High-frequency backreaction for the Einstein equations under $\mathbb U(1)$ symmetry: from Einstein-dust to Einstein-Vlasov

TL;DR

This paper demonstrates how small, symmetric solutions to the Einstein-massless Vlasov system can be approximated by high-frequency vacuum spacetimes, extending previous work from dust models to more general Vlasov matter.

Contribution

It introduces a method to approximate Einstein-massless Vlasov solutions by high-frequency vacuum spacetimes, bridging models from dust to Vlasov matter.

Findings

High-frequency vacuum spacetimes approximate Vlasov solutions.

Extension from dust to Vlasov matter in Einstein equations.

Method involves successive approximations through dust and null dust models.

Abstract

Given suitable small, localized, $\mathbb U(1)$-symmetric solutions to the Einstein-massless Vlasov system in an elliptic gauge, we prove that they can be approximated by high-frequency vacuum spacetimes. This extends previous constructions where the limiting spacetime solves the Einstein-(multiple) null dust system (i.e., where the limiting massless Vlasov field can be written as a finite sum of delta measures). The proof proceeds by first approximating solutions to the Einstein-massless Vlasov system by solutions to the Einstein-(multiple) null dust system, then approximating solutions to the Einstein-null dust system by vacuum solutions. In the process, we take the number of families of dusts to infinity.