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arxiv: 2509.24114 · v2 · submitted 2025-09-28 · 🌀 gr-qc · astro-ph.HE

Scattering of massive particles from black holes and naked singularities

Angelos Karakonstantakis , W{\l}odek Klu\'zniak , Maciek Wielgus This is my paper

Pith reviewed 2026-05-18 12:26 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.HE
keywords black holesnaked singularitiesReissner-Nordström metricgeodesicsparticle scatteringdeflection anglestidal disruption eventsevent horizon
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The pith

Black holes channel reflected particles into a narrow band of deflection angles while naked singularities scatter them in all directions.

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

The paper studies the motion of massive test particles in the Reissner-Nordström spacetime to compare scattering around black holes and naked singularities. It launches streams of particles that share the same angular momentum but cover a range of energies, solving the geodesic equations for deep encounters that mimic conditions in tidal disruption events. The central result is that an event horizon plus a centrifugal barrier keeps reflected orbits tightly grouped in deflection angle, whereas a repulsive core at a naked singularity sends particles off in widely varying directions across the orbital plane. This contrast is presented as arising directly from the different global structures of the two spacetimes. The authors note that the resulting orbital differences should produce distinct observable signatures in astrophysical events involving close passages near compact objects.

Core claim

In the Reissner-Nordström geometry, a stream of massive particles with fixed angular momentum and varying energies that encounters a black hole reflects into a family of orbits confined to a narrow interval of deflection angles. The same stream encountering a naked singularity is scattered across essentially all directions in the plane of motion. The authors trace the difference to the interplay between the centrifugal barrier located at the unstable circular orbit and either an absorbing event horizon or a repulsive core.

What carries the argument

The geodesic equation for massive particles in the Reissner-Nordström metric, with the centrifugal barrier at the unstable circular orbit interacting with either an absorbing event horizon or a repulsive core.

Load-bearing premise

The particles are treated as non-interacting test particles that follow geodesics without back-reaction on the metric or energy loss to radiation.

What would settle it

High-resolution numerical simulations of tidal disruption events that measure the distribution of deflection angles for particles reflected from a charged black hole versus a naked singularity would directly test whether the narrow versus broad scattering patterns appear.

Figures

Figures reproduced from arXiv: 2509.24114 by Angelos Karakonstantakis, Maciek Wielgus, W{\l}odek Klu\'zniak.

Figure 1
Figure 1. Figure 1: The effective potential for time-like particles in RN metric at charge-to-mass ratio 𝑞 ≡ 𝑄/𝑀 = 0.95 (BH; black line), 𝑞 = 1.05 (NkS; red line), and 𝑞 = 0 (Schwarzschild BH; blue line). Dotted vertical lines show the location of BH horizons. The specific angular momentum is fixed to ℓ = 3.5𝑀. Christoffel symbols of the second kind, which can be calculated for a general metric tensor 𝑔𝑖 𝑗 using Γ 𝑘 𝑖 𝑗 = 1 2… view at source ↗
Figure 2
Figure 2. Figure 2: The Schwarzschild effective potential for particles with different specific angular momentum values. Limit value of ℓ = 2 √ 3𝑀 is shown with the black curve together with the location of ISCO (black filled circle). attraction overcomes the centrifugal barrier. For any fixed specific angular momentum of the test particle, the BH effective potential is characterized by (at most) a single maximum correspondin… view at source ↗
Figure 3
Figure 3. Figure 3: The effective potential (top panel) and the corresponding geodesic orbits around a Schwarzschild BH (bottom panel). Time-like particles have specific angular momentum ℓ = 4.5𝑀 and their specific energy 𝜖 2 is shown with the horizontal lines in the top panel of the figure. They correspond to the trajectories of the same color solid lines of the bottom panel. and particles fall into the BH. In contrast, for … view at source ↗
Figure 4
Figure 4. Figure 4: Trajectories for Schwarzschild BH with the same initial set up as in [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The effective potential for particles with ℓ = 3.5𝑀. The black continuous curve corresponds to the BH case (𝑞 = 0.95) and the red dashed curve to the NkS (𝑞 = 1.05). The horizontal lines correspond to the values of 𝜖 2 for the trajectories plotted in [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Trajectories around a charged BH (𝑞 = 0.95, top panel) and a NkS with 𝑞 = 1.05 (bottom panel). The colors correspond to different values of 𝜖 2 shown with horizontal lines in [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Trajectories around a charged BH (𝑞 = 0.95, top panel) and a NkS with 𝑞 = 1.05 (bottom panel). Colors correspond to different values of 𝜖 2 . Dashed lines correspond to the orbits with impact parameter 𝑦0 ⪅ 20𝑀 (or 𝜖 2 > 1.027), that are captured by a BH, but reflected and scattered by a NkS. Specific angular momentum is fixed to ℓ = 3.5𝑀. the closed approach (since the radius of the potential peak decreas… view at source ↗
Figure 8
Figure 8. Figure 8: Trajectories around a NkS (𝑞 = 1.05) for orbits initialized at low values of 𝑦0, corresponding to large values of 𝜖 . The top panel presents the effective potential with horizontal lines indicating the value 𝜖 2 of orbits plotted in the bottom panel. The specific angular momentum is fixed to ℓ = 3.5𝑀. ACKNOWLEDGEMENTS This work was supported by the Polish NCN grant 2019/35/O/ST9/03965. MW is supported by a… view at source ↗
Figure 9
Figure 9. Figure 9: The effective potential for particles with different values of specific angular momentum (ℓ, represented by the various colors). The solid lines correspond to charged BH (𝑞 = 0.95) while, dashed lines show the NkS case with 𝑞 = 1.05. DATA AVAILABILITY The data used in the work presented in this article are available upon request to the corresponding author. REFERENCES Abramowicz M. A., Kluźniak W., Lasota … view at source ↗
Figure 10
Figure 10. Figure 10: Geodesic orbits around NkS for particles of the same angular momentum. From top to bottom, we compare three values of specific angular momentum ℓ/𝑀 = (3.3, 3.0, 2.7). Dashed lines show orbits with low values in impact parameter. Mummery A., Ingram A., 2024, MNRAS, 528, 2015 Nordström G., 1918, Koninklijke Nederlandse Akademie van Wetenschappen Proceedings Series B Physical Sciences, 20, 1238 Penrose R., 1… view at source ↗
read the original abstract

We performed a numerical study of the dynamics of massive particles orbiting black holes and naked singularities in the Reissner-Nordstr\"om geometry. We modeled a stream of particles with a constant angular momentum and with a range of energies. We then solved the geodesic equation of motion and compared the trajectories around black holes and naked singularities by tuning the charge parameter of the metric. The setup {allows us to explore the orbital dynamics relevant for} astrophysical scenarios such as tidal disruption events{, particularly for deep encounters}. We discussed differences and similarities in the orbital dynamics and deflection angles. We found that particles reflected by a black hole follow a stream-like family of orbits within a narrow range of deflection angles, whereas in the case of naked singularities particles are scattered in all directions on the plane of motion. We explained this behavior as an interplay between the presence of a centrifugal barrier at the location of the unstable circular orbit and an absorbing event horizon in the case of a black hole or a {repulsive core} in the case of a naked singularity. These qualitative differences are expected to impact the observable signatures of tidal disruption events.

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 / 3 minor

Summary. The paper performs a numerical integration of the geodesic equation for massive test particles in the Reissner-Nordström spacetime, holding angular momentum L fixed while varying particle energy E. By tuning the charge parameter Q/M to cross from black-hole (horizon-present) to naked-singularity (repulsive-core) regimes, the authors compare families of trajectories and resulting deflection angles. They report that black-hole scattering produces a narrow, stream-like bundle of reflected orbits confined to a limited range of deflection angles, whereas naked-singularity scattering is essentially isotropic on the orbital plane. The difference is attributed to the centrifugal barrier at the unstable circular orbit combined with horizon absorption versus core repulsion. The study is framed as relevant to deep encounters in tidal disruption events.

Significance. If the reported contrast in deflection-angle distributions is robust, the work supplies a qualitative, potentially observable signature that could help distinguish black holes from naked singularities in astrophysical scattering events. The direct numerical approach to geodesic motion for fixed L and sampled E provides a concrete illustration of how horizon versus repulsive-core boundary conditions alter orbital families. The limitation to non-back-reacting test particles is standard for such explorations but restricts immediate application to realistic, self-gravitating encounters.

major comments (1)
  1. [Numerical methods and parameter selection] The manuscript selects a specific interval of energies for the fixed angular momentum L but does not locate this interval relative to the maximum of the effective potential V_eff(r;L) (or the location of the unstable circular orbit). Without this mapping it remains unclear whether the narrow deflection-angle stream for black holes is a generic feature of the horizon-plus-barrier combination or an artifact of sampling energies just above the barrier while different regimes are sampled for the naked-singularity case. This choice is load-bearing for the central qualitative claim.
minor comments (3)
  1. [Abstract] The abstract and introduction refer to 'a range of energies' without quoting the numerical interval or the criterion used to choose it.
  2. [Numerical methods] No mention is made of step-size convergence tests, integrator tolerance, or error estimates for the numerical geodesic integration.
  3. [Discussion] The discussion of astrophysical implications for tidal disruption events would benefit from a brief statement of the regime of validity (test-particle limit, neglect of radiation reaction).

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comment on our manuscript. The concern about mapping the chosen energy interval to the effective potential is well taken, and we address it directly below with a commitment to strengthen the presentation in revision.

read point-by-point responses

  1. Referee: [Numerical methods and parameter selection] The manuscript selects a specific interval of energies for the fixed angular momentum L but does not locate this interval relative to the maximum of the effective potential V_eff(r;L) (or the location of the unstable circular orbit). Without this mapping it remains unclear whether the narrow deflection-angle stream for black holes is a generic feature of the horizon-plus-barrier combination or an artifact of sampling energies just above the barrier while different regimes are sampled for the naked-singularity case. This choice is load-bearing for the central qualitative claim.

    Authors: We agree that an explicit mapping of the sampled energies to V_eff(r;L) and the unstable circular orbit would improve clarity. In the original work the fixed L was chosen so that an unstable circular orbit (maximum of V_eff) exists for Q/M < 1; the energy interval was selected to lie above this maximum, permitting trajectories that either scatter after passing the barrier or fall into the horizon. For Q/M > 1 the effective potential instead develops a repulsive core at small r with no horizon, so the same L produces qualitatively different turning points and deflection. To resolve the ambiguity we will add, in the revised manuscript, a dedicated figure showing V_eff(r) for the adopted L, with the location of its maximum marked and horizontal lines indicating the lower and upper bounds of the sampled E range. Annotations will distinguish the black-hole and naked-singularity regimes. This addition demonstrates that the narrow deflection stream is tied to the combination of a centrifugal barrier plus an absorbing horizon, while the isotropic scattering follows from core repulsion; the qualitative contrast therefore holds for energies above the barrier in both cases. revision: yes

Circularity Check

0 steps flagged

No significant circularity in numerical geodesic scattering study

full rationale

The paper reports results from direct numerical integration of the geodesic equation for massive test particles in the Reissner-Nordström metric, holding angular momentum fixed while varying energy and tuning the charge parameter to switch between black-hole and naked-singularity regimes. The reported contrast—narrow deflection-angle streams for black holes versus isotropic scattering for naked singularities—emerges as an output of those integrations rather than any fitted parameter, self-defined quantity, or self-citation chain. The qualitative explanation invoking the centrifugal barrier, absorbing horizon, and repulsive core follows from the standard effective-potential analysis already present in the metric and is not reduced to the numerical outputs by construction. No load-bearing self-citations, ansatzes smuggled via prior work, or renaming of known results appear in the central claims, so the derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard Reissner-Nordström metric and the assumption that massive particles follow geodesics as test particles.

axioms (2)
  • domain assumption Reissner-Nordström metric is the correct spacetime for a charged, non-rotating object
    Invoked throughout the numerical study of geodesics.
  • domain assumption Test particles follow timelike geodesics without back-reaction
    Standard assumption for orbital dynamics in the abstract.

pith-pipeline@v0.9.0 · 5735 in / 1186 out tokens · 37479 ms · 2026-05-18T12:26:02.554062+00:00 · methodology

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Reference graph

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