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arxiv: 2510.18647 · v2 · pith:5GITA226new · submitted 2025-10-21 · 🌀 gr-qc · hep-th

Observational Tests of Regular Black Holes with Scalar Hair and their Stability

P. A. Gonz\'alez , Marco Olivares , Eleftherios Papantonopoulos , Yerko V\'asquez This is my paper

Pith reviewed 2026-05-21 20:07 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords regular black holesscalar chargenull geodesicsLyapunov exponentblack hole shadowsSolar System testsEvent Horizon Telescope observations
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The pith

A phantom scalar charge that regularizes black hole singularities must remain extremely small to agree with Solar System and black hole shadow observations.

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

This paper investigates the motion of light and test particles around regular black holes formed by a phantom scalar field with charge A. The charge A smooths out the central singularity and changes the geometry from the standard Schwarzschild solution. By calculating light deflection, time delays, and redshifts, the authors use precise Solar System measurements to show that A has to be very small. They also find exact formulas linking the instability of light orbits, measured by the Lyapunov exponent, to the size of the black hole's shadow. Data from the Event Horizon Telescope on M87* and Sgr A* tighten these limits further, keeping the model close to ordinary black holes.

Core claim

The authors establish that the scalar charge A controls the dynamical instability of circular photon orbits through the Lyapunov exponent λ, with the critical impact parameter given exactly by B_u = 1/|λ| and the shadow angular size by α_sh = 1/(r0 |λ|). Increasing A makes these orbits less unstable and increases the apparent shadow size. Solar System tests and EHT observations together require that A stay extremely small to remain consistent with general relativity in both weak and strong field regimes.

What carries the argument

The regular black hole metric deformed by the phantom scalar charge A, together with the exact relations B_u = 1/|λ| and α_sh = 1/(r0 |λ|) that connect photon orbit instability to observable shadow properties.

If this is right

  • Larger values of A decrease the instability of photon trajectories around the black hole.
  • The angular size of the black hole shadow grows as A increases.
  • Classical Solar System tests set very tight upper bounds on A in the weak-field limit.
  • EHT images of M87* and Sgr A* provide additional constraints that keep A close to zero.

Where Pith is reading between the lines

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

  • Future high-resolution black hole imaging could detect small deviations in shadow size if A is nonzero but still small.
  • The same regularization approach might produce observable effects in other strong-gravity systems like neutron stars.
  • These relations between Lyapunov exponents and shadow sizes could serve as a general test for modified black hole metrics.

Load-bearing premise

The spacetime is described by the specific asymptotically flat regular black hole metric sourced by a phantom scalar field with charge A that continuously deforms the Schwarzschild geometry.

What would settle it

An observation of the shadow angular size for a known black hole mass and distance that fails to satisfy the relation α_sh = 1/(r0 |λ|) for the corresponding A, or Solar System data that permits large A without violating the weak-field limits.

Figures

Figures reproduced from arXiv: 2510.18647 by Eleftherios Papantonopoulos, Marco Olivares, P. A. Gonz\'alez, Yerko V\'asquez.

Figure 1
Figure 1. Figure 1: FIG. 1: Plot of the lapse function [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Plot of the effective potential of photons. Here we [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Plot of the radial acceleration for massless particles. [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: The capture zone, trajectories can plunge into the [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Polar plot for deflection of light with [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Polar plot for second kind trajectories with [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: The critical trajectories plotted for [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Variation of the coordinate time [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: The Lyapunov exponent [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Angular diameter [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
read the original abstract

We study the geodesic structure and observable properties of asymptotically flat regular black holes sourced by a phantom scalar field characterized by a scalar charge $A$. This parameter removes the central singularity and continuously deforms the Schwarzschild geometry. The equations of motion for test particles and photons are derived, and the resulting null geodesics are analyzed, including the deflection of light, gravitational time delay, and redshift, in order to constrain $A$ using classical Solar System tests. These observations impose stringent limits on the scalar charge, confirming that $A$ must remain extremely small in the weak-field regime to ensure full consistency with general relativity. In the strong-field regime, we compute the Lyapunov exponent $\lambda$ associated with the photon sphere and establish its exact relations with the critical impact parameter $\mathcal{B}_u$ and the angular size of the shadow $\alpha_{\mathrm{sh}}$, given by $\mathcal{B}_u = 1/|\lambda|$ and $\alpha_{\mathrm{sh}} = 1/(r_{0}|\lambda|)$. These correspondences reveal that the dynamical instability of null circular orbits governs the optical appearance of the black hole. Our results show that increasing $A$ reduces the instability of photon trajectories and enlarges the angular size of the shadow, indicating that the regularization scale leaves a distinct observational imprint on the geometry of regular black holes. In addition, constraints derived from Event Horizon Telescope observations of M87* and Sgr A* further restrict the allowed range of the scalar charge, reinforcing the consistency of the model with current astrophysical observations.

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 / 2 minor

Summary. The paper examines the geodesic structure of asymptotically flat regular black holes sourced by a phantom scalar field with scalar charge A. It derives the equations of motion for test particles and photons, analyzes null geodesics for light deflection, gravitational time delay, and redshift to constrain A from Solar System tests. In the strong-field regime, it computes the Lyapunov exponent λ of the photon sphere and claims exact relations B_u = 1/|λ| and α_sh = 1/(r0 |λ|), showing that increasing A reduces photon trajectory instability and enlarges the shadow size. Additional constraints are obtained from EHT observations of M87* and Sgr A*.

Significance. If the exact relations between the Lyapunov exponent and the critical impact parameter and shadow size hold for the deformed metric, this work establishes a direct connection between the scalar charge parameter and observable black hole properties. This could provide new ways to test regular black hole models against observations. The use of both weak-field Solar System tests and strong-field EHT data is a strength, offering multi-scale constraints on A.

major comments (1)
  1. [photon sphere and Lyapunov exponent analysis] The assertion that B_u = 1/|λ| and α_sh = 1/(r0 |λ|) are exact consequences of the null geodesic equations (as stated in the abstract and derived in the photon-sphere analysis). For the standard Schwarzschild metric, this holds because the effective potential V(r) = f(r) L^2/r^2 with f(r) = 1-2M/r leads to λ = 1/B_u exactly at the unstable photon orbit. However, the regularized metric with phantom scalar introduces A-dependent modifications to f(r) and its derivatives. The manuscript must demonstrate explicitly in the derivation of λ from V''(r_ph) that these modifications do not alter the relation, or clarify if it is an approximation. This is load-bearing for the claim that increasing A reduces instability and enlarges the shadow, as well as for the subsequent EHT constraints.
minor comments (2)
  1. [Abstract] The abstract states the relations are 'exact'; this should be qualified or cross-referenced to the explicit verification in the main text if A-dependent terms are present.
  2. [model description] Provide the explicit form of the metric function f(r) including the dependence on A early in the model section to allow readers to reproduce the geodesic and Lyapunov calculations independently.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the thorough review and valuable feedback on our manuscript. We appreciate the recognition of the multi-scale constraints on the scalar charge A. Below, we provide a point-by-point response to the major comment, and we will incorporate the necessary clarifications in the revised version.

read point-by-point responses

  1. Referee: The assertion that B_u = 1/|λ| and α_sh = 1/(r0 |λ|) are exact consequences of the null geodesic equations (as stated in the abstract and derived in the photon-sphere analysis). For the standard Schwarzschild metric, this holds because the effective potential V(r) = f(r) L^2/r^2 with f(r) = 1-2M/r leads to λ = 1/B_u exactly at the unstable photon orbit. However, the regularized metric with phantom scalar introduces A-dependent modifications to f(r) and its derivatives. The manuscript must demonstrate explicitly in the derivation of λ from V''(r_ph) that these modifications do not alter the relation, or clarify if it is an approximation. This is load-bearing for the claim that increasing A reduces instability and enlarges the shadow, as well as for the subsequent EHT constraints.

    Authors: We agree with the referee that an explicit demonstration is required to confirm the exactness of the relations for the A-dependent metric. In our derivation, the effective potential is V(r) = f(r) L²/r², where f(r) incorporates the scalar charge A through the specific solution for the phantom scalar field. We compute λ as the growth rate with respect to coordinate time, λ = sqrt(-V''(r_ph)/2) * (f(r_ph)/E). Substituting the expressions and using the photon sphere condition f'(r_ph) = 2f(r_ph)/r_ph, we find that the A-dependent contributions to f''(r_ph) are such that the combination yields U''(r_ph) = -2/r_ph⁴ exactly for our model, resulting in λ = 1/B_u. We will expand the photon-sphere analysis section to include this full algebraic verification, showing the cancellation of terms involving A. This will strengthen the subsequent claims and EHT constraints without altering the results. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper derives geodesic equations for the A-deformed regular black hole metric and computes the Lyapunov exponent λ at the photon sphere from the second derivative of the effective potential. It states the relations B_u = 1/|λ| and α_sh = 1/(r0 |λ|) as exact consequences of that analysis for null circular orbits. These are then used to interpret how increasing A affects instability and shadow size, with all quantitative constraints on A obtained from independent external data (Solar System tests, EHT images of M87* and Sgr A*). No equation reduces to a prior fit or self-definition by construction; the central mapping from regularization parameter A to observables retains independent content from the metric deformation and the observational benchmarks. The derivation is self-contained against external falsifiability criteria.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The model introduces the phantom scalar field and its charge A as the mechanism that regularizes the geometry; A functions as a free parameter whose value is bounded by data rather than derived from first principles.

free parameters (1)
  • scalar charge A
    Introduced to deform Schwarzschild geometry and remove the singularity; its magnitude is constrained to be extremely small by observations.
axioms (1)
  • domain assumption The spacetime is an asymptotically flat regular black hole sourced by a phantom scalar field with charge A
    This is the defining premise of the geometry studied throughout the abstract.
invented entities (1)
  • phantom scalar field with charge A no independent evidence
    purpose: To remove the central singularity and continuously deform the Schwarzschild solution
    Postulated within the model; no independent falsifiable prediction outside the consistency with observations is supplied in the abstract.

pith-pipeline@v0.9.0 · 5826 in / 1553 out tokens · 45398 ms · 2026-05-21T20:07:12.228928+00:00 · methodology

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Forward citations

Cited by 2 Pith papers

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

  1. Anomalous Decay Rate and Greybody Factors for Regular Black Holes with Scalar Hair

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    Regular black holes with scalar hair exhibit anomalous decay rates for massive scalar perturbations, with longest-lived modes switching to lower angular momentum above a critical mass.

  2. Time Like Geodesics of Regular Black Holes with Scalar Hair

    gr-qc 2026-04 unverdicted novelty 3.0

    Timelike geodesics around asymptotically flat regular black holes with phantom scalar hair show shifted circular orbits, ISCO locations, and perihelion precession corrections proportional to the scalar charge A that c...

Reference graph

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