The adiabatic evolution of orbital parameters in the Kerr spacetime

TL;DR

This paper presents a detailed scheme for calculating the adiabatic evolution of a particle's orbital parameters in Kerr spacetime, including explicit formulas for energy, angular momentum, and Carter constant changes due to gravitational wave emission.

Contribution

We develop an explicit analytic method to evaluate the adiabatic changes of all three constants of motion, including the Carter constant, in Kerr spacetime using Mino's radiative field approach.

Findings

Derived explicit formulas for change rates of E, L, and Q.

Validated the scheme with practical evaluation of orbital evolution.

Enhanced understanding of gravitational wave emission effects in Kerr spacetime.

Abstract

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