Wataru Hikida

Analytical solutions of bound timelike geodesic orbits in Kerr spacetime

We derive the analytical solutions of the bound timelike geodesic orbits in Kerr spacetime. The analytical solutions are expressed in terms of the elliptic integrals using Mino time λ as the independent variable. Mino time decouples the radial and the polar motion of a particle and hence leads to forms more useful to estimate three fundamental frequencies, radial, polar and azimuthal motion, for the bound timelike geodesics in Kerr spacetime. This paper gives the first derivation of the analytical expressions of the fundamental frequencies. This paper also gives the first derivation of the analytical expressions of all coordinates for the bound timelike geodesics using Mino time. These analytical expressions should be useful not only to investigate physical properties of Kerr geodesics but more importantly to applications related to the estimation of gravitational waves from the extreme mass ratio inspirals.

Adiabatic evolution of orbital parameters in kerr spacetime

We investigate the adiabatic orbital evolution of a point particle in Kerr spacetime due to the emission of gravitational waves. In the case that the timescale of the orbital evolution is sufficiently smaller than the characteristic timescale of orbits, the evolution of orbits is characterized by the rates of change of three constants of motion, the energy E, the azimuthal angular momentum L, and the Carter constant Q. We can evaluate the rates of change of E and L from the fluxes of the energy and the angular momentum at infinity and on the event horizon, employing the balance argument. However, for the Carter constant, we cannot use the balance argument because we do not know the conserved current associated with it. Recently, Mino proposed a new method of evaluating the average rate of change rate of the Carter constant by using the radiative field. In a previous paper, we developed a simplified scheme for determining the evolution of the Carter constant based on Mino’s proposal. In this paper we describe our scheme in more detail and derive explicit analytic formulae for the rates of change of the energy, the angular momentum and the Carter constant.

Global structure of the Zipoy–Voorhees–Weyl spacetime and the δ = 2 Tomimatsu–Sato spacetime

We investigate the structure of the ZVW (Zipoy–Voorhees–Weyl) spacetime, which is a Weyl solution described by the Zipoy–Voorhees metric, and the δ = 2 Tomimatsu–Sato spacetime. We show that the singularity of the ZVW spacetime, which is represented by a segment ρ = 0, − σ 1. These singularities are always naked and have positive Komar masses for δ > 0. Thus, they provide a non-trivial example of naked singularities with positive mass. We further show that the ZVW spacetime has a degenerate Killing horizon with a ring singularity at the equatorial plane for δ = 2, 3 and δ ≥ 4. We also show that the δ = 2 Tomimatsu–Sato spacetime has a degenerate horizon with two components, in contrast to the general belief that the Tomimatsu–Sato solutions with even δ do not have horizons.

An Efficient Numerical Method for Computing Gravitational Waves Induced by a Particle Moving on Eccentric Inclined Orbits around a Kerr Black Hole

We develop a numerical code to compute gravitational waves induced by a particle moving on eccentric inclined orbits around a Kerr black hole. For such systems, the black hole perturbation method is applicable. The gravitational waves can be evaluated by solving the Teukolsky equation with a point like source term, which is computed from the stress-energy tensor of a test particle moving on generic bound geodesic orbits. In our previous papers, we computed the homogeneous solutions of the Teukolsky equation using a formalism developed by Mano, Suzuki and Takasugi and showed that we could compute gravitational waves efficiently and very accurately in the case of circular orbits on the equatorial plane. Here, we apply this method to eccentric inclined orbits. The geodesics around a Kerr black hole have three constants of motion: energy, angular momentum and the Carter constant. We compute the rates of change of the Carter constant as well as those of energy and angular momentum. This is the first time that the rate of change of the Carter constant has been evaluated accurately. We also treat the case of highly eccentric orbits with

Adiabatic Radiation Reaction to Orbits in Kerr Spacetime

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Adiabatic Evolution of Three ‘Constants’ of Motion for Greatly Inclined Orbits in Kerr Spacetime

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A new analytical method for self-force regularization. I - Charged scalar particles in schwarzschild spacetime

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A New Analytical Method for Self-Force Regularization. II Testing the Efficiency for Circular Orbits

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Summary of session B3: analytic approximations, perturbation methods and their applications

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Analytic approximations, perturbation methods, and their applications

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