Abstract
We provide analytical closed form solutions for the parallel transport along a bound geodesic in Kerr spacetime. This can be considered the lowest order approximation for the motion of a spinning black hole in an extreme mass-ratio inspiral. As an illustration of the usefulness of our new found expressions we scope out the locations of spin–spin resonances in quasi-circular EMRIs. All solutions are given as functions of Mino time, which facilitates the decoupling of the equations of motion. To help physical interpretation, we also provide an analytical expression for the proper time along a geodesic as a function of Mino time.
Highlights
The motion of a freely falling frame in general relativity is described by the parallel transport of a frame along a geodesic
To first approximation — ignoring all effects due to its own mass and spin — the motion of smaller black hole is described by the parallel transport of its spin along a geodesic in the Kerr spacetime
As long as the spin of the primary is aligned with the total angular momentum of the system, the solution is fairly simple; the test object will follow an equatorial geodesic, and the spin will precess around the plane spanned by the four-velocity and total angular momentum
Summary
The motion of a freely falling frame in general relativity is described by the parallel transport of a frame along a geodesic. Most of these efforts have restricted themselves to the “easy” case where the spin of the primary is aligned with the total angular momentum One reason for this is the lack of applicable closed form solutions for the parallel transport along a generic orbit. (With in recent years some clarifications being added by Bini and collaborators [17, 18]) This procedure effectively reduces the parallel transport equations to a single differential equation for the precession angle given a solution for the geodesic equation. Closed form analytic solutions for bound geodesics in Kerr spacetime were derived by Fujita and Hikida [19] in 2009 This paper takes their method an extends it to a solution for Marck’s equation for parallel transport.
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