Conserved Quantities of harmonic asymptotic initial data sets

TL;DR

This paper surveys new quasi-local and total conserved quantities in harmonic asymptotic initial data sets, providing explicit computations for a large family of such data sets and highlighting their properties.

Contribution

It introduces a large family of harmonic asymptotic initial data sets where conserved quantities can be explicitly computed, advancing understanding of their properties.

Findings

Explicit formulas for total angular momentum and center of mass.

Identification of a large family of initial data sets suitable for calculations.

Summary of key properties of the new conserved quantities.

Abstract

In the first half of this article, we survey the new quasi-local and total angular momentum and center of mass defined in [9] and summarize the important properties of these definitions. To compute these conserved quantities involves solving a nonlinear PDE system (the optimal isometric embedding equation), which is rather difficult in general. We found a large family of initial data sets on which such a calculation can be carried out effectively. These are initial data sets of harmonic asymptotics, first proposed by Corvino and Schoen to solve the full vacuum constraint equation. In the second half of this article, the new total angular momentum and center of mass for these initial data sets are computed explicitly.