On smoothness of timelike maximal cylinders in three dimensional vacuum spacetimes

TL;DR

This paper proves that timelike maximal cylinders in three-dimensional vacuum spacetimes inevitably develop singularities in finite time, with their singularities characterized by rigid or self-similar motions at infinitesimal scales.

Contribution

It demonstrates the finite-time singularity formation of timelike maximal cylinders and characterizes the nature of their singularities, extending results to non-flat backgrounds.

Findings

Timelike maximal cylinders develop singularities in finite time.

At singularities, their slices are governed by rigid or self-similar motions.

Results extend to non-flat vacuum backgrounds.

Abstract

We show that timelike maximal cylinders in $\RR^{1 + 2}$ always develop singularities in finite time and that, infinitesimally at a generic singularity, their time slices are evolved by a rigid motion or a self-similar motion. We also prove a mild generalization in non-flat backgrounds.