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Strong gravitational lensing and Quasiperiodic oscillations as a probe for an electrically charged Lorentz symmetry-violating black hole

T0 review · 3 major / 2 minor · reviewed 2026-05-10 · grok-4.3

Challenge this review Download PDF Read on arXiv ↗

Pith's one-line read Electric charge and Lorentz symmetry violation can cancel in black hole lensing observables.

desk verdict The paper applies standard lensing and QPO formulas to a charged Lorentz-violating black hole, reports a cancellation effect, and gives bounds from M87*, Sgr A*, and microquasar data. read the letter →

arxiv 2604.10572 v1 submitted 2026-04-12 gr-qc

Sohan Kumar Jha This is my paper · ORCID
classification gr-qc
keywords stronggravitationallensingquasiperiodicoscillationsLorentzsymmetryviolationchargedblackholesholeshadowsM87*SgrA*microquasars
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

The paper examines an electrically charged black hole that also breaks Lorentz symmetry and calculates how these two modifications together change light deflection in strong gravitational lensing and the frequencies of quasiperiodic oscillations near the hole. It shows that the two effects can offset each other exactly, so that certain lensing quantities remain identical to those of an ordinary Schwarzschild black hole. Shadow size measurements of the supermassive black holes M87* and Sgr A* then set limits on the Lorentz violation strength while leaving the charge parameter free. Data on quasiperiodic oscillations from the microquasars GRO J1655-40 and XTE J1550-564 yield bounds on both parameters. The study illustrates how existing astrophysical observations can test combined extensions of general relativity in the strong-field regime.

What carries the argument

The spacetime metric of the electrically charged Lorentz symmetry-violating black hole, used to compute light deflection angles in strong lensing and orbital frequencies that determine quasiperiodic oscillation periods.

What would settle it

A measured shadow angular size for M87* or Sgr A* that falls outside the range allowed by the model's bounds on the Lorentz violation parameter for every possible charge value, or a set of quasiperiodic oscillation frequencies in GRO J1655-40 or XTE J1550-564 inconsistent with the same bounds, would falsify the model.

Watch

Extended reading notes

Core claim

For the electrically charged Lorentz symmetry-violating black hole, the competing influences of electric charge and the Lorentz violation parameter cancel in selected strong-lensing quantities, reproducing the corresponding Schwarzschild values, while shadow angular sizes of M87* and Sgr A* constrain the violation parameter and quasiperiodic oscillation data from two microquasars constrain both parameters.

Load-bearing premise

The specific metric for the electrically charged Lorentz symmetry-violating black hole accurately describes the geometry, and the usual formulas for deflection angles and quasiperiodic oscillation frequencies remain valid when both charge and Lorentz symmetry breaking are present.

Editorial extensions

If this is right

  • Certain combinations of charge and Lorentz violation parameter produce identical strong-lensing deflection angles and shadow sizes to a Schwarzschild black hole.
  • The Lorentz violation parameter receives upper and lower bounds from the observed shadow angular sizes of M87* and Sgr A*.
  • The electric charge parameter cannot be constrained using only those shadow observations.
  • Quasiperiodic oscillation frequencies observed in GRO J1655-40 and XTE J1550-564 supply joint bounds on both the charge and the Lorentz violation parameters.

Reading between the lines

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

  • The cancellation mechanism might appear in additional strong-gravity observables such as the ringdown spectrum of gravitational waves from mergers.
  • Higher-resolution future shadow images could test whether the lensing and QPO constraints remain consistent with each other.
  • The model could be compared with other modified black hole solutions to determine whether the exact cancellation is unique to the combination of charge and Lorentz violation.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 2 minor

Summary. This paper examines the effects of electric charge combined with Lorentz symmetry breaking (LSB) on strong gravitational lensing and quasiperiodic oscillations (QPOs) for a black hole spacetime. It identifies regimes where the competing effects of charge and LSB cancel, resulting in observables identical to those of a Schwarzschild black hole. Bounds on the LSB parameter are obtained from the shadow angular sizes of M87* and Sgr A*, while no bounds on charge are found from these. QPO observations from the microquasars GRO J1655-40 and XTE J1550-564 are used to place bounds on both parameters.

Significance. The reported cancellation between charge and Lorentz violation parameters, leaving Schwarzschild-like strong lensing and shadow properties, is an interesting result that could have implications for testing modified gravity models. The use of both shadow observations and QPO data to constrain the model parameters is a positive aspect, providing multiple avenues for testing. If the underlying assumptions hold, this work contributes to the growing body of literature using astrophysical observations to probe deviations from general relativity in the strong-field regime.

major comments (3)
  1. [Section on strong gravitational lensing] The claim that charge and LV effects cancel to leave the deflection angle unchanged from Schwarzschild requires explicit demonstration in the relevant equations. The paper should show the condition on the parameters for which the integral for the deflection angle reduces exactly to the Schwarzschild case.
  2. [Shadow size analysis for M87* and Sgr A*] The bounds on the LV parameter are extracted by comparing the model's predicted shadow radius to the observed values. Since the charge parameter is not constrained by these observations, the analysis should address potential degeneracies between the two parameters and justify why the LV bound is reliable despite this.
  3. [QPO frequency calculations] The extraction of bounds from QPO data in GRO J1655-40 and XTE J1550-564 relies on the standard formulas for orbital and radial epicyclic frequencies. The manuscript should verify or justify that these formulas, derived in GR, remain unmodified when applied to the QKR metric, particularly if the Lorentz violation introduces a preferred frame that could affect the geodesic motion or effective potential.
minor comments (2)
  1. [Abstract] The abbreviation 'QKR BH' is introduced without expansion; it should be defined as 'electrically charged Lorentz symmetry-violating black hole' on first use.
  2. [Throughout] Ensure consistent use of notation for the Lorentz violation parameter and electric charge across equations and text.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and insightful comments on our manuscript. We have carefully reviewed each major point and provide point-by-point responses below. We agree that several clarifications and additions will strengthen the paper and plan to incorporate them in the revised version.

read point-by-point responses

  1. Referee: [Section on strong gravitational lensing] The claim that charge and LV effects cancel to leave the deflection angle unchanged from Schwarzschild requires explicit demonstration in the relevant equations. The paper should show the condition on the parameters for which the integral for the deflection angle reduces exactly to the Schwarzschild case.

    Authors: We agree that an explicit demonstration is required for rigor. In the revised manuscript, we will add a dedicated subsection deriving the deflection angle integral for the QKR metric. We will explicitly show the condition relating the electric charge parameter q and the Lorentz violation parameter (denoted l in the paper) under which the integrand reduces identically to the Schwarzschild case, confirming the cancellation analytically. revision: yes

  2. Referee: [Shadow size analysis for M87* and Sgr A*] The bounds on the LV parameter are extracted by comparing the model's predicted shadow radius to the observed values. Since the charge parameter is not constrained by these observations, the analysis should address potential degeneracies between the two parameters and justify why the LV bound is reliable despite this.

    Authors: We acknowledge the degeneracy between charge and the LV parameter arising from the cancellation in the shadow radius. The observed shadow sizes of M87* and Sgr A* are consistent with the Schwarzschild value within uncertainties, which constrains the effective combination of q and l. We will revise the shadow analysis section to explicitly discuss this degeneracy, clarify that the reported bound on the LV parameter is obtained by marginalizing over possible charge values (or equivalently, by considering the maximum LV deviation still compatible with observations), and justify its reliability as a conservative upper limit on the LV parameter. revision: partial

  3. Referee: [QPO frequency calculations] The extraction of bounds from QPO data in GRO J1655-40 and XTE J1550-564 relies on the standard formulas for orbital and radial epicyclic frequencies. The manuscript should verify or justify that these formulas, derived in GR, remain unmodified when applied to the QKR metric, particularly if the Lorentz violation introduces a preferred frame that could affect the geodesic motion or effective potential.

    Authors: This is a valid concern given the Lorentz-violating nature of the theory. However, the QKR metric remains static and spherically symmetric, allowing the standard effective-potential method for equatorial timelike geodesics to be applied directly. The orbital frequency and radial epicyclic frequency are obtained from the second derivatives of the effective potential constructed from the metric components g_tt, g_rr, and g_phiphi. We will add an appendix deriving these frequencies explicitly from the QKR line element to demonstrate that the standard expressions hold without modification for this spacetime, while noting that the preferred frame does not alter the geodesic structure for these observables. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivations apply standard GR observables to given metric

full rationale

The paper presents a metric for the charged LV black hole, computes deflection angles and QPO frequencies via direct substitution into established GR integral and frequency formulas, identifies a numerical cancellation between charge and LV terms through explicit evaluation, and constrains parameters by comparing the resulting expressions to independent observational data on shadows and microquasar QPOs. None of these steps reduce by construction to the inputs; the cancellation is a calculational outcome, and the bounds are external empirical fits rather than self-referential predictions. No self-citations or ansatzes are invoked as load-bearing justifications in the provided chain.

Assumptions & free parameters 2 free parameters · 2 assumptions · 1 invented entities

The central claim rests on an assumed modified black-hole metric that incorporates both electric charge and a Lorentz-violating term; the two parameters of this metric are then adjusted to match external observations. No independent evidence for the metric itself is supplied.

free parameters (2)
  • Lorentz violation parameter
    Fitted to shadow angular sizes and QPO frequencies to produce the reported bounds
  • electric charge
    Parameter of the black-hole model; bounded only by QPO data
assumptions (2)
  • domain assumption The spacetime geometry is described by a specific charged Lorentz-violating black-hole metric
    Invoked at the outset to define the background for all lensing and QPO calculations
  • domain assumption Standard general-relativity expressions for light deflection and orbital frequencies continue to hold after the metric is modified
    Used without further justification to compute the observables
invented entities (1)
  • Electrically charged Lorentz symmetry-violating black hole (QKR BH)
    purpose: To provide a spacetime that simultaneously includes electric charge and Lorentz violation
    The metric is postulated for this study; no independent observational signature outside the fitted parameters is given

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Cite this review

Pith. "Pith review of Strong gravitational lensing and Quasiperiodic oscillations as a probe for an electrically charged Lorentz symmetry-violating black hole." pith.science (2026). https://pith.science/paper/2604.10572

@misc{pith2026260410572,
  author       = {Pith},
  title        = {Pith review of: Strong gravitational lensing and Quasiperiodic oscillations as a probe for an electrically charged Lorentz symmetry-violating black hole},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2604.10572}},
  note         = {Machine review of arXiv:2604.10572}
}
abstract

This study examines the combined effect of electric charge and Lorentz symmetry breaking (LSB) on the observables of strong gravitational lensing (SGL) and the dynamics of quasiperiodic oscillations (QPOs) around an electrically charged, Lorentz symmetry-violating (LV) black hole (QKR BH). We first explore the SGL, which unravels an interesting effect that the two combined generate. We find cases where the competing effect of charge and LV cancels each other, leaving the underlying quantity unchanged from that of a \s BH. We find bounds on the LV parameter utilizing observations related to the shadow angular size of supermassive black holes (SMBHs) $M87^*$ and $SgrA^*$. No bound could be gleaned for the charge from these shadow observations. Observations of QPOs in microquasars provide an alternative method to probe our model and to extract bounds on its parameters. We use experimental data for the microquasars $GRO J1655-40$ and $XTE J1550-564$. Here we obtain bounds on both parameters. Our results provide deeper insights into the interplay between charge and LSB in the strong-gravity regime.

Figures

Figures reproduced from arXiv: 2604.10572 by the authors.

Figure 1
Figure 1. FIG. 1: Parameter space for the existence of BH. The coloured region is where we have a BH solution. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Variation of event horizon with charge [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Variation of shadow radius with charge [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Variation of deflection angle with impact parameter [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Variation of relative magnification with charge [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Variation of angular separation with charge [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Variation of angular radius [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Variation of angular diameter [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Variation of angular diameter [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10: Variation of ISCO radius with charge [PITH_FULL_IMAGE:figures/full_fig_p011_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11: Variation of upper frequency [PITH_FULL_IMAGE:figures/full_fig_p013_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12: Variation of lower frequency [PITH_FULL_IMAGE:figures/full_fig_p013_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13: Variation of lower and upper frequencies [PITH_FULL_IMAGE:figures/full_fig_p014_13.png]

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