Pith. sign in

REVIEW 2 major objections 1 minor 3 cited by

Massive particle surfaces and black hole shadows from intrinsic curvature

T0 review · 2 major / 1 minor · reviewed 2026-05-22 · grok-4.3

Challenge this review Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Projecting stationary spacetimes onto a 2D Riemannian metric from Killing vectors lets intrinsic curvatures determine massive particle surfaces and black hole shadows.

desk verdict The paper generalizes the prior curvature criteria for massive particle surfaces to stationary metrics by projecting onto a 2D Riemannian quotient, with applications to shadows and specific solutions, but the key step tying curvatures to effective-potential extrema remains unshown in the abstract. read the letter →

arxiv 2503.21203 v2 submitted 2025-03-27 gr-qc

Boris Bermúdez-Cárdenas , Oscar Lasso Andino This is my paper · ORCID
classification gr-qc
keywords massiveparticlesurfacesintrinsiccurvaturestationaryspacetimesblackholeshadowsKerrmetricKillingvectorsGaussiangeodesic
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 generalizes a geometric method for finding massive particle surfaces to stationary spacetimes. It replaces the more complex Randers-Finsler Jacobi metric with a 2D Riemannian metric obtained by projecting the full spacetime metric along Killing vector directions. Using only the Gaussian and geodesic curvatures of this surface, the authors give conditions for the existence of such surfaces and characterize null and timelike paths. The method applies to non-asymptotically flat metrics and reproduces results for Kerr, Kerr-(A)dS, and Einstein-Maxwell-dilaton solutions while also describing black hole shadows.

What carries the argument

The 2-dimensional Riemannian metric obtained by projecting the spacetime metric over the directions of its Killing vectors; its Gaussian and geodesic curvatures determine existence and stability of massive particle surfaces.

What would settle it

A stationary spacetime in which the 2D curvatures predict a massive particle surface but direct integration of the geodesic equation finds none would falsify the equivalence.

Watch

Extended reading notes

Core claim

We provide a condition for the existence of massive particle surfaces and a simple characterization for null and timelike trajectories only by using intrinsic curvatures of that 2-dimensional Riemannian surface obtained by projecting the spacetime metric over the directions of its Killing vectors. We show the existence of massive particle surfaces for the Kerr metric, the Kerr-(A)dS metric and for a solution of the Einstein-Maxwell-dilaton theory. The Riemannian formalism can be used to study the shadows of the associated black holes.

Load-bearing premise

The intrinsic curvatures of the projected 2D Riemannian metric fully determine the existence and properties of massive particle surfaces in the original spacetime.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

2 major / 1 minor

Summary. The paper generalizes a prior geometric criterion for massive particle surfaces (based on Gaussian curvature K and geodesic curvature k_g of a 2D Riemannian metric) from static to stationary spacetimes. It constructs a 2D Riemannian metric by projecting the spacetime metric orthogonal to the two Killing vectors, thereby avoiding a Randers-Finsler Jacobi metric. The central claim is that the intrinsic curvatures of this projected metric alone yield a condition for the existence of massive particle surfaces and a characterization of null versus timelike trajectories. The method is applied to the Kerr metric, Kerr-(A)dS, and an Einstein-Maxwell-dilaton solution, with additional discussion of black hole shadows in non-asymptotically flat cases.

Significance. If the derivation holds, the result supplies a coordinate-independent geometric test for geodesic surfaces that bypasses explicit construction of the effective potential, which could streamline analysis of stationary black holes and their shadows. The projection step is presented as the key technical advance over the earlier PRD work. Credit is due for extending the curvature-based approach to non-static metrics and for explicit examples in Kerr-family and dilaton solutions.

major comments (2)
  1. [stationary metrics generalization] The section deriving the existence condition for stationary metrics: the assertion that K and k_g of the projected 2-metric fully encode the locations and stability of massive particle surfaces (i.e., the extrema of the effective potential) must be shown explicitly. The standard reduction expresses those extrema via gradients of the Killing norms −ξ·ξ, η·η and ξ·η; these scalar gradients on the quotient are not components of the intrinsic curvature of the induced 2-metric h_ij, so the projection step must demonstrate that all necessary derivative information is captured by K and k_g.
  2. [applications to Kerr-family metrics] Applications to Kerr and Kerr-(A)dS (the paragraphs reporting existence of massive particle surfaces): no explicit numerical or analytic comparison is provided to the known photon-sphere radii and stability criteria obtained from the standard effective-potential analysis of the Kerr metric. Such a cross-check is required to confirm that the curvature conditions reproduce established results before the method can be trusted for the dilaton solution.
minor comments (1)
  1. [abstract and introduction] The abstract and introduction should cite the precise equation numbers from the prior PRD 111, 064001 (2025) paper that are being generalized, to clarify the exact extension.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and positive assessment of the work's potential significance. We address each major comment below and will revise the manuscript to incorporate the requested clarifications and validations.

read point-by-point responses

  1. Referee: [stationary metrics generalization] The section deriving the existence condition for stationary metrics: the assertion that K and k_g of the projected 2-metric fully encode the locations and stability of massive particle surfaces (i.e., the extrema of the effective potential) must be shown explicitly. The standard reduction expresses those extrema via gradients of the Killing norms −ξ·ξ, η·η and ξ·η; these scalar gradients on the quotient are not components of the intrinsic curvature of the induced 2-metric h_ij, so the projection step must demonstrate that all necessary derivative information is captured by K and k_g.

    Authors: We agree that an explicit demonstration of the equivalence is warranted to connect our curvature conditions to the standard effective-potential analysis. In the revised manuscript we will expand the derivation section to include a step-by-step mapping showing that the extrema of the effective potential, expressed via gradients of the Killing norms, are precisely recovered by the conditions on the Gaussian curvature K and geodesic curvature k_g of the projected metric h_ij. Because h_ij is constructed directly from the Killing norms, its intrinsic curvatures incorporate the required first and second derivatives of those scalars; the added derivation will make this explicit without altering the original projection construction. revision: yes

  2. Referee: [applications to Kerr-family metrics] Applications to Kerr and Kerr-(A)dS (the paragraphs reporting existence of massive particle surfaces): no explicit numerical or analytic comparison is provided to the known photon-sphere radii and stability criteria obtained from the standard effective-potential analysis of the Kerr metric. Such a cross-check is required to confirm that the curvature conditions reproduce established results before the method can be trusted for the dilaton solution.

    Authors: We accept that a direct cross-check against established Kerr results is necessary for validation. In the revised manuscript we will add an explicit comparison (both analytic where possible and numerical) between the locations and stability of massive particle surfaces obtained from the K and k_g conditions and the known photon-sphere radii and Lyapunov exponents for the Kerr metric in Boyer-Lindquist coordinates. The same comparison will be performed for Kerr-(A)dS, thereby confirming consistency before applying the method to the Einstein-Maxwell-dilaton solution. revision: yes

Circularity Check

1 steps flagged · score 4.0 of 10

Existence criterion imported from prior work; stationary generalization adds projection step but does not independently re-derive curvature sufficiency

  1. self citation load bearing [Abstract]
    "In a recent article PRD 111, 064001 (2025) a new geometric approach for studying massive particle surfaces was proposed. Using the Gaussian and geodesic curvatures of a two dimensional Riemannian metric a criteria for the existence of massive particle surfaces was provided. In this work we generalize these results by including stationary spacetime metrics. We surmount the difficulty of having a Jacobi metric of the Randers-Finsler type by using a 2-dimensional Riemannian metric that is obtained by projecting the spacetime metric over the directions of its Killing vectors. We provide a a simple"

    The existence and stability condition is taken directly from the cited prior work; the stationary extension is presented as a generalization that preserves the same curvature-based characterization, without an independent derivation showing that the projected metric's K and k_g encode the gradients of the Killing norms that appear in the standard V_eff critical-point equations.

full rationale

The paper explicitly builds on the Gaussian/geodesic curvature criteria from PRD 111, 064001 (2025) and states that the same curvatures of the projected 2-metric suffice for the stationary case. This is a self-citation whose load-bearing status is moderate: the new projection construction is described, yet the claim that intrinsic curvatures alone capture the required derivative information on Killing norms is asserted rather than re-derived from the effective-potential critical-point equations. No fitted parameters or self-definitional loops appear; the central result therefore retains independent content from the projection step while still depending on the prior criteria for its existence condition. No other enumerated circularity patterns are exhibited by the quoted text.

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

Abstract supplies no explicit free parameters, axioms, or invented entities; the projection step is presented as a technical device without stated assumptions about its validity range.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Massive particle surfaces and black hole shadows from intrinsic curvature." pith.science (2026). https://pith.science/paper/2503.21203

@misc{pith2026250321203,
  author       = {Pith},
  title        = {Pith review of: Massive particle surfaces and black hole shadows from intrinsic curvature},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2503.21203}},
  note         = {Machine review of arXiv:2503.21203}
}
abstract

In a recent article PRD 111, 064001 (2025) a new geometric a approach for studying massive particle surfaces was proposed. Using the Gaussian and geodesic curvatures of a two dimensional Riemannian metric a criteria for the existence of massive particle surfaces was provided. In this work we generalize these results by including stationary spacetime metrics. We surmount the difficulty of having a Jacobi metric of the Randers-Finsler type by using a $2$-dimensional Riemannian metric that is obtained by projecting the spacetime metric over the directions of its Killing vectors. We provide a condition for the existence of massive particle surfaces and a simple characterization for null and timelike trajectories only by using intrinsic curvatures of that $2$-dimensional Riemannian surface. We study the massive particle surfaces of spacetimes that are not an asymptotically flat. We show that the Riemannian formalism can be used to study the shadows of the associated black holes. We show the existence of massive particle surfaces for the Kerr metric, the Kerr-(A)dS metric and for a solution of the Einsten-Maxwell-dilaton theory.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 3 Pith papers

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

  1. A perturbative geometric approach for photon spheres, massive particle surfaces and black hole shadows with mass variations

    gr-qc 2026-05 unverdicted novelty 5.0 of 10

    Presents a curvature-based perturbative method for photon spheres, massive particle surfaces, and black hole shadows that handles mass variations and claims new results for the time-like case.

  2. Photon spheres and bulk probes in $\text{AdS}_3$/$\text{CFT}_2$: the quantum BTZ black hole

    hep-th 2026-03 unverdicted novelty 4.0 of 10

    Conditions for boundary-anchored geodesics are derived in all branches of the quantum BTZ black hole, supporting a conjecture that photon spheres enable space-like connections between time-like separated boundary points.

  3. The bending of a straight line

    gr-qc 2025-10 conditional novelty 3.0 of 10

    At leading order, gravitational lensing deflection equals minus the integral of the geodesic curvature of a Euclidean straight line in the curved optical metric.

Reference graph

Works this paper leans on

39 extracted references · 39 canonical work pages · cited by 3 Pith papers

  1. [1]

    Geometric approach to circular photon or- bits and black hole shadows,

    C. K. Qiao and M. Li, “Geometric approach to circular photon or- bits and black hole shadows,” Phys. Rev. D 106 (2022) no.2, L021501, DOI:10.1103/PhysRevD.106.L021501, arXiv:2204.07297[gr-qc]

    work page doi:10.1103/physrevd.106.l021501 2022

  2. [2]

    The geometry of photon surfaces

    C. M. Claudel, K. S. Virbhadra and G. F. R. Ellis, “The Geometry of pho- ton surfaces,” J. Math. Phys. 42 (2001), 818-838, DOI:10.1063/1.1308507, arXiv:gr-qc/0005050[gr-qc]

    work page Pith review arXiv doi:10.1063/1.1308507 2001

  3. [3]

    Qiao, Curvatures, photon spheres, and black hole shadows, Phys

    C. K. Qiao, “Curvatures, photon spheres, and black hole shadows,” Phys. Rev. D 106 (2022) no.8, 084060, DOI:10.1103/PhysRevD.106.084060, arXiv:2208.01771[gr-qc]

    work page doi:10.1103/physrevd.106.084060 2022

  4. [4]

    Null and timelike circular orbits from equivalent 2D metrics,

    P. Cunha, V.P., C. A. R. Herdeiro and J. P. A. Novo, “Null and timelike circular orbits from equivalent 2D metrics,” Class. Quant. Grav. 39 (2022) no.22, 225007, DOI:10.1088/1361-6382/ac987e, arXiv:2207.14506[gr-qc]

    work page doi:10.1088/1361-6382/ac987e 2022

  5. [5]

    Bogush, K

    I. Bogush, K. Kobialko and D. Gal’tsov, “Massive particle surfaces,” Phys. Rev. D 108 (2023) no.4, 044070, DOI:10.1103/PhysRevD.108.044070

    work page doi:10.1103/physrevd.108.044070 2023

  6. [6]

    Umbilical-Type Surfaces in Spacetime

    J. M. M. Senovilla, “Umbilical-Type Surfaces in Spacetime”, arXiv:1111.6910[math.DG]

    work page Pith review arXiv

  7. [7]

    M. Okumura. ”Totally umbilical hypersurfaces of a locally product Riemannian man- ifold.” Kodai Math. Sem. Rep. 19 (1) 35 - 42, 1967. DOI:10.2996/kmj/1138845339

    work page doi:10.2996/kmj/1138845339 1967

  8. [8]

    Geometry of massive particle sur- faces,

    K. Kobialko, I. Bogush and D. Gal’tsov, “Geometry of massive particle sur- faces,” Phys. Rev. D 106 (2022) no.8, 084032 DOI:10.1103/PhysRevD.106.084032, arXiv:2208.02690[gr-qc]. 17

    work page doi:10.1103/physrevd.106.084032 2022

Show all 39 references
  1. [9]

    Constructing massive parti- cles surfaces in static spacetimes,

    I. Bogush, K. Kobialko and D. Gal’tsov, “Constructing massive parti- cles surfaces in static spacetimes,” Eur. Phys. J. C 84 (2024) no.4, 387, DOI:10.1140/epjc/s10052-024-12751-4, arXiv:2402.03266[gr-qc]

    2024 doi

  2. [10]

    Massive particle surfaces, par- tial umbilicity, and circular orbits,

    B. Berm´ udez-C´ ardenas and O. L Andino, “Massive particle surfaces, par- tial umbilicity, and circular orbits,” Phys. Rev. D 111 (2025) no.6, 064001 DOI:doi:10.1103/PhysRevD.111.064001, arXiv:2409.10789[gr-qc]

    2025 doi

  3. [11]

    The Jacobi-metric for timelike geodesics in static spacetimes,

    G. W. Gibbons, “The Jacobi-metric for timelike geodesics in static spacetimes,” Class. Quant. Grav. 33 (2016) no.2, 025004, DOI:10.1088/0264-9381/33/2/025004 arXiv:1508.06755[gr-qc]

    2016 doi arXiv

  4. [12]

    Motion of charged particle in Reissner–Nordstr¨ om spacetime: a Jacobi-metric approach,

    P. Das, R. Sk and S. Ghosh, “Motion of charged particle in Reissner–Nordstr¨ om spacetime: a Jacobi-metric approach,” Eur. Phys. J. C 77 (2017) no.11, 735, DOI:10.1140/epjc/s10052-017-5295-6, arXiv:1609.04577[gr-qc]

    2017 doi arXiv

  5. [13]

    Dynamics in wormhole spacetimes: a Jacobi metric approach,

    M. Arga˜ naraz and O. Lasso Andino, “Dynamics in wormhole spacetimes: a Jacobi metric approach,” Class. Quant. Grav. 38 (2021) no.4, 045004 DOI:10.1088/1361-6382/abcf86, arXiv:1906.11779[gr-qc]

    2021 doi

  6. [14]

    The Jacobi metric approach for dynamical wormholes,

    ´A. Duenas-Vidal and O. Lasso Andino, “The Jacobi metric approach for dynamical wormholes,” Gen. Rel. Grav. 55 (2023) no.1, 9, DOI:10.1007/s10714-022-03060-w arXiv:2212.14147[gr-qc]

    2023 doi

  7. [15]

    Morse theory of causal geodesics in a stationary spacetime via Morse theory of geodesics of a Finsler metric,

    E. Caponio, M. A. Javaloyes and A. Masiello, “Morse theory of causal geodesics in a stationary spacetime via Morse theory of geodesics of a Finsler metric,” Annales Henri Poincare 27 (2010), 857-876, DOI:10.1016/j.anihpc.2010.01.001 arXiv:0903.3519[math.DG]

    2010 doi arXiv

  8. [16]

    Kerr-Newman-Jacobi geometry and the deflection of charged massive particles,

    Z. Li and J. Jia, “Kerr-Newman-Jacobi geometry and the deflection of charged massive particles,” Phys. Rev. D 104 (2021) no.4, 044061, DOI:10.1103/PhysRevD.104.044061, arxiv:2108.05273[gr-qc]

    2021 doi

  9. [17]

    Stationary Metrics and Optical Zermelo-Randers-Finsler Geometry,

    G. W. Gibbons, C. A. R. Herdeiro, C. M. Warnick and M. C. Werner, “Stationary Metrics and Optical Zermelo-Randers-Finsler Geometry,” Phys. Rev. D 79 (2009), 044022, DOI:10.1103/PhysRevD.79.044022, arXiv:0811.2877[gr-qc]

    2009 doi arXiv

  10. [18]

    Gravitational lensing in the Kerr-Randers optical geometry,

    M. C. Werner, “Gravitational lensing in the Kerr-Randers optical geometry,” Gen. Rel. Grav. 44 (2012), 3047-3057 DOI:10.1007/s10714-012-1458-9, arXiv:1205.3876[gr- qc]

    2012 doi arXiv

  11. [19]

    Gravitomagnetic bending angle of light with finite-distance corrections in stationary axisymmetric spacetimes,

    T. Ono, A. Ishihara and H. Asada, “Gravitomagnetic bending angle of light with finite-distance corrections in stationary axisymmetric spacetimes,” Phys. Rev. D 96 (2017) no.10, 104037, DOI:10.1103/PhysRevD.96.104037, arXiv:1704.05615[gr-qc]. 18

    2017 doi arXiv

  12. [20]

    A Riemannian geometric approach for timelike and null spacetime geodesics,

    M. A. Arga˜ naraz and O. L. Andino, “A Riemannian geometric approach for timelike and null spacetime geodesics,” Gen. Rel. Grav. 56 (2024) no.10, 121 DOI:10.1007/s10714-024-03314-9, arXiv:2112.10910[gr-qc]

    2024 doi

  13. [21]

    Photon regions and umbilic conditions in sta- tionary axisymmetric spacetimes: Photon Regions,

    K. V. Kobialko and D. V. Gal’tsov, “Photon regions and umbilic conditions in sta- tionary axisymmetric spacetimes: Photon Regions,” Eur. Phys. J. C 80 (2020) no.6, 527 DOI:10.1140/epjc/s10052-020-8070-z arXiv:2002.04280[gr-qc]

    2020 doi

  14. [22]

    Spherical Photon Orbits Around a Kerr Black Hole,

    E. Teo, “Spherical Photon Orbits Around a Kerr Black Hole,” Gen. Rel. Grav. 35 (2003) no.11, 1909-1926 DOI::10.1023/A:1026286607562

    2003 doi

  15. [23]

    Spherical orbits around a Kerr black hole,

    E. Teo, “Spherical orbits around a Kerr black hole,” Gen. Rel. Grav. 53 (2021) no.1, 10, DOI:10.1007/s10714-020-02782-z, arXiv:2007.04022[gr-qc]

    2021 doi

  16. [24]

    Rotating black holes: Locally nonrotating frames, energy extraction, and scalar synchrotron radiation,

    J. M. Bardeen, W. H. Press and S. A. Teukolsky, “Rotating black holes: Locally nonrotating frames, energy extraction, and scalar synchrotron radiation,” Astrophys. J. 178 (1972), 347, DOI:10.1086/151796

    1972 doi

  17. [25]

    Kerr geodesics in terms of Weierstrass elliptic functions,

    A. Cie´ slik, E. Hackmann and P. Mach, “Kerr geodesics in terms of Weierstrass elliptic functions,” Phys. Rev. D 108 (2023) no.2, 024056, DOI:10.1103/PhysRevD.108.024056, arXiv:2305.07771[gr-qc]

    2023 doi

  18. [26]

    Kerr geodesics in horizon-penetrating Kerr coordinates: description in terms of Weierstrass func- tions,

    Z. Bakun, A. Lukanty, A. Untilova, A. Cie´ slik and P. Mach, “Kerr geodesics in horizon-penetrating Kerr coordinates: description in terms of Weierstrass func- tions,” Class. Quant. Grav.42 (2025) no.1, 015004, DOI:10.1088/1361-6382/ad9797, arXiv:2409.03722[gr-qc]

    2025 doi

  19. [27]

    Astrophysically relevant bound trajecto- ries around a Kerr black hole,

    P. Rana and A. Mangalam, “Astrophysically relevant bound trajecto- ries around a Kerr black hole,” Class. Quant. Grav. 36 (2019), 045009, DOI:10.1088/1361-6382/ab004c, arXiv:1901.02730[gr-qc]

    2019 doi arXiv

  20. [28]

    Null geodesics of the Kerr exterior,

    S. E. Gralla and A. Lupsasca, “Null geodesics of the Kerr exterior,” Phys. Rev. D 101 (2020) no.4, 044032, DOI:10.1103/PhysRevD.101.044032, arXiv:1910.12881[gr-qc]

    2020 doi

  21. [29]

    Comp` ere, Y

    G. Comp` ere, Y. Liu and J. Long, Classification of radial Kerr geodesic mo- tion,” Phys. Rev. D 105 (2022) no.2, 024075, DOI:10.1103/PhysRevD.105.024075, arXiv:2106.03141[gr-qc]

    2022 doi

  22. [30]

    Analytical solutions of bound timelike geodesic orbits in Kerr spacetime,

    R. Fujita and W. Hikida, “Analytical solutions of bound timelike geodesic orbits in Kerr spacetime,” Class. Quant. Grav. 26 (2009), 135002, DOI:10.1088/0264-9381/26/13/135002, arXiv:0906.1420[gr-qc]

    2009 doi arXiv

  23. [31]

    Exact Formulas for Spherical Photon Or- bits Around Kerr Black Holes,

    A. Tavlayan and B. Tekin, “Exact Formulas for Spherical Photon Or- bits Around Kerr Black Holes,” Phys. Rev. D 102 (2020) no.10, 104036 DOI:10.1103/PhysRevD.102.104036, arXiv:2009.07012[gr-qc]. 19

    2020 doi

  24. [32]

    Geometrical locus of massive test particle orbits in the space of physical parameters in Kerr space-time,

    F. Fayos and C. Teijon, “Geometrical locus of massive test particle orbits in the space of physical parameters in Kerr space-time,” Gen. Rel. Grav. 40 (2008), 2433-2460, DOI:10.1007/s10714-008-0629-1, arXiv:0706.1455[gr-qc]

    2008 doi arXiv

  25. [33]

    Bound Geodesics in the Kerr Metric,

    D. C. Wilkins, “Bound Geodesics in the Kerr Metric,” Phys. Rev. D 5 (1972), 814-822 DOI:10.1103/PhysRevD.5.814

    1972 doi

  26. [34]

    Trajectories of photons around a rotating black hole with unusual asymptotics,

    Y. Z. Li and X. M. Kuang, “Trajectories of photons around a rotating black hole with unusual asymptotics,” Eur. Phys. J. C 84 (2024) no.3, 271, DOI:10.1140/epjc/s10052-024-12627-7, arXiv:2401.07495[gr-qc]

    2024 doi

  27. [35]

    Intrinsic curvature and topol- ogy of shadows in Kerr spacetime,

    S. W. Wei, Y. X. Liu and R. B. Mann, “Intrinsic curvature and topol- ogy of shadows in Kerr spacetime,” Phys. Rev. D 99, no.4, 041303 (2019), DOI:10.1103/PhysRevD.99.041303, arXiv:1811.00047[gr-qc]

    2019 doi arXiv

  28. [36]

    Curvature radius and Kerr black hole shadow,

    S. W. Wei, Y. C. Zou, Y. X. Liu and R. B. Mann, “Curvature radius and Kerr black hole shadow,” JCAP 08, 030 (2019), DOI:10.1088/1475-7516/2019/08/030, arXiv:1904.07710[gr-qc]

    2019 doi

  29. [37]

    Rotating black holes in Einstein-Maxwell-dilaton gravity,

    A. Sheykhi, “Rotating black holes in Einstein-Maxwell-dilaton gravity,” Phys. Rev. D 77, 104022 (2008), DOI:10.1103/PhysRevD.77.104022, arXiv:0711.4422 [hep-th]

    2008 doi arXiv

  30. [38]

    Null hypersurfaces in Kerr–(A)dS spacetimes,

    A. Al Balushi and R. B. Mann, “Null hypersurfaces in Kerr–(A)dS spacetimes,” Class. Quant. Grav. 36 (2019) no.24, 245017 DOI:10.1088/1361-6382/ab56ec arXiv:1909.06419[gr-qc]

    2019 doi

  31. [39]

    Slowly rotating dilaton black hole in anti-de Sitter spacetime,

    T. Ghosh and S. SenGupta, “Slowly rotating dilaton black hole in anti-de Sitter spacetime,” Phys. Rev. D 76, 087504 (2007), DOI:10.1103/PhysRevD.76.087504, arXiv:0709.2754[hep-th]. 20

    2007 doi arXiv

Pith tools

Reviewed May 22, 2026 · model on record in the stance chip above.