Kerr geodesics in terms of Weierstrass elliptic functions

Cieślik, Adam; Hackmann, Eva; Mach, Patryk

doi:10.1103/physrevd.108.024056

Cited by 7 publications

(4 citation statements)

References 55 publications

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“…Note that special attention is required when evaluating the logarithm in (36) in order to produce a continuous implementation of ϕðλÞ ensuring that we use the strategy discussed in [19]. Now, we consider the integral I z , which can be directly integrated using (29).…”

Section: ϕ-Motionmentioning

confidence: 99%

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Geodesic motion in a swirling universe: The complete set of solutions

Capobianco,

Hartmann,

Kunz

2024

Phys. Rev. D

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We study the geodesic motion in a space-time describing a swirling universe. We show that the geodesic equations can be fully decoupled in the Hamilton-Jacobi formalism leading to an additional constant of motion. The analytical solutions to the geodesic equations can be given in terms of elementary and elliptic functions. We also consider a space-time describing a static black hole immersed in a swirling universe. In this case, full separation of variables is not possible, and the geodesic equations have to be solved numerically.

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Section: ϕ-Motionmentioning

confidence: 99%

“…is a single pole of the above expression. A table with these and other relations can be found in [19].…”

Section: Appendix C: Integration Of Elliptic Integrals Of the Third Kindmentioning

confidence: 99%

Geodesic motion in a swirling universe: The complete set of solutions

Capobianco,

Hartmann,

Kunz

2024

Phys. Rev. D

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“…Analytical solutions for geodesics related to circular and innermost stable spherical orbits in the phase space have been obtained [36,37]. Nevertheless, compared with the classification of the equatorial geodesics in [34], some classes of orbits, such as trapped orbits associated with parameters governing bound and deflecting motion, still lack explicit analytical solutions.…”

Section: Introductionmentioning

confidence: 99%

Analytical solutions of equatorial geodesic motion in Kerr spacetime*

Liu 刘,

Sun 孙

2024

Chinese Phys. C

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The study of Kerr geodesics has a long history, particularly for those occurring within the equatorial plane, which is generally well-understood. However, upon comparison with the classification introduced by one of us \href{https://journals.aps.org/prd/abstract/10.1103/PhysRevD.105.024075}{[Phys. Rev. D 105, 024075 (2022)]}, it becomes apparent that certain classes of geodesics, such as trapped orbits, are still lacking analytical solutions. Thus, in this study, we provide explicit analytical solutions for equatorial timelike geodesics in Kerr spacetime, including solutions of trapped orbits, which capture the characteristics of special geodesics, such as the positions and conserved quantities of circular orbits, bound orbits, and deflecting orbits. Specifically, we determine the precise location at which retrograde orbits undergo a transition from counter-rotating to prograde motion due to the strong gravitational effects near the rotating black hole. Interestingly, we observe that for orbits with negative energy, the trajectory remains prograde despite the negative angular momentum. Furthermore, we investigate the intriguing phenomenon of deflecting orbits exhibiting an increased number of revolutions around the black hole as the turning point approaches the turning point of the trapped orbit. Additionally, we find that only prograde marginal deflecting geodesics are capable of traversing through the ergoregion. In summary, our findings present explicit solutions for equatorial timelike geodesics and offer insights into the dynamics of particle motion in the vicinity of a rotating black hole.

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“…The radial time-like geodesic motion of the exterior nonextremal Kerr spacetime was classified in detail in [28]. Recently, Kerr geodesics in terms of Weierstrass elliptic functions were discussed in [29]. Other works (see, for example, refs.…”

Section: Introductionmentioning

confidence: 99%