A localized construction of Kasner-like singularities

TL;DR

This paper develops a localized method for constructing Kasner-like singularities in Einstein vacuum solutions, using a hyperbolic formulation that simplifies localization and allows for detailed analysis of singular behavior.

Contribution

It introduces a new hyperbolic approach to localize Kasner-like singularity constructions, improving upon previous methods that relied on elliptic estimates.

Findings

Constructed local Kasner-like singular solutions in vacuum Einstein equations.

Proved a refined uniqueness theorem for these solutions.

Provided a method to generate general asymptotic data with expected degrees of freedom.

Abstract

We construct local, in spacetime, singular solutions to the Einstein vacuum equations that exhibit Kasner-like behavior in their past boundary. Our result can be viewed as a localization (in space) of the construction in \cite{FL}. We also prove a refined uniqueness statement and give a simple argument that generates general asymptotic data for Kasner-like singularities, enjoying all expected degrees of freedom, albeit only locally in space. The key difference of the present work with \cite{FL} is our use of a first order symmetric hyperbolic formulation of the Einstein vacuum equations, relative to the connection coefficients of a parallelly propagated orthonormal frame which is adapted to the Gaussian time foliation. This makes it easier to localize the construction, since elliptic estimates are no longer required to complete the energy argument.