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arxiv: 2503.21203 · v2 · submitted 2025-03-27 · 🌀 gr-qc

Massive particle surfaces and black hole shadows from intrinsic curvature

Boris Berm\'udez-C\'ardenas , Oscar Lasso Andino This is my paper

Pith reviewed 2026-05-22 23:27 UTC · model grok-4.3

classification 🌀 gr-qc
keywords massive particle surfacesintrinsic curvaturestationary spacetimesblack hole shadowsKerr metricKilling vectorsGaussian curvaturegeodesic curvature
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The pith

Projecting stationary spacetimes onto a 2D Riemannian metric from Killing vectors lets intrinsic curvatures determine massive particle surfaces and black hole shadows.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

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.

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.

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.

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.

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.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

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

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

specific steps
  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.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 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.

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Reference graph

Works this paper leans on

39 extracted references · 39 canonical work pages · cited by 1 Pith paper · 14 internal anchors

  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 internal anchor Pith review Pith/arXiv arXiv doi:10.1063/1.1308507 2001

  3. [3]

    Curvatures, photon spheres, and black hole shadows,

    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]

    Massive particle surfaces,

    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 internal anchor Pith review Pith/arXiv 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

  9. [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]

    work page doi:10.1140/epjc/s10052-024-12751-4 2024

  10. [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]

    work page doi:10.1103/physrevd.111.064001 2025

  11. [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]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/0264-9381/33/2/025004 2016

  12. [12]

    Motion of charged particle in Reissner-Nordstrom 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]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1140/epjc/s10052-017-5295-6 2017

  13. [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]

    work page doi:10.1088/1361-6382/abcf86 2021

  14. [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]

    work page doi:10.1007/s10714-022-03060-w 2023

  15. [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]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.anihpc.2010.01.001 2010

  16. [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]

    work page doi:10.1103/physrevd.104.044061 2021

  17. [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]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.79.044022 2009

  18. [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]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/s10714-012-1458-9 2012

  19. [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

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.96.104037 2017

  20. [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]

    work page doi:10.1007/s10714-024-03314-9 2024

  21. [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]

    work page doi:10.1140/epjc/s10052-020-8070-z 2020

  22. [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

    work page doi:10.1023/a:1026286607562 2003

  23. [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]

    work page doi:10.1007/s10714-020-02782-z 2021

  24. [24]

    M., Press W

    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

    work page doi:10.1086/151796 1972

  25. [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]

    work page doi:10.1103/physrevd.108.024056 2023

  26. [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]

    work page doi:10.1088/1361-6382/ad9797 2025

  27. [27]

    Astrophysically relevant bound trajectories 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]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/1361-6382/ab004c 2019

  28. [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]

    work page doi:10.1103/physrevd.101.044032 2020

  29. [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]

    work page doi:10.1103/physrevd.105.024075 2022

  30. [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]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/0264-9381/26/13/135002 2009

  31. [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

    work page doi:10.1103/physrevd.102.104036 2020

  32. [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]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/s10714-008-0629-1 2008

  33. [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

    work page doi:10.1103/physrevd.5.814 1972

  34. [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]

    work page doi:10.1140/epjc/s10052-024-12627-7 2024

  35. [35]

    Intrinsic curvature and topology 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]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.99.041303 2019

  36. [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]

    work page doi:10.1088/1475-7516/2019/08/030 2019

  37. [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]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.77.104022 2008

  38. [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]

    work page doi:10.1088/1361-6382/ab56ec 2019

  39. [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

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.76.087504 2007