A note on behaviour at an isotropic singularity

TL;DR

This paper investigates how Jacobi fields behave near an isotropic singularity, revealing a uniform crushing effect in all orthogonal directions, but also highlighting the importance of the singularity's precise definition.

Contribution

It demonstrates the uniform crushing of Jacobi fields at isotropic singularities and clarifies the dependence on the technical definition of such singularities.

Findings

Jacobi fields are crushed to zero length at a uniform rate

The crushing effect depends on the specific definition of isotropic singularities

Counter-example shows the importance of the technical conditions

Abstract

The behaviour of Jacobi fields along a time-like geodesic running into an isotropic singularity is studied. It is shown that the Jacobi fields are crushed to zero length at a rate which is the same in every direction orthogonal to the geodesic. We show by means of a counter-example that this crushing effect depends crucially on a technicality of the definition of isotropic singularities, and not just on the uniform degeneracy of the metric at the singularity.