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REVIEW 3 major objections 4 minor 71 references

Structure Scalars for Charged Dissipative Spherical Collapse in $f(R, T)$ Gravity

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Four Riemann-derived scalars characterize charged dissipative collapse in f(R,T) gravity.

desk verdict The paper's central claim about mass-function influencing density inhomogeneity comes from an algebraic mistake in Eq. (70). read the letter →

arxiv 2505.23605 v3 pith:NAVPRY7M submitted 2025-05-29 gr-qc hep-thmath-phmath.MP

Uttaran Ghosh , Sarbari Guha This is my paper
classification gr-qchep-thmath-phmath.MP
keywords structurescalarsf(RT)gravitysphericalgravitationalcollapsedissipativefluidsenergydensityinhomogeneitycomplexityfactorelectricchargejunctionconditions
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 aims to show that four structure scalars built from the orthogonal splitting of the Riemann tensor carry the physical fingerprint of a charged, shearing, dissipative spherical collapse in f(R,T) gravity. It derives explicit expressions for the scalars in terms of metric coefficients and matter variables, and shows that energy-density inhomogeneity is controlled by the scalar $X_{TF}$ together with the mass function, while the expansion and shear scalars are controlled by $Y_T$ and $Y_{TF}$. Charge modifies all four scalars and increases the mass-energy content. The paper also derives junction conditions for matching the charged interior to a generalized Vaidya exterior and re-expresses the energy conditions in terms of the scalars. If the framework is right, the same scalar toolkit used in general relativity can classify collapse outcomes in modified gravity.

What carries the argument

The central object is the orthogonal splitting of the Riemann tensor into the tensors $Y_{\alpha\beta}=R_{\alpha\gamma\beta\delta}u^\gamma u^\delta$ and $X_{\alpha\beta}={}^*R^*_{\alpha\gamma\beta\delta}u^\gamma u^\delta$, whose trace parts $X_T$, $Y_T$ and trace-free parts $X_{TF}$, $Y_{TF}$ are the structure scalars. These scalars bridge geometry and matter: through the f(R,T) field equations they are re-expressed in terms of effective energy density, pressure anisotropy, shear viscosity, heat flux, and charge, and they feed the evolution equations for expansion and shear via the Raychaudhuri-type identities (68) and (69). The load-bearing identity is Eq. (70), which ties the radial derivative of $X_{TF}$ to dissipation, geometry, and the mass function.

What would settle it

Take an explicit charged, shearing, dissipative interior with f(R,T)=R+\$\lambda$ T and a specified matter Lagrangian, compute both sides of Eq. (70) directly from the metric, and check equality; if any solution of the f(R,T) field equations fails the identity, or if the identity changes when $L_m$ is switched from $-\rho$ to another Lagrangian such as $p$, the claimed universal classification does not hold.

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Extended reading notes

Core claim

In f(R,T) gravity, the trace parts $X_T$, $Y_T$ and trace-free parts $X_{TF}$, $Y_{TF}$ of the tensors obtained from orthogonal splitting of the Riemann tensor determine the physical parameters of charged dissipative spherical collapse. The central relation is Eq. (70): $(X_{TF}+\rho_{eff}/(2f_R))' = -3(C'/C)X_{TF} + \cdots + 9(C'/C^4)(m - Q^2/(2C))$, which shows that in the absence of dissipation the inhomogeneity of the energy density is governed by $X_{TF}$ and the mass function, with charge appearing explicitly. The paper further shows that charge increases $X_T$ and $Y_{TF}$, decreases $X_{TF}$ and $Y_T$, contributes to the mass-energy content, and identifies $Y_{TF}$ as the complexity factor that vanishes for isotropic, homogeneous, non-dissipative configurations in the general-relativistic limit.

Load-bearing premise

The derivation of the central relations assumes the interior matter Lagrangian is $L_m=-\rho$, and f(R,T) field equations depend on this choice, so the general claims about the structure scalars are established only for that choice.

Editorial extensions

If this is right

  • In a non-dissipative charged collapse, energy-density inhomogeneity is not an independent degree of freedom: it is fixed by $X_{TF}$ and the mass function together.
  • Charge acts as a control parameter, raising $X_T$ and $Y_{TF}$, lowering $X_{TF}$ and $Y_T$, and increasing the mass-energy content, so the same collapse profile evolves differently with charge.
  • Expansion during collapse is governed by $Y_T$ and shear by $Y_{TF}$, meaning the structure scalars can serve as evolution variables instead of the metric coefficients.
  • $Y_{TF}$ is the complexity factor in this gravity theory, and the condition $Y_{TF}=0$ defines minimal-complexity collapse, reducing to the general-relativistic condition in the appropriate limit.
  • The junction conditions constrain the interior and exterior matter Lagrangians and their derivatives, so f(R,T) models with higher-order curvature terms must satisfy extra boundary conditions beyond metric matching.

Reading between the lines

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

  • The authors do not state it, but the scalar-to-matter mapping suggests the four scalars could serve as observational proxies: a reconstructed $X_{TF}$ profile from a collapse simulation would directly give the inhomogeneity and mass function without solving the full field equations.
  • Because the central derivation assumes $L_m=-\rho$, a natural test is to rerun Eqs. (54)-(57) with another matter Lagrangian such as $L_m=p$; a changed dictionary between scalars and matter variables would delimit how general the classification truly is.
  • The energy-condition reformulation hints at a practical filter: checking which f(R,T) models satisfy the scalar versions of the energy conditions could identify viable collapse candidates, though the paper only outlines the possibility.
  • A direct numerical check of Eq. (70) on an explicit charged collapsing solution, such as a charged interior matched to a generalized Vaidya exterior, would confirm the identity beyond the purely algebraic derivation.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The manuscript extends the structure-scalar formalism, previously developed in general relativity, to charged spherically symmetric dissipative collapse in f(R,T) gravity. It defines the four structure scalars via the orthogonal splitting of the Riemann tensor, relates them to the matter variables through the f(R,T) field equations, and uses one of these relations to discuss the evolution of energy-density inhomogeneity. It also presents a brief discussion of the complexity factor, the f(R,T) junction conditions with a generalized Vaidya exterior, and the energy conditions expressed partly in terms of the structure scalars. The advertised central physical result is Eq. (70), which is claimed to show that, in the absence of dissipation, the energy-density inhomogeneity is influenced by the structure scalar X_TF and by the mass function of the collapsing matter.

Significance. If the derivations are correct, the paper would provide a useful extension of the structure-scalar framework to f(R,T) gravity, including the effect of electric charge, and would give a candidate complexity factor for dissipative charged collapse. The manuscript contains many analytic expressions that reduce to known general-relativistic limits and it explicitly treats several limiting cases, which is helpful for future applications. However, the central inhomogeneity equation contains an algebraic error that invalidates a key advertised conclusion as stated; the issue is localized and fixable, but the paper cannot be accepted in its present form.

major comments (3)
  1. [§VI, Eq. (70)] Equation (70) is inconsistent with the preceding equations. Using Eq. (52) in the integrand of Eq. (63) gives I = 3(m - Q^2/(2C))/C^3, so that X_TF + rho_eff/(2 f_R) = I. Differentiating with respect to r and using Eq. (52) again yields (X_TF + rho_eff/(2 f_R))' = -3(C'/C) X_TF + (Theta+3sigma)/(2 f_R)(hat_q B + psi_q/A), with no independent 9 C'/C^4 (m - Q^2/(2C)) term. The extra term in the printed Eq. (70) double-counts the mass contribution. A direct check in the static general-relativistic limit (f = R, Q = 0, U = 0, q = 0) shows that the printed equation would force m = 0 when combined with Eq. (52), whereas the corrected equation reproduces the standard relation. Equation (72) inherits the same error. Since the abstract and the concluding bullet about energy-density inhomogeneity are based on Eq. (70), the equation must be corrected and the corresponding physical claim reworded; after correction, the mass function enters through X_TF rather than as an independent additive driver.
  2. [§V and §VIII] The field equations and all structure-scalar relations (54)-(57) and (64)-(67) are derived under the explicit choice L_m = -rho, introduced in Section V before Eq. (39). In Section VIII, however, the interior matter Lagrangian is left completely unspecified and is denoted L_m_int. In f(R,T) gravity the field equations depend explicitly on the matter Lagrangian, so the structure-scalar expressions are not universal: a different choice, such as L_m = p or a field-dependent Lagrangian, changes the equations and hence the relations among X_T, X_TF, Y_T, Y_TF and the matter variables. The paper should either extend the derivation to a general L_m before specializing, or state prominently that the structure-scalar section applies only to L_m = -rho. As written, the broad claim in the abstract that these scalars characterize charged dissipative collapse 'in f(R,T) gravity' is overbroad.
  3. [§VI, Eqs. (38), (63)-(72)] Several load-bearing steps are asserted without showing the intervening algebra. In particular, Eq. (38) for Z^2 is stated without derivation, and Eqs. (63)-(72) involve nontrivial integrations and substitutions. Given that Eq. (70) contains the algebraic error described above, the absence of these intermediate steps prevents the reader from independently verifying the remaining relations. The authors should provide the derivation, or at least a detailed appendix, for Eq. (38) and for the replacement of the integral I by the mass function in Eqs. (63)-(65).
minor comments (4)
  1. [Abstract and Introduction] There are several typographical errors, including 'spernova' in the introduction and 'enrgy momentum tensor' in Section IV; the manuscript would benefit from a careful proofreading pass.
  2. [§VI, after Eq. (70)] The bullet list following Eq. (70) states that X_TF and the mass function m together influence the energy-density inhomogeneity. After the correction of Eq. (70), this statement should be clarified: m enters through the definition of X_TF and through the combination X_TF + rho_eff/(2 f_R), not as an independent term in the evolution equation.
  3. [§VII, Eq. (110)] The condition f_R,Y Y = 0 is deduced from R_11 = 0 for the generalized Vaidya metric; it would be helpful to state that this is a coordinate-dependent component condition and to comment on whether the resulting constraint is gauge invariant in the intended matching context.
  4. [§VI, notation around Eq. (63)] The notation for the heat-flux combinations is easy to confuse: bar_q denotes q + epsilon in Eq. (48), while hat_q denotes bar_q (1 + f_T) in Eq. (63). A short note defining all hatted and barred quantities in one place would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the structure scalars are independently defined from the Riemann tensor and are related to the matter variables by direct substitution of the f(R,T) field equations, with no fitted parameter, no prediction reduced by construction, and no load-bearing self-citation.

full rationale

The derivation chain is self-contained with respect to the circularity criteria. The structure scalars X_T, X_TF, Y_T, and Y_TF are introduced in Section III from the orthogonal splitting of the Riemann tensor, independently of any f(R,T) matter content. Their later expressions in terms of the matter variables, Eqs. (54)-(57) and (64)-(67), are obtained by substituting the f(R,T) field equations, Eqs. (40)-(43), together with the definitions of the effective matter variables in Eqs. (58)-(61). These are algebraic consequences of the definitions and field equations, not predictions fitted to the same quantities. The central inhomogeneity relation, Eq. (70), is stated as a radial derivative of X_TF combined with the mass-function derivative, Eq. (52); whether the printed coefficient is algebraically correct is a separate correctness question, not a circularity. No parameter is fitted to data, no output quantity is used to define an input, and no uniqueness theorem or ansatz is imported from the authors' prior work. The self-citations [40,41] serve as background references and as consistency checks for the junction-condition limit, not as the justification of the central structure-scalar relations. The only notable issue is the internal choice L_m = -rho in Section V while Section VIII leaves L_m unspecified, but this is an assumption-consistency gap rather than a circular derivation. Therefore the appropriate circularity score is 0.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central relations rest on the f(R,T) field equations of Harko et al., the specific energy-momentum tensor for anisotropic dissipative matter, the choice L_m = -rho, and the standard Riemann splitting. No new entities are invented. The charge function Q(r) and the coupling lambda in the linear model are free inputs, not fitted.

free parameters (2)
  • lambda (coupling constant)
    Introduced in the linear f(R,T) model f(R,T)=R+lambda*T (Section VI.B) to illustrate the results; not fitted to data, but a free theory parameter.
  • Q(r) (electric charge function)
    Charge distribution inside the collapsing sphere, constrained by Maxwell equations (Eqs. 17-23); treated as an input function rather than a fitted constant.
assumptions (6)
  • domain assumption The f(R,T) field equations as given in Eq. (2)
    The paper adopts the Harko et al. formulation of f(R,T) gravity with the given G_mu_nu = (1/f_R)[...] form. This is the theoretical framework of the paper.
  • domain assumption The interior energy-momentum tensor has the anisotropic dissipative form of Eq. (12)
    The collapsing matter is modeled as an anisotropic fluid with heat flux, free-streaming radiation, and shear viscosity.
  • ad hoc to paper The matter Lagrangian L_m = -rho is chosen for the interior
    Assumed in Section V to derive the effective field equations (39)-(43); not derived from microphysics.
  • domain assumption The exterior spacetime is the generalized Vaidya metric, Eq. (95)
    Assumed for the exterior of the collapsing matter, standard for radiating collapse.
  • domain assumption The junction conditions of Rosa [63], Eqs. (125)-(131)
    Assumed for matching interior and exterior in f(R,T) gravity.
  • standard math The standard orthogonal splitting of the Riemann tensor (Bel, Gomez-Lobo) as in Section III
    The structure scalars are defined following Herrera et al. [5].

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Pith. "Pith review of Structure Scalars for Charged Dissipative Spherical Collapse in $f(R, T)$ Gravity." pith.science (2026). https://pith.science/paper/NAVPRY7M

@misc{pith2026250523605,
  author       = {Pith},
  title        = {Pith review of: Structure Scalars for Charged Dissipative Spherical Collapse in $f(R, T)$ Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NAVPRY7M}},
  note         = {Machine review of arXiv:2505.23605}
}
abstract

We examine the structure scalars constructed from the orthogonal splitting of the Riemann tensor for the spacetime metric describing the interior of a charged matter configuration undergoing dissipative collapse in the framework of $f(R,T)$ gravity (where $R$ and $T$ are the Ricci scalar and the trace of energy-momentum tensor, respectively), and also the way these quantities influence the various physical parameters of the collapsing matter. In absence of dissipation, the energy density inhomogeneity is found to be influenced by the structure scalar $X_{TF}$ and the mass-function of the collapsing matter. Further, the presence of charge affects the structure scalars and the total mass-energy content. The dependence of the various physical parameters like heat dissipation, energy density inhomogeneity, evolution of the expansion scalar, the shear scalar, effective homogeneous energy density, and pressure anisotropy on the structure scalars, have been clearly indicated along with a discussion on the complexity factor of the collapsing configuration. The $f(R,T)$ junction conditions have been presented, showing the matching conditions for the matter Lagrangian and their derivatives at the boundary. The energy conditions are also presented and the possibility of violation of the Strong Energy Condition has been discussed.

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Reference graph

Works this paper leans on

71 extracted references · 65 canonical work pages

  1. [1]

    J. R. Oppenheimer and H. Snyder, Phys. Rev. 56, 455 (1939)

    1939

  2. [2]

    Datt, Zeitschrift f ¨ur Physik 108, 314 (1938)

    B. Datt, Zeitschrift f ¨ur Physik 108, 314 (1938)

    work page 1938

  3. [3]

    E. N. Glass, Phys. Lett. A 86, 351 (1981)

    work page 1981

  4. [4]

    N. O. Santos, Mon. Not. R. Astron. Soc. 216, 403 (1985)

    1985

  5. [5]

    Herrera, J

    L. Herrera, J. Ospino, A. Di Prisco, E. Fuenmayor, and O. Troconis, Phys. Rev. D 79, 064025 (2009)

    work page 2009

  6. [6]

    A. K. G. de Oliveira, N. O. Santos, C. A. Kolassis, Mon. Not. R. Astron. Soc. 216, 1001 (1985)

    work page 1985

  7. [7]

    A. K. G. de Oliveira, J.A.F. de Pacheco, N.O. Santos, Mon. Not. R. Astron. Soc. 220, 405 (1986)

    work page 1986

  8. [8]

    Herrera, G

    L. Herrera, G. Denmat, N. O. Santos, Mon. Not. R. astr. Soc. 237, 257 (1989)

    work page 1989

Show all 71 references
  1. [9]

    R. Chan, S. Kichenassamy, G. Le Denmat, N.O. Santos, Mon. Not. R. Astron. Soc. 239, 91 (1989)

    1989

  2. [10]

    R. Chan, L. Herrera, N.O. Santos, Mon. Not. R. Astron. Soc. 265, 533 (1993)

    1993

  3. [11]

    R. Chan, L. Herrera, N. O. Santos, Mon. Not. R. astr. Soc. 267, 637 (1994)

    1994

  4. [12]

    Chan, Mon

    R. Chan, Mon. Not. R. Astron. Soc. 316, 588±604 (2000)

    2000

  5. [13]

    Herrera, N

    L. Herrera, N. O. Santos, Phys. Rev. D, Vol. 70, 084004 (2004)

    2004

  6. [14]

    Herrera, N.O

    L. Herrera, N.O. Santos, G. Le Denmat, Gen. Relativ. Gravit. 44, 1143 (2012)

    2012

  7. [15]

    A. K. G. de Oliveira, N. O. Santos, The Astrophysical Journal, 312, 640 (1987)

    1987

  8. [16]

    Bel, Ann

    L. Bel, Ann. Inst. Henri Poincare’ 17, 37 (1961)

    1961

  9. [17]

    Quantum Grav

    Alfonso Garcia-Parrado Gomez-Lobo, Class. Quantum Grav. 25, 015006 (2008)

    2008

  10. [18]

    Herrera, A

    L. Herrera, A. Di Prisco, and J. Ibanez, Phys. Rev. D 84, 107501 (2011)

    2011

  11. [19]

    Herrera, A

    L. Herrera, A. Di Prisco, J. Ospino, Gen Relativ Gravit 44, 2645 (2012)

    2012

  12. [20]

    Sharif, and M

    M. Sharif, and M. Bhatti, Gen Relativ Gravit 44, 2811 (2012)

    2012

  13. [21]

    Sharif, and M

    M. Sharif, and M. Bhatti, Mod. Phys. Lett. A, Vol. 27, 1250141 (2012)

    2012

  14. [22]

    Sharif, and M

    M. Sharif, and M. Bhatti, Astrophys Space Sci 349, 995 (2014)

    2014

  15. [23]

    Sharif, and M

    M. Sharif, and M. Bhatti, Int. Journal of Mod. Phys. D, Vol. 24, 1550014 (2015)

    2015

  16. [24]

    Sharif, and R

    M. Sharif, and R. Manzoor, Phys. Rev. D 91, 024018 (2015)

    2015

  17. [25]

    Sharif, and R

    M. Sharif, and R. Manzoor, Astrophys. Space Sci. 359, 17 (2015)

    2015

  18. [26]

    Riess et al., Astron

    Supernova Search Team Collaboration, A.G. Riess et al., Astron. J. 116, 1009 (1998)

    1998

  19. [27]

    Perlmutter, arXiv:astro-ph/9812133v1 8 Dec 1998

    S. Perlmutter, arXiv:astro-ph/9812133v1 8 Dec 1998

    1998 arXiv

  20. [28]

    De Felice, S

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

    2010

  21. [29]

    T. P. Sotiriou, V. Faraoni, Rev. of Mod. Phys., vol. 82, (2010)

    2010

  22. [30]

    Capozziello, M

    S. Capozziello, M. De Laurentis, Physics Reports, Vol. 509, 167 (2011)

    2011

  23. [31]

    Nojiri, S

    S. Nojiri, S. D. Odintsov, Physics Reports, Vol. 505, 59 (2011)

    2011

  24. [32]

    Nojiri, S

    S. Nojiri, S. D. Odintsov and V. K. Oikonomou, Phys. Rept. 692, 1 (2017)

    2017

  25. [33]

    A. V. Astashenok, K. Mosani, S. D. Odintsov and G. C. Samanta, Int. J. Geom. Meth. Mod. Phys. 16, 1950035 (2019)

    2019

  26. [34]

    Sharif, Z

    M. Sharif, Z. Yousaf, Astrophys. Space Sci. 357, 49 (2015)

    2015

  27. [35]

    M. Z. Bhatti, Z. Yousaf, Z. Tariq, Eur. Phys. J. C 81, 16 (2021)

    2021

  28. [36]

    M. Z. Bhatti, Z. Yousaf, Z. Tariq, Phys. Scr. 96, 115301 (2021)

    2021

  29. [37]

    M. Z. Bhatti, Z. Yousaf, Z. Tariq, Chinese Journal of Physics 72, 18 (2021)

    2021

  30. [38]

    Harko, F

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

    2011

  31. [39]

    P. K. Sahoo, P. H. R. S. Moraes, Parbati Sahoo, and B. K. Bishi, Eur. Phys. J. C 78, 736 (2018)

    2018

  32. [40]

    Guha, and U

    S. Guha, and U. Ghosh, Eur. Phys. J. Plus 136, 460 (2021)

    2021

  33. [41]

    Ghosh, and S

    U. Ghosh, and S. Guha, Gen Relativ Gravit 57, 47 (2025)

    2025

  34. [42]

    Yousaf, K

    Z. Yousaf, K. Bamba, and M. Z. Bhatti, Phys. Rev. D 93, 064059 (2016). 22

    2016

  35. [43]

    structure scalars

    these authors examined irregularity factors for a self-gravitating spherical star evolving in the presence of an imperfect fluid in f (R, T) gravity, where they demonstrated that, as the complexity of the matter with the anisotropic stresses increases, the inhomogeneity factor...

  36. [44]

    Yousaf, K

    Z. Yousaf, K. Bamba, and M. Z. Bhatti, Phys. Rev. D 93, 124048 (2016)

    2016

  37. [45]

    Hussain, M

    T. Hussain, M. Khurshudyan, S. Ahmed, As. Khurshudyan, Int. Journal of Mod. Phys. D, 26, 1750155 (2017)

    2017

  38. [46]

    Yousaf, M.Z

    Z. Yousaf, M.Z. Bhatti and U. Farwa, Ann. Phys. 433, 168601 (2021)

    2021

  39. [47]

    Herrera, Phys

    L. Herrera, Phys. Rev. D 97, 044010 (2018)

    2018

  40. [48]

    Herrera, A

    L. Herrera, A. Di Prisco, and J. Ospino, Phys. Rev. D 98, 104059 (2018)

    2018

  41. [49]

    Herrera, A

    L. Herrera, A. Di Prisco, and J. Ospino, Phys. Rev. D 99, 044049 (2019)

    2019

  42. [50]

    Herrera, A

    L. Herrera, A. Di Prisco, and J. Carot, Phys. Rev. D 99, 124028 (2019)

    2019

  43. [51]

    Sharif, and I

    M. Sharif, and I. I. Butt, Eur. Phys. J. C 78, 688 (2018)

    2018

  44. [52]

    Sharif and S

    M. Sharif and S. Tariq, Mod. Phys. Lett. A 35, 2050231 (2020)

    2020

  45. [53]

    Yousaf, M

    Z. Yousaf, M. Z. Bhatti, and T. Naseer, Phys. of the Dark Univ. 28, 100535 (2020)

    2020

  46. [54]

    Di Prisco, L

    A. Di Prisco, L. Herrera, G. Le Denmat, M. A. H. MacCallum and N. O. Santos, Phys. Rev. D 76, 064017 (2007)

    2007

  47. [55]

    Sharif, A

    M. Sharif, A. Waseem, Gen. Relativ. Gravit. 50, 78 (2018)

    2018

  48. [56]

    Ahmed, G

    R. Ahmed, G. Abbas, Chinese Jour. of Phys. 65, 177 (2020)

    2020

  49. [57]

    https://github.com/grtensor/grtensor

    We used GRTensor package on MAPLE software. https://github.com/grtensor/grtensor

  50. [58]

    J. S. Gagnon and J. Lesgourgues, Journ. of Cosmo. and Astroparticle Phys. 09, 026 (2011)

    2011

  51. [59]

    Golden Oldie

    A Raychaudhuri, Relativistic cosmology, I, Phys. Rev. 98, 1123 (1955). Reprinted as “Golden Oldie” in Gen. Relativ. Gravit. 32, 749 (2000)

    1955

  52. [60]

    A. Wang, Y. Wu : Gen. Relativ. Gravit. 31, 114 (1999)

    1999

  53. [61]

    S. W. Hawking, and G. F. R. Ellis, The Large Scale Structure Of Space-Time , Camridge Monographs on Mathematical Physics, Cambridge University Press, (1973)

    1973

  54. [62]

    Darmois, ‘Les equations de la gravitation Einsteinienne,’ M´ emorial de Sciences Math´ ematiques, Fascicule XXV, ch V (1927)

    G. Darmois, ‘Les equations de la gravitation Einsteinienne,’ M´ emorial de Sciences Math´ ematiques, Fascicule XXV, ch V (1927)

    1927

  55. [63]

    Israel, Nuovo Cimento 44, 1 (erratum B 49, 463)(1966)

    W. Israel, Nuovo Cimento 44, 1 (erratum B 49, 463)(1966)

    1966

  56. [64]

    J. L. Rosa, Phys. Rev. D 103, 104069 (2021)

    2021

  57. [65]

    C. A. Kolassis, N. O. Santos, and D. Tsoubelis, Class. Quantum Grav. 5, 1329 (1988)

    1988

  58. [66]

    Yousaf, M

    Z. Yousaf, M. Z. Bhatti, T. Naseer, Eur. Phys. J. Plus 135, 323 (2020)

    2020

  59. [67]

    A.V.Astashenok, S.Capozziello, S.D.Odintsov, V.K.Oikonomou, Phys. Lett. B 811, 135910 (2020)

    2020

  60. [68]

    A.V.Astashenok, S.Capozziello, S.D.Odintsov, V.K.Oikonomou, Phys. Lett. B 816, 136222 (2021)

    2021

  61. [69]

    S. A. Mardan, Maha, R. Manzoor, M. B. Riaz, Eur. Phys. J. Plus 140, (2025)

    2025

  62. [70]

    Sharif, H

    M. Sharif, H. R. Kausar, Astrophys. Space Sci. 331, 281 (2011)

    2011

  63. [71]

    Casado-Turri´ on, A

    A. Casado-Turri´ on, A. Dobado, A. dela Cruz-Dombriz, Phys. Rev. D105, 084060 (2022)

    2022

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