Black hole geodesic parallel transport and the Marck reduction procedure

The Wigner rotations arising from the combination of boosts along two different directions are rederived from a relative boost point of view and applied to study gyroscope spin precession along timelike geodesics in a Kerr spacetime. First this helps to clarify the geometrical properties of Marck’s recipe for reducing the equations of parallel transport along such world lines expressed in terms of the constants of the motion to a single differential equation for the essential planar rotation. Second this shows how to bypass Marck’s reduction procedure by direct boosting of orthonormal frames associated with natural observer families. Wigner rotations mediate the relationship between these two descriptions for reaching the same parallel transported frame along a geodesic. The comparison is particularly straightforward in the case of equatorial plane motion of a test gyroscope, where Marck’s scalar angular velocity captures the essential cumulative spin precession relative to the spherical frame locked to spatial infinity. These cumulative precession effects are computed explicitly for both bound and unbound equatorial plane geodesic orbits. The latter case is of special interest in view of recent applications to the dynamics of a two-body system with spin. Our results are consistent with the point-particle limit of such two-body results and also pave the way for similar computations in the context of gravitational self-force.

Figure 1

The initial and final rotations required to orient the spherical frame with the final Marck frame adapted to the plane of the parallel transport rotation, where the final rotation ($\mathrm{\Psi }$) takes place. The top panel shows the initial orientation of a spherical frame in the equatorial plane. The azimuthal vector $e_3$ is first rotated about the radial vector $e_1$ by the angle $\mathrm{\Upsilon }$ to align it with the direction of the angular relative velocity. The two successive boosts to the intermediate Marck frame comoving with the geodesic have relative velocities spanning the upper plane, where the resulting Wigner rotation takes place. Then $e_2$ is rotated about the new $e_3$ by the angle $\mathrm{\Xi }$ to align it with the electric part of the Killing-Yano 2-form, which is the normal to the (lower) parallel rotation plane, leading to the bottom panel, with the final triad in bold.