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Repetitive Penrose Process in Rotating 4D Einstein-Gauss-Bonnet Black Holes

T0 review · 2 major / 4 minor · reviewed 2026-07-13 · grok-4.5

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Pith's one-line read In Einstein–Gauss–Bonnet black holes the Penrose efficiency map reorganizes itself as mass falls and the dimensionless coupling grows, producing a bounded window where the modified hole beats Kerr.

desk verdict Competent extension of the Ruffini repetitive-Penrose program; the running-α effect is real, but the four-to-three efficiency topology is shown only for one hand-chosen particle set. read the letter →

arxiv 2607.08989 v1 pith:DFO36MMA submitted 2026-07-09 gr-qc

Mohammad Reza Alipour , Mohammad Ali S. Afshar , Saeed Noori Gashti This is my paper · ORCID
classification gr-qc
keywords Einstein–Gauss–BonnetblackholerepetitivePenroseprocessGauss–Bonnetcouplingenergyutilizationefficiencyirreduciblemassergospheretripleturning-pointcondition
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 shows how a quadratic-curvature correction changes the way rotational energy can be harvested from a spinning black hole when the extraction is repeated many times. Because the Gauss–Bonnet coupling itself is fixed while the black-hole mass shrinks, the dimensionless strength of the correction grows at every step; the geometry must therefore be recomputed after each decay. That self-amplification shrinks the ergosphere, lowers the maximum spin, and cuts the number of allowed extractions. Energy return falls steadily with coupling, yet utilization efficiency is non-monotonic: below a critical coupling the (coupling, radius) plane splits into four regimes that include a finite window where the modified black hole outperforms Kerr; above the critical value the window collapses into a three-regime map. No earlier charged, cosmological or accelerating extension produces this topology change, so the result isolates a purely geometric, running-parameter effect on black-hole energetics.

What carries the argument

The nonlinear iterative scheme that, after each decay under the triple turning-point condition, updates mass, angular momentum and irreducible mass while recomputing the metric with the enlarged dimensionless coupling α/M²; the closed-form conservation solution that reduces to the Kerr result at vanishing coupling.

What would settle it

If an independent numerical or analytic solution of the regularized four-dimensional Einstein–Gauss–Bonnet field equations yields horizon and ergosphere radii that differ from those of the Newman–Janis metric at the same (M,a,α), the effective potentials and the reported four-to-three-region efficiency transition would no longer hold.

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Extended reading notes

Core claim

Although the Gauss–Bonnet coupling α is a fixed constant of the action, the dimensionless ratio α/M² grows with every mass loss, self-amplifying the correction. Under the triple turning-point condition this running produces a critical coupling near 0.001 that reorganizes the energy-utilization landscape from a four-region structure (including a bounded window of superior efficiency relative to Kerr) into a three-region structure, an effect with no counterpart in Kerr, Reissner–Nordström, Kerr–de Sitter, accelerating Kerr or Kerr–Newman spacetimes.

Load-bearing premise

The entire analysis rests on treating the rotating metric generated by the modified Newman–Janis algorithm as a valid stationary solution of the regularized theory even as the dimensionless coupling grows and the geometry is recomputed at every step.

Editorial extensions

If this is right

  • At fixed decay radius near the horizon, a sufficiently large initial coupling can forbid the repetitive process entirely, while the same coupling still permits extraction at larger radii.
  • The energy return on investment always declines with coupling, so the net energy harvested per incident particle is smaller than in Kerr.
  • Irreducible-mass growth is suppressed relative to Kerr, leaving a larger fraction of the lost rotational energy available rather than locked into horizon area.
  • The particle that sets the termination threshold remains the incident particle for every coupling; the Gauss–Bonnet term only rescales the bound.

Reading between the lines

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

  • Any higher-curvature or regularized theory whose dimensionless coupling scales as 1/M² will generically produce a running-parameter reorganization of extraction efficiency once the mass is allowed to change.
  • Charged or spinning fragments would couple the electromagnetic and geometric runnings, potentially moving or destroying the critical coupling that separates four-region from three-region maps.
  • If the Newman–Janis metric is later replaced by an exact rotating solution, the same iterative bookkeeping can be re-run to test whether the efficiency island survives.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper studies the repetitive Penrose process for neutral particles in a rotating 4D Einstein–Gauss–Bonnet black hole obtained via the modified Newman–Janis algorithm. It imposes the triple turning-point condition, derives a closed-form solution of the energy, angular-momentum and radial-momentum conservation equations (Eqs. 3.5–3.6) that recovers the Kerr result at vanishing coupling, and implements a nonlinear iterative scheme that updates mass, angular momentum and the dimensionless coupling ˆα = α/M² after every decay. Because α is fixed while M decreases, ˆα self-amplifies, forcing the geometry to be recomputed at each step. Numerical sequences at fixed particle parameters show that larger ˆα lowers the extremal spin, contracts the ergosphere, reduces the number of admissible decays and terminates the process under the incident-particle critical spin. Energy return on investment falls monotonically with ˆα and irreducible-mass growth is suppressed relative to Kerr, while energy utilization efficiency is non-monotonic: below a critical coupling ˆα0,change ≈ 0.001 the (ˆα0, ˆr) plane exhibits a four-region structure containing a bounded window of EGB superiority over Kerr; above that value the structure collapses to three regions.

Significance. If the reported four-to-three reorganization of the efficiency landscape is robust, the work supplies a concrete, purely geometric example of a coupling-driven topological change in repetitive energy extraction that has no counterpart in the Kerr, RN, Kerr–dS, accelerating Kerr or Kerr–Newman cases. The closed-form conservation solution, the explicit critical-spin formulae (Eqs. 3.16–3.18) and the self-amplifying ˆα evolution are technically clean and reduce correctly to known Kerr limits, providing a usable template for other higher-curvature or regularized metrics. The result is therefore of genuine interest for the comparative study of classical energy extraction beyond general relativity, provided the topological claim survives modest variation of the free particle parameters.

major comments (2)
  1. The central claim of a coupling-driven four-to-three topological reorganization of the EUE landscape (Abstract; §4; Table 4.2; Fig. 4.3) is demonstrated only for one fixed particle set taken from the Kerr literature (ˆE0 = 1, ˆpφ1 = −19.434, ν = 0.78345, µ0 = 0.01 M0) and two discrete decay radii (ˆr = 1.2, 1.5). Because both the self-amplifying ˆα evolution (Eqs. 3.7–3.10) and the critical-spin thresholds (Eqs. 3.16–3.19) depend on the captured fragment’s energy and angular momentum, a different choice of ˆpφ1 or ν can shift or erase the crossover radii and the critical coupling itself. At least a modest scan over ˆpφ1 and ν (or an analytic argument that the merger is parameter-independent) is required before the reorganization can be asserted as a generic feature of the EGB background rather than an artifact of the comparison parameters.
  2. The entire analysis rests on the rotating metric generated by the modified Newman–Janis algorithm (Eqs. 2.1–2.3, citing Kumar & Ghosh 2020). The paper treats this geometry as a valid stationary solution of the regularized 4D EGB theory for the whole iterative sequence, including as ˆα grows and the metric is recomputed at every step (§2; §3.2). A brief statement clarifying the status of this metric (exact solution of the field equations versus approximate or seed-generated) and the range of ˆα for which it remains reliable would strengthen the foundation of the effective potentials, horizons and efficiency maps.
minor comments (4)
  1. In Table 4.1 the ˆα0 = 0 sequence lists nine physical iterations while the text (§5) states “eight in the Kerr limit”; the counting of the final admissible step should be made consistent.
  2. Figure 2.1 and the perturbative expansions (2.5)–(2.6) are useful, but the domain of validity of the small-ˆα expansions should be stated more explicitly when ˆα grows during the iteration.
  3. A short remark on why the ordering ˆamin,1 < ˆamin,2 < ˆamin,0 persists for all couplings examined (Fig. 3.1) would help the reader appreciate that the termination mechanism is robust.
  4. Typographical inconsistencies appear in the tables (e.g., spacing in ˜µ1,n and ˆE1,n columns) and in the rendering of some figure labels; a light copy-edit would improve readability.

Circularity Check

1 steps flagged · score 1.0 of 10

No load-bearing circularity: efficiency topology is a numerical output of conservation laws plus the metric, not forced by definition or self-citation.

  1. self citation load bearing [§3.3 / Fig. 3.1a; Conclusion]
    "unlike the electromagnetic coupling in the charged Kerr–Newman case [16], the Gauss–Bonnet coupling does not alter which particle governs the stopping condition"

    Minor self-citation used solely for contrast; the ordering ˆamin,1 < ˆamin,2 < ˆamin,0 is independently verified from the EGB potentials themselves and is not required for the efficiency maps. Not load-bearing for the four/three-region claim.

full rationale

The closed-form solution (Eqs. 3.5–3.6) follows directly from four-momentum conservation under the triple turning-point condition and the equatorial mass-shell relation; it reduces to the Kerr expressions when ˆα→0 by substitution of the metric functions, providing an independent consistency check rather than a tautology. Critical-spin thresholds (3.16)–(3.19) are obtained by solving the turning-point/maximum conditions on the effective potentials of the given metric; the ordering that the incident particle controls termination is verified numerically across the scanned range, not imposed by definition. The four-to-three region reorganization of Ξ (Fig. 4.3, Table 4.2) is an observed numerical outcome of the iterative updates (3.7)–(3.10) for one fixed particle set taken from the Kerr literature for comparison; the particle parameters are not fitted to produce the topology, nor is the topology used as an input. Self-citations to the authors’ Kerr–Newman work appear only as contrast (“unlike o [16]”) and do not underwrite any uniqueness claim or derivation step. The metric itself is taken from an external reference (Kumar & Ghosh). Consequently the central claim is not equivalent to its inputs by construction; residual parameter-dependence is a robustness question outside circularity.

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

The central claim rests on standard GR conservation and turning-point kinematics, the Christodoulou irreducible-mass definition adapted to the EGB horizon, the Kumar–Ghosh rotating metric obtained by modified Newman–Janis, and a handful of particle parameters fixed by hand to match the Kerr baseline. No new particles or forces are invented; the only non-standard structural input is the status of the regularized 4D EGB rotating solution itself.

free parameters (5)
  • ˆpφ1 (captured angular momentum) = -19.434
    Fixed by hand to −19.434 to match the Kerr reference [8]; controls depth of negative-energy states and therefore every efficiency number.
  • ν = μ2/μ1 (mass ratio) = 0.78345
    Fixed by hand to 0.78345 from the Kerr literature; enters the closed-form mass ratios and energy partition.
  • μ0 (incident rest mass) = 0.01 M0
    Set to 0.01 M0; scales the size of each mass/spin update and the number of iterations before termination.
  • ˆr (decay radius) = 1.2 (main); 1.5 (appendix)
    Chosen as free parameter of the process (1.2 main text, 1.5 appendix); location relative to horizon and ergosphere determines whether the process is allowed and which efficiency regime applies.
  • ˆE0 (incident energy) = 1
    Set to 1 (particle from rest at infinity) to maximize EROI following prior Kerr work; not varied.
assumptions (5)
  • domain assumption The rotating metric obtained by the modified Newman–Janis algorithm from the static 4D EGB solution is a valid stationary axisymmetric solution of the regularized theory for the range of ˆα explored.
    Invoked throughout §2 and used to recompute Δ, horizons, and effective potentials at every iteration (§3.2).
  • domain assumption Maximal extraction is realized by the triple turning-point condition ˆpr0=ˆpr1=ˆpr2=0 at the decay point.
    Derived in §3.1 from positivity of radial momentum and future-directed geodesics; closes the conservation system.
  • domain assumption α is a fixed constant of the action and is not carried by the fragments, so ˆα=α/M² increases whenever M decreases.
    Stated in Abstract and §3.2; drives the self-amplification that is the paper’s distinctive mechanism.
  • domain assumption Irreducible mass is Mirr=½√(r+²+a²) with r+ the outer root of Δ=0, and extractable energy is M−Mirr.
    Christodoulou formula adapted to the EGB horizon (§2, Eqs. 2.7–2.8); defines EUE.
  • standard math Four-momentum conservation at the decay point yields the three scalar relations (3.1a–c) for energy, axial angular momentum, and radial momentum.
    Standard Killing-vector conservation on a stationary axisymmetric background (§3).

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Pith. "Pith review of Repetitive Penrose Process in Rotating 4D Einstein-Gauss-Bonnet Black Holes." pith.science (2026). https://pith.science/paper/DFO36MMA

@misc{pith2026260708989,
  author       = {Pith},
  title        = {Pith review of: Repetitive Penrose Process in Rotating 4D Einstein-Gauss-Bonnet Black Holes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DFO36MMA}},
  note         = {Machine review of arXiv:2607.08989}
}
abstract

We investigate the repetitive Penrose process for neutral particles in a rotating four-dimensional Einstein--Gauss--Bonnet black hole obtained through the modified Newman--Janis algorithm, developing a nonlinear iterative scheme in which the mass, angular momentum, and irreducible mass are updated after each extraction event. Imposing the triple turning-point condition, we obtain a closed-form solution of the conservation equations for energy, angular momentum, and radial momentum that reduces to the Kerr result in the limit of vanishing coupling. The distinctive feature of this background is that, although the Gauss--Bonnet coupling $\alpha$ is a fixed constant of the action and is not carried by the infalling fragments, the dimensionless coupling $\hat\alpha=\alpha/M^{2}$ grows at every iteration as the mass decreases, so that the effective Gauss--Bonnet correction is self-amplified along the sequence. We find that a larger coupling lowers the extremal spin, contracts the ergosphere, and reduces the number of admissible decays, forbidding the process near the horizon at strong coupling while permitting it at larger decay radii; the termination is controlled throughout by the incident particle. The energy return on investment decreases monotonically with $\hat\alpha$ and the growth of the irreducible mass is suppressed relative to Kerr, whereas the energy utilization efficiency is non-monotonic: below a critical coupling the parameter space exhibits a four-region structure with a bounded window in which the EGB black hole is more efficient than Kerr; this four-region structure collapses into a three-region one above a critical coupling. This coupling-driven reorganization of the efficiency landscape has no analogue in the Kerr, Reissner--Nordstr\"om, Kerr--de~Sitter, accelerating Kerr, or Kerr--Newman cases.

Figures

Figures reproduced from arXiv: 2607.08989 by the authors.

Figure 2.1
Figure 2.1. Allowed parameter space for rotating four-dimensional Einstein–Gauss–Bonnet black holes in the (a/M, α/M2 ) plane. The solid curve corresponds to the extremal configuration where the two horizons coincide. which does not coincide with the event horizon except at the rotation axis. The region enclosed by the stationary limit surface and the event horizon forms the ergosphere. Since negative-energy states exist inside… view at source ↗
Figure 3.1
Figure 3.1. (a) The three critical spin thresholds aˆmin,0, aˆmin,1, and aˆmin,2 as functions of the dimensionless decay radius rˆ, for fixed Gauss–Bonnet coupling αˆ = 0.0001 and Eˆ0 = 1. (b) The critical spin threshold aˆmin,0 as a function of rˆ for three representative values of αˆ = 0, 0.0001, 0.0005. 4 Numerical Analysis of the Repetitive Penrose Process in the Rotating 4D EGB Spacetime In this section, we carry out a det… view at source ↗
Figure 4.1
Figure 4.1. a demonstrates that increasing the Gauss–Bonnet coupling αˆ0 shifts the onset of energy extraction to larger radii, near the ergosphere. Across all coupling strengths, the intermediate radial regime consistently yields the maximum total extracted energy. Concurrently, an examination of [PITH_FULL_IMAGE:figures/full_fig_p009_4_1.png] view at source ↗
Figures from the paper (2 more)
Figure 4.2
Figure 4.2. Figure 4.2: (a) Final energy utilization efficiency (EUE), Ξ, and (b) final energy return on investment (EROI), ξ, of the repetitive Penrose process as functions of the dimensionless decay radius rˆ for different values of the Gauss-Bonnet coupling parameter αˆ0. As detailed in …
Figure 4.3
Figure 4.3. Figure 4.3: (a) Final efficiency Ξnf as a function of the decay radius rˆ for αˆ0 < αˆ0,change (e.g., 0.0006), exhibiting a four-region structure. The process is null for EGB below rˆmin,RP (while remaining active for Kerr), followed by a suppressed regime (ΞGB < ΞKerr), a bound…

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Forward citations

Cited by 2 Pith papers

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

  1. Geometry-Induced Termination of the Repetitive Penrose Process in Rotating Simpson-Visser Black Holes

    gr-qc 2026-07 conditional novelty 5.0 of 10

    In rotating Simpson-Visser black holes, repeated Penrose extraction can terminate because the evolving mass/spin leaves the two-horizon regime before the spin limit is reached, an effect controlled by the regularizati...

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

    gr-qc 2026-07 conditional novelty 5.0 of 10

    In a quintessence-surrounded Rastall rotating black hole, the repetitive Penrose process stops at a spin threshold set by particle 0, and lower Rastall structure parameter values favor extraction at smaller decay radii.

Reference graph

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