Remarks on mass and angular momenta for $U(1)^2$-invariant initial data

TL;DR

This paper extends positive mass theorems and mass-angular momentum inequalities to a broad class of $U(1)^2$-invariant initial data and black hole solutions with matter fields in four-dimensional general relativity.

Contribution

It generalizes Brill's positive mass theorem and mass-angular momentum inequalities to include nonzero stress-energy tensors with symmetry invariance.

Findings

Extended positive mass theorem to $U(1)^2$-invariant data

Proved mass-angular momentum inequalities with matter fields

Applicable to simply connected four-dimensional manifolds

Abstract

We extend Brill's positive mass theorem to a large class of asymptotically flat, maximal, $U(1)^2$-invariant initial data sets on simply connected four dimensional manifolds $\Sigma$. Moreover, we extend the local mass angular momenta inequality result Ref [1] for $U(1)^2$ invariant black holes to the case with nonzero stress energy tensor with positive matter density and energy-momentum current invariant under the above symmetries.