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

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## 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

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**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 |

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**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