Mass angular momentum inequality for axisymmetric vacuum data with small trace

TL;DR

This paper proves the mass angular momentum inequality for small trace axisymmetric vacuum data by constructing a maximal surface in a perturbed spacetime, extending previous results to new initial data conditions.

Contribution

It introduces a perturbation method to establish the inequality for axisymmetric vacuum data with small trace, expanding the class of initial data for which the inequality holds.

Findings

Mass angular momentum inequality is valid for small trace axisymmetric vacuum data.

Constructed a maximal surface with the same ADM mass and angular momentum as initial data.

Extended previous results to initial data with small trace.

Abstract

In this paper, we proved the mass angular momentum inequality\cite{D1}\cite{ChrusLiWe}\cite{SZ} for axisymmetric, asymptotically flat, vacuum constraint data sets with small trace. Given an initial data set with small trace, we construct a boost evolution spacetime of the Einstein vacuum equations as \cite{ChOM}. Then a perturbation method is used to solve the maximal surface equation in the spacetime under certain growing condition at infinity. When the initial data set is axisymmetric, we get an axisymmetric maximal graph with the same ADM mass and angular momentum as the given data. The inequality follows from the known results\cite{D1}\cite{ChrusLiWe}\cite{SZ} about the maximal graph.