pith. sign in

arxiv: 2605.03366 · v1 · submitted 2026-05-05 · 🌀 gr-qc · quant-ph

Scalar bosonic oscillator fields in LV-wormholes

Omar Mustafa , Abdullah Guvendi This is my paper

Pith reviewed 2026-05-07 14:35 UTC · model grok-4.3

classification 🌀 gr-qc quant-ph
keywords Lorentz-violating wormholesscalar bosonic fieldsKlein-Gordon oscillatorconfluent Heun equationdiscrete energy spectrumcurvature effectsparticle-antiparticle symmetry
0
0 comments X

The pith

Scalar bosonic fields in Lorentz-violating wormholes reduce to a confluent Heun equation with discrete energies set by curvature and violation strength.

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

The paper investigates scalar bosonic oscillator fields in a (3+1)-dimensional Lorentz-violating wormhole spacetime that features a smooth minimal-radius throat. A nonminimally coupled vector background with the ansatz proportional to the radial function generates an effective Klein-Gordon oscillator interaction directly from the geometry. This produces a regular effective potential without singularities at the throat, and the radial spectral problem reduces to a confluent Heun equation. The resulting conditionally exact solutions yield a discrete energy spectrum governed by the wormhole curvature, the Lorentz-violation parameter, and the oscillator frequency, while preserving a relativistic particle-antiparticle symmetry deformed by curvature.

Core claim

In the Lorentz-violating wormhole geometry the Klein-Gordon dynamics for scalar bosonic oscillator fields, under the nonminimal vector coupling with ansatz F_t(x) equal to Omega times r(x), produce an effective potential that stays finite and regular across the minimal throat. The radial equation reduces to a confluent Heun structure whose solutions are conditionally exact and deliver a discrete energy spectrum controlled by curvature, Lorentz-violation strength, and oscillator frequency. The eigenvalue structure exhibits relativistic particle-antiparticle symmetry together with curvature-induced deformation and parameter-dependent confinement.

What carries the argument

The nonminimally coupled vector background ansatz F_t(x) = Omega r(x) that encodes an effective Klein-Gordon oscillator interaction intrinsically within the wormhole geometry.

Load-bearing premise

The vector background is taken proportional to the radial function r(x) specifically to produce the oscillator term from the wormhole geometry itself.

What would settle it

Numerical integration of the radial Klein-Gordon equation for chosen values of curvature and Lorentz-violation strength, followed by direct comparison of the resulting eigenvalues against the spectrum obtained from the confluent Heun equation.

Figures

Figures reproduced from arXiv: 2605.03366 by Abdullah Guvendi, Omar Mustafa.

Figure 1
Figure 1. Figure 1: FIG. 1 view at source ↗
Figure 2
Figure 2. Figure 2: reveals how spacetime curvature, Lorentz symmetry breaking, and relativistic confinement collectively determine the energy spectrum of the KG oscillator fields in the LV wormhole background. In the left panel of view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 view at source ↗
read the original abstract

We investigate the quantum dynamics of scalar bosonic oscillator fields propagating in a (3+1)-dimensional Lorentz-violating (LV) wormhole spacetime within a modified gravity framework. The underlying geometry, characterized by a smooth minimal-radius throat and a globally regular redshift sector, induces nontrivial curvature effects that significantly modify the spectral properties of the Klein-Gordon (KG) field. The field dynamics are formulated in the presence of a nonminimally coupled vector background of the form $\mathcal{F}_\mu=(\mathcal{F}_t(x),0,0,0)$, which, under the physically motivated ansatz $\mathcal{F}_t(x)=\Omega\, r(x)$, generates an effective KG-oscillator interaction intrinsically encoded by the wormhole geometry. The resulting effective potential is regular and finite at the throat, eliminating centrifugal singularities and ensuring globally well-defined propagation across the minimal-radius region. The spectral problem reduces to a confluent Heun structure, leading to conditionally exact solutions and a discrete energy spectrum governed by curvature, Lorentz-violation strength, and oscillator frequency. The associated eigenvalue structure exhibits a relativistic particle-antiparticle symmetry with curvature-induced deformation and parameter-dependent confinement. Our results demonstrate that LV wormhole spacetimes act as effective dispersive quantum gravitational media, in which spacetime topology and spontaneous Lorentz symmetry breaking jointly regulate confinement, spectral quantization, and the global evolution of scalar bosonic modes.

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

2 major / 0 minor

Summary. The paper studies the quantum dynamics of scalar bosonic oscillator fields in a (3+1)-dimensional Lorentz-violating wormhole spacetime in modified gravity. A nonminimally coupled vector background is introduced with the ansatz F_t(x) = Omega r(x) to generate an effective KG-oscillator interaction encoded by the geometry. The manuscript claims the resulting effective potential is regular and finite at the throat (eliminating centrifugal singularities), the spectral problem reduces to a confluent Heun equation, and this yields conditionally exact solutions with a discrete energy spectrum governed by curvature, LV strength, and oscillator frequency, exhibiting relativistic particle-antiparticle symmetry with curvature-induced deformation.

Significance. If the derivations are substantiated with explicit calculations, the work would illustrate how wormhole topology combined with spontaneous Lorentz symmetry breaking can regulate confinement and spectral quantization for bosonic modes, framing LV wormholes as dispersive quantum gravitational media. The reduction to confluent Heun functions and the reported symmetry structure would be of interest for exact solutions in curved-space QFT.

major comments (2)
  1. [Abstract] Abstract: the claim that the effective potential is regular at the throat and that the spectral problem reduces to a confluent Heun structure is asserted without the explicit metric, the derivation of the KG equation including the LV coupling term, or verification that no singular contributions remain after the ansatz is imposed. These steps are load-bearing for the assertions of globally well-defined propagation and the discrete spectrum.
  2. [Abstract] Abstract: the ansatz F_t(x) = Omega r(x) is presented as physically motivated to produce an oscillator interaction 'intrinsically encoded by the wormhole geometry,' yet no derivation is supplied showing why this linear form follows from the modified gravity equations or the LV sector rather than being selected to generate the desired harmonic term. The regularity, absence of singularities, and resulting eigenvalue quantization are therefore conditional on this choice.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. The comments highlight important points regarding the presentation of key derivations and the motivation for the vector-field ansatz. We address each major comment below and have made revisions to improve clarity and substantiate the claims without altering the core results.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that the effective potential is regular at the throat and that the spectral problem reduces to a confluent Heun structure is asserted without the explicit metric, the derivation of the KG equation including the LV coupling term, or verification that no singular contributions remain after the ansatz is imposed. These steps are load-bearing for the assertions of globally well-defined propagation and the discrete spectrum.

    Authors: We appreciate the referee's emphasis on explicit substantiation. The Lorentz-violating wormhole metric is stated explicitly in Section II. The Klein-Gordon equation with the nonminimal LV vector coupling is derived step by step in Section III, yielding the effective radial equation. In Section IV we substitute the ansatz directly into the effective potential and verify by explicit expansion that it remains finite and nonsingular at the throat (no centrifugal terms survive). The reduction to the confluent Heun equation, together with the quantization condition, is carried out in Section V. To make these load-bearing steps more visible from the abstract, we have added a short sentence referencing the relevant sections and the regularity verification. We believe this addresses the concern while preserving the abstract's summary character. revision: yes

  2. Referee: [Abstract] Abstract: the ansatz F_t(x) = Omega r(x) is presented as physically motivated to produce an oscillator interaction 'intrinsically encoded by the wormhole geometry,' yet no derivation is supplied showing why this linear form follows from the modified gravity equations or the LV sector rather than being selected to generate the desired harmonic term. The regularity, absence of singularities, and resulting eigenvalue quantization are therefore conditional on this choice.

    Authors: We agree that the ansatz is a modeling choice rather than a direct solution of the modified-gravity field equations for the vector field. It is selected because, within the LV framework, a radial vector background aligned with the wormhole redshift function naturally produces an effective harmonic term that is geometrically encoded. This is consistent with spontaneous Lorentz-symmetry breaking in a static, spherically symmetric background. We have expanded the introduction to clarify this rationale, to state explicitly that the ansatz is adopted to realize geometry-induced confinement, and to note that the reported regularity and discrete spectrum are conditional on this choice. No claim is made that the linear form is uniquely dictated by the field equations. revision: yes

Circularity Check

1 steps flagged

Ansatz F_t(x)=Ω r(x) imposed to generate effective KG-oscillator term presented as intrinsic to LV-wormhole geometry

specific steps
  1. other [Abstract]
    "which, under the physically motivated ansatz F_t(x)=Ω r(x), generates an effective KG-oscillator interaction intrinsically encoded by the wormhole geometry"

    The effective oscillator interaction and the subsequent Heun structure are generated only by imposing the linear ansatz on the vector background. The paper's phrasing that this interaction is 'intrinsically encoded by the wormhole geometry' therefore reduces to the ansatz choice itself rather than an independent consequence of the spacetime or LV sector.

full rationale

The central reduction to a confluent Heun equation and the discrete spectrum with particle-antiparticle symmetry both require an effective harmonic term in the potential. This term appears only after the vector background is restricted by the ansatz F_t(x)=Ω r(x). The abstract explicitly ties the oscillator interaction to this choice while labeling the result 'intrinsically encoded by the wormhole geometry,' so the claimed feature is conditional on an extra functional assumption rather than following from the metric or LV sector alone. No other load-bearing steps (self-citations, uniqueness theorems, or renamings) are visible in the supplied text.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on the assumed wormhole metric, the standard form of the non-minimally coupled KG equation in curved spacetime, and the specific ansatz for the LV vector field that is introduced to produce the oscillator interaction.

free parameters (2)
  • Omega
    Strength of the LV vector background, introduced via the ansatz F_t(x) = Omega r(x) to generate the effective oscillator term.
  • oscillator frequency
    Parameter controlling the strength of the oscillator interaction in the scalar field dynamics.
axioms (2)
  • domain assumption Klein-Gordon equation for a scalar field with non-minimal coupling to a background vector field in a curved spacetime.
    Standard starting point in QFT on curved backgrounds and modified gravity.
  • domain assumption Existence of a (3+1)-dimensional LV wormhole metric with smooth minimal-radius throat and globally regular redshift function.
    The geometry is postulated within the modified gravity framework.

pith-pipeline@v0.9.0 · 5540 in / 1547 out tokens · 79181 ms · 2026-05-07T14:35:53.930605+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

69 extracted references · 69 canonical work pages

  1. [1]

    Such constraints inherently restrict the admissible parameter space, leading to the characteris tic feature of conditional exact solvability [67]

    This truncation is implemented through the conditions Bn+1 ̸= 0 together with Bn+2 = 0 = Bn+3, where the parameter η represents the corresponding root for Bn+2 [66]. Such constraints inherently restrict the admissible parameter space, leading to the characteris tic feature of conditional exact solvability [67]. The nonvanishing cond ition Bn+1 ̸= 0 immedi...

    work page

  2. [2]

    vanishes, namely µ − (n + 2)(n + 1 + P ) = 0, leads to ν = − n2 − (β + γ + 5)n + (2β − α + 2γ + 6). 4 FIG. 2. Energies Enℓ , obtained from Eqs. (20) and (22), describing the KG oscilla tor in a L V wormhole background. For n = 0, m◦ = 1, and ℓ = 0 , 1, 2, 3, 4, the left panel shows the dependence of the relativistic en ergy eigenvalues on the wormhole thr...

    work page

  3. [3]

    T. W. B. Kibble, Lorentz invariance and the gravitational field J. Math. Phys. 2 (1961) 212–221

    work page 1961

  4. [4]

    Magueijo, L

    J. Magueijo, L. Smolin, Lorentz invariance with an invariant en- ergy scale, Phys. Rev. Lett. 88 (2002) 190403

    work page 2002

  5. [5]

    Kosteleck` y, Gravity, Lorentz violation, and the standard model, Phys

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

    work page 2004

  6. [6]

    Colladay, V.A

    D. Colladay, V.A. Kosteleck` y, Lorentz-violating extension of the standard model, Phys. Rev. D 58 (1998) 116002

    work page 1998

  7. [7]

    Clifton, P.G

    T. Clifton, P.G. Ferreira, A. Padilla, C. Skordis, Modified gravity and cosmology, Phys. Rep. 513 (2012) 1–189

    work page 2012

  8. [8]

    Carroll, et

    S.M. Carroll, et. al., Cosmology of generalized modified gravity models, Phys. Rev. D 71 (2005) 063513

    work page 2005

  9. [9]

    Nojiri, S.D

    S. Nojiri, S.D. Odintsov, Unified cosmic history in modified gravity: from F (R) theory to Lorentz non-invariant models , Phys. Rep. 505 (2011) 59–144

    work page 2011

  10. [10]

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

    work page 1989

  11. [11]

    Colladay, V

    D. Colladay, V. A. Kosteleck´ y, CPT violation and the Standard Model, Phys. Rev. D, 55 (1997) 6760

    work page 1997

  12. [12]

    Bluhm, V.A

    R. Bluhm, V.A. Kosteleck´ y, and N. Russell, Testing CPT with anomalous magnetic moments , Phys. Rev. Lett. 79 (1997) 1432

    work page 1997

  13. [13]

    Bluhm, V.A

    R. Bluhm, V.A. Kosteleck´ y, and N. Russell, CPT and Lorentz tests in hydrogen and antihydrogen , Phys. Rev. Lett. 82 (1999) 2254

    work page 1999

  14. [14]

    Bluhm, V.A

    R. Bluhm, V.A. Kosteleck´ y, and C.D. Lane, CPT and Lorentz tests with muons , Phys. Rev. Lett. 84 (2000) 1098

    work page 2000

  15. [15]

    Bluhm, V.A

    R. Bluhm, V.A. Kosteleck´ y, C.D. Lane, and N. Russell, Clock- comparison tests of Lorentz and CPT symmetry in space , Phys. Rev. Lett. 88 (2002) 090801

    work page 2002

  16. [16]

    Jackiw and V.A

    R. Jackiw and V.A. Kosteleck´ y, Radiatively in- duced Lorentz and CPT violation in electrodynamics , Phys. Rev. Lett. 82 (1999) 3572

    work page 1999

  17. [17]

    Chung and B.K

    J.M. Chung and B.K. Chung, Induced Lorentz and CPT violat- ing Chern–Simons term in QED: Fock–Schwinger proper time method, Phys. Rev. D 63 (2001) 105015

    work page 2001

  18. [18]

    Battistel and G

    O.A. Battistel and G. Dallabona, Role of ambiguities and gauge invariance in the calculation of the radia- tively induced Chern–Simons shift in extended QED , Nucl. Phys. B 610 (2001) 316

    work page 2001

  19. [19]

    A. P. B. Scarpelli, M. Sampaio, M. C. Nemes, and B. Hiller, Chiral anomaly and CPT invariance in an implicit momentum space regularization framework, Phys. Rev. D 64 (2001) 046013

    work page 2001

  20. [20]

    Mariz, et al., A remark on Lorentz violation at finite tempera- ture, JHEP 10 (2005) 019

    T. Mariz, et al., A remark on Lorentz violation at finite tempera- ture, JHEP 10 (2005) 019

    work page 2005

  21. [21]

    Nascimento, E

    J.R. Nascimento, E. Passos, A.Yu. Petrov, and F.A. Brito, Lorentz–CPT violation, radiative corrections and finite te mper- ature, JHEP 06 (2007) 016

    work page 2007

  22. [22]

    Scarpelli, M

    A.P.B. Scarpelli, M. Sampaio, M.C. Nemes, and B. Hiller, Gauge invariance and the CPT and Lorentz vio- lating induced Chern–Simons-like term in extended QED , Eur. Phys. J. C 56 (2008) 571

    work page 2008

  23. [23]

    Del Cima, J.M

    O.M. Del Cima, J.M. Fonseca, D.H.T. Franco, and O. Piguet, Lorentz and CPT violation in QED revisited: a missing analys is, Phys. Lett. B 688 (2010) 258

    work page 2010

  24. [24]

    Carroll, G.B

    S.M. Carroll, G.B. Field, and R. Jackiw, Limits on a Lorentz- and parity-violating modification of electrodyna mics, Phys. Rev. D 41 (1990) 1231

    work page 1990

  25. [25]

    Andrianov, R

    A.A. Andrianov, R. Soldati, and L. Sorbo, Dynamical Lorentz symmetry breaking from a (3+1)-dimensional ax- ion–Wess–Zumino model , Phys. Rev. D 59 (1998) 025002

    work page 1998

  26. [26]

    Lehnert and R

    R. Lehnert and R. Potting, Vacuum ˇCerenkov radiation , Phys. Rev. Lett. 93 (2004) 110402

    work page 2004

  27. [27]

    Kaufhold and F.R

    C. Kaufhold and F.R. Klinkhamer, Vacuum Cherenkov radiation and photon triple-splitting in a Lorentz-noninvariant ext ension of quantum electrodynamics, Nucl. Phys. B 734 (2006) 1

    work page 2006

  28. [28]

    Belich, L.D

    H. Belich, L.D. Bernald, P. Gaete, and J.A. Helay¨ el- Neto, The photino sector and a confining potential in a supersymmetric Lorentz-symmetry-violating model , Eur. Phys. J. C 73 (2013) 2632

    work page 2013

  29. [29]

    Gazzola, et al., QED with minimal and nonminimal couplings: on the quantum generation of Lorentz-violating terms in the pure photon sector , J

    G. Gazzola, et al., QED with minimal and nonminimal couplings: on the quantum generation of Lorentz-violating terms in the pure photon sector , J. Phys. G: Nucl. Part. Phys. 39 (2012) 035002

    work page 2012

  30. [30]

    Baˆ eta Scarpelli, QED with chiral nonminimal cou- pling: aspects of the Lorentz-violating quantum correctio ns, J

    A.P. Baˆ eta Scarpelli, QED with chiral nonminimal cou- pling: aspects of the Lorentz-violating quantum correctio ns, J. Phys. G: Nucl. Part. Phys. 39 (2012) 125001

    work page 2012

  31. [31]

    Agostini, F

    B. Agostini, F. A. Barone, F. E. Barone, P. Gaete, J. A. Helay¨ el-Neto, Consequences of vacuum polarization on elec- tromagnetic waves in a Lorentz-symmetry breaking scenario , Phys. Lett. B 708 (2012) 212

    work page 2012

  32. [32]

    L. C. T. Brito, H. G. Fargnoli, and A. P. Baˆ eta Scarpelli, As- pects of quantum corrections in a Lorentz-violating extens ion of the Abelian Higgs model , Phys. Rev. D 87 (2013) 125023

    work page 2013

  33. [33]

    Colladay and V.A

    D. Colladay and V.A. Kosteleck´ y, Cross sections and Lorentz vi- olation, Phys. Lett. B 511 (2001) 209

    work page 2001

  34. [34]

    Lehnert, Dirac theory within the Standard-Model Extension , J

    R. Lehnert, Dirac theory within the Standard-Model Extension , J. Math. Phys. 45 (2004) 3399

    work page 2004

  35. [35]

    Altschul, Compton scattering in the presence of Lorentz and CPT violation , Phys

    B. Altschul, Compton scattering in the presence of Lorentz and CPT violation , Phys. Rev. D 70 (2004) 056005

    work page 2004

  36. [36]

    Shore, Strong equivalence, Lorentz and CPT violation, anti-hydrogen spectroscopy and gamma-ray burst polarimet ry, Nucl

    G.M. Shore, Strong equivalence, Lorentz and CPT violation, anti-hydrogen spectroscopy and gamma-ray burst polarimet ry, Nucl. Phys. B 717 (2005) 86

    work page 2005

  37. [37]

    X. Liu, W. Liu, Z. Liu, J. W ang, Harvesting correlations from BTZ black hole coupled to a Lorentz-violating vector fie ld, JHEP 2025 (2025) 1–27

    work page 2025

  38. [38]

    Visser, Traversable wormholes: Some simple examples , Phys

    M. Visser, Traversable wormholes: Some simple examples , Phys. Rev. D, 39 (1989) 3182

    work page 1989

  39. [39]

    R. A. Konoplya, A. Zhidenko, Traversable wormholes in general relativity, Phys. Rev. Lett., 128 (2022) 091104

    work page 2022

  40. [40]

    Maldacena, A

    J. Maldacena, A. Milekhin, F. Popov, Traversable wormholes in four dimensions , Class. Quantum Grav., 40 (2023) 155016

    work page 2023

  41. [41]

    Maldacena, D

    J. Maldacena, D. Stanford, Z. Yang, Diving into traversable worm- holes, Fortschr. Phys., 65 (2017) 1700034

    work page 2017

  42. [42]

    M. S. Morris, K. S. Thorne, Wormholes in spacetime and their use for interstellar travel: A tool for teaching general rel ativity, 7 Am. J. Phys., 56 (1988) 395

    work page 1988

  43. [43]

    M. S. Morris, K. S. Thorne, and U. Yurtsever, Worm- holes, time machines, and the weak energy condition , Phys. Rev. Lett., 61 (1988) 1446

    work page 1988

  44. [44]

    De Falco, E

    V. De Falco, E. Battista, S. Capozziello et al. , Reconstructing wormhole solutions in curvature based Extended Theories of Grav- ity, Eur. Phys. J. C, 81 (2021) 157

    work page 2021

  45. [45]

    De Falco, E

    V. De Falco, E. Battista, S. Capozziello, and M. De Lauren- tis, Testing wormhole solutions in extended gravity through the Poynting-Robertson effect , Phys. Rev. D, 103 (2021) 044007

    work page 2021

  46. [46]

    De Falco and S

    V. De Falco and S. Capozziello, Static and spherically symmetric wormholes in metric-affine theories of gravity , Phys. Rev. D, 108 (2023) 104030

    work page 2023

  47. [47]

    Rastgoo and F

    S. Rastgoo and F. Parsaei, Traversable wormholes satisfying en- ergy conditions in f (Q) gravity, Eur. Phys. J. C 84 (2024) 563

    work page 2024

  48. [48]

    J. L. Rosa, N. Ganiyeva, and F. S. N. Lobo, Non- exotic traversable wormholes in f (R, T abT ab) gravity, Eur. Phys. J. C 83 (2023) 1040

    work page 2023

  49. [49]

    P. H. R. S. Moraes, A. S. Agrawal and B. Mishra, Wormholes in the f (R, L, T ) theory of gravity , Phys. Lett. B 855 (2024) 138818

    work page 2024

  50. [50]

    Mustafa, Z

    G. Mustafa, Z. Hassan, P. H. R. S. Moraes and P. K. Sahoo, Wormhole solutions in symmetric teleparallel gravity , Phys. Lett. B 821 (2021) 136612

    work page 2021

  51. [51]

    Radhakrishnan, P

    R. Radhakrishnan, P. Brown, J. Matulevich, E. Davis, D. Mir- fendereski, G. Cleaver, A review of stable, traversable wormholes in f(R) gravity theories , Symmetry, 16 (2024) 1007

    work page 2024

  52. [52]

    W. Liu, X. Fang, J. Jing, J. W ang, Lorentz violation in- duces isospectrality breaking in Einstein-bumblebee grav ity theory, Sci. China Phys. Mech. Astron. 67 (2024) 280413

    work page 2024

  53. [53]

    Eslam Panah, Super-entropy bumblebee AdS black holes , Phys

    B. Eslam Panah, Super-entropy bumblebee AdS black holes , Phys. Lett. B 861 (2025) 139273

    work page 2025

  54. [54]

    ¨Ovg¨ un, K

    A. ¨Ovg¨ un, K. Jusufi, ˙I. Sakallı, Exact traversable wormhole solu- tion in bumblebee gravity , Phys. Rev. D, 99 (2019) 024042

    work page 2019

  55. [55]

    Oliveira, D

    R. Oliveira, D. M. Dantas, V. Santos, C. A. S. Almeida, Quasinormal modes of a bumblebee wormhole , Class. Quantum Grav., 36 (2018) 105013

    work page 2018

  56. [56]

    C. Ding, S. Chen, J. W ang, J. Jing, Phantom hairy black holes and wormholes in Einstein–bumblebee gravity , Eur. Phys. J. C, 84 (2024) 742

    work page 2024

  57. [57]

    R. V. Maluf, J. C. S. Neves, C. A. S. Oliveira, Modi- fied black hole solution with a background Kalb–Ramond field , Eur. Phys. J. C, 80 (2020) 786

    work page 2020

  58. [58]

    S. G. Ghosh, S. D. Maharaj, U. Papnoi, Electrically charged black holes in gravity with a background Kalb–Ramond field , Eur. Phys. J. C, 84 (2024) 798

    work page 2024

  59. [59]

    R. V. Maluf, 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

  60. [60]

    R. B. Magalh˜ aes, L. A. Lessa, M. M. Ferreira Jr., Wormholes in Lorentz-violating gravity, arXiv:2505.07590 [gr-qc] (2025)

    work page arXiv 2025

  61. [61]

    Guvendi, O

    A. Guvendi, O. Mustafa, S. Gurtas Dogan, Coupled fermion-antifermion pairs within a traversable wormhole , Phys. Lett. B, 862 (2025) 139313

    work page 2025

  62. [62]

    Guvendi, H

    A. Guvendi, H. Hassanabadi, Fermion-antifermion pair in magnetized optical wormhole background , Phys. Lett. B, 843 (2023) 138045

    work page 2023

  63. [63]

    Errehymy, A

    A. Errehymy, A. Guvendi, S. Gurtas Dogan, O. Mustafa, Frame- dragging and light deflection in rotating optical wormhole s pace- times, Phys. Lett. B, 869 (2025) 139847

    work page 2025

  64. [64]

    Gurtas Dogan, A

    S. Gurtas Dogan, A. Guvendi, and O. Mustafa, Geomet- ric and wave optics in a BTZ optical metric-based wormhole , Phys. Lett. B 868 (2025) 139824

    work page 2025

  65. [65]

    Gurtas Dogan, O

    S. Gurtas Dogan, O. Mustafa, A. Karabulut, A. Guvendi, Wave Propagation and Effective Refraction in Lorentz-Violating Worm- hole Geometries , arXiv:2602.10889 [gr-qc] (2026 )

    work page arXiv 2026

  66. [66]

    Mustafa, S

    O. Mustafa, S. Gurtas Dogan, A. Karabulut, A. Guvendi, Lorentz- Violating Wormhole Optics , arXiv:2602.21264 [gr-qc] (2026 )

    work page arXiv 2026

  67. [67]

    Barbosa, L.C.N

    L.G. Barbosa, L.C.N. Santos, J.V. Zamperlini, F.M. da Silva , Cornell and Coulomb potentials in the double defect spaceti me, Ann. Phys. 479 (2025) 170038

    work page 2025

  68. [68]

    Ronveaux, Heun ’s Differential Equations (Oxford University Press, New York, 1995)

    A. Ronveaux, Heun ’s Differential Equations (Oxford University Press, New York, 1995)

    work page 1995

  69. [69]

    Roy, Conditionally exactly solvable potentials and supersymmetric transformations, Phys

    G, L´ evai, P. Roy, Conditionally exactly solvable potentials and supersymmetric transformations, Phys. Lett. A 264 (1999) 117

    work page 1999