pith. machine review for the scientific record. sign in

arxiv: 2601.07736 · v2 · submitted 2026-01-12 · 🌀 gr-qc

Recognition: 2 theorem links

· Lean Theorem

Epicyclic motion of charged particles around a weakly magnetized Kiselev black hole

Marina-Aura Dariescu , Vitalie Lungu
Authors on Pith no claims yet

Pith reviewed 2026-05-16 15:00 UTC · model grok-4.3

classification 🌀 gr-qc
keywords Kiselev black holecharged particlesepicyclic frequenciesquintessenceweak magnetic fieldperiapsis shiftLarmor precessionblack hole orbits
0
0 comments X

The pith

Weak magnetic fields combined with quintessence modify epicyclic frequencies of charged particles around Kiselev black holes.

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

The paper examines the motion of charged particles around a Kiselev black hole that includes a weak magnetic field. It derives an effective potential to identify bound orbits and stable circular paths under the combined influences. The analysis shows that quintessence and magnetic field strengths together change the radial and vertical epicyclic frequencies. It further computes the resulting periapsis shift and gravitational Larmor precession, noting explicit differences from the non-magnetized Kiselev spacetime and from the Ernst spacetime.

Core claim

In the weak magnetic field approximation the spacetime metric permits a perturbative treatment of charged particle dynamics. The effective potential obtained from the Lagrangian yields conditions for bound motion and circular orbits whose small radial and vertical oscillations have frequencies that depend on both the quintessence parameter and the magnetic field intensity. These frequencies produce a periapsis advance and a gravitational Larmor precession that deviate measurably from the predictions of the pure Kiselev solution and from the Ernst metric.

What carries the argument

The effective potential constructed from the Lagrangian of a charged test particle in the perturbed Kiselev metric that incorporates both the quintessence term and the weak uniform magnetic field.

If this is right

  • Bound and stable circular orbits exist only within specific ranges of the quintessence and magnetic-field parameters.
  • The epicyclic frequencies shift in a way that alters the conditions for orbital stability compared with either field taken alone.
  • The periapsis shift receives additive corrections from both quintessence and magnetism, producing values distinct from the Ernst or Kiselev cases.
  • Gravitational Larmor precession acquires an extra term linear in the magnetic field that is absent in the pure Kiselev spacetime.

Where Pith is reading between the lines

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

  • Models of accretion flows around supermassive black holes embedded in quintessence would need to include weak magnetic corrections to predict correct orbital periods and radiation signatures.
  • The reported differences could be used to constrain the quintessence parameter once independent measurements of the local magnetic field are available.
  • The perturbative approach naturally extends to inclined orbits, where the same effective potential would govern nodal precession rates.

Load-bearing premise

The magnetic field remains weak enough that it can be added perturbatively to the Kiselev metric without any back-reaction on the geometry itself.

What would settle it

A precise observation of the radial epicyclic frequency for a charged particle on a nearly circular orbit around a black-hole candidate that deviates from the non-magnetized Kiselev prediction but fails to follow the joint dependence on quintessence and magnetic-field strength.

Figures

Figures reproduced from arXiv: 2601.07736 by Marina-Aura Dariescu, Vitalie Lungu.

Figure 1
Figure 1. Figure 1: Plot of the surface (cyan color) f = 0 as a function of r, w and k. The horizontal planes corresponds to fixed values of k. The fixed values of k are represented by the horizontal planes, the red one corresponds to k = 0.01, blue to k = 0.05 and green one to k = 0.1. Here we used M = 1. In the figure 1, the cyan surface represents the solutions of the equation f = 0. One may observe that the number of the … view at source ↗
Figure 2
Figure 2. Figure 2: Left panel. Plot of E 2 min (solid curves) and E 2 s1 (dashed horizontal lines) as functions of b > 0 for different values of k. Right panel. Plot of E 2 min (solid curves) and E 2 s2 (dashed curves) as functions of b < 0, for different values of k. The black dots represent the critical values bcr for which E 2 min = E 2 s1,s2 . The other values of the parameters are: M = 1, w = −2/3 and L = 4. In the left… view at source ↗
Figure 3
Figure 3. Figure 3: Left panel. Plot of the effective potential (15) and the bound particle trajectory in the (x, z)− plane. The energy E 2 = 0.58 is represented by the light blue horizontal plane. The equipotential curve given by the solutions of the equation E 2 = V is represented by the red curves. The equatorial plane corresponding to z = 0 is represented by the dashed black line. Right panel. 3D plot of the bound traject… view at source ↗
Figure 4
Figure 4. Figure 4: Left panel. Plot of the effective potential (15) and the particle trajectory in the (x, z)− plane. The equipotential curve given by the solutions of the equation E 2 = V is represented by the red curve. The equatorial plane corresponding to z = 0 is represented by the dashed black line. The energy E 2 = 0.52 is represented by the light blue plane. Right panel. 3D plot of the escape trajectory of a particle… view at source ↗
Figure 5
Figure 5. Figure 5: The effective potential (22) as a function of b and r ∈ [r−, r+]. The value of the quintessence parameter is k = 0.015 in the left panel and k = 0.04 in the right panel. The other numerical values are: M = 1, w = −2/3 and L = 6. Once the model’s parameters M, b, w and k are fixed, one may have different types of trajectories, depending on the particle’s energy, specific charge and angular momentum. A detai… view at source ↗
Figure 6
Figure 6. Figure 6: Representation of the allowed range of b (greyed area) as a function of the radius rISCO. The black dot represents the radial coordinate r∗ = 11.55 for which bmin = bmax. The values of the parameters are: M = 1, w = −2/3 and k = 0.015. The key results that determine the range of bound orbits are summarized in table 2, where L0 = |b|r 2 and L + 1,2 are defined in (26) and (27). The conditions for rISCO and … view at source ↗
Figure 7
Figure 7. Figure 7: Representation of trapped states (shaded area) corresponding to the condition [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Energetics of bound orbits for b = 0.09 (left panel) and b = −0.09 (right panel). The pairs {E 2 , L} corresponding to bound orbits are situated in the shaded region. The green curve corresponds to circular orbits. The red dashed line represents the energy corresponding to the saddle points (24) and (25) respectively. The blue dotted line corresponds to E0 defined in (31) and represents the boundary betwee… view at source ↗
Figure 9
Figure 9. Figure 9: Position of ISCO in dependence on the magnetic parameter b, for different values of k, for b > 0 (left panel) and b < 0 (right panel). The other numerical values are M = 1 and w = −2/3. In view of the analysis developed in this section, one may conclude by saying that the radius rISCO is strongly depending on the model parameters b and k which have to be in their allowed ranges given in (44) and table 3. A… view at source ↗
Figure 10
Figure 10. Figure 10: Left panel. The radial profile of ωr for b = 0.09 and different values of k. Right panel. The radial profile of ωr as a function of r, for k = 0.015 and different values of b. The other numerical values are: M = 1 and w = −2/3. Even though the expressions of ω 2 θ and ωϕ are same same as for the Ernst spacetime [9], the contri￾bution of quintessence is encoded in the angular momentum (33). As it can be no… view at source ↗
Figure 11
Figure 11. Figure 11: Left panel. The expressions of ωθ (the left panel) and ωϕ (the right panel) as functions of r, for different values of k. The other numerical values are: M = 1, b = 0.09 and w = −2/3. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Left panel. The expressions of ωθ (the left panel) and ωϕ (the right panel) as functions of r, for different values of b. The other numerical values are: M = 1, k = 0.015 and w = −2/3. 6.1.2 The negative magnetic parameter Secondly, let us mention that the same analysis can be done for b < 0. Since the magnetic field and angular momentum have opposite signs, the Lorentz force is attracting the charged par… view at source ↗
Figure 13
Figure 13. Figure 13: The frequency ωr as a function of r, for different values of k and b = −0.09 (Left panel) and for different values of b and k = 0.015 (Right panel). The other numerical values are: M = 1 and w = −2/3. In the presence of quintessence, similarly to the previous case, all frequencies are decreasing as k increases (see the left panels of the figures 13, 14, 15). For k = 0, both ωr and ωϕ are approaching the L… view at source ↗
Figure 14
Figure 14. Figure 14: The frequency ωθ as a function of r, for different values of k and b = −0.09 (Left panel) and for different values of b and k = 0.015 (Right panel). The other numerical values are: M = 1 and w = −2/3. 6 8 10 12 14 16 r 0.20 0.25 0.30 0.35 ωϕ k=0 k=0.01 k=0.015 k=0.02 6 8 10 12 r 0.1 0.2 0.3 0.4 0.5 ωϕ b=-0.03 b=-0.05 b=-0.07 b=-0.09 [PITH_FULL_IMAGE:figures/full_fig_p023_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: The frequency ωϕ as a function of r, for different values of k and b = −0.09 (Left panel) and for different values of b and k = 0.015 (Right panel). The other numerical values are: M = 1 and w = −2/3. Also, by comparing the plots in the right panels of the figures 11 and 15, we see that, for b < 0, the frequency ωϕ is always positive and its values are significantly increased. 6.2 Prograde and retrograde … view at source ↗
Figure 16
Figure 16. Figure 16: The expressions of ωr, ωθ and ωϕ as functions of r. The numerical values are: M = 1, w = −2/3, k = 0.015 and b = 0.09 in the left panel and b = −0.09 in the right panel. The two vertical dashed red lines correspond to rISCO and r∗. The same analysis can be done for negative values of b. For a comparison, in the the right panel of the figure 16, we have changed the sign of b but kept the same values of the… view at source ↗
Figure 17
Figure 17. Figure 17: Left panel. rISCO (the red line) and rp (the blue line) as a functions of b for k = 0.015. Right panel. rISCO (the red plot) and rp (the blue plot) as functions of k for b = 0.05. The other numerical values are: M = 1 and w = −2/3. As we previously discussed, when the parameter b is negative, the values of rISCO are decreasing with |b| and ∆ϕ keeps the positive sign. 6.2.2 The gravitational Larmor precess… view at source ↗
Figure 18
Figure 18. Figure 18: A 3D representation of a stable quasi-circular trajectory with [PITH_FULL_IMAGE:figures/full_fig_p026_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: A 3D representation of an unstable quasi-circular trajectory with [PITH_FULL_IMAGE:figures/full_fig_p027_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: A 3D representation of a stable quasi-circular trajectory with [PITH_FULL_IMAGE:figures/full_fig_p028_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: A 3D representation of a stable quasi-circular trajectory with [PITH_FULL_IMAGE:figures/full_fig_p029_21.png] view at source ↗
read the original abstract

We investigate the motion of charged particles evolving around a magnetized Kiselev black hole, in the weak magnetic field approximation. The effective potential allows us to study the bound motion and the stable circular orbits. We analyze the impact of combined quintessence and magnetic fields on the epicyclic frequencies. Finally, we examine the periapsis shift and gravitational Larmor precession pointing out differences from the Ernst or Kiselev spacetimes.

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 / 0 minor

Summary. The manuscript investigates the motion of charged particles around a weakly magnetized Kiselev black hole using the effective potential in the weak magnetic field approximation. It analyzes bound motion and stable circular orbits, examines the combined effects of quintessence and magnetic fields on epicyclic frequencies, and computes the periapsis shift and gravitational Larmor precession, highlighting differences from the pure Ernst and pure Kiselev spacetimes.

Significance. If the perturbative ordering holds, the work provides a concrete calculation of how quintessence and weak magnetic fields jointly modify orbital frequencies and precession effects in a black-hole spacetime, offering potential benchmarks for accretion-disk modeling or tests of modified gravity with electromagnetic fields.

major comments (1)
  1. [Abstract and effective-potential section] The central results on epicyclic frequencies, periapsis shift, and Larmor precession rest on a first-order expansion in the magnetic field strength B while holding the Kiselev parameter fixed. No explicit bound on B (relative to the quintessence parameter or horizon scale) is supplied to guarantee that the magnetic correction remains sub-dominant throughout the radial domain used for the plotted frequencies and shifts; without this, the claimed separation from the pure Ernst and pure Kiselev cases cannot be verified.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the thorough review and insightful comment. We address the major point below and will revise the manuscript to strengthen the presentation of the approximation's validity.

read point-by-point responses

  1. Referee: [Abstract and effective-potential section] The central results on epicyclic frequencies, periapsis shift, and Larmor precession rest on a first-order expansion in the magnetic field strength B while holding the Kiselev parameter fixed. No explicit bound on B (relative to the quintessence parameter or horizon scale) is supplied to guarantee that the magnetic correction remains sub-dominant throughout the radial domain used for the plotted frequencies and shifts; without this, the claimed separation from the pure Ernst and pure Kiselev cases cannot be verified.

    Authors: We agree that an explicit bound on B is needed to rigorously justify the first-order expansion and to confirm the separation from the pure Ernst and pure Kiselev limits. In the revised manuscript we will insert a new paragraph immediately after the effective-potential derivation that states the perturbative ordering condition B ≪ (r_h/r)^2 (1 − 2M/r − α r) (with α the quintessence parameter and r_h the horizon radius) and verifies numerically that this inequality holds throughout the radial intervals shown in the frequency and precession plots. This addition will make the domain of validity transparent and strengthen the comparison with the limiting cases. revision: yes

Circularity Check

0 steps flagged

No circularity: direct perturbative derivation from metric and Lorentz force

full rationale

The manuscript starts from the Kiselev metric plus a weak uniform magnetic field, constructs the effective potential for charged test particles, locates circular orbits, and extracts epicyclic frequencies from the second derivatives of that potential. Periapsis shift and Larmor precession follow from the same first-order expansions. No parameter is fitted to a subset of results and then re-used as a prediction; the weak-field ordering is an explicit input assumption rather than an output. No self-citation is invoked to justify uniqueness or to smuggle an ansatz. The derivation chain therefore remains independent of its own results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The analysis rests on the Kiselev metric (quintessence parameter), the weak-field magnetic perturbation, and the standard charged-particle Lagrangian in curved spacetime; no new free parameters or invented entities are introduced beyond those already in the literature.

axioms (2)
  • domain assumption Kiselev metric with quintessence parameter is the background spacetime
    Invoked throughout the abstract as the base geometry.
  • domain assumption Weak magnetic field approximation holds
    Allows perturbative treatment of the magnetic field without back-reaction.

pith-pipeline@v0.9.0 · 5363 in / 1313 out tokens · 36419 ms · 2026-05-16T15:00:52.026190+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages · 11 internal anchors

  1. [1]

    Chandrasekhar,The Mathematical Theory of Black Holes(Oxford University Press, Oxford, 1983)

    S. Chandrasekhar,The Mathematical Theory of Black Holes(Oxford University Press, Oxford, 1983)

    work page 1983

  2. [2]

    R. D. Blandford and R. L. Znajek, Mon. Not. Roy. Astron. Soc.179, 433-456 (1977) doi:10.1093/mnras/179.3.433

    work page doi:10.1093/mnras/179.3.433 1977

  3. [3]

    R. A. Daly, Astrophys. J.886, 37 (2019) doi:10.3847/1538-4357/ab35e6 [arXiv:1905.11319 [astro- ph.HE]]

    work page doi:10.3847/1538-4357/ab35e6 2019

  4. [4]

    M. Y. Piotrovich, A. G. Mikhailov, S. D. Buliga and T. M. Natsvlishvili, Mon. Not. Roy. Astron. Soc.495, no.1, 614-620 (2020) doi:10.1093/mnras/staa1268 [arXiv:2004.07075 [astro-ph.HE]]

    work page doi:10.1093/mnras/staa1268 2020

  5. [5]

    Beck and R

    R. Beck and R. Wielebinski, doi:10.1007/978-94-007-5612-0 13 [arXiv:1302.5663 [astro-ph.GA]]

    work page doi:10.1007/978-94-007-5612-0

  6. [6]

    V. P. Frolov and A. A. Shoom, Phys. Rev. D82, 084034 (2010) doi:10.1103/PhysRevD.82.084034 [arXiv:1008.2985 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.82.084034 2010

  7. [7]

    F. J. Ernst, J. Math. Phys.17, no.1, 54-56 (1976) doi:10.1063/1.522781

    work page doi:10.1063/1.522781 1976

  8. [8]

    Y. K. Lim, Phys. Rev. D91, no.2, 024048 (2015) doi:10.1103/PhysRevD.91.024048 [arXiv:1502.00722 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.91.024048 2015

  9. [9]

    Quasi-harmonic oscillatory motion of charged particles around a Schwarzschild black hole immersed in an uniform magnetic field

    M. Koloˇ s, Z. Stuchl´ ık and A. Tursunov, Class. Quant. Grav.32, no.16, 165009 (2015) doi:10.1088/0264-9381/32/16/165009 [arXiv:1506.06799 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/0264-9381/32/16/165009 2015

  10. [10]

    A. M. Al Zahrani, V. P. Frolov and A. A. Shoom, Phys. Rev. D87, no.8, 084043 (2013) doi:10.1103/PhysRevD.87.084043 [arXiv:1301.4633 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.87.084043 2013

  11. [11]

    Kenzhebayeva, S

    S. Kenzhebayeva, S. Toktarbay, A. Tursunov and M. Koloˇ s, Phys. Rev. D109, no.6, 063005 (2024) doi:10.1103/PhysRevD.109.063005 [arXiv:2402.16529 [astro-ph.HE]]

    work page doi:10.1103/physrevd.109.063005 2024

  12. [12]

    C. L. Polo and H. S. Singh, J. Astrophys. Astron.46, no.2, 88 (2025) doi:10.1007/s12036-025- 10116-1

    work page doi:10.1007/s12036-025- 2025

  13. [13]

    K. J. Taylor and A. Ritz, Class. Quant. Grav.42, no.21, 215003 (2025) doi:10.1088/1361- 6382/ae0be3 [arXiv:2505.11629 [gr-qc]]

    work page doi:10.1088/1361- 2025

  14. [14]

    A. G. Riesset al.[Supernova Search Team], Astron. J.116, 1009-1038 (1998) doi:10.1086/300499 [arXiv:astro-ph/9805201 [astro-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1086/300499 1998

  15. [15]

    Measurements of Omega and Lambda from 42 High-Redshift Supernovae

    S. Perlmutteret al.[Supernova Cosmology Project], Astrophys. J.517, 565-586 (1999) doi:10.1086/307221 [arXiv:astro-ph/9812133 [astro-ph]]. 31

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1086/307221 1999

  16. [16]

    V. V. Kiselev, Class. Quant. Grav.20, 1187-1198 (2003) doi:10.1088/0264-9381/20/6/310 [arXiv:gr- qc/0210040 [gr-qc]]

    work page doi:10.1088/0264-9381/20/6/310 2003

  17. [17]

    M. A. Dariescu, C. Dariescu, V. Lungu and C. Stelea, Phys. Rev. D106, no.6, 064017 (2022) doi:10.1103/PhysRevD.106.064017 [arXiv:2206.12876 [gr-qc]]

    work page doi:10.1103/physrevd.106.064017 2022

  18. [18]

    Schwarzschild black hole surrounded by quintessence: Null geodesics

    S. Fernando, Gen. Rel. Grav.44, 1857-1879 (2012) doi:10.1007/s10714-012-1368-x [arXiv:1202.1502 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/s10714-012-1368-x 2012

  19. [19]

    Nariai black holes with quintessence

    S. Fernando, Mod. Phys. Lett. A28, 1350189 (2013) doi:10.1142/S0217732313501897 [arXiv:1408.5064 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1142/s0217732313501897 2013

  20. [20]

    K. K. J. Rodrigue, M. Saleh, B. B. Thomas and T. C. Kofane, Gen. Rel. Grav.50, no.5, 52 (2018) doi:10.1007/s10714-018-2367-3

    work page doi:10.1007/s10714-018-2367-3 2018

  21. [21]

    M. M. Gohain, K. Bhuyan, R. Borgohain, T. Gogoi, K. Bhuyan and P. Phukon, Nucl. Phys. B 1018, 117073 (2025) doi:10.1016/j.nuclphysb.2025.117073 [arXiv:2412.06252 [gr-qc]]

    work page doi:10.1016/j.nuclphysb.2025.117073 2025

  22. [22]

    Errehymy, Phys

    A. Errehymy, Phys. Lett. B870, 139945 (2025) doi:10.1016/j.physletb.2025.139945

    work page doi:10.1016/j.physletb.2025.139945 2025

  23. [23]

    M. A. Dariescu, V. Lungu, C. Dariescu and C. Stelea, Phys. Rev. D109, no.2, 024021 (2024) doi:10.1103/PhysRevD.109.024021 [arXiv:2311.11356 [gr-qc]]

    work page doi:10.1103/physrevd.109.024021 2024

  24. [24]

    Dariescu and V

    M. Dariescu and V. Lungu, Eur. Phys. J. Plus140, no.6, 476 (2025) doi:10.1140/epjp/s13360-025- 06435-5

    work page doi:10.1140/epjp/s13360-025- 2025

  25. [25]

    Shadow of the rotating black hole with quintessential energy in the presence of the plasma

    A. Abdujabbarov, B. Toshmatov, Z. Stuchl´ ık and B. Ahmedov, Int. J. Mod. Phys. D26, no.06, 1750051 (2016) doi:10.1142/S0218271817500511 [arXiv:1512.05206 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1142/s0218271817500511 2016

  26. [26]

    Rayimbaev, B

    J. Rayimbaev, B. Majeed, M. Jamil, K. Jusufi and A. Wang, Phys. Dark Univ.35, 100930 (2022) doi:10.1016/j.dark.2021.100930 [arXiv:2202.11509 [gr-qc]]

    work page doi:10.1016/j.dark.2021.100930 2022

  27. [27]

    On magnetized anisotropic stars

    C. Stelea, M. A. Dariescu and C. Dariescu, Phys. Rev. D97, no.10, 104059 (2018) doi:10.1103/PhysRevD.97.104059 [arXiv:1804.08075 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.97.104059 2018

  28. [28]

    Stelea, M

    C. Stelea, M. A. Dariescu and C. Dariescu, Phys. Rev. D108, no.8, 084034 (2023) doi:10.1103/PhysRevD.108.084034 [arXiv:1810.02235 [gr-qc]]

    work page doi:10.1103/physrevd.108.084034 2023

  29. [29]

    Lungu, M

    V. Lungu, M. A. Dariescu and C. Stelea, Phys. Rev. D111, no.6, 064014 (2025) doi:10.1103/PhysRevD.111.064014 [arXiv:2405.14420 [gr-qc]]

    work page doi:10.1103/physrevd.111.064014 2025

  30. [30]

    A. N. Aliev and D. V. Galtsov, Sov. Phys. Usp.32, 75 (1989) doi:10.1070/PU1989v032n01ABEH002677

    work page doi:10.1070/pu1989v032n01abeh002677 1989

  31. [31]

    L. O. Villegas, E. Ramirez-Codiz, V. Jaramillo, J. C. Degollado, C. Moreno, D. N´ u˜ nez and F. J. Romero-Cruz, JCAP08, 007 (2023) doi:10.1088/1475-7516/2023/08/007 [arXiv:2211.10464 [gr-qc]]. 32

    work page doi:10.1088/1475-7516/2023/08/007 2023

  32. [32]

    Lungu, Eur

    V. Lungu, Eur. Phys. J. Plus140, no.6, 493 (2025) doi:10.1140/epjp/s13360-025-06462-2 [arXiv:2504.04905 [gr-qc]]

    work page doi:10.1140/epjp/s13360-025-06462-2 2025

  33. [33]

    M. A. Abramowicz, T. Bulik, M. Bursa and W. Kluzniak, Astron. Astrophys.404, L21 (2003) doi:10.1051/0004-6361:20030737 [arXiv:astro-ph/0206490 [astro-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1051/0004-6361:20030737 2003

  34. [34]

    Can Lorentz invariance violation af- fect the sensitivity of deep underground neutrino experiment?

    C. Chakraborty and P. Majumdar, Eur. Phys. J. C83, no.8, 714 (2023) doi:10.1140/epjc/s10052- 023-11858-4 [arXiv:2210.17162 [gr-qc]]. 33

    work page doi:10.1140/epjc/s10052- 2023