Conserved quantities on asymptotically hyperbolic initial data sets

TL;DR

This paper studies the limits of quasi-local conserved quantities like energy, momentum, and angular momentum at infinity in asymptotically hyperbolic initial data sets, with less restrictive assumptions than previous work.

Contribution

It establishes finiteness and explicit formulas for conserved quantities at infinity under weaker asymptotic conditions in general relativity.

Findings

Limits of center of mass and angular momentum are finite for aligned foliations.

Explicit formulas relate limits to metric and second fundamental form coefficients.

Less restrictive asymptotic assumptions than previous studies.

Abstract

In this article, we consider the limit of quasi-local conserved quantities [31,9] at the infinity of an asymptotically hyperbolic initial data set in general relativity. These give notions of total energy-momentum, angular momentum, and center of mass. Our assumption on the asymptotics is less stringent than any previous ones to validate a Bondi-type mass loss formula. The Lorentz group acts on the asymptotic infinity through the exchange of foliations by coordinate spheres. For foliations aligning with the total energy-momentum vector, we prove that the limits of quasi-local center of mass and angular momentum are finite, and evaluate the limits in terms of the expansion coefficients of the metric and the second fundamental form.