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Isentropic process of Reissner-Nordstr\"om black holes: a possible excess of the entropy bound via a non-perturbative channel

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Pith's one-line read A non-perturbative quantum channel can let a charged black hole's internal entropy exceed its Bekenstein-Hawking entropy, violating the entropy bound.

desk verdict Clean classical no-go result plus a speculative quantum entropy claim that does not hold up as written. read the letter →

arxiv 2505.01663 v1 pith:OTALKVOO submitted 2025-05-03 gr-qc hep-th

Robert B. Mann , Dong-han Yeom This is my paper · ORCID
classification gr-qchep-th MSC 83C5783C4783C22 PACS 04.70.Dy04.70.-s
keywords isentropicabsorptionReissner-NordströmblackholeentropyboundBekenstein-HawkingquantumtunnelingWKBapproximationinformationlossparadoxnon-perturbativeeffects
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 argues that a charged black hole can absorb a particle without changing its horizon entropy — an isentropic absorption — even though classical gravity forbids the process. The forbidden step is that the particle must satisfy $E = qQ/r_+$, which places it at the top of a potential barrier, so no classical trajectory reaches the horizon. Quantum tunneling makes the process possible with exponentially small but nonzero probability, and if such absorptions accumulate, the black hole's internal Boltzmann entropy can exceed the Bekenstein-Hawking entropy $A/4$. The authors conclude that a non-perturbative quantum process can violate the entropy-bound relation.

What carries the argument

The load-bearing object is the radial effective potential $V_{\rm eff}(r) = -\frac{1}{m^2}(E - qQ/r)^2 + f(r)$ for a charged test particle in a static, spherically symmetric metric. The isentropic condition $E = qQ/r_+$ makes $V_{\rm eff}(r_+) = 0$ with $V_{\rm eff}'(r_+) = 4\pi T > 0$, creating a barrier immediately outside the horizon. The WKB tunneling rate is computed from the Euclidean action $S_E = \int_{r_1}^{r_2} dr/\sqrt{V_{\rm eff}(r)}$, and the enhanced tunneling in modified gravity (4D Einstein-Gauss-Bonnet) is due to a lower barrier. A second ingredient is the informational assumption that absorbing an entangled particle through this channel increases the interior-exterior entanglement entropy while the areal entropy stays constant, so the Boltzmann entropy can outgrow $A/4$.

What would settle it

Compute the full transition amplitude for the charged particle to tunnel into the Reissner-Nordström horizon under the isentropic condition and track the inside-outside entanglement entropy per absorption. If the added entanglement is bounded below the added Boltzmann entropy, or if Hawking evaporation removes mass faster than isentropic absorptions can accumulate, the Boltzmann entropy would remain below $A/4$. Concretely, evaluate numerically the condition $N e^{-\beta M} > 1$ before the black hole loses mass $\sim M$ through Hawking radiation; if this number never exceeds one, the proposed entropy excess does not materialize.

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Extended reading notes

Core claim

The central claim is that non-perturbative quantum effects can produce a black hole whose Boltzmann entropy is greater than its Bekenstein-Hawking entropy $A/4 = \pi r_+^2$. The mechanism is an isentropic absorption channel: a charged test particle with energy $E = qQ/r_+$ leaves the horizon area (and hence the areal entropy) unchanged, but the effective potential has a classically forbidden barrier just outside the horizon. Quantum mechanically the particle tunnels through this barrier with probability $\Gamma \sim e^{-2S_E} \sim e^{-\beta M}$, comparable in form to the Hawking radiation Boltzmann factor. Each such absorption increases the entanglement entropy between the black hole interior and its exterior while the areal entropy stays fixed, so repeated events can drive the Boltzmann entropy above the Bekenstein-Hawking value.

Load-bearing premise

The argument that the entropy bound must eventually be violated depends on the assumption that absorbing an entangled particle in this isentropic channel always increases the entanglement entropy between the interior and exterior, and that this entanglement entropy is never larger than the black hole's Boltzmann entropy — an informational step that the semiclassical dynamics alone does not justify.

Editorial extensions

If this is right

  • The isentropic absorption channel gives an explicit route by which a black hole's Boltzmann entropy can exceed its Bekenstein-Hawking entropy without changing the horizon area.
  • An object with more interior entropy than its area entropy is a 'monster', providing a concrete test bed for information-loss scenarios involving islands or remnants.
  • The conclusion applies to any gravity theory with $g_{00} = g_{11}^{-1}$ and electrostatic potential $Q/r$, including 4D Einstein-Gauss-Bonnet gravity, where the barrier (and hence the tunneling suppression) is smaller.
  • Entropy-bound theorems remain consistent for averaged or dominant semiclassical processes; the violation is confined to exponentially suppressed non-perturbative events, which is why earlier proofs are not overturned.
  • The process targets assumption 4 of the information-loss paradox: the area/entropy proportionality may fail once non-perturbative effects are included, changing what unitarity would require of Hawking radiation.

Reading between the lines

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

  • The same isentropic condition $E = qQ/r_+$ could be probed in analogue black-hole systems where a tunneling barrier plays the role of the horizon, offering a terrestrial test of the accumulation logic.
  • If the channel works, it suggests that the area law is a coarse-grained bound rather than a fundamental one: the interior Hilbert space would need to accommodate more states than $A/4$.
  • A natural extension is to repeat the WKB computation for rotating black holes, where the isentropic condition becomes $E = \Omega\, dJ + \Phi\, dQ$ and the barrier structure differs; the conclusion may carry over or fail depending on the ergosphere.
  • One could test the paper's informational assumption in a toy model of a one-way absorbing wall to see whether isentropic absorption generically increases entanglement entropy faster than Boltzmann entropy, which would either support or undercut the central premise.
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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 / 4 minor

Summary. The paper studies the possibility of an isentropic (constant-entropy) absorption process for a charged test particle by a Reissner-Nordström black hole. Section II derives the classical radial geodesic equation, shows that the isentropic condition E = qQ/r+ makes the effective potential vanish at the horizon with positive slope, and proves that a purely classical absorption is forbidden; the argument is extended to a modified-gravity example (4D Einstein-Gauss-Bonnet). Section III argues that quantum tunneling through the potential barrier allows the isentropic absorption, computes a WKB tunneling probability, and then claims that if such absorptions accumulate, the Boltzmann entropy of the black hole can exceed its Bekenstein-Hawking entropy, violating the entropy bound. The conclusion frames this as a non-perturbative channel that may be relevant to the black hole information loss paradox.

Significance. The classical forbiddenness proof in Section II is clear, generic, and a useful observation: the isentropic condition forces the effective potential to have a barrier near the horizon in any metric of the form ds^2 = -f dt^2 + f^{-1} dr^2 + r^2 dΩ^2 with Φ = Q/r. The modified-gravity extension is also interesting and shows the barrier persists. However, the central claim of the paper, that a non-perturbative process can violate the entropy bound, is not established by the arguments presented. The WKB action in Eq. (25) is missing a factor of the particle mass, and the entropy argument in Section III.C rests on an unproved assumption about the growth of the interior Hilbert space, which is essentially the conclusion the paper claims to derive. With the classical result and the corrected tunneling calculation, the paper would be a solid contribution to the discussion of quantum absorption channels, but the entropy-bound-violation claim is currently not supported by a derivation.

major comments (3)
  1. [III.A, Eq. (25)] The Euclidean action S_E = ∫ dr / √V_eff is dimensionally the imaginary proper time, not the action of a massive particle. The physical action for a particle of mass m should contain an overall factor of m, so the WKB exponent should be S_E = m ∫ dr / √V_eff (up to the appropriate metric factors in the conjugate momentum). As written, the exponent is independent of m, which directly affects the comparison with Hawking radiation in Eqs. (26)-(27): the claim that the tunneling probability is comparable to e^{-mM} relies on both exponents containing m. With the current formula the exponent is O(M), which is much more suppressed and changes the quantitative conclusion that the process is 'not too small to be totally neglected.' The manuscript should be corrected and the quantitative implications re-evaluated.
  2. [III.C] The argument that an isentropic absorption inevitably increases the entanglement entropy across the horizon is not derived and is in fact circular in the present context. If the black hole is already maximally entangled with its background, then S_ent = log dim H_interior = A/4. Absorbing a particle entangled with the outside cannot increase S_ent beyond A/4 unless the interior Hilbert space dimension itself grows independently of the horizon area. But the statement that dim H_interior can exceed e^{A/4} is precisely the entropy-bound violation the paper claims to prove. The additional assertion that 'the entanglement entropy must increase' also depends on the correlations between the incoming particle and the pre-existing interior/exterior state; generic entangled injections do not necessarily increase S_ent. The authors should either provide an explicit quantum state and a calculation showing S_ent increases while A/4 remains fixed, or explicitly reframe the entropy-bound violation as a conjecture rather than a conclusion.
  3. [IV] The concluding sentence 'We thus conclude that a non-perturbative quantum process can violate the entropy-bound relation' overstates what has been shown. The tunneling calculation demonstrates only that an isentropic absorption is quantum-mechanically possible with nonzero probability; it does not by itself imply that the Boltzmann entropy of the interior exceeds A/4. The logical chain from tunneling to entropy-bound violation requires the unproved Hilbert-space-growth assumption identified in Section III.C. If the authors wish to keep the strong claim, they must supply a concrete model; otherwise the conclusion should be softened to a suggestion or open possibility, with the distinction between proven results and speculation made explicit.
minor comments (4)
  1. [III.A, Eq. (21)] The Lagrangian is written as L = -f ˙t^2 + ˙r^2/f = -1, but the factor of the particle mass m is omitted. While this is acceptable for deriving geodesic equations, it is the source of the missing m in the action; the manuscript should clarify that the action is -m ∫ dτ, not ∫ dτ.
  2. [Fig. 2 caption] The caption says 'by varying E/m, α2/M' but the text and equations use α/M^2; the notation should be made consistent and the axes properly labeled.
  3. [II.D] The section is titled 'generalized quasi-topological black holes' but the example is the 4D Einstein-Gauss-Bonnet solution of Refs. [17,18]; please clarify the terminology or adjust the title to match the cited solution.
  4. [III.B] The thermodynamic stability discussion computes F and G but does not connect the signs of dF and dG to a stability criterion. If this section is intended to assess stability, an explicit statement of the relevant ensemble and stability condition would be helpful.

Circularity Check

1 steps flagged · score 6.0 of 10

The claimed entropy-bound violation rests on a Sec. III C assertion that entanglement entropy must increase, which already assumes the interior Hilbert space can exceed e^{A/4}.

  1. self definitional [Section III C, "Can the entropy-bound relation be violated?" (p. 8)]
    "However, the entanglement entropy between inside and outside the horizon must increase. The entanglement entropy is always smaller than its Boltzmann entropy. Therefore, it is inevitable to accept the possibility that the Boltzmann entropy can be greater than the Bekenstein-Hawking entropy if one allows non-perturbative effects."

    The conclusion is S_Boltzmann > A/4, but the premise already contains it. The black hole is taken to be maximally entangled with its background, so S_ent = A/4 = log dim H_interior under the standard area-entropy identification. Claiming that absorbing an entangled particle 'must increase' S_ent while the areal entropy is unchanged asserts that dim H_interior grows beyond e^{A/4}, which is exactly S_Boltzmann > A/4. The sentence 'entanglement entropy is always smaller than its Boltzmann entropy' then merely converts that asserted increase into the target inequality. No derivation of the increase from the WKB or semiclassical dynamics is provided, and no quantum state or replica computation is supplied; the increase is the entropy-bound violation stated in different words.

full rationale

The classical proof that isentropic absorption is forbidden (Secs. IIB and IIC), the WKB tunneling calculation (Sec. IIIA), the modified-gravity example, and the free-energy checks (Sec. IIIB) are self-contained computations from the RN metric, the geodesic effective potential, and standard black-hole thermodynamics; none of these steps fits a parameter to the target result or imports the entropy-bound conclusion. The circular step is confined to Sec. IIIC, where the increase of entanglement entropy is asserted rather than derived. Given the stated initial condition that the black hole is maximally entangled, S_ent = A/4, an increase of S_ent with unchanged horizon area is logically equivalent to the claimed violation S_Boltzmann > A/4. Thus the paper's headline claim reduces to the premise of that subsection. There is no load-bearing self-citation chain, and the tunneling probability itself is an independent calculation, so the circularity is partial rather than total; score 6.

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

The paper introduces no new free parameters fitted to data and no new postulated entities. It relies on standard black hole thermodynamics, the geodesic effective potential, and two domain assumptions: the validity of the WKB tunneling action and the relationship between entanglement entropy and Boltzmann entropy. The second domain assumption carries the weight of the entropy-bound violation claim.

assumptions (5)
  • standard math First law of black hole mechanics dM = T dS + Φ dQ for stationary perturbations of RN black holes.
    Used in Sec II C to define the isentropic condition TdS = dM - ΦdQ = 0.
  • standard math Radial geodesic effective potential Veff(r) = -(E - qΦ)^2/m^2 + f(r) for a charged test particle in a static spherically symmetric metric.
    Derived in Sec II A from the geodesic equation; the central tool for proving classical impossibility.
  • domain assumption WKB approximation: transmission probability through the potential barrier is Γ ≈ exp(-2 S_E) with S_E the Euclidean action.
    Invoked in Sec III A; the validity of WKB for this curved-spacetime tunneling is assumed, and the explicit form of S_E is not rigorously derived.
  • domain assumption Entanglement entropy across the horizon increases when an entangled particle is absorbed, and it is bounded above by the black hole's Boltzmann entropy.
    Invoked in Sec III C; this is the key assumption connecting the tunneling process to entropy-bound violation. No derivation is given.
  • domain assumption The generalized quasi-topological metric (20) describes a positive-temperature black hole in 4DEGB gravity.
    Used in Sec II D as an example of modified gravity; the paper relies on the cited derivation of the metric and its thermodynamic properties.

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Cite this review

Pith. "Pith review of Isentropic process of Reissner-Nordstr\"om black holes: a possible excess of the entropy bound via a non-perturbative channel." pith.science (2026). https://pith.science/paper/OTALKVOO

@misc{pith2026250501663,
  author       = {Pith},
  title        = {Pith review of: Isentropic process of Reissner-Nordstr\"om black holes: a possible excess of the entropy bound via a non-perturbative channel},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OTALKVOO}},
  note         = {Machine review of arXiv:2505.01663}
}
read the original abstract

We study the implications of an isentropic processes applied to a Reissner-Nordstr\"om black hole. This process is possible if a black hole absorbs a particle with a specific ratio of energy and charge. We show that such an absorption process is not classically allowed, not only in Einstein gravity but also in several modified gravity theories, indicating that this prohibition is quite generic. However, an isentropic absorption process is quantum mechanically allowed: the particle can penetrate the potential barrier on the event horizon. We compute the probability of this absorption process and compare it to that of semi-classical effects. Non-perturbatively, if this process is accumulated, it is possible that the entanglement entropy can be greater than its Bekenstein-Hawking entropy and violate the entropy-bound relation.

Figures

Figures reproduced from arXiv: 2505.01663 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. shows that as α increases, SE decreases, so the probability of tunneling through the barrier increases, commensurate with the height of the potential barrier decreasing. Since α has a limiting value M2 − Q2 , the Euclidean action cannot decrease indefinitely. However, the tunneling probability is approximately Γ ≃ e −βM, (26) where β is an order one constant that may depend on α. This is comparable to the Boltzmann … view at source ↗

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Forward citations

Cited by 1 Pith paper

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

  1. Nonperturbative Isentropic Processes in AdS Black Holes with Nonlinear Electrodynamics

    hep-th 2026-01 reject novelty 4.0 of 10

    For four nonlinear-electrodynamics AdS black holes, isentropic absorption of a charged particle is classically forbidden but can proceed by WKB tunneling, with a suppression that grows with black-hole size and (mostly...

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