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Diverse regular spacetimes using a parametrised density profile

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Pith's one-line read A Dekel-Zhao density profile yields a new regular black hole and a horizonless defect spacetime in general relativity, plus a thin-shell stellar model based on the defect interior.

arxiv 2504.12042 v2 pith:3T5GNMJV submitted 2025-04-16 gr-qc hep-th

Anjan Kar , Sayan Kar (Indian Institute of Technology Kharagpur , India) This is my paper · ORCID
classification gr-qchep-th
keywords regularblackdensityholesprofileconditionsdefectsdiverse
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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 works inside standard general relativity with a static, spherically symmetric metric. In this setup the Einstein equations force one relation: the radial pressure must equal minus the energy density. The authors therefore choose an energy density profile, then solve for the metric and the tangential pressure. The profile they use is the Dekel-Zhao profile, a formula that contains several famous dark-matter halo densities as special cases. This is a mathematical choice, not a claim about dark matter itself.

With the King profile, the resulting metric has finite curvature invariants at the center. For strong density parameters it has an inner and an outer horizon, giving a regular black hole: no singularity, but a trapped region. The same solution satisfies the null and weak energy conditions, while the strong energy condition fails near the center, which is typical for regular black holes. The authors compute its photon sphere and shadow, and show that for suitable length and density scales the angular shadow diameter agrees with the Event Horizon Telescope observations of M87* and Sgr A*.

With the pseudo-isothermal profile, they find a horizonless, geodesically complete spacetime with a solid angle deficit, which they call a defect. A fluid-of-strings model with a quadratic pressure-density relation sources this geometry. The defect interior is joined to a Schwarzschild exterior through a thin shell to make a gravastar-like object, and the shell's surface energy and tension are computed.

The paper is transparent about where it is incomplete: the Lagrangian that sources the black hole is not derived in the text, and several checks, such as completeness of all non-radial geodesics and strict stability of the shell, are asserted rather than demonstrated.

Extended reading notes

Core claim

For M=0, the King-density metric (3.2), f(r)=1+8πρ0R^3/sqrt(r^2+R^2)+(8πρ0R^3/r) ln[(sqrt(r^2+R^2)-r)/R], is claimed to be a family of regular black holes with finite curvature invariants and complete causal geodesics, with double horizons when 8πρ0R^2>3.448. The pseudo-isothermal metric (4.8), f=1-8πR^2ρ0+8πR^2ρ0 arctan(r/R)/(r/R), is claimed to be a regular, geodesically complete defect spacetime with a solid angle deficit, later used as a gravastar interior.

Load-bearing premise

Completeness of all causal geodesics is inferred from the smoothness of the effective potential for radial timelike geodesics in the extended radial domain (Figs. 2 and 8), following the criterion in Ref. [52]. Non-radial timelike and null geodesics are not analyzed explicitly, yet all causal geodesics complete is the stated regularity criterion. If this criterion is insufficient, the claim that the King black hole and defect spacetime are geodesically complete is not established.

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Assumptions & free parameters 6 free parameters · 5 assumptions · 1 invented entities

The main solutions are obtained by direct integration of a chosen density profile; the central existence claim is self-contained. The auxiliary matter models and the stability conclusion rely on unproved or reverse-engineered inputs counted above.

free parameters (6)
  • Dekel-Zhao profile parameters (mu, nu, alpha) = King: (3,2,0); defect subclass: (3,2,-1); other known solutions use other triples
    Chosen by hand to isolate regular subcases; they are not derived from a physical model.
  • Density scale rho0 = EHT constraints for King solution: about 10^2 kg/m^3 for M87* and 10^8 kg/m^3 for Sgr A*; otherwise arbitrary
    Density scale in the chosen profile; constrained by matching the shadow angular diameter.
  • Length scale R = About 10^12 m for M87* and 10^9 m for Sgr A*; arbitrary otherwise
    Length parameter of the DZ profile; sets the size of the regular core and the shadow.
  • Magnetic charge q_m in the NED Lagrangian = not fixed
    Free charge parameter in L(F), Eq (3.11), with delta and gamma expressed in terms of q_m; no independent value or prediction is given.
  • TOV equation of state parameters (a, b, lambda) = a=1/2, b=-3/2, lambda=5/3 for King; a=0, b=-1, lambda=2 for pseudo-isothermal; other triples for known solutions
    Chosen so the TOV integration reproduces the DZ density subclass; these are reconstruction parameters.
  • Gravastar parameters (exterior mass M, junction radius a0) = not fixed; x=2M/a0 ranges over (0,1)
    Inputs for the thin-shell model; shell quantities are scanned over the x,y parameter space.
assumptions (5)
  • domain assumption Einstein field equations (2.2) with an anisotropic fluid (2.3) and the Schwarzschild-gauge metric (2.1)
    Standard GR background assumed without modification.
  • standard math The mass function is obtained by integrating 4*pi*r^2*rho from the origin for the chosen density, with regularity conditions mu>=3, nu>0, alpha>-3
    Routine but not proved in detail in the text; the integration and limiting values are stated.
  • domain assumption Smoothness of the radial effective potential in the extended negative-r domain implies geodesic completeness for all causal geodesics, following Ref. [52]
    This is the load-bearing completeness criterion used in Sections III.A.1 and IV.A; non-radial geodesics are not checked.
  • ad hoc to paper The Lagrangian L(F) in Eq (3.11) reproduces the stress tensor of the King solution
    Asserted without derivation and delegated to self-citations [17,18].
  • ad hoc to paper A shell at fixed radius with V(a)=0 is strictly stable
    The paper's stated criterion (5.21) requires V''(a0)>0; V=0 gives V''(a0)=0, so strict stability is not inferred from the displayed conditions.
invented entities (1)
  • Magnetic monopole sourced by nonlinear electrodynamics L(F)
    purpose: Provide a matter model for the King regular black hole
    The Lagrangian is reverse-engineered, its derivation is omitted, q_m is free, and no independent observable is predicted.

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Pith. "Pith review of Diverse regular spacetimes using a parametrised density profile." pith.science (2026). https://pith.science/paper/3T5GNMJV

@misc{pith2026250412042,
  author       = {Pith},
  title        = {Pith review of: Diverse regular spacetimes using a parametrised density profile},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3T5GNMJV}},
  note         = {Machine review of arXiv:2504.12042}
}
read the original abstract

We explore the construction of diverse regular spacetimes (black holes and defects) in General Relativity (GR) using a generic parametrised density profile (the Dekel-Zhao profile), which includes, for specific parameter choices, various well-known examples usually studied in the context of dark matter halos. Our solutions, in the Schwarzschild gauge, include new regular black holes as well as non-singular solutions representing spacetime defects. For a sub-class of metrics, a TOV equation approach with a chosen equation of state works. The status of the energy conditions and the issue of geodesic completeness are explored in detail. We also provide possible Lagrangian density constructions for the matter energy-momentum tensors. Further, we study the shadow radius of the new regular black holes, and compare our findings with available observational results from the EHT collaboration. Finally, for the defect solution, we present a model for a stable star (a gravastar) by explicit use of the junction conditions and obtain relevant consequences highlighting its characteristic features.

Figures

Figures reproduced from arXiv: 2504.12042 by the authors.

Figure 1
Figure 1. Graph of the redshift function with r/R for various parameter values. positive roots of the horizon equation and [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Plot of the effective potential for radial timelike geodesic in the extended [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Plot of ρ and ρ + pt with r/R for different values of ρ0. The dashed lines and solid lines represent ρ and ρ + pt , respectively. The same coloured lines have equal ρ0. Condition (WEC) over the entire domain of the radial coordinate. This is also confirmed by [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (13 more)
Figure 4
Figure 4. Figure 4: Plot of the SEC with r/R for different values of ρ0. where F = 1 4 FµνF µν , δ = − ρ0R3 q 3/2 m and γ = R2 qm . The only nonzero component of the field strength tensor (Fµν) is Fθϕ, making the regular black hole a purely magnetic solution. The magnetic source is identi…
Figure 5
Figure 5. Figure 5: Plot depicting the shadow radius (purple line), photon sphere radius (blue line) [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 6
Figure 6. Figure 6: The different coloured lines represent the angular diameter of the theoretical [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: Plot of the redshift function of the defect geometry with [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]
Figure 8
Figure 8. Figure 8: Graph illustrating the effective potential for a radial timelike geodesic as a [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]
Figure 9
Figure 9. Figure 9: Embedding diagram of the defect geometry for 8 [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]
Figure 10
Figure 10. Figure 10: Graph of ρ and ρ + pt as a function of r/R for various values of ρ0. The dashed and solid lines denote ρ and ρ + pt , respectively. Lines of the same colour possess same ρ0 values. ρ0 = 0.5 ρ0 = 0.9 0.0 0.5 1.0 1.5 2.0 2.5 3.0 -1.5 -1.0 -0.5 0.0 r/R ρ + pr + 2 pt [PI…
Figure 11
Figure 11. Figure 11: Graph of the L. H. S. of the SEC with r/R, for various values of ρ0. B. Lagrangian model for the required matter In the GR coupled to matter scenario, we may use a fluid of strings as the possible matter model which can support such a regular defect geometry. The idea…
Figure 12
Figure 12. Figure 12: Qualitative picture of the thin shell star [PITH_FULL_IMAGE:figures/full_fig_p025_12.png]
Figure 13
Figure 13. Figure 13: A qualitative representation of µ(a0) within the parameter space (x, y) for 8πR2ρ0 = 0.5 (left) and 8πR2ρ0 = 0.9 (right). The parameter x ranges from 0 to 1, while y may assume any positive value. The graphs illustrate the areas where µ(a0) is positive (shown in orang…
Figure 14
Figure 14. Figure 14: Variation of Π(a0) within the parameter space (x, y) for 8πR2ρ0 = 0.5 (left) and 8πR2ρ0 = 0.9 (right). The parameter x ranges from 0 to 1, while y may assume any positive value [PITH_FULL_IMAGE:figures/full_fig_p032_14.png]
Figure 15
Figure 15. Figure 15: Plot of µ(a0)/Π(a0) as a function of x for various values of y. The left plot shows 8πR2ρ0 = 0.5, while the right plot displays 8πR2ρ0 = 0.9. The solid lines denote positive µ(a0), whereas the dashed lines indicate negative µ(a0) (more details are provided in the text…
Figure 16
Figure 16. Figure 16: Plot representing the allowed domain of the model parameter space. [PITH_FULL_IMAGE:figures/full_fig_p035_16.png]

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Works this paper leans on

100 extracted references · 59 canonical work pages · cited by 1 Pith paper

  1. [1]

    A smooth and continuous behaviour of the scalars over the entire domain of the radial coordinate is a necessary condition to prove that the metric is genuinely regular

    Regularity of curvature scalars and geodesic completeness To verify that the metric in Eq.(3.2) represents a regular spacetime, we examine the three independent curvature scalars explicitly. A smooth and continuous behaviour of the scalars over the entire domain of the radial coordinate is a necessary condition to prove that the metric is genuinely regula...

  2. [2]

    The energy density ( ρ) is already assumed in Eq.(3.1)

    Energy conditions Let us now examine the different energy conditions for the matter required to support such a regular geometry. The energy density ( ρ) is already assumed in Eq.(3.1). The other components of the energy-momentum tensor, as obtained assuming Einstein equations of GR hold, are the following, pr =−ρ, p t = ρ0R3(r2− 2R2) 2(r2 +R2)5/2 (3.8) It...

  3. [3]

    Similarly, we may model the required matter for the above-discussed regular black hole in terms of a magnetic monopole gov- erned by a specific nonlinear electrodynamics

    Matter source for the geometry We mentioned in the Introduction that nonlinear electrodynamics minimally coupled to gravity can source several regular black holes [8–14]. Similarly, we may model the required matter for the above-discussed regular black hole in terms of a magnetic monopole gov- erned by a specific nonlinear electrodynamics. We have found t...

  4. [4]

    The formation of black hole shadow can be understood as the result of interaction between the strong gravitational field caused by the black hole and the surrounding light rays

    Shadow radius and EHT observation Let us now study null geodesics in the new regular black hole geometry and compute the shadow radius in the equatorial plane. The formation of black hole shadow can be understood as the result of interaction between the strong gravitational field caused by the black hole and the surrounding light rays. Photons interact wi...

  5. [5]

    Hofmann and M

    S. Hofmann and M. Schneider, Phys. Rev. D 91, 125028 (2015)

    2015

  6. [6]

    Penrose, Phys

    R. Penrose, Phys. Rev. Lett. 14, 57 (1965)

    1965

  7. [7]

    S. W. Hawking and R. Penrose, Proc. Roy. Soc. Lond. A 314, 529 (1970)

    1970

  8. [8]

    In summary, the above metric char- acterises a regular compact object without a horizon and possesses a solid angle deficit

    Therefore, the spacetime is geodesically complete. In summary, the above metric char- acterises a regular compact object without a horizon and possesses a solid angle deficit. 8 π ρ0 R2 = 0.5 8 π ρ0 R2 = 0.9 -20 -10 0 10 20 0.2 0.4 0.6 0.8 1.0 r/R Veff Figure. 8: Graph illustrating the effective potential for a radial timelike geodesic as a function of r/...

Show all 100 references
  1. [9]

    J. M. M. Senovilla and D. Garfinkle, Class. Quant. Grav. 32, 124008 (2015)

    2015

  2. [10]

    G. T. Horowitz and D. Marolf, Phys. Rev. D 52, 5670 (1995)

    1995

  3. [11]

    Casals, A

    M. Casals, A. Fabbri, C. Mart´ ınez, and J. Zanelli, Phys. Rev. Lett. 118, 131102 (2017)

    2017

  4. [12]

    Borissova, S

    J. Borissova, S. Liberati and M. Visser, arXiv:2502.00548 [gr-qc]

    arXiv

  5. [13]

    Ayon-Beato and A

    E. Ayon-Beato and A. Garcia, Phys. Rev. Lett. 80, 5056 (1998)

    1998

  6. [14]

    Ayon-Beato and A

    E. Ayon-Beato and A. Garcia, Phys. Lett. B 464, 25 (1999)

    1999

  7. [15]

    Ayon-Beato and A

    E. Ayon-Beato and A. Garcia, Gen. Relativ. Gravit. 31, 629 (1999). 38

    1999

  8. [16]

    Ayon-Beato and A

    E. Ayon-Beato and A. Garcia, Phys. Lett. B 493, 149 (2000)

    2000

  9. [17]

    Ayon-Beato and A

    E. Ayon-Beato and A. Garcia, Gen. Relativ. Gravit. 37, 635 (2005)

    2005

  10. [18]

    K. A. Bronnikov, Int. J. Mod. Phys. D 27, 1841005 (2018)

    2018

  11. [19]

    Balart and E

    L. Balart and E. C. Vagenas, Phys. Rev. D 90, 124045 (2014)

    2014

  12. [20]

    Bronnikov, Phys

    K. Bronnikov, Phys. Rev. D 63, 044005 (2001)

    2001

  13. [21]

    M. B. Jahani Poshteh and N. Riazi, Int. J. Mod. Phys. D 30, 2150079 (2021)

    2021

  14. [22]

    Kar and S

    A. Kar and S. Kar, Gen. Relativ. Gravit. 56, 52 (2024)

    2024

  15. [23]

    Kar, Eur

    A. Kar, Eur. Phys. J. C 84, 1246 (2024)

    2024

  16. [24]

    K. A. Bronnikov and J. C Fabris, Phys. Rev. D 96, 251101 (2006)

    2006

  17. [25]

    A Bronnikov, Particles 1 56,(2018)

    K. A Bronnikov, Particles 1 56,(2018)

    2018

  18. [26]

    Ovalle, R

    J. Ovalle, R. Casadio and A. Giusti, Phys. Lett. B 844, 138085 (2023)

    2023

  19. [27]

    Bueno, P

    P. Bueno, P. A. Cano and R. A. Hennigar, arXiv:2403.04827 [gr-qc]

    arXiv

  20. [28]

    Dymnikova and M

    I. Dymnikova and M. Khlopov, Int. Jr. Mod. Phys. D 24, 1545002 (2015)

    2015

  21. [29]

    Zhao, Mon

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

    1996

  22. [30]

    Zhao, Mon

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

    1997

  23. [31]

    Dekel, G

    A. Dekel, G. Ishai, A. A. Dutton, A. V. Maccio, Mon. Not. Roy. Astron. Soc. 468, 1005 (2017)

    2017

  24. [32]

    Freundlich, F

    J. Freundlich, F. Jiang, A. Dekel, N. Cornuault, O. Ginzburg, R. Koskas, S. Lapiner, A. Dutton, A. V. Macci` o, Mon. Not. Roy. Astron. Soc.499 , 2912 (2020)

    2020

  25. [33]

    Salucci, Universe 11, 67 (2025)

    P. Salucci, Universe 11, 67 (2025)

    2025

  26. [34]

    Batic, J

    D. Batic, J. M. Faraji, M. Nowakowski, Eur. Phys. J. C 82, 759 (2022)

    2022

  27. [35]

    A. A. Badawi, S. Shaymatov and Y. Sekhmani, JCAP 02, 014 (2025)

    2025

  28. [36]

    Ovgun and R

    A. Ovgun and R. C. Pantig, arXiv 2501.12559 [gr-qc]

    arXiv

  29. [37]

    Barriola and A

    M. Barriola and A. Vilenkin, Phys. Rev. Lett. 63, 341 (1989)

    1989

  30. [38]

    Nucamendi, M

    U. Nucamendi, M. Salgado and D. Sudarsky, Phys. Rev. Lett. 84, 3037 (2000)

    2000

  31. [39]

    Marunovi´ c and M

    A. Marunovi´ c and M. Murkovi´ c, Class. Quantum Grav.31, 045010 (2014)

    2014

  32. [40]

    T. R. P. Caramˆ es, Phys. Rev. D108, 084002 (2023)

    2023

  33. [41]

    Vilenkin and E

    A. Vilenkin and E. P. S. Shellard, Cosmic Strings and Other Topological Defects (Cambridge University Press, Cambridge, England, 2000)

    2000

  34. [42]

    S. N. Sajadi, S. Ponglertsakul and O. Luongo, arXiv.2502.02098[gr-qc]

    arXiv

  35. [43]

    J. F. Navarro, C. S. Frenk and S. D.M. White, Astrophys. J. 462, 563 (1996). 39

    1996

  36. [44]

    King, Astron

    I. King, Astron. J. 67, 471 (1962)

    1962

  37. [45]

    Begeman, A.H

    K.G. Begeman, A.H. Broeils and R.H. Sanders, Mon. Not. Roy. Astron. Soc. 249, 523 (1991)

    1991

  38. [46]

    H. C. Plummer,Mon. Not. Roy. Astron. Soc., 71, 460-470 (1911)

    1911

  39. [47]

    Hernquist, Astrophys

    L. Hernquist, Astrophys. J. 356, 359 (1990)

    1990

  40. [48]

    Zakhary and C

    E. Zakhary and C. B. G. Mcintosh, Gen. Rel. Grav. 29, 539 (1997)

    1997

  41. [49]

    H. W. Hu, C. Lan and Y. G. Miao Lan, Eur. Phys. J. C 83, 1047 (2023)

    2023

  42. [50]

    P. T. Chru´ sciel, M. A. MacCallum, and D. B. Singleton, Phil. Trans. R. Soc. A 350, 113–141 (1995)

    1995

  43. [51]

    Godazgar and G

    M. Godazgar and G. Macaulay, Phys. Rev. D 102, 064036 (2020)

    2020

  44. [52]

    X. He, X. Wu and N. Xie, arXiv:2504.07774 [gr-qc]

  45. [53]

    C. V. Vishveshwara, J. Math. Phys. 9, 1319 (1968)

    1968

  46. [54]

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

    1988

  47. [55]

    Hawking and G.F.R

    S.W. Hawking and G.F.R. Ellis, The Large Scale Structure of Spacetime (Cambridge Univer- sity Press, Cambridge 1973)

    1973

  48. [56]

    R. M. Wald, General Relativity. Chicago Univ. Pr., Chicago, USA, 1984

    1984

  49. [57]

    Zhou and L

    T. Zhou and L. Modesto, Phys. Rev. D 107, 044016 (2023)

    2023

  50. [58]

    Carballo-Rubio, F

    R. Carballo-Rubio, F. Di Filippo, S. Liberati, and M. Visser, Phys. Rev. D 101, 084047 (2020)

    2020

  51. [59]

    Carballo-Rubio, F

    R. Carballo-Rubio, F. Di Filippo, S. Liberati, and M. Visser, J. High Energ. Phys. 02, 122 (2022)

    2022

  52. [60]

    Ansoldi, in Conference on Black Holes and Naked Singularities, Milano, Italy, 2007 , arXiv: 0802.0330 [gr-qc]

    S. Ansoldi, in Conference on Black Holes and Naked Singularities, Milano, Italy, 2007 , arXiv: 0802.0330 [gr-qc]

    2007 arXiv

  53. [61]

    O. B. Zaslavskii, Phys. Lett. B 688, 278–280 (2010)

    2010

  54. [62]

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

    2001

  55. [63]

    Bardeen, in Black Holes (Les Astres Occlus), ed

    J.M. Bardeen, in Black Holes (Les Astres Occlus), ed. C. DeWitt and B. S. DeWitt (New York: Gordon and Breach), 215 (1973)

    1973

  56. [64]

    Perlick and O

    V. Perlick and O. Y. Tsupko, Phys. Rep. 947, 1-39 (2022)

    2022

  57. [65]

    D. Chen, C. Gao, X. Liu, and C. Yu, Eur. Phys. J. C, 81, 700 (2021)

    2021

  58. [66]

    Event Horizon Telescope Collaboration et al, Astrophys. J. Lett. 875, L1 (2019)

    2019

  59. [67]

    Event Horizon Telescope Collaboration et al., Astrophys. J. Lett. 875, L5 (2019)

    2019

  60. [68]

    Event Horizon Telescope Collaboration et al., Astrophys. J. Lett. 875, L6 (2019)

    2019

  61. [69]

    Medeiros et al., Astrophys

    L. Medeiros et al., Astrophys. J. Lett. 947, L7 (2023). 40

    2023

  62. [70]

    J. P. Blakeslee et al., Astrophys. J. Lett. 694, 556–572 (2009)

    2009

  63. [71]

    S. Bird, W. E. Harris, J. P. Blakeslee, and C. Flynn, Astron. Astrophys. 524, A71 (2010)

    2010

  64. [72]

    Cantiello et al., Astrophys

    M. Cantiello et al., Astrophys. J. Lett. 854, L31 (2018)

    2018

  65. [73]

    Event Horizon Telescope Collaboration et al., Astrophys. J. Lett. 930, L12 (2022)

    2022

  66. [74]

    Event Horizon Telescope Collaboration et al., Astrophys. J. Lett. 930 L17 (2022)

    2022

  67. [75]

    Do et al., Science 365 no

    T. Do et al., Science 365 no. 6454, 664–668 (2019)

    2019

  68. [76]

    Abuter et al., Astron

    R. Abuter et al., Astron. Astrophys. 657, L12 (2022)

    2022

  69. [77]

    Abuter et al., Astron

    R. Abuter et al., Astron. Astrophys. 636, L5 (2020)

    2020

  70. [78]

    Dymnikova, Class

    I. Dymnikova, Class. Quant. Grav. 21, 4417-4429 (2004)

    2004

  71. [79]

    Z. Y. Fan and X. Wang, Phys. Rev. D 94, 124027 (2016)

    2016

  72. [80]

    Neves and A

    J.C.S. Neves and A. Saa, Phys. Lett. B 734, 44 (2014)

    2014

  73. [81]

    J. M. Bardeen, in Proceedings of International Conference GR5, Tbilisi, USSR, 1968 , p. 174

    1968

  74. [82]

    S. A. Hayward, Phys. Rev. Lett. 96, 031103 (2006)

    2006

  75. [83]

    P. S. Letelier, Phys. Rev. D 20, 1294 (1979)

    1979

  76. [84]

    P. S. Letelier, Nuov. Cim. B 63, 519–528 (1981)

    1981

  77. [85]

    H Soleng, Gen

    H. H Soleng, Gen. Rel. Grav. 27, 367–378 (1995)

    1995

  78. [86]

    L. C. N. d. Santos, arXiv:2502.15846 [gr-qc]

    arXiv

  79. [87]

    Sen, Annalen der Physik 378, 365 (1924)

    N. Sen, Annalen der Physik 378, 365 (1924)

    1924

  80. [88]

    Darmois, Memorial Des Sciences Mathematiques (Gauthier-Villars, Paris, 1927)

    G. Darmois, Memorial Des Sciences Mathematiques (Gauthier-Villars, Paris, 1927)

    1927

  81. [89]

    Israel, Nuovo Cimento B (1965-1970) 48, 463 (1967)

    W. Israel, Nuovo Cimento B (1965-1970) 48, 463 (1967)

    1967

  82. [90]

    Visser and D.L

    M. Visser and D.L. Wiltshire, Class. Quantum Gravit. 21, 1135 (2004)

    2004

  83. [91]

    Visser, Phys

    M. Visser, Phys. Rev. D 39, 3182 (1989)

    1989

  84. [92]

    Visser, Lorentzian Wormholes: From Einstein to Hawking (AIP, Woodbury, p

    M. Visser, Lorentzian Wormholes: From Einstein to Hawking (AIP, Woodbury, p. 412, 1995)

    1995

  85. [93]

    Poisson, A Relativist’s Toolkit: the mathematics of black hole mechanics , Cambridge Uni- versity Press, Cambridge, UK (2004)

    E. Poisson, A Relativist’s Toolkit: the mathematics of black hole mechanics , Cambridge Uni- versity Press, Cambridge, UK (2004)

    2004

  86. [94]

    P. M. Moruno, N. M. Garcia, F. S. N. Lobo and M. Visser, JCAP 03, 034 (2012)

    2012

  87. [95]

    N. M. Garcia, F. S. N. Lobo and M. Visser, Phys. Rev. D 86, 044026 (2012)

    2012

  88. [96]

    F. S. N. Lobo and P. Crawford, Class. Quantum Grav. 22, 4869 (2005)

    2005

  89. [97]

    Bhattacharjee and P

    D. Bhattacharjee and P. K. Chattopadhyay, J. High Energ. Astrophys. 43, 248-257 (2024)

    2024

  90. [98]

    Lanczos, Ann

    K. Lanczos, Ann. Phys. 379, 518 (1924)

    1924

  91. [99]

    Clayton, Principles of Stellar Evolution and Nucleosynthesis (McGraw-Hill, 1968)

    D.D. Clayton, Principles of Stellar Evolution and Nucleosynthesis (McGraw-Hill, 1968). 41

    1968

  92. [100]

    Sekhmani, S

    Y. Sekhmani, S. Zare, L. Nieto, H. Hassanabadi, and K. Boshkayev, J. High Energ. Astrophys. 47, 100389 (2025). 42

    2025

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