# Spherical Particle Orbits around a Rotating Black Hole in Massive Gravity

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction and Motivation

## 2. Massive Theory of Gravity and Its Static Black Hole Solution in the Cosmological Background

## 3. Spherical Particle Orbits

#### 3.1. Radii of Planar Orbits

#### Non-Monotonic Behavior of the Solutions

## 4. Analytical Solutions for the Spherical Particle Orbits

#### 4.1. The Latitudinal Motion

#### 4.2. The Azimuth Motion

#### 4.3. Explicit Examples of Non-Planar Orbits

## 5. Summary and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. The Roots of the Polynomial Equation Δ(r) = 0

## References

- Abbott, B.P.; Abbott, R.; Abbott, T.; Abernathy, M.R.; Acernese, F.; Ackley, K.; Adams, C.; Adams, T.; Addesso, P.; Adhikari, R.X.; et al. Observation of Gravitational Waves from a Binary Black Hole Merger. Phys. Rev. Lett.
**2016**, 116, 061102. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Akiyama, K.; Alberdi, A.; Alef, W.; Asada, K.; Azulay, R.; Baczko, A.K.; Ball, D.; Baloković, M.; Barrett, J.; Bintley, D.; et al. First M87 Event Horizon Telescope Results. IV. Imaging the Central Supermassive Black Hole. Astrophys. J. Lett.
**2019**, 875, L4. [Google Scholar] [CrossRef] - Akiyama, K.; Alberdi, A.; Alef, W.; Algaba, J.C.; Anantua, R.; Asada, K.; Azulay, R.; Bach, U.; Baczko, A.K.; Ball, D.; et al. First Sagittarius A* Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole in the Center of the Milky Way. Astrophys. J. Lett.
**2022**, 930, L12. [Google Scholar] [CrossRef] - Rana, P.; Mangalam, A. Astrophysically relevant bound trajectories around a Kerr black hole. Class. Quantum Gravity
**2019**, 36, 045009. [Google Scholar] [CrossRef] [Green Version] - Kapec, D.; Lupsasca, A. Particle motion near high-spin black holes. Class. Quantum Gravity
**2020**, 37, 015006. [Google Scholar] [CrossRef] [Green Version] - Gralla, S.E.; Lupsasca, A. Null geodesics of the Kerr exterior. Phys. Rev. D
**2020**, 101, 044032. [Google Scholar] [CrossRef] [Green Version] - Stein, L.C.; Warburton, N. Location of the last stable orbit in Kerr spacetime. Phys. Rev. D
**2020**, 101, 064007. [Google Scholar] [CrossRef] [Green Version] - Compère, G.; Druart, A. Near-horizon geodesics of high spin black holes. Phys. Rev. D
**2020**, 101, 084042. [Google Scholar] [CrossRef] - Rana, P.; Mangalam, A. A Geometric Origin for Quasi-periodic Oscillations in Black Hole X-ray Binaries. Astrophys. J.
**2020**, 903, 121. [Google Scholar] [CrossRef] - Carter, B. Global Structure of the Kerr Family of Gravitational Fields. Phys. Rev.
**1968**, 174, 1559–1571. [Google Scholar] [CrossRef] [Green Version] - Mino, Y. Perturbative approach to an orbital evolution around a supermassive black hole. Phys. Rev. D
**2003**, 67, 084027. [Google Scholar] [CrossRef] [Green Version] - Fujita, R.; Hikida, W. Analytical solutions of bound timelike geodesic orbits in Kerr spacetime. Class. Quantum Gravity
**2009**, 26, 135002. [Google Scholar] [CrossRef] [Green Version] - Lämmerzahl, C.; Hackmann, E. Analytical Solutions for Geodesic Equation in Black Hole Spacetimes. In 1st Karl Schwarzschild Meeting on Gravitational Physics; Nicolini, P., Kaminski, M., Mureika, J., Bleicher, M., Eds.; Springer International Publishing: Cham, Switzerland, 2016; Volume 170, pp. 43–51. [Google Scholar] [CrossRef]
- Wilkins, D.C. Bound Geodesics in the Kerr Metric. Phys. Rev. D
**1972**, 5, 814–822. [Google Scholar] [CrossRef] - Johnston, M.; Ruffini, R. Generalized Wilkins effect and selected orbits in a Kerr-Newman geometry. Phys. Rev. D
**1974**, 10, 2324–2329. [Google Scholar] [CrossRef] - Stoghianidis, E.; Tsoubelis, D. Polar orbits in the Kerr space-time. Gen. Relativ. Gravit.
**1987**, 19, 1235–1249. [Google Scholar] [CrossRef] - Hughes, S.A. Evolution of circular, nonequatorial orbits of Kerr black holes due to gravitational-wave emission. Phys. Rev. D
**2000**, 61, 084004. [Google Scholar] [CrossRef] [Green Version] - Hughes, S.A. Evolution of circular, nonequatorial orbits of Kerr black holes due to gravitational-wave emission. II. Inspiral trajectories and gravitational waveforms. Phys. Rev. D
**2001**, 64, 064004. [Google Scholar] [CrossRef] [Green Version] - Kraniotis, G.V. Precise relativistic orbits in Kerr and Kerr–(anti) de Sitter spacetimes. Class. Quantum Gravity
**2004**, 21, 4743–4769. [Google Scholar] [CrossRef] - Fayos, F.; Teijón, C. Geometrical locus of massive test particle orbits in the space of physical parameters in Kerr space–time. Gen. Relativ. Gravit.
**2008**, 40, 2433–2460. [Google Scholar] [CrossRef] [Green Version] - Hackmann, E.; Lämmerzahl, C.; Kagramanova, V.; Kunz, J. Analytical solution of the geodesic equation in Kerr-(anti-) de Sitter space-times. Phys. Rev. D
**2010**, 81, 044020. [Google Scholar] [CrossRef] [Green Version] - Grossman, R.; Levin, J.; Perez-Giz, G. Harmonic structure of generic Kerr orbits. Phys. Rev. D
**2012**, 85, 023012. [Google Scholar] [CrossRef] [Green Version] - Hod, S. Marginally bound (critical) geodesics of rapidly rotating black holes. Phys. Rev. D
**2013**, 88, 087502. [Google Scholar] [CrossRef] [Green Version] - Teo, E. Spherical orbits around a Kerr black hole. Gen. Relativ. Gravit.
**2021**, 53, 10. [Google Scholar] [CrossRef] - Tavlayan, A.; Tekin, B. Radii of spherical timelike orbits around Kerr black holes. Phys. Rev. D
**2021**, 104, 124059. [Google Scholar] [CrossRef] - Battista, E.; Esposito, G. Geodesic motion in Euclidean Schwarzschild geometry. Eur. Phys. J. C
**2022**, 82, 1088. [Google Scholar] [CrossRef] [PubMed] - Freedman, W.L.; Turner, M.S. Colloquium: Measuring and understanding the universe. Rev. Mod. Phys.
**2003**, 75, 1433–1447. [Google Scholar] [CrossRef] [Green Version] - Sotiriou, T.P.; Faraoni, V. f(R) theories of gravity. Rev. Mod. Phys.
**2010**, 82, 451–497. [Google Scholar] [CrossRef] [Green Version] - De Felice, A.; Tsujikawa, S. f(R) Theories. Living Rev. Relativ.
**2010**, 13, 3. [Google Scholar] [CrossRef] [Green Version] - Quiros, I. Selected topics in scalar–tensor theories and beyond. Int. J. Mod. Phys. D
**2019**, 28, 1930012. [Google Scholar] [CrossRef] - Fujii, Y.; Maeda, K.-i. The Scalar-Tensor Theory of Gravitation; Cambridge Monographs on Mathematical Physics; Cambridge University Press: Cambridge, UK, 2003. [Google Scholar] [CrossRef]
- Fierz, M.; Pauli, W.E.; Dirac, P.A.M. On relativistic wave equations for particles of arbitrary spin in an electromagnetic field. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci.
**1939**, 173, 211–232. [Google Scholar] [CrossRef] - de Rham, C.; Gabadadze, G. Generalization of the Fierz-Pauli action. Phys. Rev. D
**2010**, 82, 044020. [Google Scholar] [CrossRef] [Green Version] - de Rham, C.; Gabadadze, G.; Tolley, A.J. Resummation of Massive Gravity. Phys. Rev. Lett.
**2011**, 106, 231101. [Google Scholar] [CrossRef] [PubMed] [Green Version] - De Rham, C. Massive Gravity. Living Rev. Relativ.
**2014**, 17, 7. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Ghosh, S.G.; Tannukij, L.; Wongjun, P. A class of black holes in dRGT massive gravity and their thermodynamical properties. Eur. Phys. J. C
**2016**, 76, 119. [Google Scholar] [CrossRef] [Green Version] - Panpanich, S.; Burikham, P. Fitting rotation curves of galaxies by de Rham-Gabadadze-Tolley massive gravity. Phys. Rev. D
**2018**, 98, 064008. [Google Scholar] [CrossRef] [Green Version] - Ashtekar, A. Implications of a positive cosmological constant for general relativity. Rep. Prog. Phys.
**2017**, 80, 102901. [Google Scholar] [CrossRef] [Green Version] - Boonserm, P.; Ngampitipan, T.; Simpson, A.; Visser, M. Innermost and outermost stable circular orbits in the presence of a positive cosmological constant. Phys. Rev. D
**2020**, 101, 024050. [Google Scholar] [CrossRef] [Green Version] - Rincón, A.; Panotopoulos, G.; Lopes, I.; Cruz, N. ISCOs and OSCOs in the Presence of a Positive Cosmological Constant in Massive Gravity. Universe
**2021**, 7, 278. [Google Scholar] [CrossRef] - Berezhiani, L.; Chkareuli, G.; de Rham, C.; Gabadadze, G.; Tolley, A.J. On black holes in massive gravity. Phys. Rev. D
**2012**, 85, 044024. [Google Scholar] [CrossRef] [Green Version] - Cai, Y.F.; Easson, D.A.; Gao, C.; Saridakis, E.N. Charged black holes in nonlinear massive gravity. Phys. Rev. D
**2013**, 87, 064001. [Google Scholar] [CrossRef] [Green Version] - Newman, E.T.; Janis, A.I. Note on the Kerr Spinning-Particle Metric. J. Math. Phys.
**1965**, 6, 915–917. [Google Scholar] [CrossRef] - Azreg-Aïnou, M. Generating rotating regular black hole solutions without complexification. Phys. Rev. D
**2014**, 90, 064041. [Google Scholar] [CrossRef] [Green Version] - Bardeen, J.M.; Press, W.H.; Teukolsky, S.A. Rotating Black Holes: Locally Nonrotating Frames, Energy Extraction, and Scalar Synchrotron Radiation. Astrophys. J.
**1972**, 178, 347–370. [Google Scholar] [CrossRef] - Bardeen, J. Timelike and null geodesics in the Kerr metric. In Les Houches Summer School of Theoretical Physics: Black Holes; American Astronomical Society: Les Houches, France, 1973; pp. 215–240. [Google Scholar]
- Chandrasekhar, S. The Mathematical Theory of Black Holes; Oxford Classic Texts in the Physical Sciences; Oxford University Press: Cambridge, UK, 1998. [Google Scholar]
- Kraniotis, G.V. Gravitational redshift/blueshift of light emitted by geodesic test particles, frame-dragging and pericentre-shift effects, in the Kerr–Newman–de Sitter and Kerr–Newman black hole geometries. Eur. Phys. J. C
**2021**, 81, 147. [Google Scholar] [CrossRef] - Dokuchaev, V.I.; Nazarova, N.O. Visible Shapes of Black Holes M87* and SgrA*. Universe
**2020**, 6, 154. [Google Scholar] [CrossRef] - Song, Y. The evolutions of the innermost stable circular orbits in dynamical spacetimes. Eur. Phys. J. C
**2021**, 81, 875. [Google Scholar] [CrossRef] - Chen, C.Y.; Yang, H.Y.K. Curved accretion disks around rotating black holes without reflection symmetry. Eur. Phys. J. C
**2022**, 82, 307. [Google Scholar] [CrossRef] - Ospino, J.; Hernández-Pastora, J.L.; Núñez, L.A. All analytic solutions for geodesic motion in axially symmetric space-times. Eur. Phys. J. C
**2022**, 82, 591. [Google Scholar] [CrossRef] - Cao, W.; Liu, W.; Wu, X. Integrability of Kerr-Newman spacetime with cloud strings, quintessence and electromagnetic field. Phys. Rev. D
**2022**, 105, 124039. [Google Scholar] [CrossRef] - Bogush, I.; Gal’tsov, D.; Gyulchev, G.; Kobialko, K.; Nedkova, P.; Vetsov, T. Photon surfaces, shadows, and accretion disks in gravity with minimally coupled scalar field. Phys. Rev. D
**2022**, 106, 024034. [Google Scholar] [CrossRef] - Baines, J.; Berry, T.; Simpson, A.; Visser, M. Constant-r geodesics in the Painlevé–Gullstrand form of Lense–Thirring spacetime. Gen. Relativ. Gravit.
**2022**, 54, 79. [Google Scholar] [CrossRef] - Dymnikova, I.; Dobosz, A. Orbits of Particles and Photons around Regular Rotating Black Holes and Solitons. Symmetry
**2023**, 15, 273. [Google Scholar] [CrossRef] - Bogush, I.; Kobialko, K.; Gal’tsov, D. Glued massive particles surfaces. arXiv
**2023**, arXiv:2306.12888. [Google Scholar] - Fathi, M.; Olivares, M.; Villanueva, J.R. Spherical photon orbits around a rotating black hole with quintessence and cloud of strings. Eur. Phys. J. Plus
**2023**, 138, 7. [Google Scholar] [CrossRef]

**Figure 1.**The region plot of ${\delta}_{\Delta}>0$ for $\Lambda ={10}^{-6}\phantom{\rule{0.166667em}{0ex}}{M}^{-2}$ and $0<a<1$ is shown. According to the diagrams, for each value of a, the positivity of the discriminant is guaranteed inside a particular domain ${\gamma}_{min}<\gamma <{\gamma}_{max}$. In this case, the shaded region represents the domain corresponding to $a=0.9M$. For all values of a, it holds that ${\gamma}_{min}<0$, and the width of the region ${\delta}_{\Delta}>0$ significantly increases as the spin parameter decreases. It is worth noting that, for the exact values of ${\gamma}_{min}$ and ${\gamma}_{max}$, ${\delta}_{\Delta}=0$, and possessing these values results in extremal black holes.

**Figure 2.**The behavior of $\Delta \left(r\right)$ for (

**a**) a slow- and (

**b**) a fast-rotating black hole plotted for different values of the $\gamma $-parameter. The thick and thin solid curves represent, respectively, the negative and positive values, and the dashed curves correspond to the extremal cases.

**Figure 3.**The radial profile of ${g}_{tt}$ plotted for $\theta =\pi /4$ for the two cases of (

**a**) $a=0.3M$ and (

**b**) $a=0.9M$, for the same values of the $\gamma $-parameter as in Figure 2. The first two roots of ${g}_{tt}=0$ are shown for all of the cases, and for the extremal black holes, only two static limits are available.

**Figure 4.**The $x-k$ diagrams for the real parts of the solutions ${x}_{i}$ of Equation (29) plotted for $h=0.08$, $l={10}^{-6}$, and $u=0.3$. In panel (

**a**), the whole range of k is shown, whereas the range at which ${x}_{7}$ and ${x}_{8}$ are real is magnified in panel (

**b**). The color coding of the solutions ${x}_{i}$ used here is also applied in all of the forthcoming diagrams within the paper.

**Figure 5.**The $x-k$ diagram for the real parts of the solutions ${x}_{i}$ plotted for $h=0.08$, $l={10}^{-6}$, and $u=0.9$.

**Figure 6.**The $x-u$ diagrams for the real parts of the solutions ${x}_{i}$ plotted for $h=0.08$ and $l={10}^{-6}$: (

**a**) $k=0.003$ and (

**b**) $k=3.2$.

**Figure 7.**The u-profile of ${R}^{\u2033}\left(x\right)$ plotted for ${x}_{7}$ and ${x}_{8}$ in accordance with Figure 6a.

**Figure 8.**Some examples of spherical particle orbits in accordance with the data presented in Table 2. The sphere indicates the closure of points swept by the radii ${x}_{i}$, which is cut into halves by a circle on the $\theta =\pi /2$ surface.

**Table 1.**The radii of prograde and retrograde orbits outside the event horizon for different energies and spin parameters obtained by assuming $h=0.08$ and $l={10}^{-6}$.

k | u | ${\mathit{x}}_{\mathbf{prograde}}$ | ${\mathit{x}}_{\mathbf{retrograde}}$ | k | u | ${\mathit{x}}_{\mathbf{prograde}}$ | ${\mathit{x}}_{\mathbf{retrograde}}$ | k | u | ${\mathit{x}}_{\mathbf{prograde}}$ | ${\mathit{x}}_{\mathbf{retrograde}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|

0 | 0.1 | 1.762 | 2.213, 2.517 | 1 | 0.1 | – | 4.047 | 2 | 0.1 | 2.918 | – |

0 | 0.2 | 1.652 | 2.295, 2.533 | 1 | 0.2 | – | 4.096 | 2 | 0.2 | 2.677 | – |

0 | 0.3 | 1.562 | 2.357, 2.549 | 1 | 0.3 | – | 4.144 | 2 | 0.3 | 2.473 | – |

0 | 0.4 | 1.480 | 2.408, 2.564 | 1 | 0.4 | – | 4.193 | 2 | 0.4 | 2.289 | – |

0 | 0.5 | 1.403 | 2.452, 2.578 | 1 | 0.5 | – | 4.242 | 2 | 0.5 | 2.115 | – |

0 | 0.6 | 1.328 | 2.491, 2.593 | 1 | 0.6 | – | 4.290 | 2 | 0.6 | 1.944 | – |

0 | 0.7 | 1.252 | 2.526, 2.606 | 1 | 0.7 | – | 4.338 | 2 | 0.7 | 1.769 | – |

0 | 0.8 | 1.175 | 2.559, 2.619 | 1 | 0.8 | 1.730 | 2.269, 4.386 | 2 | 0.8 | 1.578 | – |

0 | 0.9 | 1.092 | 2.590, 2.632 | 1 | 0.9 | 1.335 | 2.437, 4.432 | 2 | 0.9 | 1.345 | – |

0 | 1.0 | 1.0 | 2.618, 2.644 | 1 | 1.0 | – | 2.529, 4.478 | 2 | 1.0 | – | – |

**Table 2.**The information for the exemplary cases outside the event horizon considered for $l={10}^{-6}$.

Name | u | $\mathit{\nu}$ | h | k | ${\mathit{x}}_{\mathit{i}}$ | ${\mathit{\xi}}_{\mathit{c}}\left({\mathit{x}}_{\mathit{i}}\right)$ |
---|---|---|---|---|---|---|

(a) | 0.3 | 0.1 | 0.08 | 0.003 | 2.342 | 7.093 |

(b) | 0.9 | 0.1 | 0.08 | 0.890 | 1.921 | 2.208 |

(c) | 0.85 | 0.3 | 0.04 | 0.91 | 5.038 | 6.590 |

(d) | 0.5 | 0.4 | 0.004 | 1.1 | 2.533 | 3.938 |

(e) | 0.6 | 0.6 | 0.0004 | 0.93 | 5.803 | 7.916 |

(f) | 0.8 | 0.7 | 0.01 | 1.3 | 4.109 | 2.711 |

(g) | ${u}_{\mathrm{ext}}=0.990$ | 0.9 | 0.001 | 0.9 | 2.922 | 1.067 |

(h) | 0.85 | 1 (polar) | 0.0001 | 1 | 3.497 | 0.0045 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Fathi, M.; Villanueva, J.R.; Cruz, N.
Spherical Particle Orbits around a Rotating Black Hole in Massive Gravity. *Symmetry* **2023**, *15*, 1485.
https://doi.org/10.3390/sym15081485

**AMA Style**

Fathi M, Villanueva JR, Cruz N.
Spherical Particle Orbits around a Rotating Black Hole in Massive Gravity. *Symmetry*. 2023; 15(8):1485.
https://doi.org/10.3390/sym15081485

**Chicago/Turabian Style**

Fathi, Mohsen, José R. Villanueva, and Norman Cruz.
2023. "Spherical Particle Orbits around a Rotating Black Hole in Massive Gravity" *Symmetry* 15, no. 8: 1485.
https://doi.org/10.3390/sym15081485