Pith. sign in

REVIEW 4 major objections 5 minor 52 references

A compact extra dimension enters the magnetic Casimir energy as an effective fermion mass.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-01 23:52 UTC pith:M7GS5WYV

load-bearing objection A legitimate new combination, but the master formula Eq. (III.15) doesn't follow from the derivation and fails a dimension check; everything built on it inherits the error. the 4 major comments →

arxiv 2607.15212 v1 pith:M7GS5WYV submitted 2026-07-16 hep-th

Magnetic corrections to the fermionic Casimir effect with Lorentz symmetry violation and a compactified extra dimension

Andrea Erdas This is my paper
classification hep-th
keywords Casimir effectfermionic vacuum energyLorentz symmetry violationcompactified extra dimensionszeta function regularizationMIT bag boundary conditionsmagnetic fieldKaluza-Klein modes
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper derives closed-form analytic expressions for the Casimir energy and pressure of a massive charged fermion that violates Lorentz symmetry and lives in a spacetime with one compactified circular extra dimension, placed between two parallel plates with a constant magnetic field perpendicular to them. The vacuum energy is evaluated with the zeta-function technique, treating timelike Lorentz violation and spacelike violation in three distinct directions. The central qualitative result is that the compact extra dimension enters the physics only through the effective mass sqrt(m^2 + k0^2), where k0 is the lowest Kaluza-Klein momentum, and that the Lorentz-violating parameter either strengthens or weakens the attractive pressure depending on its direction. The closed expressions make the magnetic-field, plate-separation, and compactification-size dependence explicit, so they can be compared directly with numerical evaluation of the original mode sums.

Core claim

For non-vanishing compactification parameter, the paper claims the vacuum energy for timelike Lorentz violation is E_VAC = -L^2 eB / (4 pi^(3/2) a) (m^2 + k0^2)^(1/4) exp(-2a sqrt(m^2+k0^2)) coth( a eB / sqrt(m^2+k0^2) ), with analogous expressions for the three spacelike directions. The derivation starts from sums over Landau levels, plate modes, and Kaluza-Klein modes; zeta regularization makes the sums well defined, Poisson resummation renders the plate-mode sum exponentially convergent, and the assumption that the compactification length q is much smaller than the plate separation a keeps only the lowest Kaluza-Klein mode. A saddle-point integration then produces the exponential form. Th

What carries the argument

The zeta-function technique: the vacuum energy is written as E = -lim L^2 eB/(2 pi) sum over ell,n,j [m^2 + (2j+1±1)eB + (k_n/a)^2 + k_ell^2]^{-s}, with the parameter s = -(1+epsilon)/2 and the limit epsilon -> 0 taken after analytic continuation. Two identities do the main work: the Landau-level sum collapses to coth(eBt), and Poisson resummation turns the half-integer plate-mode sum into a sum of exponentials. The q << a assumption truncates the Kaluza-Klein tower to the lowest mode k0 = 2 pi beta / q, and the final z-integral is evaluated by the saddle-point method. This chain turns an intractable triple sum into a single exponential containing only m, a, eB, and k0.

Load-bearing premise

The formulas rely on the compact extra dimension being much smaller than the plate separation (q << a); if that inequality fails, the Kaluza-Klein tower cannot be truncated, the exponential saddle-point expressions break down, and the closed-form energies no longer hold.

What would settle it

Numerically evaluate the exact sums in Eqs. (II.10)-(II.13) for fixed q and eB over a range of plate separations a, and compare with the analytic expressions of Section III: if the numerical energy differs from Eq. (III.15) when q is not much smaller than a, the paper's closed formulas are restricted to the stated regime, not universal.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

Share X Bluesky LinkedIn Reddit HN

If this is right

  • For beta != 0 the magnetic Casimir pressure is attractive and exponentially suppressed in the plate separation; the compact-dimension size q sets the suppression scale through k0 = 2 pi beta / q.
  • For beta = 0 and light fermions the leading-order energy reproduces the known Dirac-fermion result with a logarithmic magnetic-field term, while the extra dimension contributes only an exponentially small correction of relative order exp(-2a k1).
  • When Lorentz violation is directed along the plate normal, the pressure is strengthened relative to the timelike case; when it is directed perpendicular to the magnetic field, the pressure is weakened.
  • When Lorentz violation points along the compactified dimension, the parameter lambda enters only in the extra-dimensional part of the energy and reduces the effective-mass shift from sqrt(m^2+k0^2) to sqrt(m^2 + (1-lambda)^2 k0^2).
  • All cases examined yield attractive pressures, so the magnetic field and the extra dimension modify the strength but not the sign of the fermionic Casimir force.

Where Pith is reading between the lines

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

  • A direct numerical check is suggested by the derivation itself: evaluating the original sums (II.10)-(II.13) without the q << a truncation should reproduce Eq. (III.15) when q << a and deviate from it when a approaches q; that crossover has not been computed in the paper.
  • Because the paper works with a single compact dimension and a field perpendicular to the plates, the same technique with tilted magnetic fields or multiple compact dimensions would likely produce products of coth factors and vector-valued effective masses; the paper does not explore these.
  • If the compact dimension is large enough to be detected, the exponential suppression exp(-2a sqrt(m^2+k0^2)) offers a sharper experimental signature than the power-law dependence of ordinary Casimir theory, since the pressure depends on q through the exponent as well as the prefactor.
  • The result that Lorentz violation along the extra dimension only affects the Kaluza-Klein terms suggests a possible way to disentangle Lorentz-violation parameters from compactification effects in precision Casimir measurements, though the paper does not make this suggestion.

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

4 major / 5 minor

Summary. The paper computes the Casimir energy and pressure for a massive, charged Dirac fermion with aether-like Lorentz violation in R^4 x S^1, with MIT bag boundary conditions on two parallel plates and a constant magnetic field perpendicular to the plates. Using zeta-function regularization, Poisson resummation, and a saddle-point approximation, the author derives exponentially suppressed analytic expressions for timelike and three spacelike Lorentz-violation directions, under the assumption that the compactification radius q is much smaller than the plate separation a. The paper claims that the compact extra dimension effectively shifts the fermion mass, m -> sqrt(m^2 + k_l^2), and that the Lorentz-violation parameter suppresses or enhances the pressure depending on its direction.

Significance. The physical setup is topical and well-defined, and the zeta-function/saddle-point route is a standard tool in Casimir calculations. If the final formulas were correct, the paper would provide useful analytic approximations for magnetic corrections in a Lorentz-violating extra-dimensional model. However, the central master formula is dimensionally inconsistent and does not follow from the preceding equations, so the paper's quantitative results are not supported. The qualitative features (exponential suppression and an effective mass shift from the compact dimension) are plausible and may survive a corrected derivation, but the manuscript as written cannot be accepted without a substantial recalculation of Sections III and IV.

major comments (4)
  1. [Section III A, Eq. (III.8)] The Poisson-resummed identity is printed with a sqrt(pi t) multiplying the bracket. The correct identity is sum_{n=1}∞ exp[-(k_n/a)^2 t] = (a/sqrt(pi t)) [1/2 + sum (-1)^n exp(-n^2 a^2/t)]. As printed, Eq. (III.8) is inconsistent with Eq. (III.9), which uses t^{s-3/2} and therefore corresponds to the a/sqrt(pi t) form. This sign/power error is load-bearing for the subsequent t-exponent and prefactor.
  2. [Section III A, Eq. (III.11)] The variable change t = z n a / sqrt(m^2 + k_l^2) gives a prefactor (n a / sqrt(m^2+k_l^2))^{s-1/2}, not (n a sqrt(m^2+k_l^2))^{s-1/2}. The printed version changes the powers of a and M by a factor (a^2M^2)^{s-1/2} relative to the correct expression. This error propagates into Eqs. (III.12)-(III.15) and into all spacelike and pressure results derived by substitution.
  3. [Section III A, Eqs. (III.13)-(III.15)] The saddle-point evaluation is incorrect. For I = ∫ dz z^{s-3/2} exp[-A(z+1/z)] coth(C z), the saddle gives I ≈ exp(-2A) sqrt(pi/A) coth(C), not exp(-2A) A^{-1/4} coth(C). Taking the limit s→-1/2 in Eq. (III.14) does not produce Eq. (III.15); Eq. (III.15) has mass dimension [M]^{5/4}, whereas the original mode sum (II.10) has dimension [M]. An independent evaluation using the integral representation ∫ t^{ν-1} e^{-αt-β/t} dt = 2(β/α)^{ν/2} K_ν(2√(αβ)) gives, at leading order for large a sqrt(m^2+k_0^2), E_VAC = -L^2 eB/(4π^{3/2}) sqrt((m^2+k_0^2)/a) exp[-2a sqrt(m^2+k_0^2)] coth(aeB/sqrt(m^2+k_0^2)). Thus the central master formula and its consequences are unsupported.
  4. [Section III A, below Eq. (III.11)] The truncation to the ℓ=0 mode for β≠0 requires not merely q << a but a |k_0| >> 1, i.e., β a/q >> 1. For β small but nonzero, the ℓ = ±1 modes are not exponentially suppressed and the simple formulas fail. This condition should be stated explicitly, as it restricts the validity of all the paper's compactification results.
minor comments (5)
  1. [Section V] Several of the equation references in the discussion are incorrect: the text repeatedly cites 'Eq. (III.15)' when referring to spacelike or β=0 results (e.g., after Eq. (III.24)); these should be Eqs. (III.23), (III.27), (III.31), etc.
  2. [Throughout] The notation √π^3 is ambiguous; it should be written π^{3/2} consistently, especially in Eqs. (III.9)-(III.14).
  3. [Section II, Eqs. (II.5)-(II.8)] For the timelike case (II.5), the vacuum energy has no explicit λ dependence, although the Lagrangian (II.1) contains a timelike aether term. If this is a result inherited from Ref. [18], it should be stated and justified; otherwise the dispersion relation appears to require a (1+λ)^{-1} factor.
  4. [Section III A] The regime of validity should be stated in terms of all parameters: the saddle-point and n=1 truncation require a sqrt(m^2+k_l^2) >> 1, not only q << a. The condition a|k_0| >> 1 for β≠0 is especially important.
  5. [References] Reference [39] has an incomplete page number ('20 (2021)'); other entries should be checked for completeness.

Circularity Check

0 steps flagged

No significant circularity: the timelike-LV master formula is derived from the mode sum, and the spacelike cases are explicit rescalings; one minor self-citation (Ref. [36]) is used for the beta=0 leading term but is not the paper's central claim.

full rationale

The central derivation is self-contained. The paper starts from the vacuum-energy mode sum Eq. (II.10), which is obtained by standard Landau-level quantization of the non-magnetic expressions Eqs. (II.5)-(II.8). It then applies the zeta-function representation, the coth identity, Poisson resummation, and a saddle-point approximation to arrive at the master formula Eq. (III.15). No parameter is fitted and no target result is assumed in the derivation; the approximations q<<a and keeping only the lowest Kaluza-Klein and plate modes are stated, not circular. The spacelike-LV results in Sec. III B are obtained by the paper's own explicit substitutions (e.g., a -> a/(1-lambda), eB -> (1-lambda)eB, k_l -> (1-lambda)k_l), as acknowledged in the text; this is a legitimate derivational shortcut, not a redefinition of the target. The only self-citation used as an input is Ref. [36] for the beta=0, extra-dimension-independent leading term Eq. (III.19), doubled for Dirac fermions. That term is a prior independent calculation of the standard four-dimensional magnetic Casimir energy and is not the paper's novel extra-dimension/LV claim, so it does not constitute load-bearing circularity. The apparent prefactor and dimensional inconsistencies in Eqs. (III.8)-(III.15) are correctness concerns, not circularity, and do not change this assessment.

Axiom & Free-Parameter Ledger

3 free parameters · 6 axioms · 0 invented entities

The calculation introduces no fitted parameters; lambda and beta are model inputs. The heavy lifting is the zeta-function machinery and the q << a truncation.

free parameters (3)
  • lambda (Lorentz-violation parameter)
    Dimensionless model parameter in the Lagrangian (II.1), taken lambda << 1; not fitted to any data.
  • beta (compactification parameter)
    Phase in the compactification condition (II.3), 0 <= beta < 1; model input controlling the lowest Kaluza-Klein momentum.
  • q (compactification radius)
    Circumference of the S^1; physical input with the assumed hierarchy q << a.
axioms (6)
  • domain assumption Aether-like CPT-even Lorentz-violating fermion Lagrangian (II.1) from Ref. [18] with lambda << 1.
    The target model; if the LV model is not physical, the calculation describes a different theory.
  • domain assumption MIT bag boundary conditions ina gamma^a psi = psi on the plates (II.4).
    Determines the discrete k3 spectrum; other boundary conditions would give different Casimir forces.
  • standard math Zeta-function regularization: E = -lim_{s -> -1/2} sum omega^{-s} and analytic continuation.
    Used throughout to define divergent sums; standard but scheme-dependent for cutoff terms.
  • standard math Poisson resummation identity (III.7) and saddle-point approximation for large a sqrt(m^2 + k_ell^2).
    Converts the mode sums into exponentially suppressed analytic forms; valid for large arguments.
  • domain assumption q << a hierarchy and retention of only the lowest ell mode (ell=0 for beta != 0; ell=-1,0,1 for beta=0) and n=1 term.
    The central approximation; if q is comparable to a, the exponential saddle-point results fail.
  • domain assumption Leading-order beta=0 magnetic Casimir term for a Dirac fermion is twice the Majorana result of Ref. [36].
    Borrowed from the author's earlier work; assumed to be correct for the Dirac case.

pith-pipeline@v1.3.0-alltime-deepseek · 3395 in / 4032 out tokens · 272325 ms · 2026-08-01T23:52:08.556164+00:00 · methodology

0 comments
Cite this review

Pith. "Pith review of Magnetic corrections to the fermionic Casimir effect with Lorentz symmetry violation and a compactified extra dimension." pith.science (2026). https://pith.science/paper/M7GS5WYV

@misc{pith2026260715212,
  author       = {Pith},
  title        = {Pith review of: Magnetic corrections to the fermionic Casimir effect with Lorentz symmetry violation and a compactified extra dimension},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M7GS5WYV}},
  note         = {Machine review of arXiv:2607.15212}
}
Share X Bluesky LinkedIn Reddit HN
read the original abstract

In this work I study the effect of a magnetic field on the Casimir effect due to a massive, charged fermion field that violates Lorentz invariance in an aether-like manner while maintaining CPT, in a space with one toroidally compactified extra dimension. I take the fermion field to obey MIT bag boundary conditions on two identical square parallel plates. I do this investigation using the zeta function technique that enables me to calculate the Casimir energy and pressure in the presence of a constant magnetic field perpendicular to the plates. I examine the cases of timelike Lorentz violation, of spacelike Lorentz violation in the direction of the magnetic field, of spacelike Lorentz violation in the direction perpendicular to the magnetic field, and of spacelike Lorentz violation in the direction of the compactified extra dimension. In all scenarios, I find simple and accurate analytic expressions of the magnetic field dependent Casimir energy and pressure.

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

52 extracted references · 1 linked inside Pith

  1. [1]

    Notice thatk 3 is discretized due to the presence of the Casimir plates andk 4 is discretized because this dimension is compactified

    When the Lorentz violation is timelike,u a = (1,0,0,0,0), EV AC=−C L2 2π2 Z dk1dk2 ∞X ℓ=−∞ ∞X n=1 " m2 +k 2 1 +k 2 2 + kn a 2 +k 2 ℓ #1/2 ,(II.5) whereC= 2 is the spin degeneracy factor,k n = (n− 1 2 )π, andk ℓ = 2π q (ℓ+β). Notice thatk 3 is discretized due to the presence of the Casimir plates andk 4 is discretized because this dimension is compactified

  2. [2]

    When the Lorentz violation is spacelike and in thex 3-direction,u a = (0,0,0,1,0), EV AC=−C L2 2π2 Z dk1dk2 ∞X ℓ=−∞ ∞X n=1 " m2 +k 2 1 +k 2 2 + (1−λ) 2 kn a 2 +k 2 ℓ #1/2 .(II.6)

  3. [3]

    When the Lorentz violation is spacelike and in thex 1 −x 2 direction,u a = (0, 1√ 2 , 1√ 2 ,0,0), EV AC=−C L2 2π2 Z dk1dk2 ∞X ℓ=−∞ ∞X n=1 " m2 + 1− λ 2 2 (k2 1 +k 2

  4. [4]

    + kn a 2 +k 2 ℓ #1/2 .(II.7)

  5. [5]

    The presence of the magnetic field removes the spin degeneracy and, instead of real-valuedk 1 and k2, discrete Landau levels appear

    When the Lorentz violation is spacelike and in the direction of the compactified extra dimension,u a = (0,0,0,0,1), EV AC=−C L2 2π2 Z dk1dk2 ∞X ℓ=−∞ ∞X n=1 " m2 +k 2 1 +k 2 2 + kn a 2 + (1−λ) 2k2 ℓ #1/2 .(II.8) The novelty of this work is the presence of a constant magnetic fieldBperpendicular to the plates, thus pointing in thex 3-direction. The presence...

  6. [6]

    when the Lorentz violation is timelike E± =− L2eB 2π ∞X ℓ=−∞ ∞X n=1 ∞X j=0 " m2 + (2j+ 1±1)eB+ kn a 2 +k 2 ℓ #1/2 ,(II.10) wherej= 0,1,2,· · ·labels the Landau levels, (2j+ 1)eBis the energy of thej-th Landau level,±eBis the spin contribution, and L2eB 2π takes into account the degeneracy of the Landau levels

  7. [7]

    When the Lorentz violation is spacelike in thex 3-direction E± =− L2eB 2π ∞X ℓ=−∞ ∞X n=1 ∞X j=0 " m2 + (2j+ 1±1)eB+ (1−λ) 2 kn a 2 +k 2 ℓ #1/2 ,(II.11)

  8. [8]

    when the Lorentz violation is spacelike in thex 1 −x 2 direction E± =− L2eB 2π ∞X ℓ=−∞ ∞X n=1 ∞X j=0 " m2 + 1− λ 2 2 (2j+ 1±1)eB+ kn a 2 +k 2 ℓ #1/2 ,(II.12)

  9. [9]

    CASIMIR ENERGY In this section, I evaluate the Casimir energy of the fieldψ, using the zeta function technique [41]

    when the Lorentz violation is spacelike in the direction of the compactified extra dimension E± =− L2eB 2π ∞X ℓ=−∞ ∞X n=1 ∞X j=0 " m2 + (2j+ 1±1)eB+ kn a 2 + (1−λ) 2k2 ℓ #1/2 .(II.13) 4 III. CASIMIR ENERGY In this section, I evaluate the Casimir energy of the fieldψ, using the zeta function technique [41]. In subsection III A, I obtain the Casimir energy ...

  10. [10]

    aeBp m2 + (1−λ) 2k2 1 # ,(IV.14) valid whenam≫1. Notice how, in this case, the mass

    Forβ= 0, I find PC =− 7 8 π2 120a4 + m2 48a2 − (eB)2 12π2 ln a √ eB + 9 8 − eB√ π3a m2 +k 2 1 3/4 ×e −2a √ m2+k2 1 coth aeBp m2 +k 2 1 ! ,(IV.3) 8 valid whenam≪1. Dropping terms which are negligible in the small mass approximation, Eq. (IV.3) becomes PC =− 7 8 π2 120a4 − (eB)2 12π2 ln a √ eB + 9 8 − eB√ π3a k3/2 1 e−2ak1 coth aeB k1 ,(IV.4) and it is attr...

  11. [11]

    It is also interesting that, in the case of Lorentz violation in the direction of the compactified extra dimension, the dependence on the Lorentz violating parameter,λ, appears only in the part of the Casimir energy and pressure that depends on the size of the compactified extra dimension, thus diminishing the ”increased” mass effect

  12. [12]

    H. B. G. Casimir, Indag. Math.10, 261 (1948) [Kon. Ned. Akad. Wetensch. Proc.51, 793 (1948)] [Front. Phys.65, 342 (1987)] [Kon. Ned. Akad. Wetensch. Proc.100N3-4, 61 (1997)]

    1948

  13. [13]

    M. J. Sparnaay, Physica24, 751 (1958)

    1958

  14. [14]

    Bordag, U

    M. Bordag, U. Mohideen, and V. M. Mostepanenko, Phys. Rep.353, 1 (2001). 10

    2001

  15. [15]

    Bordag, G

    M. Bordag, G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko, International series of monographs on physics. 145

  16. [16]

    T. H. Boyer, Phys. Rev. A9, 2078 (1974)

    2078

  17. [17]

    T. H. Boyer, Phys. Rev.174, 1764 (1968)

    1968

  18. [18]

    Ambjorn and S

    J. Ambjorn and S. Wolfram, Annals Phys.147, 33 (1983)

    1983

  19. [19]

    Chodos, R

    A. Chodos, R. L. Jaffe, K. Johnson, C. B. Thorn and V. F. Weisskopf, Phys. Rev. D9, 3471-3495 (1974)

    1974

  20. [20]

    Johnson, Acta Phys

    K. Johnson, Acta Phys. Polon. B6, 865 (1975)

    1975

  21. [21]

    Horava, Phys

    P. Horava, Phys. Rev. D79, 084008 (2009)

    2009

  22. [22]

    Ferrari, H

    A. Ferrari, H. Girotti, M. Gomes, A. Petrov and A. da Silva, Mod. Phys. Lett. A28, 1350052 (2013)

    2013

  23. [23]

    Morales Ulion, E

    I. Morales Ulion, E. Bezerra de Mello and A. Petrov, Int. J. Mod. Phys. A30, no.36, 1550220 (2015)

    2015

  24. [24]

    M. B. Cruz, E. R. Bezerra de Mello and A. Y. Petrov, Phys. Rev. D96, no.4, 045019 (2017)

    2017

  25. [25]

    M. B. Cruz, E. R. Bezerra De Mello and A. Y. Petrov, Mod. Phys. Lett. A33, no.20, 1850115 (2018)

    2018

  26. [26]

    D. R. da Silva, M. B. Cruz and E. R. Bezerra de Mello, Int. J. Mod. Phys. A34, no.20, 1950107 (2019)

    2019

  27. [27]

    Gomes, J

    M. Gomes, J. R. Nascimento, A. Y. Petrov and A. J. da Silva, Phys. Rev. D81, 045018 (2010)

    2010

  28. [28]

    M. B. Cruz, E. R. Bezerra de Mello and A. Y. Petrov, Phys. Rev. D99, no.8, 085012 (2019)

    2019

  29. [29]

    Rohim and A

    A. Rohim and A. Salim Adam, PTEP2025, no.12, 123B11 (2025)

    2025

  30. [30]

    K. E. L. de Farias, M. A. Anacleto, E. Passos, I. Brevik, H. Mota and J. R. L. Santos, Phys. Rev. D109, no.12, 125010 (2024)

    2024

  31. [31]

    Kaluza, Sitzungsber

    T. Kaluza, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys. )1921, 966-972 (1921)

    1921

  32. [32]

    Klein, Z

    O. Klein, Z. Phys.37, 895-906 (1926)

    1926

  33. [33]

    Arkani-Hamed, S

    N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, Phys. Lett. B429, 263-272 (1998)

    1998

  34. [34]

    Antoniadis, N

    I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, Phys. Lett. B436, 257-263 (1998)

    1998

  35. [35]

    Arkani-Hamed, S

    N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, Phys. Rev. D59, 086004 (1999)

    1999

  36. [36]

    Randall and R

    L. Randall and R. Sundrum, Phys. Rev. Lett.83, 3370-3373 (1999)

    1999

  37. [37]

    E. G. Adelberger, J. H. Gundlach, B. R. Heckel, S. Hoedl and S. Schlamminger, Prog. Part. Nucl. Phys.62, 102-134 (2009)

    2009

  38. [38]

    Murata and S

    J. Murata and S. Tanaka, Class. Quant. Grav.32, no.3, 033001 (2015)

    2015

  39. [39]

    T. G. Rizzo, JHEP09, 036 (2005)

    2005

  40. [40]

    Colladay and V

    D. Colladay and V. A. Kostelecky, Phys. Rev. D58, 116002 (1998)

    1998

  41. [41]

    S. M. Carroll and H. Tam, Phys. Rev. D78, 044047 (2008)

    2008

  42. [42]

    Bellucci and A

    S. Bellucci and A. A. Saharian, Phys. Rev. D80, 105003 (2009)

    2009

  43. [43]

    M. V. Cougo-Pinto, C. Farina, M. R. Negrao and A. C. Tort, J. Phys. A32, no.24, 4457-4462 (1999)

    1999

  44. [44]

    M. V. Cougo-Pinto, C. Farina, M. R. Negrao and A. Tort, [arXiv:hep-th/9810033 [hep-th]]

    Pith/arXiv arXiv

  45. [45]

    Erdas and K

    A. Erdas and K. P. Seltzer, Phys. Rev. D88, 105007 (2013)

    2013

  46. [46]

    Erdas and K

    A. Erdas and K. P. Seltzer, Int. J. Mod. Phys. A29, 1450091 (2014)

    2014

  47. [47]

    Erdas, Phys

    A. Erdas, Phys. Rev. D83, 025005 (2011)

    2011

  48. [48]

    Erdas, Int

    A. Erdas, Int. J. Mod. Phys. A35, no.31, 2050209 (2020)

    2020

  49. [49]

    Erdas, Int

    A. Erdas, Int. J. Mod. Phys. A36, no.20, 20 (2021)

    2021

  50. [50]

    Erdas, Int

    A. Erdas, Int. J. Mod. Phys. A40, no.19, 2550061 (2025)

    2025

  51. [51]

    Erdas, Int

    A. Erdas, Int. J. Mod. Phys. A38, no.22n23, 2350117 (2023)

    2023

  52. [52]

    S. W. Hawking, Commun. Math. Phys.55, 133 (1977)

    1977