Analytical solutions of bound timelike geodesic orbits in Kerr spacetime

One of the ways to investigate the properties of a Kerr black hole spacetime is to study geodesic motion in this background. Detailed works on the geodesic motion in black hole spacetimes are summarized in Chandrasekhar [2]. In the weak field regime, at large distances from the black hole, the orbits of a particle are almost the same as that in Newtonian gravity. In the strong field regime, however, the orbits become more complicated and it is difficult to compare the orbits with that in Newtonian gravity. For the case of bound geodesics, this can be explained by mismatches between the fundamental frequencies of radial, Ω r , polar, Ω θ and azimuthal-motion, Ω φ . For example, Ω φ -Ω θ shows the precession of the orbital plane and Ω φ -Ω r shows the precession of the orbital ellipse. Differences between the fundamental frequencies become larger as the particle goes into the strong gravity region around black hole horizon or separatrix, which is the boundary between stable and unstable orbits. These relativistic effects have been studied for some cases and some examples of extreme phenomena are found as follows.

Wilkins [3] derived the analytical expressions for the ratio of the azimuthal frequency and the polar frequency, Ω φ /Ω θ , when a particle moves on both circular and non-equatorial orbits around the extreme Kerr black hole. He then showed that the ratio becomes larger as the particle approaches the horizon and found that the particle traces out a helix-like orbit on a sphere around the black hole. He also pointed out that there exist horizon-skimming orbits which have the same radius as the horizon. Horizon-skimming orbits are also studied by numerical calculations including the effects of the emission of gravitational waves from a particle for circular and non-equatorial orbits [4] and for generic orbits [5] around near-extremal Kerr black holes. Glampedakis and Kennefick [6] numerically investigated the ratio of the azimuthal frequency and the radial frequency, Ω φ /Ω r , when a particle moves both on eccentric and equatorial orbits around the Kerr black hole. They found that the ratio becomes larger as the particle approaches the separatrix and the particle traces out a quasi-circular orbit around the periapsis before going back to the apoapsis. These orbits are called zoom-whirl orbits.

The above results show that the fundamental frequencies play an important role in understanding bound geodesic orbits. However, the coupling of the r and θ motions in the geodesic equation has prevented one from deriving the fundamental frequencies, Ω r , Ω θ and Ω φ , for general bound geodesic orbits until recently. Using the elegant Hamilton-Jacobi formalism, Schmidt [7] derived the fundamental frequencies without discussing the coupling of the r and θ motions. Although his results show that we can expand an arbitrary function of the particle’s orbit in a Fourier series, we can not estimate the Fourier components because of the coupling of the r and θ-motion. Mino [8] showed that we can separate r and θ-motion if we use new time parameter λ and derived the integral forms of the periods of both r and θ-motion with respect to λ, which is called Mino time. Combining Schmidt’s method with Mino time, Drasco and Hughes [9] derived the fundamental frequencies and showed how the Fourier components of arbitrary functions of orbits with respect to Mino time can be computed because of the decoupling of both r and θ motions. They also showed how from these results using Mino time, the Fourier components with respect to coordinate time can also be derived. Thanks to these results, one can compute gravitational waves from binary systems in which a stellar mass compact star is moving on a general bound geodesic orbit around a supermassive black hole, the so-called extreme mass ratio inspirals(EMRIs) [10]. Gravitational waves from EMRIs are one of the main targets for space-based Laser Interferometer Space Antenna (LISA) [11].

In this paper, we derive analytical expressions for bound timelike geodesic orbits in Kerr spacetime using Mino time as the independent variable. Despite a lot of works on geodesic motion [2], the analytical expressions of null or timelike geodesics in Kerr spacetime are still important subjects. Fast and accurate computation of null geodesics in Kerr spacetime is required to study radiation which pass near black holes in accretion systems such as active galactic nuclei and X-ray binaries (see, for example, [12,13] and references therein). Fast and accurate computation of timelike geodesics is also required