On the center of mass of asymptotically hyperbolic initial data sets

TL;DR

This paper introduces a new definition of the total center of mass for asymptotically hyperbolic initial data in general relativity, aligning with established concepts for asymptotically Euclidean spaces.

Contribution

It unifies Hamiltonian charge and CMC-foliation approaches to define center of mass in asymptotically hyperbolic spacetimes, ensuring proper transformation and evolution properties.

Findings

Center of mass transforms correctly under coordinate changes.

Center of mass evolves along linear momentum under Einstein equations.

Unified approach applicable to asymptotically anti-de Sitter spacetimes.

Abstract

We define the (total) center of mass for suitably asymptotically hyperbolic time-slices of asymptotically anti-de Sitter spacetimes in general relativity. We do so in analogy to the picture that has been consolidated for the (total) center of mass of suitably asymptotically Euclidean time-slices of asymptotically Minkowskian spacetimes (isolated systems). In particular, we unite -- an altered version of -- the approach based on Hamiltonian charges with an approach based on CMC-foliations near infinity. The newly defined center of mass transforms appropriately under changes of the asymptotic coordinates and evolves in the direction of an appropriately defined linear momentum under the Einstein evolution equations.