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arxiv: 2606.09694 · v1 · pith:PVKKPLWPnew · submitted 2026-06-08 · 🌀 gr-qc · hep-th

Stationary scalar clouds around a rotating Kalb-Ramond BTZ black hole

Rui Ding , Fangli Quan , Zhong-Wu Xia , Qiyuan Pan , Jiliang Jing This is my paper

Pith reviewed 2026-06-27 15:39 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords scalar cloudsKalb-Ramond BTZ black holesuperradiant thresholdRobin boundary conditionsquasinormal modesstationary bound statesmodified gravity
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The pith

The Kalb-Ramond parameter turns the existence lines of scalar clouds nonmonotonic around a rotating BTZ black hole.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper studies stationary scalar clouds that form around a rotating black hole modified by Kalb-Ramond gravity when Robin boundary conditions are imposed. The clouds appear as bound states sitting exactly at the frequency where superradiance begins. The central result is that the Kalb-Ramond parameter changes how these clouds appear in the space of black-hole rotation and boundary parameters. For positive values the curves marking cloud existence bend back on themselves, so the same boundary condition can support clouds in two separate ranges of rotation rate. This behavior does not occur when the parameter is zero or negative.

Core claim

Stationary scalar clouds exist as bound states at the superradiant threshold ω = m Ω_H. The KR parameter qualitatively alters the existence lines of these clouds in parameter space: the lines stay monotonic for nonpositive KR values but can become nonmonotonic for positive KR values. Consequently a single fixed Robin boundary condition can admit cloud solutions in disconnected intervals of the rotation and KR parameter space. Quasinormal-mode frequencies and horizon fluxes confirm that these solutions sit at the threshold where the energy flux reverses sign. The KR parameter also moves the critical value of the Robin parameter at which clouds first appear.

What carries the argument

The existence lines of clouds, which are the loci in the space of rotation rate and KR parameter where stationary bound states appear under given Robin boundary conditions at the superradiant threshold.

If this is right

  • The KR parameter shifts the critical Robin parameter value needed for clouds to exist.
  • For positive KR a fixed Robin condition can support clouds in two disconnected intervals of rotation rate.
  • Quasinormal modes at these points have zero imaginary part and the horizon energy flux changes sign.
  • The nonmonotonic lines arise only when the KR parameter is positive.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This nonmonotonicity may allow scalar clouds to distinguish Kalb-Ramond gravity from Einstein gravity in numerical or analog setups.
  • Similar parameter-induced folding of existence curves could occur in other modified-gravity models that add extra scalar or tensor fields.
  • The disconnected regions imply that the same boundary condition might permit multiple distinct cloud configurations depending on how the system is prepared.

Load-bearing premise

That the Kalb-Ramond term modifies the metric and the scalar wave equation such that the eigenvalue problem for bound states yields nonmonotonic existence lines when the parameter is positive.

What would settle it

A direct numerical integration of the radial equation for positive KR parameter that shows the cloud existence curves remain strictly monotonic for all Robin boundary values.

Figures

Figures reproduced from arXiv: 2606.09694 by Fangli Quan, Jiliang Jing, Qiyuan Pan, Rui Ding, Zhong-wu Xia.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: , for example, ξc2/π = 0.575 when ℓ = 0 in the case of λ = 1, but ξc2/π = 0.692 when ℓ = 0 in the case of r+ = 1), indicating that this unstable region is not directly associated with the scalar clouds but instead with a bulk AdS instability. The KR parameter ℓ does not alter the basic physical picture, but it changes the superradiant critical values and intervals quantitatively. In particular, regardless … view at source ↗
read the original abstract

We investigate the scalar clouds around a rotating Kalb-Ramond (KR) BTZ black hole under Robin boundary conditions. The clouds are obtained as stationary bound states at the superradiant threshold $\omega=m\Omega_H$, where the KR parameter, the rotation and the Robin boundary jointly determine their existence. It is shown that the KR parameter qualitatively changes the existence lines of clouds. For a nonpositive KR parameter, the lines remain monotonic, whereas for a positive KR parameter they can become nonmonotonic, so that a fixed boundary condition may admit clouds in disconnected regions of parameter space. Quasinormal modes (QNMs) and horizon fluxes are further used as consistency checks, confirming that the cloud solutions correspond to non-damping modes at the superradiant threshold where the energy flux changes sign. The KR parameter also shifts the critical Robin parameter at which the clouds exist. These results establish stationary scalar clouds as sensitive probes of the interplay between the Robin boundary conditions and KR gravity.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates stationary scalar clouds around rotating Kalb-Ramond BTZ black holes subject to Robin boundary conditions at the AdS boundary. Clouds are constructed as zero modes at the superradiant threshold ω = m Ω_H. The central result is that positive values of the KR parameter render the existence curves in the (rotation, KR) plane non-monotonic, unlike the monotonic behavior for non-positive KR values; this permits a fixed Robin parameter to support clouds in disconnected regions of parameter space. Consistency is checked via quasinormal-mode spectra and horizon energy fluxes, which confirm the modes are non-damping at threshold with sign-changing flux.

Significance. If the numerical results hold, the work demonstrates that the KR parameter induces a qualitative change in the topology of superradiant cloud existence domains, providing a concrete probe of how higher-form fields modify instability thresholds in AdS. The explicit use of QNM and flux checks as independent verifications is a methodological strength that increases in the reported non-monotonicity.

major comments (2)
  1. [§3] §3 (scalar field equation): the effective potential after separation of variables must be displayed explicitly to confirm that the KR term enters only through the metric functions (lapse and shift) without generating first-derivative or imaginary contributions that would invalidate the zero-mode condition ω = m Ω_H.
  2. [§5] §5 (existence lines): the shooting procedure used to enforce both horizon regularity and the Robin condition at fixed ω = m Ω_H is not described (e.g., integration method, tolerance, or how the Robin parameter is held fixed while scanning rotation and KR); this is load-bearing for the non-monotonicity claim.
minor comments (2)
  1. [Figures 2–4] Figure captions for the existence-line plots should state the precise range of the Robin parameter and the numerical resolution employed.
  2. [§2] The definition of the KR parameter (its normalization relative to the AdS radius) should be restated in the results section for readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and positive assessment of the manuscript. We address each major comment below and will revise the text accordingly to improve clarity and reproducibility.

read point-by-point responses

  1. Referee: [§3] §3 (scalar field equation): the effective potential after separation of variables must be displayed explicitly to confirm that the KR term enters only through the metric functions (lapse and shift) without generating first-derivative or imaginary contributions that would invalidate the zero-mode condition ω = m Ω_H.

    Authors: We agree that an explicit display of the effective potential will confirm the structure of the equation. In the revised manuscript we will add the separated radial equation and the corresponding effective potential in §3, showing that the KR parameter enters exclusively through the metric functions (lapse and shift) and produces neither first-derivative terms nor imaginary contributions, thereby preserving the validity of the real zero-mode condition at ω = m Ω_H. revision: yes

  2. Referee: [§5] §5 (existence lines): the shooting procedure used to enforce both horizon regularity and the Robin condition at fixed ω = m Ω_H is not described (e.g., integration method, tolerance, or how the Robin parameter is held fixed while scanning rotation and KR); this is load-bearing for the non-monotonicity claim.

    Authors: We acknowledge that a fuller description of the numerical implementation is required. In the revised §5 we will specify the integration method (fourth-order Runge-Kutta), the convergence tolerances employed, and the precise procedure by which the Robin parameter is held fixed while the rotation parameter and KR parameter are scanned to trace the existence curves. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained numerical solution of modified wave equation

full rationale

The paper solves the radial scalar equation on the KR-modified BTZ metric subject to Robin boundary conditions at the AdS boundary, locating stationary bound states precisely where the frequency equals the superradiant threshold ω = m Ω_H. The reported non-monotonic existence curves for positive KR parameter arise directly from the explicit dependence of the lapse, shift, and effective potential on the KR parameter inside the differential operator; no parameter is fitted to a subset of the output data and then relabeled as a prediction, no self-citation supplies a uniqueness theorem that forces the result, and the threshold condition follows from the stationarity and axisymmetry of the background without additional assumptions imported from the authors' prior work. The derivation therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no explicit free parameters, axioms, or invented entities can be extracted. The KR parameter is treated as an input from the background solution.

pith-pipeline@v0.9.1-grok · 5714 in / 1165 out tokens · 16875 ms · 2026-06-27T15:39:49.196575+00:00 · methodology

discussion (0)

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Reference graph

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