Conformal Parameterizations of Slices of Flat Kasner Spacetimes

TL;DR

This paper explores the conformal method for constructing slices of flat Kasner spacetimes, revealing new non-CMC solutions and proposing modifications to improve the detection of these solutions.

Contribution

It provides a full description of non-CMC slices in flat Kasner spacetimes and introduces modifications to the conformal method to better identify these solutions.

Findings

Most data sets produce a single Kasner spacetime

Some data sets generate one-parameter families of slices

Non-CMC families differ from CMC families in key ways

Abstract

The Kasner metrics are among the simplest solutions of the vacuum Einstein equations, and we use them here to examine the conformal method of finding solutions of the Einstein constraint equations. After describing the conformal method's construction of constant mean curvature (CMC) slices of Kasner spacetimes, we turn our attention to non-CMC slices of the smaller family of flat Kasner spacetimes. In this restricted setting we obtain a full description of the construction of certain $U^{n-1}$ symmetric slices, even in the far-from-CMC regime. Among the conformal data sets generating these slices we find that most data sets construct a single flat Kasner spacetime, but that there are also far-from-CMC data sets that construct one-parameter families of slices. Although these non-CMC families are analogues of well-known CMC one-parameter families, they differ in important ways. Most significantly, unlike the CMC case, the condition signaling the appearance of these non-CMC families is not naturally detected from the conformal data set itself. In light of this difficulty, we propose modifications of the conformal method that involve a conformally transforming mean curvature.