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arxiv: 2604.10572 · v1 · submitted 2026-04-12 · 🌀 gr-qc

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

Sohan Kumar Jha This is my paper

Pith reviewed 2026-05-10 15:58 UTC · model grok-4.3

classification 🌀 gr-qc
keywords strong gravitational lensingquasiperiodic oscillationsLorentz symmetry violationcharged black holesblack hole shadowsM87*Sgr A*microquasars
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The pith

Electric charge and Lorentz symmetry violation can cancel in black hole lensing observables.

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

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.

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.

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.

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.

Where Pith is reading between the lines

These are 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.

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.

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.

Figures

Figures reproduced from arXiv: 2604.10572 by Sohan Kumar Jha.

Figure 1
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. 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. Figure 3: FIG. 3: Variation of shadow radius with charge [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Variation of deflection angle with impact parameter [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Variation of relative magnification with charge [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Variation of angular separation with charge [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Variation of angular radius [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Variation of angular diameter [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Variation of angular diameter [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Variation of ISCO radius with charge [PITH_FULL_IMAGE:figures/full_fig_p011_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: Variation of upper frequency [PITH_FULL_IMAGE:figures/full_fig_p013_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: Variation of lower frequency [PITH_FULL_IMAGE:figures/full_fig_p013_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13: Variation of lower and upper frequencies [PITH_FULL_IMAGE:figures/full_fig_p014_13.png] view at source ↗
read the original 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.

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

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

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.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 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
axioms (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) no independent evidence
    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

pith-pipeline@v0.9.0 · 5513 in / 1777 out tokens · 43420 ms · 2026-05-10T15:58:58.248963+00:00 · methodology

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Reference graph

Works this paper leans on

99 extracted references · 99 canonical work pages

  1. [1]

    Kostelecky and S

    V.A. Kostelecky and S. Samuel, Spontaneous breaking of Lorentz symmetry in string theory, Phys. Rev. D 39 (1989) 683

    work page 1989

  2. [2]

    Alfaro et al., Loop quantum gravity and light propagation, Phys

    J. Alfaro et al., Loop quantum gravity and light propagation, Phys. Rev. D 65 (2002) 103509

    work page 2002

  3. [3]

    Horava, Quantum Gravity at a Lifshitz Point, Phys

    P. Horava, Quantum Gravity at a Lifshitz Point, Phys. Rev. D 79 (2009) 084008

    work page 2009

  4. [4]

    Carroll et al., Noncommutative field theory and Lorentz violation, Phys

    S.M. Carroll et al., Noncommutative field theory and Lorentz violation, Phys. Rev. Lett. 87 (2001) 141601

    work page 2001

  5. [5]

    Jacobson and D

    T. Jacobson and D. Mattingly, Gravity with a dynamical preferred frame, Phys. Rev. D 64 (2001) 024028

    work page 2001

  6. [6]

    Dubovsky et al., Massive graviton as a testable cold dark matter candidate, Phys

    S.L. Dubovsky et al., Massive graviton as a testable cold dark matter candidate, Phys. Rev. Lett. 94 (2005) 181102

    work page 2005

  7. [7]

    Bengochea and R

    G.R. Bengochea and R. Ferraro, Dark torsion as the cosmic speed-up, Phys. Rev. 79 (2009) 124019

    work page 2009

  8. [8]

    Cohen and S.L

    A.G. Cohen and S.L. Glashow, Very special relativity, Phys. Rev. Lett. 97 (2006) 021601

    work page 2006

  9. [9]

    Kostelecky, Gravity, Lorentz violation, and the standard model, Phys

    V.A. Kostelecky, Gravity, Lorentz violation, and the standard model, Phys. Rev. D 69 (2004) 105009

    work page 2004

  10. [10]

    Kostelecky and S

    V.A. Kostelecky and S. Samuel, Gravitational Phenomenology in Higher Dimensional Theories and Strings, Phys. Rev. D 40 (1989) 1886

    work page 1989

  11. [11]

    Kostelecky and S

    V.A. Kostelecky and S. Samuel, Phenomenological gravitational constraints on strings and higher dimensional theories, Phys. Rev. Lett. 63 (1989) 224

    work page 1989

  12. [12]

    Bailey and V.A

    Q.G. Bailey and V.A. Kostelecky, Signals for Lorentz violation in post-Newtonian gravity, Phys. Rev. D 74 (2006) 045001

    work page 2006

  13. [13]

    Bluhm et al., Constraints and stability in vector theories with spontaneous Lorentz violation, Phys

    R. Bluhm et al., Constraints and stability in vector theories with spontaneous Lorentz violation, Phys. Rev. D 77 (2008) 125007

    work page 2008

  14. [14]

    Altschul et al., Lorentz violation with an antisymmetric tensor, Phys

    B. Altschul et al., Lorentz violation with an antisymmetric tensor, Phys. Rev. D 81, 065028 (2010)

    work page 2010

  15. [15]

    Casana et al., An exact Schwarzschild-like solution in a bumblebee gravity model, Phys

    R. Casana et al., An exact Schwarzschild-like solution in a bumblebee gravity model, Phys. Rev. D 97, 104001 (2018)

    work page 2018

  16. [16]

    Lessa et al., Modified black hole solution with a background Kalb-Ramond field, Eur

    L.A. Lessa et al., Modified black hole solution with a background Kalb-Ramond field, Eur. Phys. J. C 80 (2020) 335

    work page 2020

  17. [17]

    Ke Yang et al., Static and spherically symmetric black holes in gravity with a background Kalb-Ramond field: Phys. Rev. D 108, 124004 (2023)

    work page 2023

  18. [18]

    Ovg¨ un et al., Gravitational lensing under the effect of Weyl and bumblebee gravities: Applications of Gauss-Bonnet theorem, Annals Phys

    A. Ovg¨ un et al., Gravitational lensing under the effect of Weyl and bumblebee gravities: Applications of Gauss-Bonnet theorem, Annals Phys. 399 (2018) 193

    work page 2018

  19. [19]

    Kanzi and ˙I

    S. Kanzi and ˙I. Sakallı, GUP modified hawking radiation in bumblebee gravity, Nucl. Phys. B 946 (2019) 114703

    work page 2019

  20. [20]

    Yang et al., Effects of Lorentz breaking on the accretion onto a Schwarzschild-like black hole, Commun

    R.-J. Yang et al., Effects of Lorentz breaking on the accretion onto a Schwarzschild-like black hole, Commun. Theor. Phys. 71 (2019) 568

    work page 2019

  21. [21]

    Cai and R.-J

    Z. Cai and R.-J. Yang, Accretion of the Vlasov gas onto a Schwarzschild-like black hole, Phys. Dark Univ. 42 (2023) 101292

    work page 2023

  22. [22]

    Oliveira et al., Quasinormal frequencies for a black hole in a bumblebee gravity, EPL 135 (2021) 10003

    R. Oliveira et al., Quasinormal frequencies for a black hole in a bumblebee gravity, EPL 135 (2021) 10003

    work page 2021

  23. [23]

    Maluf and J.C.S

    R.V. Maluf and J.C.S. Neves, Black holes with a cosmological constant in bumblebee gravity, Phys. Rev. D 103 (2021) 044002

    work page 2021

  24. [24]

    Xu et al., Static spherical vacuum solutions in the bumblebee gravity model, Phys

    R. Xu et al., Static spherical vacuum solutions in the bumblebee gravity model, Phys. Rev. D 107 (2023) 024011

    work page 2023

  25. [25]

    Mai et al., Extended thermodynamics of the bumblebee black holes, Phys

    Z.-F. Mai et al., Extended thermodynamics of the bumblebee black holes, Phys. Rev. D 108 (2023) 024004

    work page 2023

  26. [26]

    Xu et al., Bumblebee black holes in light of event horizon telescope observations, Astrophys

    R. Xu et al., Bumblebee black holes in light of event horizon telescope observations, Astrophys. J. 945 (2023) 148

    work page 2023

  27. [27]

    Liang et al., Probing vector hair of black holes with extreme-mass-ratio inspirals, Phys

    D. Liang et al., Probing vector hair of black holes with extreme-mass-ratio inspirals, Phys. Rev. D 107 (2023) 044053

    work page 2023

  28. [28]

    Ding et al., Exact Kerr-like solution and its shadow in a gravity model with spontaneous Lorentz symmetry breaking, Eur

    C. Ding et al., Exact Kerr-like solution and its shadow in a gravity model with spontaneous Lorentz symmetry breaking, Eur. Phys. J. C 80 (2020) 178

    work page 2020

  29. [29]

    Ding and X

    C. Ding and X. Chen, Slowly rotating Einstein-bumblebee black hole solution and its greybody factor in a Lorentz violation model, Chin. Phys. C 45 (2021) 025106

    work page 2021

  30. [30]

    Wang and S.-W

    H.-M. Wang and S.-W. Wei, Shadow cast by Kerr-like black hole in the presence of plasma in Einstein-bumblebee gravity, Eur. Phys. J. Plus 137 (2022) 571

    work page 2022

  31. [31]

    Liu et al., Thin accretion disk around a rotating Kerr-like black hole in Einstein-bumblebee gravity model, [arXiv:1910.13259]

    C. Liu et al., Thin accretion disk around a rotating Kerr-like black hole in Einstein-bumblebee gravity model, [arXiv:1910.13259]

    work page arXiv 1910

  32. [32]

    Liu et al., QNMs of slowly rotating Einstein-Bumblebee black hole, Eur

    W. Liu et al., QNMs of slowly rotating Einstein-Bumblebee black hole, Eur. Phys. J. C 83 (2023) 83

    work page 2023

  33. [33]

    Wang et al., Constraint on parameters of a rotating black hole in Einstein-bumblebee theory by quasi-periodic oscilla- tions, Eur

    Z. Wang et al., Constraint on parameters of a rotating black hole in Einstein-bumblebee theory by quasi-periodic oscilla- tions, Eur. Phys. J. C 82 (2022) 528

    work page 2022

  34. [34]

    Ding et al., Rotating BTZ-like black hole and central charges in Einstein-bumblebee gravity, Eur

    C. Ding et al., Rotating BTZ-like black hole and central charges in Einstein-bumblebee gravity, Eur. Phys. J. C 83 (2023) 573

    work page 2023

  35. [35]

    Chen et al., Quasinormal modes of a scalar perturbation around a rotating BTZ-like black hole in Einstein-bumblebee gravity, Phys

    C. Chen et al., Quasinormal modes of a scalar perturbation around a rotating BTZ-like black hole in Einstein-bumblebee gravity, Phys. Lett. B 846 (2023) 138186

    work page 2023

  36. [36]

    G¨ ull¨ u and A.¨Ovg¨ un, Schwarzschild-like black hole with a topological defect in bumblebee gravity, Annals Phys

    ˙I. G¨ ull¨ u and A.¨Ovg¨ un, Schwarzschild-like black hole with a topological defect in bumblebee gravity, Annals Phys. 436 (2022) 168721

    work page 2022

  37. [37]

    Zhang et al., Quasinormal modes and late time tails of perturbation fields on a Schwarzschild-like black hole with a global monopole in the Einstein-bumblebee theory, Sci

    X. Zhang et al., Quasinormal modes and late time tails of perturbation fields on a Schwarzschild-like black hole with a global monopole in the Einstein-bumblebee theory, Sci. China Phys. Mech. Astron. 66 (2023) 100411

    work page 2023

  38. [38]

    Lin et al., Quasinormal modes of the spherical bumblebee black holes with a global monopole, Eur

    R.-H. Lin et al., Quasinormal modes of the spherical bumblebee black holes with a global monopole, Eur. Phys. J. C 83 (2023) 720

    work page 2023

  39. [39]

    Ding et al., Einstein-Gauss-Bonnet gravity coupled to bumblebee field in four dimensional spacetime, Nucl

    C. Ding et al., Einstein-Gauss-Bonnet gravity coupled to bumblebee field in four dimensional spacetime, Nucl. Phys. B 975 (2022) 115688. 16

    work page 2022

  40. [40]

    Jha and A

    S.K. Jha and A. Rahaman, Bumblebee gravity with a Kerr-Sen-like solution and its Shadow, Eur. Phys. J. C 81 (2021) 345

    work page 2021

  41. [41]

    Ding et al., High dimensional AdS-like black hole and phase transition in Einstein-bumblebee gravity, Chin

    C. Ding et al., High dimensional AdS-like black hole and phase transition in Einstein-bumblebee gravity, Chin. Phys. C 47 (2023) 045102

    work page 2023

  42. [42]

    ¨Ovg¨ un et al., Exact traversable wormhole solution in bumblebee gravity, Phys

    A. ¨Ovg¨ un et al., Exact traversable wormhole solution in bumblebee gravity, Phys. Rev. D 99 (2019) 024042

    work page 2019

  43. [43]

    Liang et al., Polarizations of Gravitational Waves in the Bumblebee Gravity Model, Phys

    D. Liang et al., Polarizations of Gravitational Waves in the Bumblebee Gravity Model, Phys. Rev. D 106 (2022) 124019

    work page 2022

  44. [44]

    Amarilo et al., Gravitational waves effects in a Lorentz-violating scenario, [arXiv:2307.10937]

    K.M. Amarilo et al., Gravitational waves effects in a Lorentz-violating scenario, [arXiv:2307.10937]

    work page arXiv

  45. [45]

    Kalb and P

    M. Kalb and P. Ramond, Classical direct interstring action, Phys. Rev. D 9 (1974) 2273

    work page 1974

  46. [46]

    Kao et al., Induced Einstein-Kalb-Ramond theory and the black hole, Phys

    W.F. Kao et al., Induced Einstein-Kalb-Ramond theory and the black hole, Phys. Rev. D 53 (1996) 2244

    work page 1996

  47. [47]

    Kar et al., Static spherisymmetric solutions, gravitational lensing and perihelion precession in Einstein-Kalb-Ramond theory, Phys

    S. Kar et al., Static spherisymmetric solutions, gravitational lensing and perihelion precession in Einstein-Kalb-Ramond theory, Phys. Rev. D 67 (2003) 044005

    work page 2003

  48. [48]

    Chakraborty and S

    S. Chakraborty and S. SenGupta, Strong gravitational lensing — a probe for extra dimensions and Kalb-Ramond field, JCAP 07 (2017) 045

    work page 2017

  49. [49]

    Nair and A.M

    K.K. Nair and A.M. Thomas, Kalb-Ramond field-induced cosmological bounce in generalized teleparallel gravity, Phys. Rev. D 105 (2022) 103505

    work page 2022

  50. [50]

    Fu et al., Q-form fields on p-branes, JHEP 10 (2012) 060

    C.-E. Fu et al., Q-form fields on p-branes, JHEP 10 (2012) 060

    work page 2012

  51. [51]

    Chakraborty and S

    S. Chakraborty and S. SenGupta, Solutions on a brane in a bulk spacetime with Kalb-Ramond field, Annals Phys. 367 (2016) 258

    work page 2016

  52. [52]

    Atamurotov et al., Particle dynamics and gravitational weak lensing around black hole in the Kalb-Ramond gravity, Eur

    F. Atamurotov et al., Particle dynamics and gravitational weak lensing around black hole in the Kalb-Ramond gravity, Eur. Phys. J. C 82 (2022) 659

    work page 2022

  53. [53]

    Kumar et al., Gravitational deflection of light and shadow cast by rotating Kalb-Ramond black holes, Phys

    R. Kumar et al., Gravitational deflection of light and shadow cast by rotating Kalb-Ramond black holes, Phys. Rev. D 101 (2020) 104001

    work page 2020

  54. [54]

    Lessa et al., Traversable wormhole solution with a background Kalb-Ramond field, Annals Phys

    L.A. Lessa et al., Traversable wormhole solution with a background Kalb-Ramond field, Annals Phys. 433 (2021) 168604

    work page 2021

  55. [55]

    Jha, Black hole surrounded by perfect fluid dark matter with a background Kalb-Ramond field, JCAP 09(2025)069

    S.Kr. Jha, Black hole surrounded by perfect fluid dark matter with a background Kalb-Ramond field, JCAP 09(2025)069

    work page 2025

  56. [56]

    Maluf and C.R

    R.V. Maluf and C.R. Muniz, Exact solution for a traversable wormhole in a curvature-coupled antisymmetric background field, Eur. Phys. J. C 82 (2022) 445

    work page 2022

  57. [57]

    Maluf and J.C.S

    R.V. Maluf and J.C.S. Neves, Bianchi type I cosmology with a Kalb-Ramond background field, Eur. Phys. J. C 82 (2022) 135

    work page 2022

  58. [58]

    Z. Q. Duan et al., Electrically charged black holes in gravity with a background Kalb-Ramond field, Eur. Phys. J. C 84 (2024), 798

    work page 2024

  59. [59]

    Darwin, The gravity field of a particle, Proc

    C. Darwin, The gravity field of a particle, Proc. R. Soc. A 249 (1959) 180

    work page 1959

  60. [60]

    K. S. Virbhadra and G. F. R. Ellis, Schwarzschild black hole lensing, Phys. Rev. D 62 (2000) 084003

    work page 2000

  61. [61]

    Bozza, S

    V. Bozza, S. Capozziello, G. Iovane G. Scarpetta, Strong Field Limit of Black Hole Gravitational Lensing, Gen. Rel. Grav. 33 (2001) 1535

    work page 2001

  62. [62]

    Bozza, Gravitational lensing in the strong field limit, Phys

    V. Bozza, Gravitational lensing in the strong field limit, Phys. Rev. D 66 (2002) 103001

    work page 2002

  63. [63]

    Bozza and L

    V. Bozza and L. Mancini, Time Delay in Black Hole Gravitational Lensing as a Distance Estimator, Gen. Rel. Grav. 36 (2004) 435

    work page 2004

  64. [64]

    E. F. Eiroa and D. F. Torres, Strong field limit analysis of gravitational retro-lensing, Phys. Rev. D 69 (2004) 063004

    work page 2004

  65. [65]

    Whisker, Strong gravitational lensing by braneworld black holes, Phys

    R. Whisker, Strong gravitational lensing by braneworld black holes, Phys. Rev. D 71 (2005) 064004

    work page 2005

  66. [66]

    E. F. Eiroa, Braneworld black hole gravitational lens: Strong field limit analysis, Phys. Rev. D 71 (2005) 083010

    work page 2005

  67. [67]

    Bhadra, Gravitational lensing by a charged black hole of string theory, Phys

    A. Bhadra, Gravitational lensing by a charged black hole of string theory, Phys. Rev. D 67 (2003) 103009

    work page 2003

  68. [68]

    Shaikh et al., Analytical approach to strong gravitational lensing from ultracompact objects, Phys

    R. Shaikh et al., Analytical approach to strong gravitational lensing from ultracompact objects, Phys. Rev. D 99 (2019) 104040

    work page 2019

  69. [69]

    E. F. Eiroa and C. M. Sendra, Gravitational lensing by a regular black hole, Class. Quant. Grav. 28 (2011) 085008

    work page 2011

  70. [70]

    Kumar et al., Testing Strong Gravitational Lensing Effects of Supermassive Compact Objects with Regular Spacetimes, Astrophys

    J. Kumar et al., Testing Strong Gravitational Lensing Effects of Supermassive Compact Objects with Regular Spacetimes, Astrophys. J. 938 (2022) 104

    work page 2022

  71. [71]

    S. K. Jha and A. Rahaman, Strong gravitational lensing in hairy Schwarzschild background, Eur. Phys. J. Plus 138 (2023) 86

    work page 2023

  72. [72]

    Feleppa et al., Strong deflection limit analysis of black hole lensing in inhomogeneous plasma, Phys

    F. Feleppa et al., Strong deflection limit analysis of black hole lensing in inhomogeneous plasma, Phys. Rev. D 110 (2024) 064031

    work page 2024

  73. [73]

    Vachher et al., Probing dark matter via strong gravitational lensing by black holes, Phys

    A. Vachher et al., Probing dark matter via strong gravitational lensing by black holes, Phys. Dark Univ. 44 (2024) 101493

    work page 2024

  74. [74]

    U.Molla et al., Strong gravitational lensing bySgrA ∗ andM87 ∗ black holes embedded in dark matter halo exhibiting string cloud and quintessential field, Eur

    N. U.Molla et al., Strong gravitational lensing bySgrA ∗ andM87 ∗ black holes embedded in dark matter halo exhibiting string cloud and quintessential field, Eur. Phys. J. C 84 (2024) 574

    work page 2024

  75. [75]

    Chen-Kai Qiao and Mi Zhou, Gravitational Lensing of Schwarzschild and Charged Black Holes Immersed in Perfect Fluid Dark Matter Halo, JCAP 12 (2023) 005

    work page 2023

  76. [76]

    E. L. B. Junior et al., Gravitational lensing of a Schwarzschild-like black hole in Kalb-Ramond gravity, Phys. Rev. D 110 (2024) 024077

    work page 2024

  77. [77]

    Akiyama et al., First M87 Event Horizon Telescope Results

    K. Akiyama et al., First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole, Astrophys. J. 875 (2019) L1

    work page 2019

  78. [78]

    First SagittariusA ∗ Event Horizon Telescope Results

    Kazunori Akiyama et al. First SagittariusA ∗ Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole in the Center of the Milky Way. Astrophys. J. Lett., 930(2):L12, 2022

    work page 2022

  79. [79]

    Gillessen et al., AN UPDATE ON MONITORING STELLAR ORBITS IN THE GALACTIC CENTER, Astrophys

    S. Gillessen et al., AN UPDATE ON MONITORING STELLAR ORBITS IN THE GALACTIC CENTER, Astrophys. J. 837 (2017) 30. 17

    work page 2017

  80. [80]

    M. E. Beer and P. Podsiadlowski, The quiescent light curve and evolutionary state of gro J1655-40, Mon. Not. Roy. Astron. Soc. 331 (2002) 351

    work page 2002

Showing first 80 references.