Pith. sign in

REVIEW 5 major objections 5 minor 1 cited by

Black hole with global monopole charge in self-interacting Kalb-Ramond field

T0 review · 5 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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

Pith's one-line read The paper claims that a static, spherically symmetric black hole in a self-interacting Kalb-Ramond field with a global monopole has a lapse function with linear and quadratic terms controlled by a Lorentz-violating parameter $\ell$ and…

desk verdict The metric is a real extension of the KR black hole, but the solar-system constraints in the abstract are contradicted by the paper's own time-delay formula. read the letter →

arxiv 2501.09899 v2 pith:LSOFHJOJ submitted 2025-01-17 gr-qc astro-ph.HEhep-thmath-phmath.MP

Mohsen Fathi , Ali Övgün This is my paper · ORCID
classification gr-qcastro-ph.HEhep-thmath-phmath.MP PACS 95.30.Sf04.70.-s97.60.Lf04.50.Kd
keywords blackholesmodifiedgravityKalb-RamondfieldglobalmonopoleLorentzsymmetryviolationholethermodynamicssolarsystemtestsShapirotimedelay
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

This paper asks what happens when two beyond-Schwarzschild ingredients are switched on together: a Kalb-Ramond field whose vacuum expectation value breaks Lorentz symmetry, and a global monopole from a spontaneously broken global O(3) symmetry. It derives the exterior metric of the resulting black hole, showing that the combination generates terms that grow linearly and quadratically with radius, mimicking a cosmological constant without one. The central claim is that the lapse function has the form $A(r)\approx \frac{1}{1-\ell}-\frac{2M}{r}+\frac{\ell M\eta^2}{2(1-\ell)}r-\frac{\ell\eta^2}{6(1-\ell)^2}r^2$, with $\ell$ the Lorentz-violating parameter and $\eta$ the monopole charge. The paper then shows how horizons, thermodynamics, and the four classic solar-system tests respond to these parameters and concludes that observations force $10^{-9}\leq |\ell|\leq 10^{-4}$ and $10^{-9}\leq \eta\leq 10^{-6}\,\mathrm{m}^{-1}$. This matters because it turns an abstract string-motivated background field and a cosmological defect into a testable set of deviations from Schwarzschild gravity.

What carries the argument

The load-bearing object is the Kalb-Ramond field, a rank-two antisymmetric tensor field whose nonzero vacuum expectation value spontaneously breaks Lorentz symmetry, combined with a global monopole described by a scalar triplet with symmetry-breaking scale $\eta$. The paper parametrizes the KR effect by the dimensionless combination $\ell=\xi_2 b^2/2$ and writes the monopole configuration as $\phi^a=\eta f(r)x^a/r$, then freezes $f(r)=1$ so the monopole contributes only through the charge $\eta$. The argument is carried by the assumption $V'(Y)=0$, which removes all potential-derivative terms from the gravitational field equations; subtracting two of the resulting equations forces $A(r)=B(r)^{-1}$ and leads to the exact and then approximate metric functions. This mechanism converts the KR field and monopole charge into effective $r$ and $r^2$ curvature terms, which in turn control the horizon structure, the thermodynamic corrections, and the weak-field observables.

What would settle it

Re-solve the same static, spherically symmetric field equations with a small nonzero $V'(Y)$; if the lapse function no longer matches Eq. (13) at order $\eta^2$, the solution is an artifact of the minimum condition. Observationally, a radar time-delay measurement accurate enough to separate the $-2\ell(r_1+r_2)$ term from the $\ell\eta^2$ terms would decide whether the $\ell\sim10^{-9}$ band from Shapiro delay is the real signature.

Watch

Extended reading notes

Core claim

The central discovery is an exterior black hole solution that unites spontaneous Lorentz violation from a Kalb-Ramond field with a global monopole. For a static, spherically symmetric ansatz and with the self-interaction potential at its minimum, $V'(Y)=0$, the field equations imply $A(r)=B(r)^{-1}$ and produce an exact solution involving the imaginary error function; expanded to second order in the monopole charge it becomes the approximate lapse function above. For $\eta=0$ it reduces to the known Kalb-Ramond black hole, and for $\ell=0$ to Schwarzschild. The extra $r$ and $r^2$ terms mimic quintessence and a cosmological constant but originate from the KR-monopole interaction. For $\ell\leq 0$ the spacetime always has one event horizon, while for $\ell>0$ the horizons can merge into an extremal black hole or vanish, leaving a naked singularity. The thermodynamics show an entropy with a logarithmic correction to the area law, and a specific heat that turns positive for sufficiently large black holes, while the Gibbs free energy stays negative, indicating global stability. The weak-field version of the same metric, applied to perihelion precession, gravitational redshift, light deflection, and radar time delay, yields the quoted parameter bands.

Load-bearing premise

The derivation assumes the Kalb-Ramond self-interaction potential sits exactly at its local minimum, $V'(Y)=0$, so all potential-derivative terms drop out of the field equations; if that condition fails, the metric, horizons, thermodynamics, and all four solar-system bounds must be recomputed.

Editorial extensions

If this is right

  • The event horizon is not generally at $2M$: for $\ell<0$ the black hole is larger than Schwarzschild, for $\ell>0$ it is smaller, and for positive $\ell$ the $r^2$ term can produce a second, Cauchy-type horizon.
  • The area law is violated by the monopole: entropy picks up a logarithmic term, so area alone no longer fixes the entropy.
  • Large asymptotically AdS-like black holes ($\ell\leq 0$) can be locally stable, with positive specific heat, while small ones are unstable just like Schwarzschild.
  • The apparent cosmological-constant-like behavior in the metric comes from the KR-monopole interaction, so phenomena usually attributed to a cosmological constant could in principle be mimicked by these parameters.
  • The solar-system bounds are parameter-specific: perihelion and redshift allow $|\ell|\sim10^{-4}$ to $10^{-6}$, while deflection and radar delay push $\ell$ toward $10^{-9}$.

Reading between the lines

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

  • The paper fixes the monopole profile at $f(r)=1$, which ignores the monopole core; solving the full radial profile could show whether the exterior metric remains valid all the way to the horizon or needs a cutoff.
  • The four tests constrain different combinations of $\ell$ and $\eta$, and lensing prefers negative $\ell$ while Shapiro delay prefers positive $\ell$; combining all four with a shadow measurement could break the degeneracy and test the theory more sharply.
  • Because every result rests on $V'(Y)=0$, a small deviation from the potential minimum is the most direct stress test; if the metric changes at order $\eta^2$ when $V'(Y)\neq 0$, the constraints reported here apply only at the exact minimum.
  • The authors mention black-hole shadow constraints as future work; a natural extension would be to compute whether the allowed bands from shadow radius are compatible with the solar-system bands or exclude the parameter space.
Share X Bluesky LinkedIn Reddit HN

Formalized claims in Lean

  1. Claim #1: The central discovery is an exterior black hole solution that unites spontaneous Lorentz violation from a Kalb-Ramond field with a global monopole. For a static, spherically symmetric ansatz and with the self-interaction potential at its minimum, $V'(Y)=0$, the field equations imply $A(r)=B(r)^{-1}$ and produce an exact solution involving the imaginary error function; expanded to second order in t

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 5 minor

Summary. This paper studies a static, spherically symmetric black hole solution in Einstein gravity with a Kalb-Ramond field that acquires a non-zero vacuum expectation value and a global monopole as the matter source. It presents a lapse function (13) depending on a dimensionless Lorentz-violating parameter ℓ and a monopole charge η, discusses horizon structure and black hole thermodynamics, and derives weak-field solar-system observables. The abstract concludes with parameter ranges 10^-9 ≤ |ℓ| ≤ 10^-4 and 10^-9 ≤ η ≤ 10^-6 m^-1.

Significance. The solution extends the known neutral Kalb-Ramond black hole of Ref. [74] by including a global monopole, and the paper is explicit about its starting action, energy-momentum tensors, and the reduction to the Schwarzschild/KR limits. The closed-form weak-field observables are a useful feature. If the constraints were correct, they would provide phenomenological bounds on Lorentz violation and monopole charge. However, several load-bearing claims are not supported by the paper's own equations: the time-delay constraint is internally inconsistent, the independent constraint on η is not identifiable from the observables, and there are technical errors in the field definitions, the extremal condition, and the entropy expansion. The claimed parameter ranges therefore cannot be accepted as stated.

major comments (5)
  1. [Section II, Eq. (1)] The action in Eq. (1) contains H_μνρ = ∂_[μ B_νρ] and a potential V(B_μν B^μν), which requires B_μν to be the antisymmetric Kalb-Ramond field. The sentence defining B_μν = ∂_μ B_ν − ∂_ν B_μ as the field strength of a vector field B_μ is incompatible with this structure, and no such vector field appears elsewhere in the action. This ambiguity propagates into Eq. (4) and the field equations (9)–(11), so the central gravitational equations are not unambiguously derived from the stated action.
  2. [Section IV.D, Eq. (64)] For the Viking geometry, Eq. (64) gives Δt_⊙ ≈ 4M[1+ln(4r1r2/r0^2)] − 2ℓ(r1+r2) − (1/2)ℓMη^2[r1^2+r2^2−r0^2 ln(...)]. With r1 ≈ 1.5×10^11 m, r2 ≈ 2.3×10^11 m, and the quoted 10 ns error corresponding to about 3 m in c=1 units, the ℓ-term alone is |2ℓ(r1+r2)| ≈ 7.6×10^11 |ℓ| m. Setting ℓ ∼ 10^-9 would therefore give a discrepancy of roughly 760 m, more than two orders of magnitude above the quoted error. The stated Viking constraint ℓ ∼ 10^-9 is thus contradicted by the paper's own formula; the formula implies |ℓ| ≲ 4×10^-12 unless additional unstated priors are imposed.
  3. [Section IV, Eqs. (41), (46), (55), (64)] In all four solar-system observables, the monopole charge η appears only in products with ℓ, such as ℓη^2 or ℓMη^2. The perihelion correction is proportional to ℓη^2M^2, the redshift constraint involves ℓMη^2, the deflection correction is ℓη^2M^2, and the time delay involves ℓMη^2. Consequently the data constrain the product ℓη^2, not η separately. The abstract's independent range 10^-9 ≤ η ≤ 10^-6 m^-1 is therefore ill-posed: if ℓ = 0, no bound on η follows. The authors should present joint constraints on ℓ and ℓη^2, or state explicitly the additional assumptions needed to bound η alone.
  4. [Section II, Eq. (21)] The extremal condition η^2_* is dimensionally inconsistent as written. In Eq. (21), the numerator contains the dimensionless number 3 together with terms such as 4√3 M(1−ℓ)^2 and √3 inside the square root, while the denominator 2M^2(1−ℓ)ℓ has dimension length^2. Therefore η^2_* cannot have the required dimension length^-2 unless factors are suppressed or units with M=1 are understood. This affects the discriminant analysis, Fig. 3, and the claimed existence of extremal black holes and naked singularities for ℓ > 0.
  5. [Section III, Eq. (25)] The entropy expression S = −4π(1−ℓ) ln(η^2ℓ r_+^2 + 4 − 4ℓ)/(η^2ℓ) is singular in the limit η → 0, contradicting the claim immediately after Eq. (25) that η = 0 reduces to S = A_+/4. The reduction requires starting from a different integral rather than taking the limit of the displayed formula. In addition, the expansion at O(r_+^3) is incorrect: after the constant term the expansion is quadratic, S ≈ −4π(1−ℓ) ln(4−4ℓ)/(η^2ℓ) − π r_+^2 + O(r_+^4). This affects the subsequent Smarr-formula discussion and the thermodynamic conclusions.
minor comments (5)
  1. [Section IV.D] The sentence about 'massless energy carriersneutrinos and gravitational wavesproviding further support for the presence of dark matter [106]' is garbled and the cited reference does not support that claim; this passage should be corrected or removed.
  2. [Keywords] The keyword 'Kalb-Ramon' should be 'Kalb-Ramond'.
  3. [Section IV.A] The phrase 'which funds our mathematical approach' appears to be a typo for 'which forms our mathematical approach'.
  4. [Figure 6 caption] The caption contains '(b,s)' in the description of panel (b); this should presumably read '(b,d)'.
  5. [Section II, after Eq. (11)] The assumption V'(Y)=0 is stated only parenthetically, but it is essential to the entire solution. The paper should state explicitly that all subsequent metric, thermodynamic, and observational results are obtained under this condition, and that the shape of the self-interaction potential is not probed by the solar-system bounds.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the metric is derived from the explicit action with stated assumptions, and the solar-system constraints are comparisons to data rather than fitted predictions.

full rationale

The derivation chain is self-contained in the relevant sense. The central lapse function A(r), Eq. (13), follows from solving the gravitational field equations (9)-(11), which are obtained by varying the explicit action (1) with the global-monopole matter Lagrangian (5). The assumption V'(Y)=0 is a stated physical condition (following Ref. [74]) that removes potential-derivative terms; it does not presuppose the final metric, and the solution (12)-(13) is then obtained by integration. The identification c1/2 = M is a standard mass normalization imported from an external reference [74], not from the authors' own prior work, and it is not the claim being tested. Thermodynamic quantities and solar-system results are computed from the metric and then compared with observational data; no fitted parameter is renamed as a prediction. The self-citations that appear, e.g., Refs. [15] and [86], are contextual or auxiliary and are not load-bearing justifications for the central solution. Even if the solar-system constraints are internally inconsistent (e.g., the time-delay formula in Eq. (64) implies a tighter bound on |ell| than the abstract's range), that is a correctness or self-consistency issue, not a circularity, and therefore does not increase the circularity score.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The solution introduces two physical parameters, ℓ and η, plus the mass M and the VEV norm b^2. The monopole self-coupling λ appears in the Lagrangian but drops out after the f(r)=1 ansatz. The paper does not fit these to data; it claims to constrain them, but the constraints are not supported by the equations.

free parameters (3)
  • ℓ (dimensionless KR coupling) = constrained to 10^-9 to 10^-4 (claimed)
    Appears in the metric (13) and in the field equations via ℓ ≡ ξ2 b^2/2; the paper claims solar system tests bound it, but the bounds are inconsistent.
  • η (monopole charge) = constrained to 10^-9 to 10^-6 m^-1 (claimed)
    Symmetry-breaking scale of the global monopole, with dimension length^-1; appears in the metric (13); the paper claims constraints, but η enters observables only via ℓη^2 or ℓM^2η^2 products, so the constraints are not supported.
  • M (mass) = set from horizon condition (Eq. 22)
    Integration constant identified as mass via c1/2 = M from Ref. [74]; in thermodynamics, M is re-expressed in terms of r+.
assumptions (5)
  • domain assumption The KR potential is at its minimum, V'(Y)=0, so only the constant VEV norm b^2 matters.
    Invoked after Eq. (11) to eliminate potential-derivative terms; without it, the field equations and the solution (13) would change.
  • domain assumption The monopole scalar field is in the hedgehog configuration f(r)=1 outside the core.
    Used to simplify T^(GM) to diag(-η^2 A/r^2, η^2 B/r^2, 0, 0); standard for global monopole exteriors, but it ignores core backreaction.
  • domain assumption The VEV has constant norm b^{μν}b_{μν} = ∓b^2.
    Required to impose spontaneous Lorentz violation; stated in Section II before Eq. (4).
  • domain assumption c1/2 = M, as shown in Ref. [74].
    The integration constant in (12) is identified with the black hole mass by reference to Ref. [74]; the present paper does not re-derive it.
  • domain assumption Extended phase-space thermodynamics with pressure P = -l_eff/(8π) is valid.
    Introduced in Section III to define thermodynamic volume and the first law; this is a modeling choice following AdS black hole chemistry.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Black hole with global monopole charge in self-interacting Kalb-Ramond field." pith.science (2026). https://pith.science/paper/LSOFHJOJ

@misc{pith2026250109899,
  author       = {Pith},
  title        = {Pith review of: Black hole with global monopole charge in self-interacting Kalb-Ramond field},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LSOFHJOJ}},
  note         = {Machine review of arXiv:2501.09899}
}
abstract

In this study, we explore a static, spherically symmetric black hole solution in the context of a self-interacting Kalb-Ramond field coupled with a global monopole. By incorporating the effects of Lorentz-violating term $\ell$ and the monopole charge $\eta$ in the KR field, we derive the modified gravitational field equations and analyze the resulting black hole spacetime. The obtained solution exhibits deviations from the Schwarzschild metric with topological defect, as it is influenced by the monopole charge and self-interaction potential. We investigate the thermodynamic properties of the black hole, including its Hawking temperature, entropy, and specific heat, revealing novel stability conditions. Additionally, we perform solar system tests such as perihelion precession, gravitational redshift, light deflection, and time delay of signals to impose constraints on the Lorentz-violating parameter and monopole charge. Our findings suggest that these parameters have to be significantly small, although there are different constraints imposed by individual tests, ranging from $10^{-9}\leq|\ell|\leq 10^{-4}$ and $10^{-9}\leq\eta\leq 10^{-6}\, \mathrm{m}^{-1}$.

Figures

Figures reproduced from arXiv: 2501.09899 by the authors.

Figure 1
Figure 1. FIG. 1. Radial profile of [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Profiles of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Relationship between [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: FIG. 4. The behavior of the Hawking temperature for the AdS-like black hole within [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Entropy behavior for the AdS-like black hole with [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. The behavior of the specific heat within [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. The behavior of the Gibbs free energy within [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Constraints on [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. The confidence intervals for [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Parameter constraints on [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Constraints on [PITH_FULL_IMAGE:figures/full_fig_p015_11.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Charged black holes in Kalb-Ramond gravity: Weak Deflection Angle, Shadow cast, Quasinormal Modes and Neutrino annihilation

    gr-qc 2025-05 reject novelty 4.0 of 10

    The Kalb-Ramond black hole phenomenology is mostly an extension of known results, and its shadow formula is internally inconsistent, invalidating the EHT-based constraints.

Reference graph

Works this paper leans on

108 extracted references · 76 canonical work pages · cited by 1 Pith paper

  1. [74]

    Bianchi type I cosmology with a Kalb-Ramond background field

    R. Maluf and J. Neves, “Bianchi type i cosmology with a kalb-ramond background field,” Eur. Phys. J. C , vol. 82, p. 135, 2022. arXiv:2111.13165

    work page Pith review arXiv 2022

  2. [1]

    r r − r0 r + r0 + 2 ln r + p r2 − r2 0 r0 !# , (61a) tℓ(r, r0) = −ℓ q r2 − r2 0, (61b) tη(r, r0) = − 1 2 ℓM η2r2 0

    Thus, the coordinate time is given by t(r, r0) = Z r r0 dr A(r) q 1 − r2 0 A(r0) A(r) r2 , (59) for the segment of the journey between r0 and r. To first-order corrections, this reduces to t(r, r0) ≈ q r2 − r2 0 + tM (r, r0) + tℓ(r, r0) + tη(r, r0), (60) where: tM (r, r0) = M "r r − r0 r + r0 + 2 ln r + p r2 − r2 0 r0 !# , (61a) tℓ(r, r0) = −ℓ q r2 − r2 0...

  3. [2]

    Spontaneous breaking of lorentz symmetry in string theory,

    V. Kostelecky and S. Samuel, “Spontaneous breaking of lorentz symmetry in string theory,” Phys. Rev. D , vol. 39, p. 683, 1989

    1989

  4. [3]

    Loop quantum gravity and light propagation,

    J. Alfaro, H. Morales-Tecotl, and L. Urrutia, “Loop quantum gravity and light propagation,” Phys. Rev. D , vol. 65, p. 103509, 2002

    2002

  5. [4]

    Quantum gravity at a lifshitz point,

    P. Horava, “Quantum gravity at a lifshitz point,” Phys. Rev. D , vol. 79, p. 084008, 2009

    2009

  6. [5]

    Noncommutative field theory and lorentz violation,

    S. Carroll, J. Harvey, V. Kostelecky, C. Lane, and T. Okamoto, “Noncommutative field theory and lorentz violation,” Phys. Rev. Lett., vol. 87, p. 141601, 2001

    2001

  7. [6]

    Gravity with a dynamical preferred frame,

    T. Jacobson and D. Mattingly, “Gravity with a dynamical preferred frame,” Phys. Rev. D , vol. 64, p. 024028, 2001

    2001

  8. [7]

    Massive graviton as a testable cold dark matter candidate,

    S. Dubovsky, P. Tinyakov, and I. Tkachev, “Massive graviton as a testable cold dark matter candidate,” Phys. Rev. Lett. , vol. 94, p. 181102, 2005

    2005

Show all 108 references
  1. [8]

    Dark torsion as the cosmic speed-up,

    G. Bengochea and R. Ferraro, “Dark torsion as the cosmic speed-up,” Phys. Rev. D , vol. 79, p. 124019, 2009

    2009

  2. [9]

    Very special relativity,

    A. Cohen and S. Glashow, “Very special relativity,” Phys. Rev. Lett. , vol. 97, p. 021601, 2006

    2006

  3. [10]

    CPT and Lorentz-symmetry breaking: A Review,

    R. Lehnert, “CPT and Lorentz-symmetry breaking: A Review,” Frascati Phys. Ser. , vol. 43, pp. 131–154, 2007

    2007

  4. [11]

    Lorentz violation in Dirac and Weyl semimetals,

    V. A. Kosteleck´ y, R. Lehnert, N. McGinnis, M. Schreck, and B. Seradjeh, “Lorentz violation in Dirac and Weyl semimetals,” Phys. Rev. Res. , vol. 4, no. 2, p. 023106, 2022

    2022

  5. [12]

    The Cerenkov effect in Lorentz-violating vacua,

    R. Lehnert and R. Potting, “The Cerenkov effect in Lorentz-violating vacua,” Phys. Rev. D , vol. 70, p. 125010, 2004. [Erratum: Phys.Rev.D 70, 129906 (2004)]

    2004

  6. [13]

    Relativity violations and beta decay,

    J. S. D ´ ıaz, A. Kosteleck´ y, and R. Lehnert, “Relativity violations and beta decay,”Phys. Rev. D , vol. 88, no. 7, p. 071902, 2013

    2013

  7. [14]

    Massive photons and Lorentz violation,

    M. Cambiaso, R. Lehnert, and R. Potting, “Massive photons and Lorentz violation,” Phys. Rev. D , vol. 85, p. 085023, 2012

    2012

  8. [15]

    A Monopole solution in a Lorentz-violating field theory,

    M. D. Seifert, “A Monopole solution in a Lorentz-violating field theory,” Phys. Rev. Lett. , vol. 105, p. 201601, 2010

    2010

  9. [16]

    Schwarzschild-like black hole with a topological defect in bumblebee gravity,

    I. G¨ ull¨ u and A.¨Ovg¨ un, “Schwarzschild-like black hole with a topological defect in bumblebee gravity,” Annals Phys. , vol. 436, p. 168721, 2022

    2022

  10. [17]

    Exact traversable wormhole solution in bumblebee gravity,

    A. ¨Ovg¨ un, K. Jusufi, and I. Sakallı, “Exact traversable wormhole solution in bumblebee gravity,” Phys. Rev. D , vol. 99, no. 2, p. 024042, 2019

    2019

  11. [18]

    Quasinormal modes of bumblebee wormhole,

    R. Oliveira, D. M. Dantas, V. Santos, and C. A. S. Almeida, “Quasinormal modes of bumblebee wormhole,” Class. Quant. Grav., vol. 36, no. 10, p. 105013, 2019

    2019

  12. [19]

    Weak field deflection angle and analytical parameter estimation of the Lorentz-violating Bumblebee parameter through the black hole shadow using EHT data,

    G. Lambiase, R. C. Pantig, and A. ¨Ovg¨ un, “Weak field deflection angle and analytical parameter estimation of the Lorentz-violating Bumblebee parameter through the black hole shadow using EHT data,” EPL, vol. 148, no. 4, p. 49001, 2024

    2024

  13. [20]

    Energy extraction via magnetic reconnection in Lorentz breaking Kerr–Sen and Kiselev black holes,

    A. Carleo, G. Lambiase, and L. Mastrototaro, “Energy extraction via magnetic reconnection in Lorentz breaking Kerr–Sen and Kiselev black holes,” Eur. Phys. J. C , vol. 82, no. 9, p. 776, 2022

    2022

  14. [21]

    Lorentz violation and generalized uncertainty principle,

    G. Lambiase and F. Scardigli, “Lorentz violation and generalized uncertainty principle,” Phys. Rev. D , vol. 97, no. 7, p. 075003, 2018

    2018

  15. [22]

    An exact stationary axisymmetric vacuum solution within a metric-affine bumblebee gravity,

    A. A. Ara´ ujo Filho, J. R. Nascimento, A. Y. Petrov, and P. J. Porf ´ ırio, “An exact stationary axisymmetric vacuum solution within a metric-affine bumblebee gravity,” JCAP, vol. 07, p. 004, 2024

    2024

  16. [23]

    Vacuum solution within a metric-affine bumblebee gravity,

    A. A. A. Filho, J. R. Nascimento, A. Y. Petrov, and P. J. Porf ´ ırio, “Vacuum solution within a metric-affine bumblebee gravity,” Phys. Rev. D , vol. 108, no. 8, p. 085010, 2023

    2023

  17. [24]

    Derivative four-fermion model, effective action and bumblebee generation,

    R. Araujo, T. Mariz, J. R. Nascimento, and A. Y. Petrov, “Derivative four-fermion model, effective action and bumblebee generation,” Eur. Phys. J. C , vol. 84, no. 10, p. 1034, 2024

    2024

  18. [25]

    On metric-affine bumblebee model coupled to scalar matter,

    J. R. Nascimento, G. J. Olmo, A. Y. Petrov, and P. J. Porfirio, “On metric-affine bumblebee model coupled to scalar matter,” Nucl. Phys. B , vol. 1004, p. 116577, 2024

    2024

  19. [26]

    Scattering effects of bumblebee gravity in metric-affine formalism,

    N. Heidari, C. F. B. Macedo, A. A. A. Filho, and H. Hassanabadi, “Scattering effects of bumblebee gravity in metric-affine formalism,” Eur. Phys. J. C , vol. 84, no. 11, p. 1221, 2024

    2024

  20. [27]

    Shadows, greybody factors, emission rate, topological charge, and phase transitions for a charged black hole with a Kalb-Ramond field background,

    F. Hosseinifar, A. A. A. Filho, M. Y. Zhang, H. Chen, and H. Hassanabadi, “Shadows, greybody factors, emission rate, topological charge, and phase transitions for a charged black hole with a Kalb-Ramond field background,” 7 2024

    2024

  21. [28]

    Particle creation and evaporation in Kalb-Ramond gravity,

    A. A. Ara´ ujo Filho, “Particle creation and evaporation in Kalb-Ramond gravity,” 11 2024

    2024

  22. [29]

    Antisymmetric tensor propagator with spontaneous Lorentz violation,

    R. V. Maluf, A. A. Ara´ ujo Filho, W. T. Cruz, and C. A. S. Almeida, “Antisymmetric tensor propagator with spontaneous Lorentz violation,” EPL, vol. 124, no. 6, p. 61001, 2018

    2018

  23. [30]

    Gravity, lorentz violation, and the standard model,

    V. Kostelecky, “Gravity, lorentz violation, and the standard model,” Phys. Rev. D , vol. 69, p. 105009, 2004

    2004

  24. [31]

    Gravitational phenomenology in higher dimensional theories and strings,

    V. Kostelecky and S. Samuel, “Gravitational phenomenology in higher dimensional theories and strings,” Phys. Rev. D , vol. 40, p. 1886, 1989

    1989

  25. [32]

    Phenomenological gravitational constraints on strings and higher dimensional theories,

    V. Kostelecky and S. Samuel, “Phenomenological gravitational constraints on strings and higher dimensional theories,” Phys. Rev. Lett. , vol. 63, p. 224, 1989

    1989

  26. [33]

    Signals for lorentz violation in post-newtonian gravity,

    Q. Bailey and V. Kostelecky, “Signals for lorentz violation in post-newtonian gravity,” Phys. Rev. D , vol. 74, p. 045001, 2006

    2006

  27. [34]

    Constraints and stability in vector theories with spontaneous lorentz violation,

    R. Bluhm, N. Gagne, R. Potting, and A. Vrublevskis, “Constraints and stability in vector theories with spontaneous lorentz violation,” Phys. Rev. D , vol. 77, p. 125007, 2008

    2008

  28. [35]

    Exact schwarzschild-like solution in a bumblebee gravity model,

    R. Casana, A. Cavalcante, F. P. Poulis, and E. B. Santos, “Exact schwarzschild-like solution in a bumblebee gravity model,” Phys. Rev. D , vol. 97, p. 104001, 2018. 17

    2018

  29. [36]

    Gravitational lensing under the effect of weyl and bumblebee gravities: Applications of gauss-bonnet theorem,

    A. ¨Ovg¨ un, K. Jusufi, and I. Sakalli, “Gravitational lensing under the effect of weyl and bumblebee gravities: Applications of gauss-bonnet theorem,” Annals Phys. , vol. 399, p. 193, 2018

    2018

  30. [37]

    Probing Schwarzschild-like black holes in metric-affine bumblebee gravity with accretion disk, deflection angle, greybody bounds, and neutrino propagation,

    G. Lambiase, L. Mastrototaro, R. C. Pantig, and A. Ovgun, “Probing Schwarzschild-like black holes in metric-affine bumblebee gravity with accretion disk, deflection angle, greybody bounds, and neutrino propagation,” JCAP, vol. 12, p. 026, 2023

    2023

  31. [38]

    Strong gravitational lensing and shadow constraint from M87* of slowly rotating Kerr-like black hole,

    X.-M. Kuang and A. ¨Ovg¨ un, “Strong gravitational lensing and shadow constraint from M87* of slowly rotating Kerr-like black hole,” Annals Phys. , vol. 447, p. 169147, 2022

    2022

  32. [39]

    Probing the Lorentz Invariance Violation via Gravitational Lensing and Analytical Eigenmodes of Perturbed Slowly Rotating Bumblebee Black Holes,

    M. Mangut, H. G¨ ursel, S. Kanzi, and I. Sakallı, “Probing the Lorentz Invariance Violation via Gravitational Lensing and Analytical Eigenmodes of Perturbed Slowly Rotating Bumblebee Black Holes,” Universe, vol. 9, no. 5, p. 225, 2023

    2023

  33. [40]

    Testing black holes with cosmological constant in Einstein-bumblebee gravity through the black hole shadow using EHT data and deflection angle,

    R. C. Pantig, S. Kala, A. ¨Ovg¨ un, and N. J. L. S. Lobos, “Testing black holes with cosmological constant in Einstein-bumblebee gravity through the black hole shadow using EHT data and deflection angle,” 10 2024

    2024

  34. [41]

    Strange Quark Stars and Condensate Dark Stars in Bumblebee Gravity,

    G. Panotopoulos and A. ¨Ovg¨ un, “Strange Quark Stars and Condensate Dark Stars in Bumblebee Gravity,” 9 2024

    2024

  35. [42]

    Modified Hawking radiation of Schwarzschild-like black hole in bumblebee gravity model,

    I. Sakallı and E. Y¨ or¨ uk, “Modified Hawking radiation of Schwarzschild-like black hole in bumblebee gravity model,” Phys. Scripta, vol. 98, no. 12, p. 125307, 2023

    2023

  36. [43]

    Quasinormal frequencies for a black hole in a bumblebee gravity,

    R. Oliveira, D. M. Dantas, and C. A. S. Almeida, “Quasinormal frequencies for a black hole in a bumblebee gravity,” EPL, vol. 135, p. 10003, 2021

    2021

  37. [44]

    Black holes with a cosmological constant in bumblebee gravity,

    R. V. Maluf and J. C. S. Neves, “Black holes with a cosmological constant in bumblebee gravity,” Phys. Rev. D , vol. 103, p. 044002, 2021

    2021

  38. [45]

    Static spherical vacuum solutions in the bumblebee gravity model,

    R. Xu, D. Liang, and L. Shao, “Static spherical vacuum solutions in the bumblebee gravity model,” Phys. Rev. D , vol. 107, p. 024011, 2023

    2023

  39. [46]

    Exact kerr-like solution and its shadow in a gravity model with spontaneous lorentz symmetry breaking,

    C. Ding, C. Liu, R. Casana, and A. Cavalcante, “Exact kerr-like solution and its shadow in a gravity model with spontaneous lorentz symmetry breaking,” Eur. Phys. J. C , vol. 80, p. 178, 2020

    2020

  40. [47]

    Slowly rotating einstein-bumblebee black hole solution and its greybody factor in a lorentz violation model,

    C. Ding and X. Chen, “Slowly rotating einstein-bumblebee black hole solution and its greybody factor in a lorentz violation model,” Chin. Phys. C , vol. 45, p. 025106, 2021

    2021

  41. [48]

    Polarizations of gravitational waves in the bumblebee gravity model,

    D. Liang, R. Xu, X. Lu, and L. Shao, “Polarizations of gravitational waves in the bumblebee gravity model,” Phys. Rev. D , vol. 106, p. 124019, 2022

    2022

  42. [49]

    Gravitational waves effects in a lorentz-violating scenario,

    K. M. Amarilo, M. B. F. Filho, A. A. A. Filho, and J. A. A. S. Reis, “Gravitational waves effects in a lorentz-violating scenario,”

  43. [50]

    Lorentz violation with an antisymmetric tensor,

    B. Altschul, Q. G. Bailey, and V. A. Kosteleck, “Lorentz violation with an antisymmetric tensor,” Physical Review D , vol. 81, p. 065028, Mar. 2010

    2010

  44. [51]

    Classical direct interstring action,

    M. Kalb and P. Ramond, “Classical direct interstring action,” Phys. Rev. D , vol. 9, p. 2273, 1974

    1974

  45. [52]

    Induced einstein-kalb-ramond theory and the black hole,

    W. Kao, W. Dai, S.-Y. Wang, T.-K. Chyi, and S.-Y. Lin, “Induced einstein-kalb-ramond theory and the black hole,” Phys. Rev. D, vol. 53, p. 2244, 1996

    1996

  46. [53]

    Static spherisymmetric solutions, gravitational lensing and perihelion precession in einstein- kalb-ramond theory,

    S. Kar, S. SenGupta, and S. Sur, “Static spherisymmetric solutions, gravitational lensing and perihelion precession in einstein- kalb-ramond theory,” Phys. Rev. D , vol. 67, p. 044005, 2003. arXiv:hep-th/0210176

    2003 arXiv

  47. [54]

    Strong gravitational lensing a probe for extra dimensions and kalb-ramond field,

    S. Chakraborty and S. SenGupta, “Strong gravitational lensing a probe for extra dimensions and kalb-ramond field,” JCAP, vol. 07, p. 045, 2017. arXiv:1611.06936

    2017 arXiv

  48. [55]

    Gravitational lensing of a Schwarzschild-like black hole in Kalb-Ramond gravity,

    E. L. B. Junior, J. T. S. S. Junior, F. S. N. Lobo, M. E. Rodrigues, D. Rubiera-Garcia, L. F. D. da Silva, and H. A. Vieira, “Gravitational lensing of a Schwarzschild-like black hole in Kalb-Ramond gravity,” Phys. Rev. D , vol. 110, no. 2, p. 024077, 2024

    2024

  49. [56]

    Electric Penrose, circular orbits and collisions of charged particles near charged black holes in Kalb–Ramond gravity,

    M. Zahid, J. Rayimbaev, N. Kurbonov, S. Ahmedov, C. Shen, and A. Abdujabbarov, “Electric Penrose, circular orbits and collisions of charged particles near charged black holes in Kalb–Ramond gravity,” Eur. Phys. J. C , vol. 84, no. 7, p. 706, 2024

    2024

  50. [57]

    Circular motion and QPOs near black holes in Kalb–Ramond gravity,

    S. Jumaniyozov, S. U. Khan, J. Rayimbaev, A. Abdujabbarov, S. Urinbaev, and S. Murodov, “Circular motion and QPOs near black holes in Kalb–Ramond gravity,” Eur. Phys. J. C , vol. 84, no. 9, p. 964, 2024

    2024

  51. [58]

    Thermal chemistry of Anti-de-Sitter black holes in Kalb-Ramond gravity,

    A. Ditta, F. Javed, A. Bouzenada, G. Mustafa, A. Mahmood, F. Atamurotov, and V. Khamidov, “Thermal chemistry of Anti-de-Sitter black holes in Kalb-Ramond gravity,” JHEAp, vol. 45, pp. 62–74, 2025

    2025

  52. [59]

    Geodesics structure and deflection angle of electrically charged black holes in gravity with a background Kalb–Ramond field,

    A. al Badawi, S. Shaymatov, and I. Sakallı, “Geodesics structure and deflection angle of electrically charged black holes in gravity with a background Kalb–Ramond field,” Eur. Phys. J. C , vol. 84, no. 8, p. 825, 2024

    2024

  53. [60]

    Energy extraction and Keplerian fundamental frequencies in the Kalb–Ramond gravity,

    D. Ortiqboev, F. Javed, F. Atamurotov, A. Abdujabbarov, and G. Mustafa, “Energy extraction and Keplerian fundamental frequencies in the Kalb–Ramond gravity,” Phys. Dark Univ. , vol. 46, p. 101615, 2024

    2024

  54. [61]

    Spontaneous Lorentz symmetry-breaking constraints in Kalb–Ramond gravity,

    E. L. B. Junior, J. T. S. S. Junior, F. S. N. Lobo, M. E. Rodrigues, D. Rubiera-Garcia, L. F. D. da Silva, and H. A. Vieira, “Spontaneous Lorentz symmetry-breaking constraints in Kalb–Ramond gravity,” Eur. Phys. J. C , vol. 84, no. 12, p. 1257, 2024

    2024

  55. [62]

    Quantum Gravity Evolution of the Kalb–Ramond like black hole,

    R. Ali, R. Babar, M. Asgher, and G. Mustafa, “Quantum Gravity Evolution of the Kalb–Ramond like black hole,” Chin. J. Phys. , vol. 86, pp. 269–279, 2023

    2023

  56. [63]

    Fermionic greybody factors and quasinormal modes of black holes in Kalb–Ramond gravity,

    A. Al-Badawi and A. Kraishan, “Fermionic greybody factors and quasinormal modes of black holes in Kalb–Ramond gravity,” Annals Phys. , vol. 458, p. 169467, 2023

    2023

  57. [64]

    Deflection of massive body around wormholes in Einstein–Kalb–Ramond spacetime,

    F. Rahaman, A. Aziz, T. Manna, A. Islam, N. A. Pundeer, and S. Islam, “Deflection of massive body around wormholes in Einstein–Kalb–Ramond spacetime,” Phys. Dark Univ. , vol. 42, p. 101287, 2023

    2023

  58. [65]

    Quasinormal modes and bounding greybody factors of GUP-corrected black holes in Kalb–Ramond gravity,

    A. Baruah, A. ¨Ovg¨ un, and A. Deshamukhya, “Quasinormal modes and bounding greybody factors of GUP-corrected black holes in Kalb–Ramond gravity,” Annals Phys. , vol. 455, p. 169393, 2023

    2023

  59. [66]

    Kalb-ramond field-induced cosmological bounce in generalized teleparallel gravity,

    K. Nair and A. Thomas, “Kalb-ramond field-induced cosmological bounce in generalized teleparallel gravity,” Phys. Rev. D , vol. 105, p. 103505, 2022. arXiv:2112.11945

    2022

  60. [67]

    Q-form fields on p-branes,

    C.-E. Fu, Y.-X. Liu, K. Yang, and S.-W. Wei, “Q-form fields on p-branes,” JHEP, vol. 10, p. 060, 2012. arXiv:1207.3152

    2012 arXiv

  61. [68]

    Solutions on a brane in a bulk spacetime with kalb-ramond field,

    S. Chakraborty and S. SenGupta, “Solutions on a brane in a bulk spacetime with kalb-ramond field,” Annals Phys. , vol. 367, p. 258, 2016. arXiv:1412.7783

    2016 arXiv

  62. [69]

    Modified black hole solution with a background kalb-ramond field,

    L. Lessa, J. Silva, R. Maluf, and C. Almeida, “Modified black hole solution with a background kalb-ramond field,” Eur. Phys. J. C, vol. 80, p. 335, 2020. arXiv:1911.10296. 18

    2020 arXiv

  63. [70]

    Particle dynamics and gravitational weak lensing around black hole in the kalb-ramond gravity,

    F. Atamurotov, D. Ortiqboev, A. Abdujabbarov, and G. Mustafa, “Particle dynamics and gravitational weak lensing around black hole in the kalb-ramond gravity,” Eur. Phys. J. C , vol. 82, p. 659, 2022

    2022

  64. [71]

    Gravitational deflection of light and shadow cast by rotating kalb-ramond black holes,

    R. Kumar, S. Ghosh, and A. Wang, “Gravitational deflection of light and shadow cast by rotating kalb-ramond black holes,” Phys. Rev. D , vol. 101, p. 104001, 2020. arXiv:2001.00460

    2020 arXiv

  65. [72]

    Traversable wormhole solution with a background kalb-ramond field,

    L. Lessa, R. Oliveira, J. Silva, and C. Almeida, “Traversable wormhole solution with a background kalb-ramond field,” Annals Phys., vol. 433, p. 168604, 2021. arXiv:2010.05298

    2021 arXiv

  66. [73]

    Exact solution for a traversable wormhole in a curvature-coupled antisymmetric background field,

    R. Maluf and C. Muniz, “Exact solution for a traversable wormhole in a curvature-coupled antisymmetric background field,” Eur. Phys. J. C , vol. 82, p. 445, 2022. arXiv:2110.12202

    2022 arXiv

  67. [75]

    Static and spherically symmetric black holes in gravity with a background Kalb-Ramond field,

    K. Yang, Y.-Z. Chen, Z.-Q. Duan, and J.-Y. Zhao, “Static and spherically symmetric black holes in gravity with a background Kalb-Ramond field,” Physical Review D , vol. 108, p. 124004, Dec. 2023

    2023

  68. [76]

    Static neutral black holes in Kalb-Ramond gravity,

    W. Liu, D. Wu, and J. Wang, “Static neutral black holes in Kalb-Ramond gravity,” Journal of Cosmology and Astroparticle Physics, vol. 2024, p. 017, Sept. 2024

    2024

  69. [77]

    Electrically charged black holes in gravity with a background KalbRamond field,

    Z.-Q. Duan, J.-Y. Zhao, and K. Yang, “Electrically charged black holes in gravity with a background KalbRamond field,” The European Physical Journal C , vol. 84, p. 798, Aug. 2024

    2024

  70. [78]

    Gravitational field of vacuum domain walls and strings,

    A. Vilenkin, “Gravitational field of vacuum domain walls and strings,” Phys. Rev. D , vol. 23, pp. 852–857, 1981

    1981

  71. [79]

    Cosmological string theories,

    A. Vilenkin, “Cosmological string theories,” Nucl. Phys. B , vol. 196, pp. 240–258, 1982

    1982

  72. [80]

    Gravitational field of a global monopole,

    M. Barriola and A. Vilenkin, “Gravitational field of a global monopole,” Physical Review Letters , vol. 63, pp. 341–343, July 1989

    1989

  73. [81]

    Schwarzschild black hole with global monopole charge,

    N. Dadhich, K. Narayan, and U. A. Yajnik, “Schwarzschild black hole with global monopole charge,” Pramana, vol. 50, pp. 307–314, 1998

    1998

  74. [82]

    Black holes with a nonlinear electromagnetic field,

    K. Bronnikov, B. Meierovich, and E. Podolyak, “Black holes with a nonlinear electromagnetic field,” J. Exp. Theor. Phys. , vol. 95, p. 392, 2002

    2002

  75. [83]

    Gauge-invariant extension of the quantum chromodynamics,

    J. Preskill, “Gauge-invariant extension of the quantum chromodynamics,” Phys. Rev. Lett. , vol. 43, p. 1365, 1979

    1979

  76. [84]

    Gravitational monopoles and solitons,

    M. Barriola and A. Vilenkin, “Gravitational monopoles and solitons,” Phys. Rev. Lett. , vol. 63, p. 341, 1989

    1989

  77. [85]

    Modified black hole solution with a background KalbRamond field,

    L. A. Lessa, J. E. G. Silva, R. V. Maluf, and C. A. S. Almeida, “Modified black hole solution with a background KalbRamond field,” The European Physical Journal C , vol. 80, p. 335, Apr. 2020

    2020

  78. [86]

    Spontaneous Lorentz and diffeomorphism violation, massive modes, and gravity,

    R. Bluhm, S.-H. Fung, and V. A. Kosteleck, “Spontaneous Lorentz and diffeomorphism violation, massive modes, and gravity,” Physical Review D , vol. 77, p. 065020, Mar. 2008

    2008

  79. [87]

    Study of null and time-like geodesics in the exterior of a Schwarzschild black hole with quintessence and cloud of strings,

    M. Fathi, M. Olivares, and J. R. Villanueva, “Study of null and time-like geodesics in the exterior of a Schwarzschild black hole with quintessence and cloud of strings,” The European Physical Journal C , vol. 82, p. 629, July 2022

    2022

  80. [88]

    Exact vacuum solution to conformal Weyl gravity and galactic rotation curves,

    P. D. Mannheim and D. Kazanas, “Exact vacuum solution to conformal Weyl gravity and galactic rotation curves,” The Astrophysical Journal , vol. 342, p. 635, July 1989

    1989

  81. [89]

    The four laws of black hole mechanics,

    J. M. Bardeen, B. Carter, and S. W. Hawking, “The four laws of black hole mechanics,” Communications in Mathematical Physics, vol. 31, pp. 161–170, June 1973

    1973

  82. [90]

    Enthalpy and the mechanics of AdS black holes,

    D. Kastor, S. Ray, and J. Traschen, “Enthalpy and the mechanics of AdS black holes,” Classical and Quantum Gravity , vol. 26, p. 195011, Oct. 2009

    2009

  83. [91]

    Insight into the Microscopic Structure of an AdS Black Hole from a Thermodynamical Phase Transition,

    S.-W. Wei and Y.-X. Liu, “Insight into the Microscopic Structure of an AdS Black Hole from a Thermodynamical Phase Transition,” Physical Review Letters , vol. 115, p. 111302, Sept. 2015

    2015

  84. [92]

    Black hole chemistry: thermodynamics with Lambda,

    D. Kubizk, R. B. Mann, and M. Teo, “Black hole chemistry: thermodynamics with Lambda,” Classical and Quantum Gravity , vol. 34, p. 063001, Mar. 2017

    2017

  85. [93]

    Kinetics of a phase transition for a Kerr-AdS black hole on the free-energy landscape,

    S.-J. Yang, R. Zhou, S.-W. Wei, and Y.-X. Liu, “Kinetics of a phase transition for a Kerr-AdS black hole on the free-energy landscape,” Physical Review D , vol. 105, p. 084030, Apr. 2022

    2022

  86. [94]

    Reverse Hawking-Page phase transition in de Sitter black holes,

    S. Mbarek and R. B. Mann, “Reverse Hawking-Page phase transition in de Sitter black holes,” Journal of High Energy Physics , vol. 2019, p. 103, Feb. 2019

    2019

  87. [95]

    Elementary derivation of the advance of the perihelion of a planetary orbit,

    S. Cornbleet, “Elementary derivation of the advance of the perihelion of a planetary orbit,” American Journal of Physics , vol. 61, no. 7, pp. 650–651, 1993

    1993

  88. [96]

    Ryder, Introduction to General Relativity

    L. Ryder, Introduction to General Relativity . Cambridge University Press, 2009

    2009

  89. [97]

    Test of Relativistic Gravitation with a Space-Borne Hydrogen Maser,

    R. F. C. Vessot et al. , “Test of Relativistic Gravitation with a Space-Borne Hydrogen Maser,” Phys. Rev. Lett. , vol. 45, pp. 2081–2084, 1980

    1980

  90. [98]

    Gravitational deflection angle of light: Definition by an observer and its application to an asymptotically nonflat spacetime,

    K. Takizawa, T. Ono, and H. Asada, “Gravitational deflection angle of light: Definition by an observer and its application to an asymptotically nonflat spacetime,” Physical Review D , vol. 101, p. 104032, May 2020

    2020

  91. [99]

    Schwarzschild black hole lensing,

    K. S. Virbhadra and G. F. R. Ellis, “Schwarzschild black hole lensing,” Phys. Rev. D , vol. 62, p. 084003, 2000

    2000

  92. [100]

    Applications of the Gauss-Bonnet theorem to gravitational lensing,

    G. W. Gibbons and M. C. Werner, “Applications of the Gauss-Bonnet theorem to gravitational lensing,” Class. Quant. Grav. , vol. 25, p. 235009, 2008

    2008

  93. [101]

    Gravitational bending angle of light for finite distance and the Gauss-Bonnet theorem,

    A. Ishihara, Y. Suzuki, T. Ono, T. Kitamura, and H. Asada, “Gravitational bending angle of light for finite distance and the Gauss-Bonnet theorem,” Phys. Rev. D , vol. 94, no. 8, p. 084015, 2016

    2016

  94. [102]

    Circular Orbit of a Particle and Weak Gravitational Lensing,

    Z. Li, G. Zhang, and A. ¨Ovg¨ un, “Circular Orbit of a Particle and Weak Gravitational Lensing,” Phys. Rev. D , vol. 101, no. 12, p. 124058, 2020

    2020

  95. [103]

    Study of gravitational deflection of light ray,

    S. Roy and A. K. Sen, “Study of gravitational deflection of light ray,” Journal of Physics: Conference Series , vol. 1330, p. 012002, Oct. 2019

    2019

  96. [104]

    Fourth Test of General Relativity,

    I. I. Shapiro, “Fourth Test of General Relativity,” Physical Review Letters , vol. 13, pp. 789–791, Dec. 1964. 19

    1964

  97. [105]

    Fourth Test of General Relativity: Preliminary Results,

    I. I. Shapiro, G. H. Pettengill, M. E. Ash, M. L. Stone, W. B. Smith, R. P. Ingalls, and R. A. Brockelman, “Fourth Test of General Relativity: Preliminary Results,” Physical Review Letters , vol. 20, pp. 1265–1269, May 1968

    1968

  98. [106]

    Time delay and magnification centroid due to gravitational lensing by black holes and naked singularities,

    K. S. Virbhadra and C. R. Keeton, “Time delay and magnification centroid due to gravitational lensing by black holes and naked singularities,” Phys. Rev. D , vol. 77, p. 124014, 2008

    2008

  99. [107]

    GW170817 falsifies dark matter emulators,

    S. Boran, S. Desai, E. Kahya, and R. Woodard, “GW170817 falsifies dark matter emulators,” Physical Review D , vol. 97, p. 041501, Feb. 2018

    2018

  100. [108]

    Viking relativity experiment - Verification of signal retardation by solar gravity,

    R. D. Reasenberg, I. I. Shapiro, P. E. MacNeil, R. B. Goldstein, J. C. Breidenthal, J. P. Brenkle, D. L. Cain, T. M. Kaufman, T. A. Komarek, and A. I. Zygielbaum, “Viking relativity experiment - Verification of signal retardation by solar gravity,” The Astrophysical Journal , ...

    1979

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.