Black hole geodesic parallel transport and the Marck reduction procedure

Abstract

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 gyroscope spin precession along timelike geodesics in a Kerr spacetime, clarifying the geometrical properties of Marck's recipe for describing parallel transport along such world lines expressed in terms of the constants of the motion. His final angular velocity isolates the cumulative spin precession angular velocity independent of the spacetime tilting required to keep the spin 4-vector orthogonal to the gyro 4-velocity. As an explicit example the cumulative precession effects are computed for a test gyroscope moving along both bound and unbound equatorial plane geodesic orbits.

Gödel spacetime: Planar geodesics and gyroscope precession

“…III analyzes the timelike geodesics on the planar surfaces orthogonal to the cylindrical symmetry axes, in order to study gyroscopic precession in Sec. IV, completing previous results [9]. The signature of the metric is − + ++ and the units are chosen so that c = 1 = G. ric when expressed in cylindrical coordinates (t, r, φ, z)…”

“…We have shown that for these elliptic-like orbits there is no periastron advance, unlike the black hole case. Furthermore, we have discussed the geometric properties of a parallel propagated frame along these geodesics, recently found in a previous paper [9]. Finally, we expect that this work will be relevant for the study of perturbations to this spacetime induced by a particle moving along an elliptic-like geodesic.…”

“…By doing so we have 4) mapped the standard timelike geodesic parameter space described by the conserved energy (E) and angular momentum (L) onto p and e. Furthermore, along those closed eccentric orbits which encircle the cylindrical symmetry axis, we have 5) introduced two physical frames: a Frenet-Serret frame and a parallel transported frame (recently obtained in Appendix B of Ref. [9]), allowing us to discuss test gyroscope precession (a novel result of the present paper) together with its limit of the known circular orbit case [5,6,10].…”

Spin Dynamics of Moving Bodies in Rotating Black Hole Spacetimes

“…The above transformation is a special case of the Wigner‐rotation, [ 107 ] which was discussed recently in ref. [108]. However explicit expressions for the rotation between the frames which we denote $$E_{\bm {\alpha }}(f,U)$$ and $$E_{ \bm {\alpha }}(e,U)$$ were not presented in ref.…”

Analytic solutions for parallel transport along generic bound geodesics in Kerr spacetime

“…An elegant procedure for solving the parallel transport equations along a geodesic in Kerr spacetime was set out by Marck [16] in 1983. (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.…”