Proof of the Einstein quadrupole formula for solutions of the Einstein-Vlasov system close to Minkowski spacetime

TL;DR

This paper rigorously derives the Einstein quadrupole formula for solutions of the Einstein-Vlasov system near Minkowski spacetime, providing explicit error estimates and emphasizing the importance of no-incoming-radiation conditions.

Contribution

It offers a rigorous derivation of the quadrupole formula with explicit error bounds for the Einstein-Vlasov system, advancing beyond prior post-Newtonian approaches.

Findings

Explicit error estimates depend on solution properties

Validation of the quadrupole formula for global solutions

Highlighting the role of no-incoming-radiation condition

Abstract

We rigorously derive the quadrupole formula within the context of the Einstein-Vlasov system. The main contribution of this work is an estimate of the remainder terms, derived from well-defined assumptions, with explicitly stated error terms that depend on the solution's boundedness and decay properties, and the distance to the source. The assumptions are linked to established properties of global solutions of the Einstein-Vlasov system as in \cite{LT}. Prior derivations of the quadrupole formula have relied on post-Newtonian analysis and lacked comparisons with global solution properties. The importance of the no-incoming-radiation condition is emphasized underscoring the need for solutions satisfying this condition. This work thus addresses the limitations of existing results and provides motivation for further research on global solution properties of the Einstein-Vlasov system.