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Lower bound on the proper lengths of stationary bound-state charged massive scalar clouds

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Pith's one-line read This paper proves that stationary bound-state charged massive scalar clouds around Kerr-Newman black holes have a universal minimum proper size: the distance from the horizon to the cloud's radial maximum always exceeds…

desk verdict A new universal no-short-hair bound for charged scalar clouds on Kerr-Newman, probably correct but with one 'cumbersome' inequality that needs explicit proof before I'd fully trust it. read the letter →

arxiv 2507.03079 v1 pith:M5FNJ4U4 submitted 2025-07-03 gr-qc astro-ph.HEhep-th

Shahar Hod This is my paper
classification gr-qcastro-ph.HEhep-th MSC 83C57 PACS 04.70.-s
keywords Kerr-Newmanblackholeschargedscalarcloudsno-short-hairtheoremproperlengthboundKlein-GordonequationsuperradiantstatesradialTeukolskyuniversallower
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

Charged massive scalar fields in stationary bound states around spinning, charged Kerr-Newman black holes cannot form clouds that are arbitrarily compact. The paper proves analytically that, for every allowed value of the black-hole spin and charge and of the scalar field's charge, mass, and angular mode numbers, the proper distance from the horizon to the radial maximum of the cloud satisfies $\ell_{\max}/M > \ln(3+\sqrt{8}) \approx 1.7627$. This universal constant is parameter-free: it does not depend on the black hole or on the field. The result matters because earlier work showed these clouds can violate the null-geodesic no-short-hair bound, so it was not known whether they could be squeezed to zero size; this theorem closes that gap by establishing a finite floor.

What carries the argument

The central object is the effective binding potential $V(r; M, a, Q, \mu, q, l, m)$ of the radial Schrodinger-like equation $d^2\psi/dy^2 - V\psi = 0$, obtained from the radial Teukolsky equation via $\psi = rR$ and $dy = r^2/\Delta\, dr$. The load-bearing identity is the lower bound on $V$ which, combined with the inflection-point condition $V(r_0)=0$, yields lower bounds on $r_0$ and on the maximum $r_{\max}$. The extremal limit $\tau \to 0$ with $s \to 1$ reduces the final proper-length integral to $M\cosh^{-1}(3) = M\ln(3+\sqrt{8})$.

What would settle it

Compute the proper distance from the horizon to the radial maximum for a stationary bound-state charged scalar cloud around a Kerr-Newman black hole, using regular-at-horizon and decaying-at-infinity boundary conditions; a solution with $\ell_{\max}/M \le \ln(3+\sqrt{8}) \approx 1.7627$ would refute the claim. A more targeted check is to evaluate the paper's potential inequality (32) against numerically computed angular eigenvalues across a fine grid of the allowed parameters.

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Extended reading notes

Core claim

The paper's central claim is that a linearized charged massive scalar field in a stationary bound state around a Kerr-Newman black hole necessarily has a radial profile that peaks at a finite distance from the horizon, and that this proper distance is bounded below by $M\ln(3+\sqrt{8})$. The argument rewrites the radial Klein-Gordon equation as a Schrodinger-like equation with an effective potential $V(r)$, and shows that the near-horizon solution is positive, increasing, and convex while the far-region solution decays exponentially, forcing an inflection point $r_0$ between the horizon and the maximum $r_{\max}$. Using the angular eigenvalue lower bound $K_{lm} \ge m^2 - a^2(\mu^2 - \omega_c^2)$ and a chain of inequalities on the effective potential, the paper locates $r_0$ and $r_{\max}$, converts the coordinate distance to proper length, and minimizes over the black-hole parameters to obtain the constant $\ln(3+\sqrt{8})$.

Load-bearing premise

The proof depends on two inequalities about the angular eigenvalues and the effective potential; if either fails in some allowed corner of the parameter space, the universal bound would not follow.

Editorial extensions

If this is right

  • Charged scalar clouds in Kerr-Newman spacetimes cannot be made arbitrarily compact; their proper radial extent is at least about 1.76 times the black-hole mass.
  • The bound holds across the full physical parameter range of Kerr-Newman black holes, up to extremal spin and sub-extremal charge, and for all scalar-field charges, masses, and angular mode numbers, so any shorter cloud would require leaving the linearized stationary bound-state regime.
  • The result extends the no-short-hair idea from spherically symmetric static hairy black holes to non-spherically symmetric charged configurations, showing that violating the null-geodesic bound does not imply arbitrarily short clouds.
  • Because the proof uses only the radial maximum, other effective-length definitions, such as the radius containing half of the cloud's mass, are left open and would need numerical treatment.

Reading between the lines

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

  • The linearized, test-field nature of the derivation means nonlinear backreaction could in principle alter the constant; a numerical evolution of the coupled Einstein-Maxwell-charged-scalar system would settle that.
  • The bound is minimized in the extremal limit $\tau \to 0$ with $s \to 1$, so searching for near-extremal configurations whose peak approaches $\ln(3+\sqrt{8})$ would test whether the constant is tight.
  • The inflection-point technique may transfer to other bosonic clouds, such as charged vector (Proca) fields, yielding analogous universal proper-length bounds, though the paper does not address those cases.
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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 manuscript claims a universal no-short-hair theorem for stationary, bound-state charged massive scalar clouds supported by Kerr-Newman black holes. The proof rewrites the radial Teukolsky equation as a Schrödinger-like equation, uses the near-horizon Bessel behavior and exponential decay at infinity to establish a non-monotonic radial profile, and locates an inflection point where the effective potential vanishes. A lower bound on the inflection-point radius is obtained via a sequence of potential inequalities, converted into a parameter-independent bound on the location of the radial maximum, and finally integrated against the proper radial metric to obtain the dimensionless bound ℓmax/M > ln(3+√8), claimed to hold for all allowed black-hole and field parameters.

Significance. If the proof is completed as claimed, the result is a clean and striking extension of the no-short-hair theorem to charged scalar clouds around Kerr-Newman black holes. The bound is universal, parameter-free, and falsifiable: it predicts that such clouds cannot be made arbitrarily compact even though they can violate the older r>r_null bound. The paper's strategy is mostly analytical and uses no fitting parameters. Its main input ingredients are the angular eigenvalue bound of Bardeen-Press-Teukolsky and the previously established existence of charged clouds; the near-horizon Bessel analysis and the integral estimate for the proper length are transparent and checkable. The significance is somewhat limited by the fact that the analysis concerns linearized test fields rather than fully back-reacted configurations, but within its stated scope the claimed theorem would be a valuable addition to the black-hole hair literature.

major comments (2)
  1. [§IV, Eq. (32)] The inequality (32) is the load-bearing step of the paper: it is the only input that converts the angular eigenvalue bound (11) and the explicit form of the effective potential (22) into the quantitative inflection-point bound (43), which ultimately yields the universal constant (60). However, the passage from Eqs. (11), (14)-(18), and (22) to (32) is not shown; the phrase 'one finds' with footnotes [38,39] is not a derivation. Since (32) is a pointwise inequality in r and is described as 'rather cumbersome,' the proof needs either a complete algebraic derivation or a certified numerical verification over the allowed parameter range. Without this, the central claim is not established.
  2. [§V, Eqs. (44)-(49)] The minimization of F(β) is performed as if β could range over (0,∞), but the physical parameter space is constrained. Since γ=qQs/m≥0, one has β≤1/s^2, and the Kerr-Newman condition a^2+Q^2≤M^2 translates into s^2+τ≤1. The paper should state this domain and verify that βmax of Eq. (48) lies inside it, or discuss the endpoint cases, before asserting that (49) gives the global minimum. This check is absent; without it the derivation of (49) and hence of the subsequent bound (52) is not fully justified.
minor comments (4)
  1. [Eq. (5)] The horizon formula has a sign typo: it should read r±=M±√(M^2-a^2-Q^2). The correct relation is used later in Eq. (34), so this typo does not affect the proof, but it should be corrected.
  2. [§V, Eq. (44)] The admissible range of β should be stated explicitly when F(β) is introduced. In particular, (42) gives β>τ/(2s^2), and the assumption Q≥0 (footnote [13]) gives β≤1/s^2; making these bounds explicit would greatly clarify the minimization step.
  3. [§VI, Eqs. (57)-(58)] The statement that the right-hand side of (57) 'can be minimized' by the limiting values τ→0+ and s→1− should be formulated as an infimum. The paper proves a lower bound but does not construct a sequence of cloud solutions that attains it; the limiting argument is sufficient for the stated inequality.
  4. [References and footnotes] In Eq. (11), the citation [27] appears to refer to a footnote rather than a numbered reference, which makes the source of the eigenvalue bound confusing; the reference list should be aligned with the citation markers.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the universal proper-length bound is derived algebraically from the radial equation and an external angular-eigenvalue bound, with no fitted parameter and no use of the target inequality as an input.

full rationale

The derivation chain starts from the Klein-Gordon radial equation (10), the angular eigenvalue bound (11) attributed to Bardeen, Press, and Teukolsky (ref. [26]), and the resonance condition (14)-(15) for stationary clouds. The paper then transforms the effective potential into inequality (32), simplifies it to (36), obtains the inflection-point lower bound (43), minimizes the function F(beta) analytically (45)-(51), and converts the radial bound into the proper-length integral (53)-(58). None of these steps is equivalent to the target inequality (60). The final constant ln(3+sqrt(8)) is obtained by evaluating the proper-length integral in the extremal limits s -> 1, tau -> 0; it is not fitted and it is not used as an input. Although the paper cites the author's earlier works (e.g., refs. [3,4,6,8]), those citations supply the existence of charged clouds and the near-horizon Bessel asymptotics; the near-horizon behavior is re-derived in Eqs. (23)-(27), and the existence statement is a hypothesis of the theorem rather than the conclusion. The angular eigenvalue bound is an external result, not a restatement of the present claim. The most substantive concern is the unshown algebraic step leading to inequality (32), and footnote [47] concedes that the non-linear regime remains untested; both are correctness or rigor concerns, not circularity. No load-bearing step reduces by construction to its own input.

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

The theorem is analytical and contains no fitted numbers and no invented entities. The universal constant ln(3+√8) emerges from minimizing a closed-form expression over the black-hole parameters (τ → 0, s → 1). The main unpaid inputs are the existence of the bound-state clouds, the angular eigenvalue bound, the near-horizon form of the potential, and the chosen definition of 'effective proper length.'

assumptions (5)
  • domain assumption Kerr-Newman geometry with Δ = r² − 2Mr + a² + Q² and standard horizon roots r± = M ± √(M² − a² − Q²)
    Used throughout as the fixed background metric (Eqs. 3-5); note Eq. (5) prints the wrong sign for Q².
  • domain assumption Stationary bound-state charged scalar clouds exist and satisfy resonance ω = ω_c = mΩ_H + qΦ_H and ω² < μ²
    The theorem is conditional on the existence of these linearized configurations, established in refs [3,4]; Eqs. (14) and (18).
  • standard math Angular eigenvalue bound K_lm ≥ m² − a²(μ² − ω_c²)
    Cited from refs [26,27] and used to derive the crucial potential inequality (32); Eq. (11).
  • domain assumption Near-horizon approximation Eq. (25) with the I0 solution (26) and positivity properties (27)
    Used to establish that the radial function is convex near the horizon, forcing a maximum and an inflection point; valid for x << τ.
  • ad hoc to paper Effective length is defined as proper distance to the radial maximum rmax, not a mass-concentration radius
    Section I; the abstract's 'effective proper lengths' refers to this specific peak-distance definition.

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Pith. "Pith review of Lower bound on the proper lengths of stationary bound-state charged massive scalar clouds." pith.science (2026). https://pith.science/paper/M5FNJ4U4

@misc{pith2026250703079,
  author       = {Pith},
  title        = {Pith review of: Lower bound on the proper lengths of stationary bound-state charged massive scalar clouds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M5FNJ4U4}},
  note         = {Machine review of arXiv:2507.03079}
}
abstract

It has recently been revealed that charged scalar clouds, spatially regular matter configurations which are made of linearized charged massive scalar fields, can be supported by spinning and charged Kerr-Newman black holes. Using analytical techniques, we establish a no-short hair theorem for these stationary bound-state field configurations. In particular, we prove that the effective proper lengths of the supported charged massive scalar clouds are bounded from below by the remarkably compact dimensionless relation $\ell/M>\ln(3+\sqrt{8})$, where $M$ is the mass of the central supporting black hole. Intriguingly, this lower bound is universal in the sense that it is valid for all Kerr-Newman black-hole spacetimes [that is, in the entire regime $\{a/M\in(0,1],Q/M\in[0,1)\}$ of the dimensionless spin and charge parameters that characterize the central supporting black holes] and for all values of the physical parameters (electric charge $q$, proper mass $\mu$, and angular harmonic indexes $\{l,m\}$) that characterize the supported stationary bound-state scalar fields.

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Works this paper leans on

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Reviewed August 6, 2026 · model on record in the stance chip above.