Parallel-propagated frame along null geodesics in higher-dimensional black hole spacetimes

TL;DR

This paper extends the construction of parallel-transported frames along null geodesics to higher-dimensional Kerr-NUT-(A)dS spacetimes, providing explicit methods and exploring principal null directions.

Contribution

It demonstrates a modified construction for null geodesics in higher-dimensional spacetimes with conformal Killing-Yano tensors, including explicit examples in 4D to 6D.

Findings

Explicit parallel-transported frames along null geodesics in D=4,5,6 Kerr-NUT-(A)dS spacetimes

Analysis of parallel transport along principal null directions

Extension of the method to 4D Plebanski-Demianski background

Abstract

In [arXiv:0803.3259] the equations describing the parallel transport of orthonormal frames along timelike (spacelike) geodesics in a spacetime admitting a non-degenerate principal conformal Killing-Yano 2-form h were solved. The construction employed is based on studying the Darboux subspaces of the 2-form F obtained as a projection of h along the geodesic trajectory. In this paper we demonstrate that, although slightly modified, a similar construction is possible also in the case of null geodesics. In particular, we explicitly construct the parallel-transported frames along null geodesics in D=4,5,6 Kerr-NUT-(A)dS spacetimes. We further discuss the parallel transport along principal null directions in these spacetimes. Such directions coincide with the eigenvectors of the principal conformal Killing-Yano tensor. Finally, we show how to obtain a parallel-transported frame along null geodesics in the background of the 4D Plebanski-Demianski metric which admits only a conformal generalization of the Killing-Yano tensor.