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Regular black holes with flat, Minkowskian cores produce shadows and accretion-disk images nearly identical to those of Schwarzschild and Kerr.

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2026-08-02 07:09 UTC pith:6WQAZ6DI

load-bearing objection A new Minkowskian-core regular black hole family whose advertised shadow degeneracy is real but built into the ansatz; the current metric has a coefficient error and the WEC claim fails, so the numerical results need correction. the 4 major comments →

arxiv 2607.10713 v2 pith:6WQAZ6DI submitted 2026-07-12 gr-qc hep-th

Regular black holes with Minkowskian cores: causal structure and observational degeneracy

Francisco Tello-Ortiz , Y. G\'omez-Leyton , Alejandro Rueda , Kazuharu Bamba , Manuel Gonz\'alez-Espinoza This is my paper
classification gr-qc hep-th MSC 83C5783C1583C20
keywords regular black holesMinkowskian coregravitational decouplingshadowphoton ringKerr spacetimeaccretion disk ray-tracingCauchy horizon
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.

This paper constructs a new family of regular black holes—static and rotating—whose central singularity is replaced by a flat, Minkowskian core, rather than the de Sitter core used in most regular models. Using gravitational decoupling with a Kerr–Schild constraint and an exponentially localized matter profile, the authors obtain an asymptotically flat two-horizon geometry that is an exact deformation of Schwarzschild. Their central claim is that the optical appearance—the shadow boundary and the full ray-traced images of a thin accretion disk—is nearly indistinguishable from that of Schwarzschild and Kerr with the same mass and spin. This matters because it shows that even a dramatically different interior does not show up in current horizon-scale observations, and that regularity and the presence of a Cauchy horizon are logically independent in this class of solutions.

Core claim

The paper's central discovery is a family of static and rotating regular black holes whose deep interior is locally Minkowskian rather than de Sitter, produced by gravitational decoupling from a Schwarzschild seed under the Kerr–Schild condition e^ν = e^{−λ} and an exponential energy-density profile that vanishes smoothly at the origin. Once the ADM mass is fixed by regularity, the metric approaches flat space as r→0 with O(r^4) corrections, and the solution develops two horizons for deformation parameter α above an extremal threshold. Ray-tracing the rotating geometry through a thin, optically thin accretion disk, the authors find that the shadow boundary and the full images—direct emission

What carries the argument

The exponential density ansatz (39), κE = (α r²/ℓ⁴)e^{−r/ℓ}, combined with the Kerr–Schild constraint (28), e^ν = e^{−λ}, which decouples the gravitational-decoupling matter sector and reduces the static metric to the simple form (46). The exponential profile is the load-bearing device: it confines all curvature modification to a shell deep inside the photon sphere, so the exterior geometry, photon-sphere radius, and shadow are fixed by the ADM mass and the spin alone. The rotating extension is the Gürses–Gürsey metric (61) built from the same radial mass function (47), and the ray-tracing computation of geodesics in this metric produces the claimed degeneracy.

Load-bearing premise

The construction rests on the specific exponential density profile (39) combined with the Kerr–Schild constraint e^ν=e^{−λ}; if that profile is not the correct effective matter description, the nearly exact shadow and image degeneracy with Schwarzschild and Kerr no longer follows.

What would settle it

Resolve the second-order photon ring of a horizon-scale black hole with a future very-long-baseline interferometer: the paper's images show that this ring would be identical to Kerr's at current resolution, so any statistically significant deviation in its diameter, shape, or brightness would falsify the degeneracy claim.

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

If this is right

  • Current horizon-scale imaging cannot distinguish these regular black holes from Kerr; the shadow and thin-disk images encode only the exterior geometry.
  • The degeneracy is not limited to the shadow boundary: ray-traced images including direct emission, the photon ring, and the Doppler-boosted crescent are also practically identical to Kerr at the same spin and mass.
  • The construction separates two features usually tied together in regular black holes: a nonsingular core does not force an effective de Sitter region, and the inner Cauchy horizon arises from the localized deformation rather than from the core.
  • The model supplies an analytically tractable template for testing how much internal structure can be hidden from external observers, and for comparing future high-resolution observations against regular alternatives to Kerr.

Where Pith is reading between the lines

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

  • The paper asserts the weak energy condition holds throughout, but its Eq. (58) is negative for r < M/(6α), so that specific claim is not supported as written; the shadow-degeneracy result depends only on the exterior geometry and so stands apart.
  • Because the deformation is exponentially localized inside the photon sphere, the same degeneracy should extend to gravitational-wave ringdown and quasinormal-mode frequencies—an extension the paper does not compute but that follows from the exterior-only nature of the observable.
  • By decoupling regularity from the presence of an inner horizon, the construction points toward regular-black-hole models with no Cauchy horizon at all; the solution presented here still contains one, and its dynamical stability is left unexplored.

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 constructs a family of asymptotically flat, spherically symmetric regular black holes with Minkowskian cores using gravitational decoupling, and extends them to a rotating Gürses-Gürsey metric by promoting the mass function to a radial profile. The authors claim the static and rotating solutions have two horizons, satisfy the weak energy condition, and produce shadow boundaries and ray-traced thin-disk images that are virtually indistinguishable from Schwarzschild and Kerr. The main evidence is an analytic metric (Eqs. 41 and 46), horizon and curvature analyses, and numerical ray tracing with the Gradus.jl package.

Significance. If established, the construction would provide a new analytic family of regular black holes with Minkowskian rather than de Sitter cores, and would illustrate how strongly the interior geometry can vary while leaving exterior observables essentially unchanged. The paper's use of explicit metric functions, numerical horizon analysis, and public ray-tracing software is a strength. However, the advertised conclusions are compromised by an algebraic inconsistency between the central metric formulas, a false weak-energy-condition claim, and the fact that the observational degeneracy is, to a large extent, a mathematical consequence of the exponentially localized deformation ansatz. These issues are local and fixable, but they affect the validity of the numerical results and the physical interpretation, so the paper requires substantial revision before the claims can be accepted.

major comments (4)
  1. [§III.C, Eqs. (41)–(47)] Eq. (46) is not algebraically equivalent to Eq. (41). Substituting ℓ = M/(12α) into Eq. (41) gives e^ν = 1 − 2M/r + e^{−12αr/M}[2M/r + 24α + 144α²r/M + 576α³r²/M² + ...], whereas Eq. (46) contains the term 72α instead of 24α. Consequently, the metric in Eq. (46) has the limit e^ν → 1 + 48α as r → 0, not the advertised Minkowskian-core behavior 1 + O(r⁴). Since Eqs. (46) and (47) are used for the horizon analysis, shadow computation, and ray-traced images, the numerical results presented in Figs. 1–9 do not correspond to the regular spacetime the paper claims to study. This error must be corrected and the numerical computations rerun.
  2. [§III.B, Eqs. (39) and (58)] The weak energy condition analysis is internally inconsistent. For the chosen density E = (α/κ) r² ℓ^{−4} e^{−r/ℓ}, the derivative is E′ = (α/κ) r ℓ^{−4} e^{−r/ℓ}(2 − r/ℓ), which is positive for r < 2ℓ. This contradicts the statement in Sec. III.B that E′ < 0 is satisfied. Correspondingly, Eq. (58) is negative for r < M/(6α) = 2ℓ because the factor (6αr − M) is negative there. The claim that the static solution satisfies the weak energy condition — and the later claim in the Conclusions that the rotating solution does as well — is therefore unsupported. The profile must be modified or the claim retracted, and the energy conditions in the rotating case must be checked explicitly.
  3. [§IV–V, observational degeneracy] The central conclusion that the shadows and images are indistinguishable from Schwarzschild/Kerr is baked into the ansatz rather than being an emergent property of the model. Because the deformation enters as e^{−r/ℓ} = e^{−12αr/M}, at the photon sphere r ≈ 3M the deviation is suppressed by e^{−36α}; for α ≥ 0.35, this is ≲ 3 × 10^{−6}. Thus the shadow degeneracy is a parametric consequence of the exponential localization of the deformation, not a generic feature of Minkowskian-core black holes. The paper should state this explicitly and, if the claim is to be nontrivial, examine regimes with weaker suppression or compare at the level of the corrected metric.
  4. [§IV and Conclusions] The rotating metric (61) is introduced by simply substituting the static mass function (47) into the Gürses-Gürsey metric. No field equations, conservation laws, or energy conditions are verified for the rotating case, yet the conclusions assert that the model satisfies the weak energy condition 'even in the rotating regime.' This claim is unsupported. The authors should either provide and check the effective stress-energy tensor for the rotating metric or explicitly limit the energy-condition statements to the static case.
minor comments (5)
  1. [Abstract] The abstract states that the solution 'satisfies the weak energy condition,' which is contradicted by Eq. (58). Please revise after correcting the energy-condition analysis.
  2. [Figs. 1, 3, 5] The extremal parameter values (α_E ≈ 0.348, and the rotating curves) and the shadow comparisons are computed from Eq. (46) and should be updated once Eq. (46) is corrected. In Fig. 5, the dashed red line is not visibly distinct from the black Kerr-shadow line; a residual plot or zoom would be helpful.
  3. [§III.C, Eqs. (56)–(58)] The coefficients A, B, and C in Eqs. (56)–(58) are never specified. Please provide their values, especially after the metric coefficient is corrected, so that the regularity and energy-condition statements can be checked.
  4. [§V.A, Eq. (70)] The phrase 'Johnson Standard Unbound distribution' appears to be a naming error; the intended reference is likely the 'Johnson Unbounded' distribution used in the GLM emission models. Please check the terminology.
  5. [§IV, Eq. (62)–(64)] The transition from the static metric (46) to the rotating metric (61) is made without detailing the coordinate basis or the method used to obtain the Gürses-Gürsey form. Clarify whether a Newman-Janis-type algorithm is applied and whether the resulting metric is meant to solve the field equations with the decoupled θ-sector.

Circularity Check

0 steps flagged

No significant circularity: shadow degeneracy is a computed consequence of the stated exponential ansatz, not an input or a self-citation.

full rationale

The paper's derivation chain is not circular. It begins from a Schwarzschild seed, imposes the Kerr-Schild condition e^nu=e^{-lambda} (Eq. 28), chooses the exponentially decaying density (Eq. 39) to satisfy the weak-energy inequalities (Eqs. 34-38), solves Eq. (37) for h(r), enforces regularity via c1=24alpha ell (Eq. 43), and then computes photon-sphere, shadow, and ray-traced observables from the resulting metric. Each of these is a well-defined mathematical consequence of the stated ansatz rather than an assumption of the shadow-degeneracy conclusion. The fact that the exponential falloff makes the metric essentially Schwarzschild/Kerr outside the photon sphere is indeed what makes the shadows nearly identical, but that is a legitimate derivation from the model, not a fitted parameter renamed as a prediction. The GD and rotating-extension machinery is taken from independent prior work by other authors, and the ray tracing uses the external Gradus.jl package with direct comparison to Kerr, so no load-bearing self-citation chain is present. No specific equation reduces to another by construction, no fitted input is relabeled as a prediction, and no imported uniqueness theorem forces the result. Separate from circularity, the paper has serious internal-consistency problems: Eq. (46) is not algebraically equivalent to Eq. (41) (the linear-r coefficient is 72alpha instead of 24alpha), and Eq. (58) is negative for r<M/(6alpha), contradicting the claimed WEC. These are correctness issues that should be fixed, but they are not circularity.

Axiom & Free-Parameter Ledger

2 free parameters · 5 axioms · 1 invented entities

The entire model is generated by two hand-chosen ingredients: the Kerr-Schild condition (28) and the exponential density profile (39). The resulting horizon structure, Minkowskian core, and observational degeneracy all follow from these choices. The paper provides no independent microphysical support for the θμν sector, and the WEC violation near r<2ℓ contradicts the stated physical-consistency claims.

free parameters (2)
  • α (deformation parameter)
    One-parameter family of deformations; after imposing regularity M=12αℓ, α (or ℓ) is free. For M=1 the BH branch requires α>α_E≈0.348; the entire horizon structure and optical appearance depend on α.
  • GLM emission parameters μ, σ, γ = GLM1: μ=x0, σ=1.5, γ=-1.5; GLM2: μ=x0, σ=0.5, γ=0; GLM3: μ=17x0/6, σ=0.25, γ=-2
    Taken from Gralla-Lupsasca-Marrone (ref [78]) for the disk emission model; they affect the intensity images but are not fitted to new data.
axioms (5)
  • domain assumption Gravitational decoupling split of the action into two matter sectors with metric deformations (16)-(17)
    Framework from Ovalle [72,73]; the paper reviews but does not re-derive the validity of the decoupling procedure.
  • ad hoc to paper Kerr-Schild condition e^ν=e^{-λ} (Eq. 28) to close the θ-sector
    Chosen to simplify the horizon structure and yields P_r=-E; without this, the system of equations is underdetermined. The choice shapes the two-horizon causal structure.
  • ad hoc to paper Energy density profile E=α r²/(κℓ⁴)e^{-r/ℓ} (Eq. 39)
    Central ansatz chosen to produce a regular Minkowskian core. The paper asserts E'<0 for all r to satisfy WEC, but E' is actually positive for r<2ℓ, so the WEC claim fails.
  • domain assumption Rotating extension is the Gurses-Gursey metric (61) with the static mass function (47)
    The paper assumes, following ref [63], that this is a valid stationary axisymmetric solution of the decoupled system; it does not verify conservation or energy conditions in the rotating case.
  • domain assumption Geometrically thin, optically thin, monochromatic disk model (Eq. 71) with GLM intensity profiles
    Standard toy accretion model from refs [76-78]; used for all synthetic images and intensity profiles.
invented entities (1)
  • θμν source (additional matter sector in gravitational decoupling) no independent evidence
    purpose: Sources the metric deformation that removes the central singularity and creates the Minkowskian core
    No microphysical Lagrangian or particle content is provided; it is an effective stress tensor with an ad hoc density profile (39), and it violates the weak energy condition near the center.

pith-pipeline@v1.3.0-alltime-deepseek · 15466 in / 20280 out tokens · 192889 ms · 2026-08-02T07:09:58.575717+00:00 · methodology

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read the original abstract

We construct a new family of asymptotically flat static and rotating regular black holes characterized by a Minkowskian core and a finite two-horizon structure. The solutions are obtained within the framework of gravitational decoupling and provide an analytically tractable realization of non-singular black hole geometries with a regular interior and well-defined asymptotic properties. We investigate the optical appearance of the rotating spacetime through shadow observables and ray-traced images of geometrically thin, optically thin accretion disks. Despite the substantial differences between the interior geometry of these solutions and that of singular black holes, their optical signatures are found to remain remarkably close to those of Schwarzschild and Kerr spacetimes with the same asymptotic parameters. Our results show that markedly different black hole interiors may lead to nearly indistinguishable optical appearances, highlighting the challenges of probing the internal structure of compact objects using current shadow and imaging observations.

Figures

Figures reproduced from arXiv: 2607.10713 by Alejandro Rueda, Francisco Tello-Ortiz, Kazuharu Bamba, Manuel Gonz\'alez-Espinoza, Y. G\'omez-Leyton.

Figure 1
Figure 1. Figure 1: FIG. 1. The inverse radial metric potential versus the radial [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The BH region showing the admissible values for the [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The parameter space [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Observed Intensity for [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Observational appearance for [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Observational appearance of an inclined observer at [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Observational appearance of the rotating case for an inclined observer at [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗

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Works this paper leans on

97 extracted references · 69 linked inside Pith · cited by 1 Pith paper

  1. [1]

    S. W. Hawking and G. F. R. Ellis,The Large Scale Struc- ture of Space-Time, Cambridge Monographs on Mathe- matical Physics (Cambridge University Press, 2011)

    2011

  2. [2]

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

    1984

  3. [3]

    Penrose, Riv

    R. Penrose, Riv. Nuovo Cim.1, 252 (1969)

    1969

  4. [4]

    Earman,Bangs, crunches, whimpers, and shrieks: Singularities and acausalities in relativistic space-times (1995)

    J. Earman,Bangs, crunches, whimpers, and shrieks: Singularities and acausalities in relativistic space-times (1995)

    1995

  5. [5]

    Salazar, A

    I. Salazar, A. Garcia, and J. Plebanski, J. Math. Phys. 28, 2171 (1987)

    1987

  6. [6]

    Ayon-Beato and A

    E. Ayon-Beato and A. Garcia, Phys. Rev. Lett.80, 5056 (1998), arXiv:gr-qc/9911046

    Pith/arXiv arXiv 1998

  7. [7]

    K. A. Bronnikov, Phys. Rev. D63, 044005 (2001), arXiv:gr-qc/0006014

    Pith/arXiv arXiv 2001

  8. [8]

    Dymnikova, Class

    I. Dymnikova, Class. Quant. Grav.21, 4417 (2004), arXiv:gr-qc/0407072

    Pith/arXiv arXiv 2004

  9. [9]

    Balart and E

    L. Balart and E. C. Vagenas, Phys. Rev. D90, 124045 (2014), arXiv:1408.0306 [gr-qc]

    Pith/arXiv arXiv 2014

  10. [10]

    Toshmatov, B

    B. Toshmatov, B. Ahmedov, A. Abdujabbarov, and Z. Stuchlik, Phys. Rev. D89, 104017 (2014), arXiv:1404.6443 [gr-qc]

    Pith/arXiv arXiv 2014

  11. [11]

    Fan and X

    Z.-Y. Fan and X. Wang, Phys. Rev. D94, 124027 (2016), arXiv:1610.02636 [gr-qc]

    Pith/arXiv arXiv 2016

  12. [12]

    Allahyari, M

    A. Allahyari, M. Khodadi, S. Vagnozzi, and D. F. Mota, JCAP2002, 003 (2020), arXiv:1912.08231 [gr-qc]

    Pith/arXiv arXiv 2020

  13. [13]

    Heidari, A

    N. Heidari, A. A. Ara´ ujo Filho, V. Vertogradov, and A. ¨Ovg¨ un, Phys. Lett. B879, 140606 (2026), arXiv:2412.05072 [gr-qc]

    Pith/arXiv arXiv 2026

  14. [14]

    A. A. Ara´ ujo Filho, E. L. B., Junior., J. T. S. S., Ju- nior., F. S. N. Lobo, J. A. A. Ramos, M. E. Rodrigues, D. Rubiera-Garcia, L. F. D. da Silva, and H. A. Vieira, Phys. Dark Univ.53, 102396 (2026), arXiv:2604.20066 [gr-qc]

    Pith/arXiv arXiv 2026

  15. [15]

    M. Y. Khlopov, R. V. Konoplich, S. G. Rubin, and A. S. Sakharov, Grav. Cosmol.6, 153 (2000)

    2000

  16. [16]

    M. Y. Khlopov, Res. Astron. Astrophys.10, 495 (2010), arXiv:0801.0116 [astro-ph]

    Pith/arXiv arXiv 2010

  17. [17]

    Khodadi and R

    M. Khodadi and R. Pourkhodabakhshi, Phys. Rev. D 106, 084047 (2022), arXiv:2210.06861 [gr-qc]

    Pith/arXiv arXiv 2022

  18. [18]

    Khodadi and J

    M. Khodadi and J. T. Firouzjaee, Phys. Lett. B857, 138986 (2024), arXiv:2408.12873 [gr-qc]

    Pith/arXiv arXiv 2024

  19. [19]

    Khodadi, Phys

    M. Khodadi, Phys. Lett. B870, 139922 (2025), arXiv:2509.20483 [gr-qc]

    arXiv 2025

  20. [20]

    R. V. Konoplich, S. G. Rubin, A. S. Sakharov, and M. Y. Khlopov, Phys. Atom. Nucl.62, 1593 (1999)

    1999

  21. [21]

    Dymnikova and M

    I. Dymnikova and M. Khlopov, Int. J. Mod. Phys. D24, 1545002 (2015), arXiv:1510.01351 [gr-qc]

    Pith/arXiv arXiv 2015

  22. [22]

    V. P. Frolov, Phys. Rev. D94, 104056 (2016), arXiv:1609.01758 [gr-qc]

    Pith/arXiv arXiv 2016

  23. [23]

    S. A. Hayward, Phys. Rev. Lett.96, 031103 (2006), arXiv:gr-qc/0506126

    Pith/arXiv arXiv 2006

  24. [24]

    Bonanno, A.-P

    A. Bonanno, A.-P. Khosravi, and F. Saueressig, Phys. Rev. D103, 124027 (2021), arXiv:2010.04226 [gr-qc]

    Pith/arXiv arXiv 2021

  25. [25]

    Carballo-Rubio, F

    R. Carballo-Rubio, F. Di Filippo, S. Liberati, C. Pacilio, and M. Visser, JHEP09, 118 (2022), arXiv:2205.13556 [gr-qc]

    Pith/arXiv arXiv 2022

  26. [26]

    Franzin, S

    E. Franzin, S. Liberati, J. Mazza, and V. Vellucci, (2022), arXiv:2207.08864 [gr-qc]

    Pith/arXiv arXiv 2022

  27. [27]

    Bonanno, A.-P

    A. Bonanno, A.-P. Khosravi, and F. Saueressig, (2022), arXiv:2209.10612 [gr-qc]

    Pith/arXiv arXiv 2022

  28. [28]

    Casadio, A

    R. Casadio, A. Giusti, and J. Ovalle, Phys. Rev. D105, 124026 (2022), arXiv:2203.03252 [gr-qc]. 13

    Pith/arXiv arXiv 2022

  29. [29]

    Ovalle, R

    J. Ovalle, R. Casadio, and A. Giusti, Physics Letters B 844, 138085 (2023)

    2023

  30. [30]

    Ovalle, Phys

    J. Ovalle, Phys. Rev. D107, 104005 (2023), arXiv:2305.00030 [gr-qc]

    Pith/arXiv arXiv 2023

  31. [31]

    Dafermos, Ann

    M. Dafermos, Ann. Math158, 875 (2003)

    2003

  32. [32]

    B. P. Abbottet al.(LIGO Scientific, Virgo), Phys. Rev. Lett.116, 061102 (2016), arXiv:1602.03837 [gr-qc]

    Pith/arXiv arXiv 2016

  33. [33]

    B. P. Abbottet al.(LIGO Scientific, Virgo), Phys. Rev. Lett.119, 161101 (2017), arXiv:1710.05832 [gr-qc]

    Pith/arXiv arXiv 2017

  34. [34]

    R. e. a. Abbott (LIGO Scientific, Virgo, and KAGRA), Phys. Rev. X11, 021053 (2021), arXiv:2111.03606

    Pith/arXiv arXiv 2021

  35. [35]

    Akiyamaet al.(Event Horizon Telescope), Astrophys

    K. Akiyamaet al.(Event Horizon Telescope), Astrophys. J. Lett.875, L1 (2019), arXiv:1906.11238 [astro-ph.GA]

    Pith/arXiv arXiv 2019

  36. [36]

    Akiyamaet al.(Event Horizon Telescope), Astrophys

    K. Akiyamaet al.(Event Horizon Telescope), Astrophys. J. Lett.875, L6 (2019), arXiv:1906.11243 [astro-ph.GA]

    Pith/arXiv arXiv 2019

  37. [37]

    Akiyamaet al.(Event Horizon Telescope), Astrophys

    K. Akiyamaet al.(Event Horizon Telescope), Astrophys. J. Lett.875, L3 (2019), arXiv:1906.11240 [astro-ph.GA]

    Pith/arXiv arXiv 2019

  38. [38]

    Akiyamaet al.(Event Horizon Telescope), Astrophys

    K. Akiyamaet al.(Event Horizon Telescope), Astrophys. J. Lett.875, L5 (2019), arXiv:1906.11242 [astro-ph.GA]

    Pith/arXiv arXiv 2019

  39. [39]

    Akiyamaet al.(Event Horizon Telescope), Astrophys

    K. Akiyamaet al.(Event Horizon Telescope), Astrophys. J. Lett.875, L4 (2019), arXiv:1906.11241 [astro-ph.GA]

    Pith/arXiv arXiv 2019

  40. [40]

    Akiyamaet al.(Event Horizon Telescope), Astrophys

    K. Akiyamaet al.(Event Horizon Telescope), Astrophys. J. Lett.875, L2 (2019), arXiv:1906.11239 [astro-ph.IM]

    Pith/arXiv arXiv 2019

  41. [41]

    Johannsen, Classical and Quantum Gravity33, 124001 (2016), arXiv:1602.07694

    T. Johannsen, Classical and Quantum Gravity33, 124001 (2016), arXiv:1602.07694

    Pith/arXiv arXiv 2016

  42. [42]

    P. V. P. Cunha and C. A. R. Herdeiro, Gen. Rel. Grav. 50, 42 (2018), arXiv:1801.00860 [gr-qc]

    Pith/arXiv arXiv 2018

  43. [43]

    Psaltis, Gen

    D. Psaltis, Gen. Rel. Grav.51, 137 (2019), arXiv:1806.09740 [astro-ph.HE]

    Pith/arXiv arXiv 2019

  44. [44]

    Vagnozziet al., Class

    S. Vagnozziet al., Class. Quant. Grav.40, 165007 (2023), arXiv:2205.07787 [gr-qc]

    Pith/arXiv arXiv 2023

  45. [45]

    P. V. P. Cunha, C. A. R. Herdeiro, and E. Radu, Phys. Rev. D96, 024039 (2017), arXiv:1705.05461 [gr-qc]

    Pith/arXiv arXiv 2017

  46. [46]

    Ovalle, R

    J. Ovalle, R. Casadio, R. da Rocha, A. Sotomayor, and Z. Stuchlik, EPL124, 20004 (2018), arXiv:1811.08559 [gr-qc]

    Pith/arXiv arXiv 2018

  47. [47]

    Contreras, A

    E. Contreras, A. Rinc´ on, and P. Bargue˜ no, Eur. Phys. J. C79, 216 (2019), arXiv:1902.02033 [gr-qc]

    Pith/arXiv arXiv 2019

  48. [48]

    Da Rocha and A

    R. Da Rocha and A. A. Tomaz, Eur. Phys. J. C79, 1035 (2019), arXiv:1905.01548 [hep-th]

    Pith/arXiv arXiv 2019

  49. [49]

    Heydarzade, M

    Y. Heydarzade, M. Misyura, and V. Vertogradov, Phys. Rev. D108, 044073 (2023), arXiv:2307.04556 [gr-qc]

    Pith/arXiv arXiv 2023

  50. [50]

    Ovalle, R

    J. Ovalle, R. Casadio, R. d. Rocha, A. Sotomayor, and Z. Stuchlik, Eur. Phys. J.C78, 960 (2018), arXiv:1804.03468 [gr-qc]

    Pith/arXiv arXiv 2018

  51. [51]

    R. T. Cavalcanti, K. d. S. Alves, and J. M. Hoff da Silva, Universe8, 363 (2022), arXiv:2207.03995 [gr-qc]

    Pith/arXiv arXiv 2022

  52. [52]

    R. T. Cavalcanti, R. C. de Paiva, and R. da Rocha, Eur. Phys. J. Plus137, 1185 (2022), arXiv:2203.08740 [gr-qc]

    Pith/arXiv arXiv 2022

  53. [53]

    Meert and R

    P. Meert and R. da Rocha, Eur. Phys. J. C82, 175 (2022), arXiv:2109.06289 [hep-th]

    Pith/arXiv arXiv 2022

  54. [54]

    Sultana, Symmetry13, 1598 (2021)

    J. Sultana, Symmetry13, 1598 (2021)

    2021

  55. [55]

    Ovalle, R

    J. Ovalle, R. Casadio, E. Contreras, and A. Sotomayor, Phys. Dark Univ.31, 100744 (2021), arXiv:2006.06735 [gr-qc]

    Pith/arXiv arXiv 2021

  56. [56]

    da Rocha and A

    R. da Rocha and A. A. Tomaz, Eur. Phys. J. C80, 857 (2020), arXiv:2005.02980 [hep-th]

    Pith/arXiv arXiv 2020

  57. [57]

    Fernandes-Silva, A

    A. Fernandes-Silva, A. J. Ferreira-Martins, and R. da Rocha, Phys. Lett.B791, 323 (2019), arXiv:1901.07492 [hep-th]

    Pith/arXiv arXiv 2019

  58. [58]

    Ovalle, Eur

    J. Ovalle, Eur. Phys. J. C82, 170 (2022), arXiv:2202.12037 [gr-qc]

    Pith/arXiv arXiv 2022

  59. [59]

    Estrada, Annals Phys.439, 168792 (2022), arXiv:2106.02166 [gr-qc]

    M. Estrada, Annals Phys.439, 168792 (2022), arXiv:2106.02166 [gr-qc]

    Pith/arXiv arXiv 2022

  60. [60]

    Ovalle, R

    J. Ovalle, R. Casadio, and A. Giusti, (2023), 10.1016/j.physletb.2023.138085, arXiv:2304.03263 [gr- qc]

    arXiv 2023

  61. [61]

    Zhang, M

    C.-M. Zhang, M. Zhang, and D.-C. Zou, Chin. Phys. C 47, 015106 (2023), arXiv:2208.06830 [gr-qc]

    Pith/arXiv arXiv 2023

  62. [62]

    M. R. Khosravipoor and M. Farhoudi, Eur. Phys. J. C 83, 1045 (2023), arXiv:2311.02456 [gr-qc]

    Pith/arXiv arXiv 2023

  63. [63]

    Contreras, J

    E. Contreras, J. Ovalle, and R. Casadio, Phys. Rev. D 103, 044020 (2021), arXiv:2101.08569 [gr-qc]

    Pith/arXiv arXiv 2021

  64. [64]

    Ramos, C

    A. Ramos, C. Arias, R. Avalos, and E. Contreras, Annals Phys.431, 168557 (2021), arXiv:2107.01146 [gr-qc]

    Pith/arXiv arXiv 2021

  65. [65]

    Ovalle, E

    J. Ovalle, E. Contreras, and Z. Stuchlik, Phys. Rev. D 103, 084016 (2021), arXiv:2104.06359 [gr-qc]

    Pith/arXiv arXiv 2021

  66. [66]

    P. J. Arias, P. Bargue˜ no, E. Contreras, and E. Fuen- mayor, Astronomy1, 2 (2022), arXiv:2203.00661 [gr-qc]

    Pith/arXiv arXiv 2022

  67. [67]

    Avalos, P

    R. Avalos, P. Bargue˜ no, and E. Contreras, Fortsch. Phys. 2023, 2200171 (2023), arXiv:2303.04119 [gr-qc]

    Pith/arXiv arXiv 2023

  68. [68]

    Avalos and E

    R. Avalos and E. Contreras, Eur. Phys. J. C83, 155 (2023), arXiv:2302.09148 [gr-qc]

    Pith/arXiv arXiv 2023

  69. [69]

    Casadio, A

    R. Casadio, A. Giusti, and J. Ovalle, (2023), arXiv:2303.02713 [gr-qc]

    Pith/arXiv arXiv 2023

  70. [70]

    Contreras and P

    E. Contreras and P. Bargue˜ no, Eur. Phys. J. C78, 985 (2018), arXiv:1809.09820 [gr-qc]

    Pith/arXiv arXiv 2018

  71. [71]

    Estrada and R

    M. Estrada and R. Prado, Eur. Phys. J. C80, 799 (2020), arXiv:2003.13168 [gr-qc]

    Pith/arXiv arXiv 2020

  72. [72]

    Ovalle, Phys

    J. Ovalle, Phys. Rev. D95, 104019 (2017)

    2017

  73. [73]

    Ovalle, Phys

    J. Ovalle, Phys. Lett.B788, 213 (2019), arXiv:1812.03000 [gr-qc]

    Pith/arXiv arXiv 2019

  74. [74]

    Simpson and M

    A. Simpson and M. Visser, Universe6, 8 (2019), arXiv:1911.01020 [gr-qc]

    Pith/arXiv arXiv 2019

  75. [75]

    Simpson and M

    A. Simpson and M. Visser, JCAP02, 042 (2019), arXiv:1812.07114 [gr-qc]

    Pith/arXiv arXiv 2019

  76. [76]

    Luminet, Astron

    J.-P. Luminet, Astron. Astrophys.75, 228 (1979)

    1979

  77. [77]

    S. E. Gralla, D. E. Holz, and R. M. Wald, Phys. Rev. D 100, 024018 (2019)

    2019

  78. [78]

    S. E. Gralla, A. Lupsasca, and D. P. Marrone, Phys. Rev. D102, 124004 (2020), arXiv:2008.03879 [astro-ph.HE]

    Pith/arXiv arXiv 2020

  79. [79]

    Martin-Moruno and M

    P. Martin-Moruno and M. Visser, Fundam. Theor. Phys. 189, 193 (2017), arXiv:1702.05915 [gr-qc]

    Pith/arXiv arXiv 2017

  80. [80]

    J. M. Bardeen, Proceedings of the International Confer- ence GR5 (1968)

    1968

Showing first 80 references.