Geodesic motion in the Kundt spacetimes and the character of envelope singularity

TL;DR

This paper analyzes geodesic behavior and curvature properties of Kundt type N spacetimes, revealing that envelope singularities are non-scalar curvature singularities despite vanishing scalar invariants.

Contribution

It provides explicit calculations of curvature components and characterizes the envelope singularity as a non-scalar curvature singularity in Kundt spacetimes.

Findings

Envelope singularity is a non-scalar curvature singularity.

Spacetimes exhibit an inherent rotation of wave propagation.

Scalar invariants of the Riemann tensor vanish at the singularity.

Abstract

We investigate geodesics in specific Kundt type N (or conformally flat) solutions to Einstein's equations. Components of the curvature tensor in parallelly transported tetrads are then explicitly evaluated and analyzed. This elucidates some interesting global properties of the spacetimes, such as an inherent rotation of the wave-propagation direction, or the character of singularities. In particular, we demonstrate that the characteristic envelope singularity of the rotated wave-fronts is a (non-scalar) curvature singularity, although all scalar invariants of the Riemann tensor vanish there.