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arxiv: 2511.10307 · v2 · submitted 2025-11-13 · 🌀 gr-qc · hep-th

Gravitational Atoms from Topological Stars

Ibrahima Bah , Emanuele Berti , Bogdan Ganchev , David Pere\~niguez , Nicholas Speeney This is my paper

Pith reviewed 2026-05-17 22:43 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords topological starsgravitational atomsbound statesscalar fieldsnormal modesCompton wavelengthhorizonless objectsKaluza-Klein momentum
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0 comments X

The pith

A massive scalar field forms stable bound states around topological stars, creating genuine gravitational atoms distinct from black holes.

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

The paper examines bound states of a massive scalar field on the background of a topological star, a horizonless compact object. It establishes that these states are strictly normal modes with purely real frequencies, unlike the decaying modes that occur around black holes. The spectrum depends on the ratio of the field's Compton wavelength to the star's size: a hydrogen-like cloud forms for large wavelengths, localization along timelike paths occurs for small wavelengths, and a richer structure appears when the scales match. This construction yields a gravitational atom that can carry electric charge or Kaluza-Klein momentum. The result offers a concrete way to differentiate horizonless objects from black holes through their response to scalar perturbations.

Core claim

We study the bound states of a massive scalar field around a topological star, and show that these are strictly normal modes. This yields a genuine gravitational atom, sharply distinguishing horizonless objects from black holes. We show that the modes are controlled by the field's Compton wavelength compared to the size of the star. When the Compton wavelength is large, the field forms a cloud with a hydrogen-like spectrum, while in the opposite regime it is localized along timelike trajectories. When the two scales are comparable the spectrum becomes richer, and we characterize it in detail allowing the field to carry electric charge and Kaluza--Klein momentum.

What carries the argument

The ratio of the scalar field's Compton wavelength to the topological star radius, which sets the effective potential and selects between hydrogen-like clouds, localized trajectories, and richer charged or Kaluza-Klein spectra.

If this is right

  • For large Compton wavelength the field forms a cloud with a hydrogen-like spectrum.
  • For small Compton wavelength the field localizes along timelike trajectories.
  • When the scales are comparable the spectrum includes richer modes with electric charge and Kaluza-Klein momentum.
  • The normal-mode character sharply distinguishes topological stars from black holes, which support only decaying quasinormal modes.

Where Pith is reading between the lines

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

  • If such stars exist, the bound states could produce long-lived, narrow-line signals in gravitational-wave or electromagnetic observations.
  • The same wavelength-to-size comparison may apply to other horizonless geometries and yield analogous atomic spectra.
  • Including backreaction in future calculations would test whether the gravitational atom remains stable once the scalar field sources the metric.

Load-bearing premise

The topological star metric is treated as a fixed background with no backreaction from the scalar field.

What would settle it

Finding imaginary frequency components in the scalar field spectrum around a topological star would show the modes are not strictly normal.

Figures

Figures reproduced from arXiv: 2511.10307 by Bogdan Ganchev, David Pere\~niguez, Emanuele Berti, Ibrahima Bah, Nicholas Speeney.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic energy spectrum of normal modes at fixed [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Schematic representation of the spatial distribution [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Surface [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Schematic representation of the scalar potential [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: , as anticipated by the analysis in the main text (as well as the different range on the horizontal axis). energies decrease, faster for the TS modes in comparison to the cloud ones (for mixed modes the rate depends on which type dominates the “mixture”). We expect that for large overtones, n ≫ 1, the binding energy will tend to zero, γ → 0. This picture holds qualitatively for all scalar masses µ˜ and for… view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Top Panels: Scalar wavefunctions on the TS background with parameters as in Fig. [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Left: Wavefunction and potential for the fundamental mode, using the same parameters as in Fig. [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Fundamental normal mode spectrum in the regime [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Comparison of normal mode spectra for various [PITH_FULL_IMAGE:figures/full_fig_p016_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p018_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Field amplitudes [PITH_FULL_IMAGE:figures/full_fig_p020_14.png] view at source ↗
read the original abstract

We study the bound states of a massive scalar field around a topological star, and show that these are strictly normal modes. This yields a genuine gravitational atom, sharply distinguishing horizonless objects from black holes. We show that the modes are controlled by the field's Compton wavelength compared to the size of the star. When the Compton wavelength is large, the field forms a cloud with a hydrogen-like spectrum, while in the opposite regime it is localized along timelike trajectories. When the two scales are comparable the spectrum becomes richer, and we characterize it in detail allowing the field to carry electric charge and Kaluza--Klein momentum.

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

1 major / 1 minor

Summary. The paper studies the bound states of a massive scalar field on the fixed background of a topological star. It claims to demonstrate that these states are strictly normal modes with real frequencies, forming a genuine gravitational atom that sharply distinguishes horizonless objects from black holes. The spectrum is analyzed in three regimes determined by the ratio of the field's Compton wavelength to the star size: hydrogen-like clouds for large wavelengths, localization along timelike trajectories for small wavelengths, and a richer spectrum when the scales are comparable. Extensions to electrically charged fields and those with Kaluza-Klein momentum are included.

Significance. If the normal-mode conclusion is robust, the work supplies a concrete, analytically tractable example of discrete real-frequency modes around a horizonless compact object. This offers a clear contrast to the quasinormal-mode spectra of black holes and could inform searches for exotic compact objects via gravitational-wave or scalar-field observations. The regime-dependent characterization, including the hydrogen-like limit, provides an intuitive and potentially falsifiable structure.

major comments (1)
  1. [Abstract and the derivation of normal modes (around the Klein-Gordon analysis)] The central claim that the modes are 'strictly normal' rests on solving the massive Klein-Gordon equation on the fixed topological-star metric with regularity imposed at the minimal radius and exponential decay at infinity. No estimate of scalar-field backreaction on the metric or solution of the coupled Einstein-scalar system is provided to confirm that the effective potential remains free of horizon-like features that would convert the spectrum into quasinormal modes with nonzero imaginary part. This approximation is load-bearing for the distinction from black holes.
minor comments (1)
  1. [Section on mode solutions] Clarify in the text how the boundary conditions are numerically or analytically verified to produce purely real frequencies, addressing any potential approximations in the mode extraction.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive feedback. We address the major comment below.

read point-by-point responses

  1. Referee: [Abstract and the derivation of normal modes (around the Klein-Gordon analysis)] The central claim that the modes are 'strictly normal' rests on solving the massive Klein-Gordon equation on the fixed topological-star metric with regularity imposed at the minimal radius and exponential decay at infinity. No estimate of scalar-field backreaction on the metric or solution of the coupled Einstein-scalar system is provided to confirm that the effective potential remains free of horizon-like features that would convert the spectrum into quasinormal modes with nonzero imaginary part. This approximation is load-bearing for the distinction from black holes.

    Authors: We thank the referee for raising this point. Our analysis is performed in the test-field (probe) approximation on the fixed topological-star background, which is a regular, horizonless geometry. The massive Klein-Gordon equation is solved subject to regularity at the minimal radius and exponential decay at infinity, yielding a discrete set of real frequencies. This boundary-value problem is of Sturm-Liouville type and therefore admits only real eigenvalues; the absence of a horizon precludes the dissipative boundary conditions responsible for complex quasinormal frequencies in black-hole spacetimes. The same test-field approach is routinely used to study gravitational atoms around black holes, where the resulting spectrum is nevertheless quasinormal. We have added a clarifying paragraph to the discussion section that explicitly states the regime of validity of the linear approximation (small scalar amplitude, backreaction entering only at second order) and notes that a fully coupled Einstein-scalar analysis, while interesting, lies beyond the present scope. revision: partial

Circularity Check

0 steps flagged

No circularity: normal modes obtained by direct solution of wave equation on fixed background

full rationale

The paper solves the Klein-Gordon equation for a massive scalar on the given topological-star metric, imposing regularity at the minimal radius and exponential decay at infinity to extract discrete real frequencies. These frequencies are presented as strictly normal modes forming a gravitational atom, with the spectrum characterized in different regimes of Compton wavelength versus star size. This computation follows directly from the differential equation and boundary conditions without any fitted parameters renamed as predictions, without self-citation chains supporting the core result, and without ansatze or uniqueness theorems imported from prior author work. The distinction from black holes arises from the horizonless nature of the background metric itself, which is an input rather than a derived output. The fixed-background approximation is an explicit modeling choice whose validity is not claimed to be proven within the derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The analysis assumes a fixed background metric for the topological star and linear scalar perturbations on it. No new entities are introduced.

axioms (2)
  • domain assumption The topological star is a stable, horizonless solution of the higher-dimensional Einstein equations with the given topology.
    Invoked when treating the metric as a fixed background for the scalar wave equation.
  • domain assumption The scalar field is a test field with negligible backreaction on the geometry.
    Required to keep the background fixed while solving the wave equation.

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Reference graph

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