Solving parallel transport equations in the higher-dimensional Kerr-NUT-(A)dS spacetimes

TL;DR

This paper derives a method to solve parallel transport equations in higher-dimensional Kerr-NUT-(A)dS spacetimes by reducing them to first-order ODEs, enabling explicit solutions through separation of variables.

Contribution

It introduces a novel reduction of parallel transport equations using eigenspaces of a projected conformal Killing-Yano tensor, applicable to higher-dimensional rotating black hole spacetimes.

Findings

Eigenvalue eigenspaces are at most 2D and independently parallel-propagated.

Parallel transport equations reduce to first-order ODEs for rotation angles.

Separation of variables allows explicit solutions in 3-5 dimensional Kerr-NUT-(A)dS spacetimes.

Abstract

We obtain and study the equations describing the parallel transport of orthonormal frames along geodesics in a spacetime admitting a non-degenerate principal conformal Killing-Yano tensor h. We demonstrate that the operator F, obtained by a projection of h to a subspace orthogonal to the velocity, has in a generic case eigenspaces of dimension not greater than 2. Each of these eigenspaces are independently parallel-propagated. This allows one to reduce the parallel transport equations to a set of the first order ordinary differential equations for the angles of rotation in the 2D eigenspaces. General analysis is illustrated by studying the equations of the parallel transport in the Kerr-NUT-(A)dS metrics. Examples of three, four, and five dimensional Kerr-NUT-(A)dS are considered and it is shown that the obtained first order equations can be solved by a separation of variables.