Pith. sign in

REVIEW 2 major objections 4 minor 90 references

Possible wormholes in generalized geometry-matter coupling gravity induced by the Dekel-Zhao dark matter profile

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read Traversable wormhole solutions are derived from the Dekel-Zhao dark matter halo profile in f(R,L_m,T) gravity, with a shape function that can satisfy traversability and energy conditions in some parameter ranges.

desk verdict The reader's chi-cancellation objection is wrong; the real flaw is that the 'ordinary matter' branch violates the flaring-out condition at the throat, so the central claim is unsupported. read the letter →

arxiv 2505.21081 v1 pith:FKBRAAX5 submitted 2025-05-27 gr-qc

A. Errehymy , O. Donmez , A. Syzdykova , K. Myrzakulov , S. Muminov , A. Dauletov , J. Rayimbaev This is my paper · ORCID
classification gr-qc MSC 83C1583D0583C57 PACS 04.20.Jb04.50.Kd95.35.+d
keywords traversablewormholesf(RL_mT)gravityDekel-ZhaodarkmatterprofileshapefunctionenergyconditionsgravitationallensingTolman-Oppenheimer-Volkoffequilibrium
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper sets out to show that a realistic dark matter halo profile can seed a traversable wormhole in an extended theory of gravity. Using the Dekel-Zhao double power-law density distribution, it derives a wormhole shape function and checks the Morris-Thorne traversability conditions. The authors report that the resulting wormholes can satisfy the relevant energy conditions in some parameter regions and violate them in others, so the solutions can be supported by ordinary matter as well as exotic matter. If correct, this would mean wormholes could form inside dark matter halos without necessarily requiring exotic matter, and it would also give observational signatures through gravitational lensing.

What carries the argument

The central object is the wormhole shape function S(r), constructed by integrating the Dekel-Zhao double power-law dark matter density profile through the simplified field equations (38)-(40) under a constant redshift. This shape function simultaneously determines the metric and the matter content of the wormhole, and it is what carries the traversability, energy condition, and lensing analyses.

What would settle it

Substitute the explicit matter expressions (46)-(48) and the shape function (44) into the full field equations (34)-(36) with \chi=0.3, r_0=0.85, and r_c=0.5, and check whether the equations are satisfied identically; any nonzero residual term contradicts the claim that these are solutions of the stated f(R,L_m,T) theory.

Watch

Extended reading notes

Core claim

For the additive model f(R,L_m,T)=R+\$\lambda$ L_m+\chi T with a constant redshift function, the paper derives a family of traversable wormhole solutions whose shape function is obtained directly from the Dekel-Zhao dark matter density profile. The central result is the explicit shape function, Eq. (44), which satisfies the throat condition S(r_0)=r_0, the flaring-out condition, and asymptotic flatness for suitable couplings. The associated matter content can respect or violate the null and dominant energy conditions depending on the parameters, leading the authors to conclude that wormholes in this gravity theory can sustain both exotic and ordinary matter.

Load-bearing premise

The load-bearing premise is that the simplified field equations (38)-(40) are equivalent to the full field equations (34)-(36) for the parameter values used, including \chi=0.3, even though the reduction appears to require \chi=0.

Editorial extensions

If this is right

  • Wormhole solutions can be embedded naturally in dark matter halos described by realistic density profiles such as Dekel-Zhao.
  • The energy conditions can be satisfied in certain parameter regions, suggesting that ordinary matter may suffice to support these wormholes.
  • The deflection angle can be negative, implying a repulsive gravitational force that could be probed through lensing observations.
  • The Tolman-Oppenheimer-Volkoff equilibrium condition is satisfied, supporting the stability of the solutions.
  • The coupling parameters \lambda and \chi are constrained by requiring the shape function to meet the traversability conditions.

Reading between the lines

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

  • The same construction could be applied to other dark matter profiles, such as Navarro-Frenk-White or Einasto, to test whether traversable wormholes are generic features of halo models in this gravity theory.
  • If the negative deflection angle is a robust prediction, high-resolution lensing observations of galaxy halos could in principle distinguish such wormholes from ordinary compact objects.
  • A direct check of whether the simplified field equations (38)-(40) exactly solve the full field equations (34)-(36) for the plotted value \chi=0.3 would settle whether the solutions satisfy the theory as stated.
  • The parameter regions where ordinary matter supports the wormhole could be used to constrain the allowed values of \lambda and \chi in f(R,L_m,T) gravity.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper constructs static, constant-redshift wormhole solutions in f(R,L_m,T) gravity with f = R + λL_m + χT, using the Dekel-Zhao dark matter density profile as input. By integrating Eq. (38) (S' = ω r² ρ) with a double power-law density, the authors obtain a shape function S(r), impose S(r0)=r0, and then evaluate the energy conditions, gravitational lensing, and the TOV balance equation for the resulting matter content. The abstract claims that the construction yields traversable wormholes that can be supported by both exotic and ordinary matter, depending on parameters (e.g., the scale radius rc). The paper also claims stability from the TOV equation and reports repulsive gravitational lensing for positive couplings.

Significance. If the central claim were sound, the existence of traversable wormhole solutions supported by ordinary matter (i.e., matter satisfying the null energy condition) in a well-defined extension of GR would be a noteworthy result, and the analytic shape function derived from an observationally motivated dark-matter profile would be a useful addition to the wormhole literature. The paper provides explicit analytic expressions and parameter studies, which is a strength. However, the central claim fails: as shown in the major comments, the branch of solutions that satisfies the NEC at the throat violates the Morris-Thorne flaring-out condition S'(r0)<1, so that branch does not describe a traversable wormhole. In addition, the energy-condition results are effectively imposed by the chosen input density rather than being dynamical predictions of the gravity theory, and the TOV 'stability' check is an identity. I also verified that the reduction from Eqs. (34)-(36) to Eqs. (38)-(40) is algebraically exact (the residual χ-terms vanish identically because ρ+Pr+2Pt=0 for these solutions), so the submitted reader-report concern about the χ-reduction is not a valid defect.

major comments (2)
  1. [Sec. IV, Eq. (32) with Eqs. (38)-(40)] The ordinary-matter branch cannot satisfy the Morris-Thorne throat condition. From (38) and (39), at the throat r0, where S(r0)=r0, one obtains ρ+Pr = (S' − S/r)/(ω r²), so at r0 the sign of ρ+Pr equals the sign of S'(r0)−1 for the positive coupling ω used in the plots. The paper reports in Sec. IV that ρ+Pr is satisfied for rc > 2.492 and violated for rc < 2.492. In the satisfied ('ordinary matter') branch, S'(r0)>1, which violates the flaring-out condition (32), S' < S/r, at the throat. Thus the NEC-satisfying branch is not a traversable wormhole throat; only the NEC-violating branch is traversable. This contradicts the abstract's claim that the solutions can 'sustain both exotic as well as ordinary matter.'
  2. [Sec. IV, derivation of the shape function from Eq. (38)] The energy-condition results are not independent predictions of the modified gravity theory. The shape function is obtained by integrating Eq. (38), S' = ω r² ρ, where ρ is the externally assumed Dekel-Zhao density (42) [or (46)]. Therefore the signs of ρ+Pr and ρ+Pt follow directly from the chosen input profile, not from the dynamics of f(R,L_m,T). In particular, the dichotomy between a NEC-satisfying branch and a flaring-out-satisfying branch is a kinematic consequence of the throat geometry, as shown in the previous comment. The paper should state this circularity explicitly and should not present the energy-condition plots as evidence that the gravity theory can support traversable wormholes with ordinary matter.
minor comments (4)
  1. [Sec. IV, Eq. (42)] Equation (42) as printed has positive exponents and grows as r^{3.5} at large radius, which is inconsistent with the Dekel-Zhao halo profile and with the density expression (46) actually used in the paper; the intended Zhao profile presumably has negative exponents in (42).
  2. [Sec. III, references] The citation [75] appears in the text of Sec. III but is missing from the reference list, and the reference entry [75] in the bibliography is empty.
  3. [Sec. VI, Eqs. (56)-(58)] With constant redshift, the TOV equation (56) reduces to dPr/dr + (2/r)(Pr−Pt) = 0, which is identically satisfied by the field equations (39)-(40). Thus the statement that the solutions are stable is an overstatement; the computation is a consistency check, not a stability analysis.
  4. [Sec. IV, SEC discussion] The observation that ρ+Pr+2Pt=0 is an algebraic identity following from (38)-(40); presenting it as showing that the SEC is satisfied is correct only in the marginal sense of equality, and it carries no independent physical content.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the wormhole solution is constructed from the assumed Dekel-Zhao density profile, and the energy-condition results are consequences of that input rather than disguised inputs.

full rationale

The paper's derivation chain is a standard solution-generating exercise. It assumes the Dekel-Zhao form for rho (Eq. 42), substitutes it into the constant-redshift field equations (38)-(40), integrates to obtain the shape function S(r) (Eq. 44), fixes the integration constant with the throat condition S(r0)=r0, and then evaluates energy conditions and lensing as consequences. The simplified equations (38)-(40) are exactly equivalent to the full system (34)-(36): substituting the simplified expressions into (34)-(36) cancels the chi terms identically because the simplified equations imply rho+Pr+2Pt=0. Thus there is no hidden reduction of the field equations. No parameter is fitted to a target dataset and then renamed a prediction; no uniqueness theorem from the authors' prior work is invoked; the f(R,L_m,T) framework is cited to Haghani & Harko (2021), which has no author overlap with the present paper. The self-citation [53] (Errehymy et al. 2023) is a literature pointer in the introduction and is not load-bearing. The central objects (S, rho, Pr, Pt) are outputs determined by the stated input profile and couplings, not restatements of the conclusions. A separate internal-consistency concern exists: the branch with rho+Pr >= 0 at the throat is incompatible with the Morris-Thorne flaring-out inequality S'(r0)<1, since Eqs. (38)-(40) imply rho+Pr = (S'-1)/(omega r0^2). That is a physical/correctness problem, not a circularity, and therefore does not raise the circularity score.

Assumptions & free parameters 6 free parameters · 7 assumptions · 0 invented entities

The construction rests on the adopted f(R,L_m,T) equations, the L_m=-rho choice, an assumed dark-matter density profile, constant redshift, and an unstated simplification of the field equations that is invalid for nonzero chi. Six free parameters are used for the displayed solutions. No new particles or forces are introduced.

free parameters (6)
  • r0 (throat radius) = 0.85 km
    Chosen by hand for the displayed solutions; the Morris-Thorne throat condition S(r0)=r0 is imposed at this value.
  • rc (scale radius) = varied over [0,6.5] km, e.g. 0.5
    Free parameter of the Dekel-Zhao profile; energy-condition plots scan over it.
  • rho_ch (characteristic density) = 0.9 km^-2
    Free normalization of the Dekel-Zhao profile, chosen for the plots.
  • a (inner slope) = 0.2
    Free slope parameter of the Dekel-Zhao profile, chosen for the plots.
  • lambda = 0.2
    Coupling strength on the matter Lagrangian L_m in f=R+lambda L_m+chi T; chosen for the plots.
  • chi = 0.3
    Coupling strength on T in f=R+lambda L_m+chi T; chosen for the plots, and inconsistently dropped in the simplified field equations.
assumptions (7)
  • domain assumption The Haghani-Harko f(R,L_m,T) field equations (Eq. 9) are the correct equations of motion.
    The paper adopts Eq. (9) from [47] without independent derivation and builds all solutions on it.
  • domain assumption The matter Lagrangian is L_m = -rho, with rho independent of metric derivatives.
    This choice produces Eq. (8) and the simplified field equations; a different L_m would change Theta_mu_nu and all subsequent results.
  • ad hoc to paper Equations (38)-(40) are equivalent to the field equations (34)-(36) for the plotted couplings.
    The paper silently drops the chi terms; the equivalence holds only for chi=0, contradicting the plotted chi=0.3.
  • domain assumption The Dekel-Zhao double power-law profile (Eq. 42) is the actual matter density of the wormhole halo.
    The shape function is obtained by integrating this assumed density, so the traversability and energy-condition results inherit this assumption.
  • domain assumption The wormhole matter is an anisotropic perfect fluid with diagonal stress-energy tensor (Eq. 15).
    The pressure split into radial and tangential parts is assumed, as is standard in wormhole constructions.
  • domain assumption The redshift function is constant.
    Used to set the gravitational force to zero in the TOV equation and to simplify the lensing formula; not derived from a microphysical model.
  • standard math Morris-Thorne conditions (30)-(33) define traversability and flaring-out.
    These are the standard geometric criteria from [61], used to validate the constructed shape function.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Possible wormholes in generalized geometry-matter coupling gravity induced by the Dekel-Zhao dark matter profile." pith.science (2026). https://pith.science/paper/FKBRAAX5

@misc{pith2026250521081,
  author       = {Pith},
  title        = {Pith review of: Possible wormholes in generalized geometry-matter coupling gravity induced by the Dekel-Zhao dark matter profile},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FKBRAAX5}},
  note         = {Machine review of arXiv:2505.21081}
}
abstract

In the late 1980s, Morris and Thorne led in theoretical physics by creating solutions to wormholes and formulating the crucial requirements for safe traversability of wormholes. They found that exotic matter must meet the requirement $P_r + \rho < 0$, where $P_r$ is radial pressure and $\rho$ is energy density. This is a rudimentary grasp of our understanding of general relativity. In this paper, we continue their excellent work by looking at how to build traversable wormhole solutions in an extended theory of gravity. We adopt a process of linearly modifying the matter Lagrangian and the energy-momentum tensor with some coupling strengths $\lambda$ and $\chi$. This may be considered as a special case of linear $f(R, T)$ gravity with matter coupling variability or as an additively separable simple $f(R, L_m, T)$ model. We undertake a detailed analysis of static wormhole solutions with a constant redshift function. This allows us to present our results as a first-order approximation in the $f(R, L_m, T)$ scenario. We derive the wormhole shape function from the Dekel-Zhao dark matter distribution in such a way that our solutions satisfy the needed conditions for traversability as well as the requirement of exotic matter. This is particularly exciting as it shows that wormholes in $f(R, L_m, T)$ gravity can sustain both exotic as well as ordinary matter. To ensure that the shape function meets the requirement of flaring-out and is asymptotically flat, we place some constraints on the couplings. We also examine the gravitational lensing effects, which exhibit a repulsive gravitational force that appears in our extended gravity for positive couplings.

Figures

Figures reproduced from arXiv: 2505.21081 by the authors.

Figure 1
Figure 1. FIG. 1: The shape function for DZ profile halos features a consistent throat radius of [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: This diagram illustrates the matter density, [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: This diagram illustrates the NEC, [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4: This diagram illustrates the NEC, [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 7
Figure 7. Figure 7: FIG. 7: This diagram illustrates the deflection angle, [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: This diagram illustrates the anisotropic force, [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

90 extracted references · 72 canonical work pages

  1. [1]

    NEC: The sum of pressure pi and energy density ρ must be greater than or equal to zero: (pi + ρ) ≥ 0. (23)

  2. [2]

    WEC: The energy density ρ must be non-negative, and the sum of pressure and energy density must also be non-negative: ρ ≥ 0, (pi + ρ) ≥ 0. (24)

  3. [3]

    DEC: The difference between energy density and the absolute value of pressure must be non- negative: (ρ − |pi|) ≥ 0, (25)

  4. [4]

    fictitious

    SEC: The sum of energy density and the total pres- sure must be non-negative, along with the previous condition: (ρ + 3X i=1 pi) ≥ 0, (pi + ρ) ≥ 0. (26) where i refers to spatial indices. 5 When we encounter violations of energy conditions, it often points to the existence of exotic matter. For instance, wormhole configurations specifically require a brea...

  5. [5]

    At the throat r0, the shape function should equal the radial coordinate: ˆS(r0) = r0. (30)

  6. [6]

    The rate at which the shape function changes at this point must not exceed 1: ˆS ′(r0) ≤ 1. (31) (Here, the prime symbol indicates we’re talking about the derivative with respect to the radial coordinate.) Additionally, there’s an important inequality that the shape function must satisfy: ˆS ′(r) < ˆS(r) r . (32) This condition ensures that the ratio of t...

  7. [7]

    These visualizations not only provide a clear picture of the wormholes but also illustrate the complex relation- ships between different regions of spacetime

    This figure includes embedding diagrams that show the wormholes with DZ profile halos, generated by ro- tating them around the z-axis through a full 2 π angle. These visualizations not only provide a clear picture of the wormholes but also illustrate the complex relation- ships between different regions of spacetime. In Fig. 1, we can observe the embeddin...

  8. [8]

    Riess, et al., Astron

    A.G. Riess, et al., Astron. J. 116, 1009, (1998)

    1998

Show all 90 references
  1. [9]

    Perlmutter, et al., Astrophys

    S. Perlmutter, et al., Astrophys. J. 517, 565, (1999)

    1999

  2. [10]

    Hobson, et al., General Relativity: an Introduction for Physicists, Cambridge University Press, Cambridge, UK, (2006)

    M.P. Hobson, et al., General Relativity: an Introduction for Physicists, Cambridge University Press, Cambridge, UK, (2006)

    2006

  3. [11]

    Weinberg, Rev

    S. Weinberg, Rev. Mod. Phys. 61, 1, (1989)

    1989

  4. [12]

    Salucci, Astron

    P. Salucci, Astron. Astrophys. Rev. 27, 2, (2019)

    2019

  5. [13]

    Limousin, et al., Astron

    M. Limousin, et al., Astron. Astrophys. 664, A90, (2022)

    2022

  6. [14]

    Montes, I

    M. Montes, I. Trujillo, Mon. Not. R. Astron. Soc. 482, 2838, (2019)

    2019

  7. [15]

    Smith, J.D

    P.F. Smith, J.D. Lewin, Phys. Rep. 187, 203, (1990)

    1990

  8. [16]

    C. G. Boehmer, et al., Phys. Rev. Lett. 92, 101301, (2004)

    2004

  9. [17]

    Mayet, Phys

    F. Mayet, Phys. Rep. 627, 1, (2016)

    2016

  10. [18]

    Astrophys

    Planck Collaboration, Astron. Astrophys. 641, A6, (2020)

    2020

  11. [19]

    Klypin, et al., Astrophys

    A. Klypin, et al., Astrophys. J. 522, 82, (1999)

    1999

  12. [20]

    Moore, et al., Astrophys

    B. Moore, et al., Astrophys. J. Lett. 524, L19, (1999)

    1999

  13. [21]

    Dubinski, R.G

    J. Dubinski, R.G. Carlberg, Astrophys. J. 378, 496, (1991)

    1991

  14. [22]

    Walker, J

    M.G. Walker, J. Pe˜ narrubia, Astrophys. J. 742, 20, (2011)

    2011

  15. [23]

    Bull, et al., Phys

    P. Bull, et al., Phys. Dark Universe 12, 56, (2016)

    2016

  16. [24]

    De Felice, S

    A. De Felice, S. Tsujikawa, Living Rev. Relativ. 13, 3, (2010)

    2010

  17. [25]

    Amendola, et al., Phys

    L. Amendola, et al., Phys. Rev. D 75, 083504, (2007)

    2007

  18. [26]

    Nojiri, S.D

    S. Nojiri, S.D. Odintsov, Phys. Rev. D 74, 086005, (2006)

    2006

  19. [27]

    Capozziello, et al., Phys

    S. Capozziello, et al., Phys. Rev. D 71, 043503, (2005)

    2005

  20. [28]

    Capozziello, M

    S. Capozziello, M. De Laurentis, Ann. Astrophys. 524, 545, (2012)

    2012

  21. [29]

    Boehmer, et al., Astropart

    C.G. Boehmer, et al., Astropart. Phys. 29, 386, (2008)

    2008

  22. [30]

    Casado-Turrion, et al., Phys

    A. Casado-Turrion, et al., Phys. Rev. D 108, 064006, (2023)

    2023

  23. [31]

    H. Aman, Z. Yousaf, JHEAp 47, 100380 (2025)

    2025

  24. [32]

    Khan and Z

    S. Khan and Z. Yousaf, Phys. Scripta 99, no.9, 095304 (2024)

    2024

  25. [33]

    S. Khan, A. Adeel, Z. Yousaf, Eur. Phys. J. C 84, no.6, 572 (2024)

    2024

  26. [34]

    A. M. Albalahi, M. Z. Bhatti, A. Ali, S. Khan, Eur. Phys. J. C 84, no.3, 293 (2024)

    2024

  27. [35]

    Yousaf, S

    Z. Yousaf, S. Khan, N. B. Turki, T. Suzuki, Chin. J. Phys. 89, 1595-1610 (2024)

    2024

  28. [36]

    Yousaf, Ind

    Z. Yousaf, Ind. J. Phys. 99, 1-13 (2025)

    2025

  29. [37]

    Yousaf, K

    Z. Yousaf, K. Bamba, B. Almutairi, Y. Hashimoto and S. Khan, Phys. Dark Univ. 46, 101629 (2024)

    2024

  30. [38]

    Yousaf, Phys

    Z. Yousaf, Phys. Dark Univ. 48, 101884 (2025)

    2025

  31. [39]

    M. Z. Bhatti, Z. Yousaf and Z. Tariq, Chin. J. Phys. 72, 18-37 (2021)

    2021

  32. [40]

    Malik, T

    A. Malik, T. Naz, A. Qadeer, M. F. Shamir and Z. Yousaf, Eur. Phys. J. C 83, no.6, 522 (2023)

    2023

  33. [41]

    F. S. N. Lobo and M. A. Oliveira, Phys. Rev. D 80, 104012 (2009)

    2009

  34. [42]

    Bahamonde, M

    S. Bahamonde, M. Jamil, P. Pavlovic and M. Sossich, Phys. Rev. D 94, no.4, 044041 (2016)

    2016

  35. [43]

    Karakasis, E

    T. Karakasis, E. Papantonopoulos and C. Vlachos, Phys. Rev. D 105, no.2, 024006 (2022)

    2022

  36. [44]

    Capozziello and M

    S. Capozziello and M. De Laurentis, Phys. Rept. 509, 167-321 (2011)

    2011

  37. [45]

    S. E. Perez Bergliaffa, Phys. Lett. B 642, 311-314 (2006)

    2006

  38. [48]

    N. M. Garcia and F. S. N. Lobo, Phys. Rev. D 82, 104018 (2010)

    2010

  39. [49]

    N. M. Garcia, T. Harko, F. S. N. Lobo and J. P. Mimoso, Phys. Rev. D 83, 104032 (2011)

    2011

  40. [50]

    Harko, F

    T. Harko, F. S. N. Lobo, S. Nojiri and S. D. Odintsov, Phys. Rev. D 84, 024020 (2011)

    2011

  41. [51]

    S. D. Odintsov and D. S´ aez-G´ omez, Phys. Lett. B725, 437-444 (2013)

    2013

  42. [52]

    Liu and M

    D. Liu and M. J. Reboucas, Phys. Rev. D 86, 083515 (2012) 12

    2012

  43. [53]

    Baccetti, P

    V. Baccetti, P. Martin-Moruno and M. Visser, JHEP 08, 148 (2012)

    2012

  44. [54]

    Haghani, T

    Z. Haghani, T. Harko, Eur. Phys. J. C 81, 615, (2021)

    2021

  45. [55]

    Morris, K.S

    M.S. Morris, K.S. Thorne, Am. J. Phys. 56, 395, (1988)

    1988

  46. [56]

    Curiel, A primer on energy conditions, Springer, New York, (2017)

    E. Curiel, A primer on energy conditions, Springer, New York, (2017)

    2017

  47. [57]

    Luis Rosa, P.M

    J. Luis Rosa, P.M. Kull, Eur. Phys. J. C82, 1154, (2022)

    2022

  48. [58]

    Carr, et al., Class

    B.J. Carr, et al., Class. Quantum Gravity 27, 183101, (2010)

    2010

  49. [59]

    Bronnikov, et al., Symmetry 12, 1306, (2020)

    K.A. Bronnikov, et al., Symmetry 12, 1306, (2020)

    2020

  50. [60]

    Errehymy, et al., Phys

    A. Errehymy, et al., Phys. Dark Univ. 41, 101258 (2023)

    2023

  51. [61]

    Nandi, et al., Phys

    K.K. Nandi, et al., Phys. Rev. D 95, 104011, (2017)

    2017

  52. [62]

    Ohgami, N

    T. Ohgami, N. Sakai, Phys. Rev. D 91, 124020, (2015)

    2015

  53. [63]

    Paul, et al., JCAP 03, 055, (2020)

    S. Paul, et al., JCAP 03, 055, (2020)

    2020

  54. [64]

    Harko, Phys

    T. Harko, Phys. Rev. D 90, no.4, 044067 (2014)

    2014

  55. [65]

    S. I. dos Santos, G. A. Carvalho, P. H. R. S. Moraes, C. H. Lenzi and M. Malheiro, Eur. Phys. J. Plus 134, no.8, 398 (2019)

    2019

  56. [66]

    Hochberg, M

    D. Hochberg, M. Visser, Phys. Rev. D 58, 044021 (1998)

    1998

  57. [67]

    Hochberg, D., Visser, M.: Phys. Rev. Lett. 81 746 (1998)

    1998

  58. [68]

    M. S. Morris, K. S. Thorne, Am. J. Phys. 56, 395-412 (1988)

    1988

  59. [69]

    Capozziello, F

    S. Capozziello, F. S. N. Lobo and J. P. Mimoso, Phys. Lett. B 730, 280-283 (2014)

    2014

  60. [70]

    Santos, J

    J. Santos, J. S. Alcaniz, M. J. Reboucas and F. C. Car- valho, Phys. Rev. D 76, 083513 (2007)

    2007

  61. [71]

    Bertolami and M

    O. Bertolami and M. C. Sequeira, Phys. Rev. D 79, 104010 (2009)

    2009

  62. [72]

    Akramov, et al., Europhysics Letters, 147, 62001 (2024)

    M. Akramov, et al., Europhysics Letters, 147, 62001 (2024)

    2024

  63. [73]

    Rakhmanov, et al., Optical high harmonic generation in Dirac materials

    S. Rakhmanov, et al., Optical high harmonic generation in Dirac materials. The Eur. Phys. J. B, 98, 35 (2025)

    2025

  64. [74]

    Turimov, et al., Phys

    B. Turimov, et al., Phys. Dark Univ. 48, 101876 (2025)

    2025

  65. [75]

    Razina, et al., Eur

    O. Razina, et al., Eur. Phys. J. Plus 126, 85 (2011)

    2011

  66. [76]

    Kumar, et al., Eur

    A. Kumar, et al., Eur. Phys. J. Plus 138, no.12, 1071 (2023)

    2023

  67. [77]

    F. J. Tipler, Phys. Rev. D 17, 2521-2528 (1978)

    1978

  68. [78]

    M. S. Morris, K. S. Thorne and U. Yurtsever, Phys. Rev. Lett. 61, 1446-1449 (1988)

    1988

  69. [79]

    Wang and K

    J. Wang and K. Liao, Class. Quant. Grav. 29, 215016 (2012)

    2012

  70. [80]

    Atazadeh, et al., Int

    K. Atazadeh, et al., Int. J. Mod. Phys. D 18, 1101-1111 (2009)

    2009

  71. [81]

    F. D. Albareti, J. A. R. Cembranos, A. de la Cruz- Dombriz and A. Dobado, JCAP 07, 009 (2013) [75]

    2013

  72. [82]

    Chatterjee, D

    S. Chatterjee, D. A. Easson and M. Parikh, Class. Quant. Grav. 30, 235031 (2013)

    2013

  73. [83]

    Zhao, Mon

    H. Zhao, Mon. Not. Roy. Astron. Soc. 278, 488-496 (1996)

    1996

  74. [84]

    Zhao, Mon

    H. Zhao, Mon. Not. Roy. Astron. Soc. 287, 525 (1997)

    1997

  75. [85]

    Freundlich et al., Mon

    J. Freundlich et al., Mon. Not. Roy. Astron. Soc. 499, 2912-2933 (2020)

    2020

  76. [86]

    Shaikh et al., JCAP 07, 028 (2019)

    R. Shaikh et al., JCAP 07, 028 (2019)

    2019

  77. [87]

    Virbhadra, G.F.R

    K.S. Virbhadra, G.F.R. Ellis, Phys. Rev. D 62, 084003 (2000)

    2000

  78. [88]

    Virbhadra, G.F.R

    K.S. Virbhadra, G.F.R. Ellis,, Phys. Rev. D 65, 103004 (2002)

    2002

  79. [89]

    Claudel, K.S

    C.M. Claudel, K.S. Virbhadra, G.F.R. Ellis, J. Math. Phys. 42, 818-838 (2001)

    2001

  80. [90]

    Oppenheimer, G.M

    J.R. Oppenheimer, G.M. Volkoff, Phys. Rev. 55, 374 (1939)

    1939

  81. [91]

    Gorini et al., Phys

    V. Gorini et al., Phys. Rev. D 78, 064064 (2008)

    2008

  82. [92]

    Kuhfittig, Fundamental J

    P.K.F. Kuhfittig, Fundamental J. Mod. Phys. 14, 23-31 (2020)

    2020

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.