Construction of oscillatory singularities

TL;DR

This paper explores the construction of solutions to Einstein's equations that exhibit oscillatory singularities, advancing understanding of complex spacetime behaviors near singularities through recent mathematical theorems.

Contribution

It presents new theorems demonstrating the existence of spatially homogeneous solutions with oscillatory singularities, using dynamical systems and heteroclinic chain techniques.

Findings

Existence of solutions with oscillatory singularities proven

Application of heteroclinic chains in general relativity

Framework for future generalizations of oscillatory singularities

Abstract

One way to understand more about spacetime singularities is to construct solutions of the Einstein equations containing singularities with prescribed properties. The heuristic ideas of the BKL picture suggest that oscillatory singularities should be very common and give a detailed picture of how these could look. The more straightforward case of singularities without oscillations is reviewed and existing results on that subject are surveyed. Then recent theorems proving the existence of spatially homogeneous solutions with oscillatory singularities of a specific type are presented. The proofs of these involve applications of some ideas concerning heteroclinic chains and their stability. Some necessary background from the theory of dynamical systems is explained. Finally some directions in which this research might be generalized in the future are pointed out.