning-point idealization: at the splitting radius the radial momenta of the incident particle and both fragments vanish, so each particle sits at a turning point of its effective potential and the conservation equations (energy, angular momentum, radial momentum) admit a closed analytic solution. That solution is embedded in an iterative loop that updates the black hole mass, angular momentum, and the dimensionless Rastall structure parameter after each decay, recomputes the horizon and irreducible mass, and checks a set of stopping conditions. The decisive object is the minimum spin threshold, a-hat_min, of each decay particle; for E-hat_0 = 1 the largest of th

What would settle it

Simulate successive two-fragment decays in the same Rastall rotating metric with generic (non-vanishing) radial momenta—say, particle 0 approaching with a small inward radial velocity—and compare where the iteration terminates with the Particle 0 threshold from Eq. (3.4) and with the paper's tables. If the spin at termination, the number of iterations, or the energy utilization efficiency differs, the claim that Particle 0 uniquely controls the stopping condition is falsified.

Watch

Extended reading notes

Core claim

The core claim is that the stopping criterion for the repetitive Penrose process in this spacetime is fixed by Particle 0. Working at unit incident energy (E-hat_0 = 1) and imposing vanishing radial momenta for all three particles at each decay, the authors derive analytic expressions for decay products and iterate the black hole's mass, spin, irreducible mass, and structure parameter. They find the ordering a-hat_min,1 < a-hat_min,2 < a-hat_min,0 throughout the parameter space, so the highest threshold—the one that halts extraction—belongs to Particle 0 and coincides with the co-rotating marginally bound orbit. This threshold rises slowly from one iteration to the next; when the evolving sp

Load-bearing premise

The argument rests on the assumption that at every decay the incident particle and both fragments all have zero radial momentum at the splitting point, so the analytic turning-point solution applies at each iteration; if real decays occur away from a radial turning point, the reported stopping spins, efficiencies, and iteration counts would no longer be the ones that occur.

Editorial extensions

If this is right

For unit-energy incident particles, repeated Penrose decay cannot push the black hole below Particle 0's spin threshold; a residual extractable-energy reservoir always survives, and at small decay radii it can be larger than in previous repetitive-Penrose studies.
Energy utilization efficiency can exceed 50% in the Rastall background at larger decay radii and larger parameter values, unlike the Kerr repetitive process reported in the paper's comparison table.
Smaller initial values of the Rastall structure parameter make the process most efficient near the horizon and stop it after very few iterations—sometimes after a single decay—leaving a large untouched reservoir.
The Rastall coupling parameter alpha shifts the peak extracted energy and lowers total extractable energy, but its effect on the stopping threshold is negligible, so N-hat_s, not alpha, controls termination.
Because irreducible mass grows monotonically at each step, the generalized second law is respected and the process is self-limiting; a local decrease is never observed in the reported iterations.

Reading between the lines

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

If the triple turning-point condition is relaxed, the stopping threshold may no longer be set by Particle 0; a full geodesic treatment of off-turning-point decays would show whether the analytic threshold remains the controlling one or is only a special-case bound.
Since alpha barely moves the thresholds, the repetitive Penrose process is a weak probe of the Rastall coupling but a comparatively strong probe of the structure parameter; disentangling Rastall effects from plain Kerr would need a measurement of the final spin and residual extractable energy together.
The residual extractable energy left after termination suggests a natural two-stage scenario: the repetitive process extracts down to the Particle 0 threshold, after which a different mechanism, such as an electromagnetic process, could mine the remaining reservoir—a testable combined energy budget for high-energy astrophysical sources.
The reported efficiency gains could be tested in numerical-relativity simulations of particle splitting in a Rastall metric with evolving mass and spin, checking whether the iteration counts and stopping radii match the paper's tables.

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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

Short version: this is a straightforward extension of the repetitive Penrose process to the Rastall rotating metric with quintessence. It iterates the decay equations from the Kerr literature, tracks mass, spin, and irreducible mass, and maps how N-hat_s and alpha shift efficiency, stopping spin, and remaining extractable energy. I checked the first table's mass and angular-momentum updates against p_phi1 = -19.434 and mu0 = 0.01M; they are internally consistent. The limit N_s -> 0 reduces to the Kerr case. The core calculation is competent.

What is actually new is the parameter study: the claim that for E-hat_0 = 1, Particle 0 sets the stopping threshold across this parameter space, and that smaller N-hat_s boosts efficiency at smaller decay radii while alpha matters little. The trend is consistent with the effective-potential behavior shown.

Weak spots. First, Table V claims EUE > 50%, but every tabulated maximum is 43.7% or less. That is an overclaim or a typo. It needs to be fixed, not justified. Second, the conclusion drops the E-hat_0 = 1 qualifier and says Particle 0 always controls the stopping spin; the text itself says for E-hat_0 > 1, Particle 2 is the controller. The abstract and conclusion overstate the scope. Third, the promised N_s = 0 comparison with Ref. [48] is mentioned but never shown. That is missing evidence, not a fatal flaw. Fourth—and the one that needs the most work in review—the whole calculation leans on the triple turning-point assumption: at every split, particles 0 and 2 sit on the right of their effective-potential peaks and particle 1 on the left. The authors state this condition, then Tables I–IV assert it is satisfied, but they do not verify it by direct substitution for this metric. Since the stopping thresholds are derived from the same peak-coincidence limit, the numerical results stand or fall on that check. It is an inherited assumption from the prior framework and likely true for the parameter choices shown, but the paper does not demonstrate it.

No code or data is provided. For this kind of parameter study that is common, but it makes the tables hard to audit beyond the first step.

Who this is for: people working on black-hole energy extraction in modified gravity. It is an incremental but useful data point. I would send it out rather than desk-reject. The core derivation seems solid; the issues are correctable. If I worked on Penrose processes in modified spacetimes, I would cite it for the Rastall parameter dependence.

Recommendation: send to peer review, and ask the authors to verify the turning-point inequalities for the tabulated rows, fix the EUE > 50% statement, restore the E-hat_0 = 1 qualifier, and show the promised Kerr comparison.

Referee Report

4 major / 5 minor

Summary. This manuscript applies the recently developed repetitive Penrose process to a rotating Rastall black hole surrounded by quintessence. The authors adopt the standard triple turning-point framework, import the analytic solution of Refs. [18,48], and formulate iterative updates for the black hole mass, spin, irreducible mass, and extractable energy. They identify Particle 0 as the particle controlling the stopping threshold for the E_hat_0 = 1 case and numerically study how the Rastall structure parameter N_hat_s and the Rastall coupling α affect the energy return on investment, energy utilization efficiency, extracted energy, and remaining extractable energy. The paper reports four sample iteration tables and three-dimensional parameter scans.

Significance. If the underlying triple turning-point configuration is valid for the Rastall metric, the paper would extend the repetitive Penrose literature to a modified-gravity setting and quantify how Rastall parameters alter the iterative energy extraction. The explicit iteration tables are a useful feature; for example, Table I is internally consistent with the quoted p_phi1 = -19.434, mu1/mu0 ~ 0.02, and mu0 = 0.01M. However, the manuscript is a direct application of equations imported from earlier work rather than a new formalism, and its main quantitative claims currently rest on unverified side-of-peak inequalities. Several internal contradictions involving EUE > 50% and the unconditional Particle-0 claim must also be resolved before the results can be accepted.

major comments (4)

[Sec. 3, Eq. (3.2); Tables I-IV] The entire iterative analysis rests on the triple turning-point condition: at each decay radius r_p, particles 0, 1, and 2 are all at radial turning points, with particle 0 and 2 on the right of their effective-potential peaks and particle 1 on the left. The text asserts in Sec. 4 that 'All data in TABLE I satisfied the iteration conditions,' but no row is checked against these side-of-peak inequalities. The analytic solution (2.9)-(2.13) alone does not prove that such a real triple turning-point configuration exists for the Rastall metric with the chosen parameters. Since a_critical is computed from the limiting peak-coincidence case (3.2), the reported iteration counts, ξ, Ξ, and the Particle-0 control claim all inherit this gap. Please provide a direct numerical verification of the inequalities at each iteration or clearly limit the claims to the cases where the check has been perform
[Table V, last row; Table IV] Table V states, in the Physical Interpretation column for the Rastall rotating black hole, 'resulting EUE>50%.' This is directly contradicted by the manuscript's own numbers: Table IV gives Ξ = 0.436857 (43.69%) in the terminal iteration, and all tabulated maxima in Figs. 4, 5, and 7 are at or below roughly 44%. The text immediately after Table IV also quotes 43.68 (presumably percent). This overclaim appears in the central comparison table and must be corrected; if the intended statement is that EUE can approach 50% in some parameter region, that region must be identified with a concrete table row or figure panel.
[Sec. 3, after Eq. (3.2); Abstract; Conclusion] Section 3 states explicitly that for E_hat_0 = 1 the stopping spin lower limit is controlled by Particle 0, while for E_hat_0 > 1 it is governed instead by Particle 2, with its minimum spin boundary at the co-rotating photon sphere radius. The Abstract and Conclusion, however, claim without qualification that the termination of the repetitive Penrose process is 'consistently governed by Particle 0.' Since all numerical work in Sec. 4 uses E_hat_0 = 1, the unconditional formulation is unsupported. Please restrict the claim to E_hat_0 = 1 or extend the analysis to E_hat_0 > 1.
[Sec. 2, after Eq. (2.16)] The manuscript states, 'Crucially, the repetitive Penrose process is modelled under the assumption of a constant Rastall structure parameter N_s.' This is a strong modeling assumption: N_s is not a conserved charge of the black hole, and its constancy during mass accretion is not derived or justified physically. This assumption directly controls the iterative updates of N_hat_s and therefore affects all subsequent stopping thresholds and efficiency results. Please clarify the physical status of this assumption and, if possible, test the sensitivity of the main conclusions to alternative evolution prescriptions for N_s.

minor comments (5)

[Sec. 4, text after Table I] The text says the minimum spin lower bound 'marginally increases with successive iterations,' but the acritical column in Table I decreases monotonically from 0.976754 (n=1) to 0.976491 (n=7). Please correct the description.
[Sec. 4, text after Table III] The sentence 'after the termination of iteration there still exists a relative large amount of extractable energy, typically not less than 0.1M' is ambiguous and appears inconsistent with the final row of Table III, where E_extractable/M0 = 0.0393461. Clarify whether this comparison refers to the Kerr case or to the present Rastall case.
[Sec. 4, text after Table I] The phrase 'applying the mass deficit relation (25)' references an equation number that does not exist in the manuscript; the equations are numbered (2.1)-(2.20). Please update the reference.
[Tables I-IV, column headings] The column headed 'a' is not defined explicitly in the table captions; it appears to be a/M (or a_n/M_n). Please define it and state the initial value consistently (the text says a=M, but Table I starts with a = 0.990000 rather than 1).
[Sec. 4, text after Table IV] The phrase 'the optimal amount of the energy utilization energy Ξ = 43.68' should read 'Ξ = 43.68%' or 'Ξ = 0.4368' to avoid mixing fractions and percentages.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the imported iterative equations are external algebraic inputs, and the Rastall-specific results are direct evaluations, not fitted predictions.

full rationale

The derivation chain is not circular. The Rastall metric (2.1) and its Kerr/quintessence limits are taken from external prior work and are not fitted to the paper's outputs. The repetitive-process conservation equations and the analytic solution (2.6)-(2.13) are imported from Ref. [18], and the effective potential (2.8) is also cited from [18]; although one present author (K. Wang) co-authored [18], these are algebraic geodesic-splitting equations with explicitly stated assumptions, and the paper's target results are direct evaluations on the Rastall background rather than quantities used to determine those equations. The stopping thresholds are computed independently from the marginal-bound-orbit and photon-sphere conditions (3.4)-(3.5) and then compared; the reported ordering a_min,1 < a_min,2 < a_min,0 is a numerical outcome, not an imposed constraint. The fixed inputs p_phi1 = -19.434, nu = 0.78345, and mu0 = 0.01M are adopted from Ref. [48] for comparability, not fitted to extracted energies, and the Ns = 0 limit is explicitly checked against [48]. The triple turning-point idealization is an inherited physical assumption and may be restrictive, but an unverified assumption or missing existence proof is a correctness/robustness concern, not circularity. No fitted parameter is relabeled as a prediction, and no uniqueness claim rests on a self-citation chain. Accordingly, the circularity score is 0.

Assumptions & free parameters 8 free parameters · 6 assumptions · 0 invented entities

All analytical input is imported from earlier papers: the metric from [40], the turning-point Penrose solution from [18], and the repetitive-process setup from [48]. The paper's own contribution is the iteration, stopping analysis, and parameter scan. The numerical output is therefore conditional on the imported metric, the imported area law, and the imported turning-point/splitting parameter choices.

free parameters (8)

p_phi1 (dimensionless angular momentum of infalling fragment) = -19.434
Fixed by hand and inherited from Ref. [48]; used at every iteration and controls spin-down.
nu = mu2/mu1 = 0.78345
Mass ratio of decay products from Ref. [48]; controls the mass-deficit condition (3.1).
E-hat_0 = 1
Incident specific energy chosen to maximize return on investment; selects Particle 0 as the stopping governor.
mu0/M = 0.01
Incident particle mass scale; sets the absolute energy scale in Tables I-IV.
a-hat_0 = 0.99
Initial dimensionless spin used in all numerical iterations.
omega_s (quintessence equation-of-state parameter) = -2/3
Fixed throughout; defines zeta = 3/(1-alpha).
r-hat_p (decay radius) = 1.3-1.9 (scanned)
Independent variable in the parameter scans; all metric functions are evaluated there.
N-hat_s and alpha = 0-0.07 and 0-0.8 (scanned)
The two studied Rastall parameters; chosen values are not fitted to data.

assumptions (6)

domain assumption Metric (2.1)-(2.2) is the rotating Rastall black hole surrounded by a quintessence field.
Taken from Ref. [40] with no independent derivation; all Penrose calculations inherit this geometry.
domain assumption Bekenstein-Hawking area law applies, giving M_irr = 1/2 sqrt(r_+^2+a^2).
Eq. (2.4); not proven for Rastall gravity. If entropy/irreducible mass differ, E_extractable changes.
ad hoc to paper N_s remains constant during extraction, so N-hat_s updates only through M_n.
Stated in Sec. 2 after Eq. (2.16); no physics is given for why the surrounding field is unaffected by mass and angular-momentum loss.
domain assumption All decays occur at turning points in the equatorial plane with zero radial momenta (Eqs. 2.8, 3.2).
Makes the Penrose equations analytically solvable; not generically true for realistic decays.
ad hoc to paper Four-momentum conservation (2.6) plus fixed p-hat_phi1, nu, and E-hat_0 across iterations.
Carried over from Ref. [48]; decay products are assumed identical at every step.
domain assumption Irreducible mass must be non-decreasing; the process stops when a < a_critical.
Enforced as a stopping condition to respect the second law; central to the termination claim.

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Cite this review

Pith. "Pith review of Repetitive Penrose Process in Rastall Rotating Black Holes Immersed in Quintessence Dark Energy." pith.science (2026). https://pith.science/paper/KO6DZZZD

@misc{pith2026260719541,
  author       = {Pith},
  title        = {Pith review of: Repetitive Penrose Process in Rastall Rotating Black Holes Immersed in Quintessence Dark Energy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KO6DZZZD}},
  note         = {Machine review of arXiv:2607.19541}
}

abstract

We investigate the repetitive Penrose process in the spacetime of a Rastall rotating black hole surrounded by a quintessence dark energy field. After reviewing the fundamental properties of the black hole geometry, we formulate the repetitive Penrose process by deriving the conservation equations governing particle splitting within the ergoregion, along with the corresponding iterative evolution equations. The physical conditions required for terminating the energy extraction iterations are established, and the minimum spin thresholds of the decay particles are analyzed to identify the critical stopping criterion. Our analysis reveals that the termination of the repetitive Penrose process is consistently governed by Particle~$0$, which possesses the highest minimum spin threshold among all decay products. Numerical results further demonstrate that the dimensionless Rastall structure parameter $\hat{N}_s$ and the Rastall coupling parameter $\alpha$ significantly influence the evolution of the energy extraction process. At the same decay radii increasing initial values of both parameters boosts the energy utilization efficiency and energy return on investment. Specifically, smaller values of $\hat{N}_s$ enhances the energy utilization efficiency at lower decay radii, shifts the maximum extracted energy toward lower decay radii, and accelerates the depletion of the remaining extractable energy reservoir. This indicates that the repetitive Penrose process is highly favored at lower decay radii. Smaller initial values of $\hat{N}_s$ yield a larger maximum energy return on investment. Similarly, increasing $\alpha$ enhances the energy utilization efficiency, alters the location of the peak extracted energy, and reduces the total extractable energy. But the effects of $\alpha$ on these energetics are very small as compared to $\hat{N}_s$.

Figures

Figures reproduced from arXiv: 2607.19541 by the authors.

**Figure 2.** FIG. 2: Variation of the minimum spin lower limits with the decay radius ˆr [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 3.** FIG. 3: Comparison of the minimum spin lower limits for the three particles for different values [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: FIG. 4: Panels (a)-(d) illustrate the variation of key energetic parameters with the decay radius [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]

**Figure 5.** Figure 5: FIG. 5: Variation of [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]

**Figure 6.** Figure 6: FIG. 6: The maximum energy return on investment [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]

**Figure 7.** Figure 7: FIG. 7: The maximum energy utilization efficiency Ξ as a function of [PITH_FULL_IMAGE:figures/full_fig_p021_7.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

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INTRODUCTION Black holes are among the most remarkable predictions of Einstein’s theory of general relativity (GR) and play a central role in modern relativistic astrophysics. A wide range of high-energy as- trophysical phenomena, such as active galactic nuclei, relativistic jets, quasars, gamma-ray bursts, and X-ray binaries, reveal that rotating black h...

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THE PENROSE PROCESS IN RAST ALL ROT A TING BLACK HOLES The metric for rotating Rastall black hole in the Boyer-Lindquist coordinates is defined as [40] ds2 =− 1− 2M r+Nsrζ Σ dt2 − 2asin 2 θ 2M r+Nsrζ Σ dt dϕ+ Σ ∆ dr2 + Σdθ2 + sin 2 θ " r2 +a 2 + a2 sin2 θ 2M r+Nsrζ Σ # dϕ2,(2.1) where Σ =r 2 +a 2 cos2 θand ∆ =r 2 −2M r+a 2 −N srζ,(2.2) in whichMis the bla...

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ITERA TIVE STOPPING CONSTRAINTS For the successful execution of the repetitive energy extraction process, it should satisfy certain conditions as described in [16–19]. For instance, initially, the mass deficit must obey the constraint as 1−¯µ1 −¯µ2 >0.(3.1) Second, the energy of particle 1 must remain strictly negative throughout the process, which requir...

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In this per- spective, the authors in [48] interpret that in the case of ˆE0 >1, the energy return on investment is lower than in the case of ˆE0 = 1

THE REPETITIVE PENROSE PROCESS IN THE RAST ALL ROT A TING BLACK HOLE Now, we are going to investigate the dynamics of the repetitive Penrose process. In this per- spective, the authors in [48] interpret that in the case of ˆE0 >1, the energy return on investment is lower than in the case of ˆE0 = 1. Therefore, we choose ˆE0 = 1 to maximize the energy retu...

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To verify the numerical consistency of our analysis, we consider the limiting case 21 (a) ˆrp = 1.3 (b) ˆrp = 1.5 (c) ˆrp = 1.7 (d) ˆrp = 1.9 FIG

CONCLUSION In this work, we have investigated the repetitive Penrose process for neutral particles in a Rastall rotating black hole surrounded by a quintessence field, treating every decay self-consistently through the triple turning-point condition and updating the mass, angular momentum, and irreducible mass at each step. To verify the numerical consist...

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