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REVIEW 3 major objections 5 minor 61 references

Penrose and super-Penrose energy extraction from a Reissner-Nordstr\"om black hole spacetime with a cosmological constant through the BSW mechanism: Full story

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read In an extremal charged black hole, a BSW collision between a critical and a usual particle near the horizon can eject a particle with arbitrarily large but finite Killing energy, a super-Penrose process.

desk verdict A clean, self-contained classification of BSW/Penrose energy extraction for extremal RN with a cosmological constant, but the 'arbitrarily large E3' claim holds only within the test-particle limit and needs a global mass-budget caveat. read the letter →

arxiv 2411.14528 v1 pith:6WDSRI35 submitted 2024-11-21 gr-qc astro-ph.HEhep-th

Duarte Feiteira , José P. S. Lemos , Oleg B. Zaslavskii This is my paper · ORCID
classification gr-qcastro-ph.HEhep-th
keywords PenroseprocessBSWmechanismsuper-PenroseReissner-Nordströmblackholeelectricergospherecosmologicalconstantenergyextractionhigher-dimensionalgravity
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 claims that in an extremal Reissner-Nordström black hole in d dimensions with a cosmological constant, a collision near the horizon between a specially tuned 'critical' particle and a 'usual' particle can produce an escaping particle 3 whose energy has no finite upper bound, while a companion particle 4 falls into the hole with negative energy and negative charge, living inside its own electric ergosphere. The process is a collisional Penrose process built from the BSW mechanism: the center-of-mass energy diverges at the horizon, and the paper shows when that divergence translates into real energy extraction. The result is a 'super-Penrose process' when the mass of the incoming critical particle is very small: the lower bound E3b on the emitted energy can be arbitrarily large, and E3 itself can be arbitrarily large but never infinite. The paper classifies the outcomes into four cases (OUT±, IN±) determined by the emitted mass m3 relative to a threshold ΔE1, and shows the energy bounds depend on the cosmological constant and dimension only through the factor sqrt(g(r+))/(d-3). If correct, this extends the known d=4, Λ=0 super-Penrose result to arbitrary d and to AdS, flat, and dS backgrounds.

What carries the argument

The carrying object is the critical particle, defined by matching its electric charge to the critical charge e_{ic} = r_+^{d-3}(d-3)E_i/Q so that X(r+) = 0 at the horizon. The argument proceeds by expanding all particle quantities near the horizon in powers of $\sqrt$(f(r)), reducing the conservation laws of energy, radial momentum, and electric charge to a single relation that fixes the emitted charge perturbation Δe3/e3 in terms of E3 and a lower bound E3b (Eq. 34). The sign of E3 - E3b selects the four outcome cases OUT± and IN±, and the threshold ΔE1 = (d-3)/$\sqrt$(g(r+)) [E1 - $\sqrt$($E1^{2}$ - $m1^{2}$ g(r+)/(d-3)^2)] decides whether the emitted particle leaves directly or first falls inward and turns back.

What would settle it

Compute the gravitational self-force on particle 3 as E3 grows; if the maximum achievable E3 becomes finite (or if the geodesic equations acquire horizon-crossing obstructions) once the particle's own field is included, the super-Penrose claim fails. Alternatively, a direct check of Eq. (34) against a full two-body collision simulation in the extremal RN-AdS/dS metric would settle whether E3b is indeed the sharp lower bound.

Watch

Extended reading notes

Core claim

The central claim is that the energy of the emitted particle 3 satisfies E3b ≤ E3 < ∞ in the OUT+ and IN+ cases, where E3b is a lower bound built from the masses and energy of the incoming particles, with no finite upper bound. When m1 is very small, E3b can be arbitrarily large but not infinite, and E3 can be arbitrarily large but not infinite as well, characterizing a super-Penrose process. In the same events, particle 4 necessarily carries E4 < 0 and e4 < 0, so it moves in its own electric ergosphere until it is absorbed by the horizon. For zero cosmological constant the bounds are independent of dimension d; for AdS or dS they acquire a weak d-dependence through the factor sqrt(g(r+))/(d-3).

Load-bearing premise

The test-particle approximation: the four particles are treated as charged test bodies in a fixed extremal Reissner-Nordström background, ignoring backreaction, self-force, and radiation reaction, even though the central claim lets E3 grow without bound.

Editorial extensions

If this is right

  • In asymptotically flat spacetimes (k=0), the energy bounds are dimension-independent, so the d=4 result extends to all d≥4.
  • In AdS (k=-1) the lower bound E3b is larger than in dS (k=+1) when m3 < m1, equal when m3=m1, and smaller when m3>m1, for fixed r+/l.
  • Particle 4 always has negative energy and negative charge whenever extraction occurs, so the electric ergosphere is an essential part of the process, not an optional feature.
  • The emitted particle 3 can be superheavy (large m3) in the IN+ case, allowing for emission of superheavy particles as well as high-energy ones.
  • Direct detection of the outgoing particle at infinity is possible because it is near-critical and carries a large charge, giving a distinctive charge-counter signal.

Reading between the lines

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

  • If the super-Penrose claim holds, then charged black holes in AdS could in principle act as high-energy particle sources in holographic settings, where near-horizon collisions map to boundary CFT processes; this is an inference, not in the paper.
  • The test-particle breakdown scale could be estimated by requiring the emitted particle's own charge-radius or self-gravity to be negligible; this might place an observable upper bound on E3 that the paper does not discuss.
  • The same conservation-law structure might apply to other extremal charged geometries with a horizon, e.g., higher-dimensional or Gauss-Bonnet extensions, since only the near-horizon factorization f ~ (r-r+)^2 g(r) is used.
  • The four-case classification suggests a recipe for searching for super-Penrose signatures: detectors should look for large-energy charged particles arriving from a direction consistent with a black hole, since a usual (non-fine-tuned) escaping particle cannot escape from the immediate horizon.
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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

3 major / 5 minor

Summary. The paper analyzes a collisional Penrose process in a d-dimensional extremal Reissner-Nordström spacetime with cosmological constant, combining the BSW mechanism with the Penrose process. The authors consider a collision between an ingoing critical charged particle and an ingoing usual charged particle near the horizon, impose energy, radial-momentum, and charge conservation (Eqs. (23)-(25)), and derive constraints on the outgoing particle 3, including the lower bound E3b (Eq. (34)) and the sign condition (Eq. (36)). They classify outcomes into four cases (OUT±, IN±), claim that in the OUT+ and IN+ cases E3 can be arbitrarily large but finite (a super-Penrose process), and show that particle 4 has negative energy and charge and lies in its own electric ergosphere. They also examine how the bounds depend on the spacetime dimension d and on the sign and magnitude of the cosmological constant.

Significance. The derivation is self-contained and algebraically transparent: the critical charge condition, the near-horizon expansions, and the bounds on E3 and e3 follow from the stated conservation laws without fitted parameters. The paper extends previous d=4, Λ=0 results to arbitrary dimension and to AdS and dS asymptotics, and the dimension dependence encoded in Eq. (37) is a useful and clean result. However, the central physical claim of unbounded extracted energy is drawn from a fixed-background test-particle calculation and conflicts with global conservation of the spacetime's total mass and charge. Once the global bound is imposed, the result is still a legitimate extension of earlier work, but the 'super-Penrose with no finite upper bound' conclusion must be significantly qualified.

major comments (3)
  1. [Sec. IV.B (OUT+ and IN+ cases, around Eqs. (34)-(36))] The claim that in cases OUT+ and IN+ one has E3b ≤ E3 < ∞, with E3 'arbitrarily large but not infinite', is inconsistent with global energy conservation in the asymptotically flat (k=0) case and, in the corresponding static-patch sense, for k=±1. The total conserved energy before the process is M + E1 + E2; after particle 4 is absorbed and particle 3 escapes, energy conservation gives the final black hole mass M′ = M + E1 + E2 − E3 (neglecting radiation). Requiring M′ > 0 bounds E3 < M + E1 + E2. Thus the unbounded energy range contains values for which the final black hole would have negative mass, and the scale of the breakdown is set by M, not by an unspecified self-force threshold. For sufficiently small m1, E3b can even exceed M + E1 + E2, in which case the OUT+ and IN+ cases would cease to exist altogether. The authors should state this bound explicitly and restrict the super-Penrose claim to E3b ≤ E3 < M + E1 + E2, or alternatively label the unbounded statement as a formal property of the test-particle equations with the background held fixed.
  2. [Sec. IV.B, Eqs. (23)-(25) and Eq. (35)] A second, related global constraint comes from electric charge. For large E3 the emitted particle is near-critical, so by Eq. (11) e3 ≈ (d−3) r_+^{d−3} E3 / Q; hence e3 grows with E3 and Eq. (25) gives e4 = e1 + e2 − e3 of order −E3. Absorbing particle 4 changes the black hole charge to Q′ = Q + e4. The final state must satisfy the extremality or cosmic-censorship condition |Q′| ≤ M′ (in the normalization of Eq. (2)), which further restricts E3. In d=4 with k=0 and E1+E2 ≪ M, this gives E3 ≲ M/2 before the negative-mass bound is reached. The paper never computes this global charge bound, so the allowed interval E3b ≤ E3 < ∞ includes values that cannot correspond to any physical black hole spacetime. The authors should include this bound or justify why it is not relevant to their interpretation.
  3. [Sec. II.C.1 (equations of motion) and Sec. IV.B (super-Penrose claim)] The calculation is a test-particle calculation in a fixed extremal Reissner-Nordström background. Once E3 and hence e3 are allowed to grow without bound, the particle's own gravitational and electromagnetic fields, and the change in the black hole parameters, can no longer be neglected. The paper should state the regime of validity of the test-particle approximation, for example E3 ≪ M and |e3| ≪ Q in appropriate units, and should not present the unbounded limit as a property of a self-consistent process without such a disclaimer. This comment is closely connected to the two previous ones, but an explicit statement is needed because the manuscript currently offers no estimate of where the approximation breaks down.
minor comments (5)
  1. [Sec. II.C.1] The text contains a repeated typo: 'Reissner-Nodtsröm' should be 'Reissner-Nordström'.
  2. [Abstract and Sec. V] There are duplicated words and misspellings: 'the the parameters' in the Abstract and 'arbitary' in Sec. V should be corrected.
  3. [Appendix, first paragraph] The line 'm ≡ m1 = m2 = m3 = m3' appears to contain a typo; it should presumably read 'm1 = m2 = m3 = m4'.
  4. [Sec. IV.B, after Eq. (34)] The statement that 'any information about particle 2 has disappeared from the formulas above' is confusing, since the energy-extraction condition E3 > E1 + E2 depends on E2; the authors should clarify that E2 enters through the collision kinematics and the properties of particle 4 rather than through E3b.
  5. [Sec. IV.C.1] The phrase 'g(r+)/(d−3)^2 increases in the presence of the cosmological constant' would be clearer as 'is larger than its k=0 value for fixed r_+/l'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is self-contained, with prior self-citations used only as technical references or consistency checks.

full rationale

The core derivation is not circular. The paper defines critical particles through the condition X_i(r_+) = 0, which gives e_ic = (d-3) r_+^{d-3} E_i / Q, and then uses the conservation laws (23)-(25) plus the near-horizon expansions (26) and (31) to solve for the emitted particle's parameters. The bound E3b in Eq. (34) is derived algebraically from those equations, not fitted to any data and not identical to any input. The claimed range E3b <= E3 < infinity and the super-Penrose limit follow from taking m1 small in Eq. (34) and from the algebraic structure of Eq. (35); this is a mathematical consequence of the stated test-particle model, not a prediction that reduces to its own definition. The citations to the same authors' earlier work, especially [17] for the delta expansion, are non-load-bearing: the expansion delta = (Delta e3 / e3) sqrt(f) is an ordering assumption verified by the consistency of the resulting equations, and the d=4, k=0 agreement with [17] is presented as a consistency check rather than as a premise. The skeptic concern about the global ADM energy budget and backreaction is a physical correctness caveat about whether the test-particle idealization remains valid for arbitrarily large E3, not a circularity in the derivation. Accordingly, no circular step is exhibited and the score is 0.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

No parameters are fitted to data. The only tuned quantity is the limit m1 -> 0 that produces arbitrarily large emitted energies, and this is an initial-condition choice rather than a fit. The derivation rests on four idealizations: exact extremality, pure radial motion, test particles, and the near-critical near-horizon ansatz. No new particles, fields, or dimensions are introduced.

free parameters (1)
  • m1 (incoming critical particle mass) = limit m1 -> 0
    The super-Penrose limit E3 -> infinity for fixed E1 is attained by sending m1 to zero. This is an initial-condition choice used to make the derivation's outcome extreme, not a constant fitted to data. At exactly m1=0 the process degenerates: particle 3 has no turning point and the energy falls into the hole.
assumptions (5)
  • domain assumption Local conservation of energy (X), radial momentum (epsilon Z), and electric charge across the two-to-two collision (Eqs. 23-25).
    Assumed in Sec. IV.A. Standard for instantaneous collisions of test particles, but no interaction Lagrangian or scattering cross-section is given.
  • domain assumption Test particles move in the fixed extremal RN-Tangherlini background with Coulomb force; no backreaction, radiation, or self-force.
    Introduced in Sec. II.C via the charged-particle Lagrangian; the entire BSW/Penrose analysis uses these equations of motion.
  • domain assumption The black hole is exactly extremal, so f(r) has a double root (Eq. 7).
    Sec. II.B. The divergent center-of-mass energy in Eq. (20) requires the double-root factorization of the extremal metric.
  • ad hoc to paper Particle 3 is near-critical with charge deviation scaling as delta = (Delta e3/e3) sqrt(f) (Eq. 30), with Delta e3/e3 of order one.
    This near-horizon ansatz is introduced in Sec. IV.B to make the expansion tractable; it restricts the analysis to a fine-tuned regime and is not proven to cover all possible collision outcomes.
  • domain assumption All particles move purely radially, with angular momentum set to zero.
    Sec. II.C.1 assumes u^a = u^t delta^a_t + u^r delta^a_r; the four-case classification applies only to head-on radial collisions.

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Pith. "Pith review of Penrose and super-Penrose energy extraction from a Reissner-Nordstr\"om black hole spacetime with a cosmological constant through the BSW mechanism: Full story." pith.science (2026). https://pith.science/paper/6WDSRI35

@misc{pith2026241114528,
  author       = {Pith},
  title        = {Pith review of: Penrose and super-Penrose energy extraction from a Reissner-Nordstr\"om black hole spacetime with a cosmological constant through the BSW mechanism: Full story},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6WDSRI35}},
  note         = {Machine review of arXiv:2411.14528}
}
abstract

The Penrose process, a process that transfers energy from a black hole to infinity, together with the BSW mechanism, which uses collisions of ingoing particles at the event horizon of a black hole to locally produce large amounts of energy, is studied in a combined description for a $d$ dimensional extremal Reissner-Nordstr\"om black hole spacetime with negative, zero, or positive cosmological constant, i.e., for an asymptotically anti-de Sitter (AdS), flat, or de Sitter (dS) spacetime. In an extremal Reissner-Nordstr\"om black hole background, in the vicinity of the horizon, several types of radial collisions between electrically charged particles can be considered. The most interesting one is between a critical particle, with its electric charge adjusted in a specific way, and a usual particle, as it gives a divergent center of mass frame energy locally, this being a favorable but not sufficient condition to extract energy from the black hole. To understand whether energy can be extracted in such a collisional Penrose process, we investigate in detail a collision between ingoing particles 1 and 2, from which particles 3 and 4 emerge, with the possibility that particle 3 can carry energy far out from the black hole horizon. One finds that the mass, energy, electric charge, and initial direction of motion of particle 3 can have different values, depending on the collision internal process, but these values lie within some range. Moreover, the energy of particle 3 can be arbitrarily high but not infinite, characterizing a super-Penrose process. It is also shown that particle 4 has negative energy, living in its own electric ergosphere before being engulfed by the event horizon. For zero cosmological constant the results do not depend on the number of dimensions, but they do for nonzero cosmological constant, which also introduces differences in the lower bound for the energy extracted.

Figures

Figures reproduced from arXiv: 2411.14528 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Schematic picture for an electrically charged BSW collision. The electrically charged particle 1 collides with the [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗

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Reference graph

Works this paper leans on

61 extracted references · 49 canonical work pages

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    Thus, e3 ≤ e3c ≤ e30(r) and ε3 = −1, the particle goes in immediately after the collision and then continues the motion entering down the black hole

    IN −: ∆ E1 < m3 < ∞ and m3 √ g(r+) d−3 ≤ E3 ≤ E3b. Thus, e3 ≤ e3c ≤ e30(r) and ε3 = −1, the particle goes in immediately after the collision and then continues the motion entering down the black hole. We have seen that for the case m1 ≤ q E1 E3 m3 there is no turning point for the ingoing particle. So, small masses m1 which yield high E3b have no turning ...

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    Thus, e3c < e3 < e30(r) and ε3 = +1, the particle goes directly out after the collision

    OUT +: 0 ≤ m3 < ∆E1 and E3b ≤ E3 < ∞. Thus, e3c < e3 < e30(r) and ε3 = +1, the particle goes directly out after the collision. This case, can yield a Penrose process with energy extraction and it is not restricted, but there is an upper bound in m3, i.e., not any mass can be emitted. Moreover, clearly, when m1 is very small then E3b can be arbitrarily lar...

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    To obtain the equations of motion for a charged particle it is useful to resort to the Lagrangian of the particle and its Euler-Lagrange equations of motion

    Equations of motion for particles Neutral particles follow geodesics in a Reissner-Nodtsr¨ om spacetime, but electrically charged particles do not. To obtain the equations of motion for a charged particle it is useful to resort to the Lagrangian of the particle and its Euler-Lagrange equations of motion. Consider a particle with mass m and specific charge...

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    The electric ergosphere Given the forward in time condition, ˙t >0, one has X > 0 for any radii outside the horizon. From Eq. (9) this implies E − eQ (d−3)rd−3 > 0. Given a black hole electric charge Q, with Q positive as assumed, it is clear from the latter expression that one can have negative energy states E for the particle, as long as its electric ch...

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    Critical, near-critical, and usual particles We nominate each particle as particle i, in general i = 1, 2, 3, 4. Each particle has attributes like its energy Ei, its mass mi, its electric charge ei, and so on. In what follows, definitions of critical, near-critical, and usual particles will be important. Thus, we establish here definitions for thee differ...

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    Particle collision and energy at the center of mass frame To study the energy generated from the BSW effect, a collision between two ingoing particles is assumed to occur in a d dimensional extremal Reissner-Nordstr¨ om black hole spacetime with horizon radiusr+, electric charge Q, and cosmological constant kΛ. The mass of particle i is denoted as mi, and...

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    Since at r, p2 is an invariant, one gets −(E2 CM) = −2m2 + 2m2gabua 1ub 2, i.e., E2 CM 2m2 = 1 − gabua 1ub

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    For a pure radial collision one has ua 1 = ut 1δa t + ur 1δa r and ua 2 = ut 2δa t + ur 2δa r , so that E2 CM 2m2 = 1 − gttut 1ut 2 − grrur 1ur

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    Using the equations of motion given in Eq

    Since ut 1 = ˙t1, ur 1 = ˙r1, ut 2 = ˙t2, ur 2 = ˙r2, gtt = −f , and grr = 1 f , one finds E2 CM 2m2 = 1 + f ˙t1 ˙t2 − ˙r1 ˙r2 f . Using the equations of motion given in Eq. (8), one finds that E2 CM(r) 2m2 = 1 + X1(r)X2(r) − Z1(r)Z2(r) m2f (r) , (15) where Xi and Zi, i = 1, 2...

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    So we assume that particle 1 is exactly critical and particle 2 is exactly critical

    Collision between two critical particles Here we study a collision between two ingoing critical particles in the vicinity of the extremal black hole event horizon, at some radius r which is near or at r+. So we assume that particle 1 is exactly critical and particle 2 is exact...

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    So, we assume that particle 1 is exactly critical and particle 2 is usual

    Collision between a critical and a usual particle Here we study a collision between an ingoing critical particle and an ingoing usual particle in the vicinity of the extremal black hole event horizon, at some radius r which is near or at r+. So, we assume that particle 1 is ex...

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    The energy at the center of mass frame for collisions happening at the black hole event horizon, Eq

    Collision between a near-critical and a usual particle We now consider a collision between an ingoing near-critical particle, which for definiteness is chosen to be particle 1, and an ingoing usual particle, which is particle 2. The energy at the center of mass frame for colli...

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    m3 √ g(r+) d−3 ≤ E3, e3 ≤ e3c ≤ e30(r), and ε3 = −1. The collision happens in the allowed region, just near the horizon, where e3 ≤ e3c ≤ e30(r). Particle 3 is near critical or critical, since in this case the collision can happen at the horizon itself. Since ε3 = −1, the part...

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    (36) If ∆E1 > m3 we deduce from Eq. (36) that ε3 = +1, particle 3 goes out from the point of collision. If ∆ E1 < m3 we deduce from Eq. (36) that ε3 = −1, particle 3 goes in from the point of collision, in particular when m1 is sufficiently small and so ∆ E1 is small, which co...

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    The dependence on the cosmological constant through k and on the dimension d The bounds for the energy extracted, i.e., the energy of particle 3, depend on the factor √ g(r+) d−3 , see Eq. (34). Using the factorization of f (r) obtained in Eq. (7), it follows that g(r+) (d−3)2...

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    Such particles can be obtained by fine-tuning their parameters in their initial state, i.e., in the state previous to the collision

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    2024 arXiv

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.