Asymptotic Behavior of Polarized and Half-Polarized U(1) symmetric Vacuum Spacetimes

TL;DR

This paper investigates the asymptotic behavior near singularities of U(1) symmetric vacuum spacetimes, demonstrating AVTD behavior for polarized and half-polarized solutions using the Fuchsian algorithm, and exploring foliation independence.

Contribution

It shows that polarized and half-polarized U(1) symmetric vacuum solutions exhibit AVTD behavior near singularities, extending understanding of their asymptotic dynamics.

Findings

Polarized solutions are AVTD near singularity.

Half-polarized solutions also exhibit AVTD behavior.

AVTD behavior is consistent across various harmonic time choices.

Abstract

We use the Fuchsian algorithm to study the behavior near the singularity of certain families of U(1) Symmetric solutions of the vacuum Einstein equations (with the U(1) isometry group acting spatially). We consider an analytic family of polarized solutions with the maximum number of arbitrary functions consistent with the polarization condition (one of the ``gravitational degrees of freedom'' is turned off) and show that all members of this family are asymptotically velocity term dominated (AVTD) as one approaches the singularity. We show that the same AVTD behavior holds for a family of ``half polarized'' solutions, which is defined by adding one extra arbitrary function to those characterizing the polarized solutions. (The full set of non-polarized solutions involves two extra arbitrary functions). We begin to address the issue of whether AVTD behavior is independent of the choice of time foliation by showing that indeed AVTD behavior is seen for a wide class of choices of harmonic time in the polarized and half-polarized (U(1) Symmetric vacuum) solutions discussed here. \