# On center of mass and foliations by constant spacetime mean curvature surfaces for isolated systems in General Relativity

## TL;DR

This paper introduces a new foliation by constant spacetime mean curvature surfaces in asymptotically flat spacetimes, enabling a covariant definition of total center of mass that behaves like a point particle under Einstein evolution.

## Contribution

It constructs a unique STCMC-foliation near infinity, providing a new, covariant way to define the total center of mass in General Relativity, improving upon previous CMC-based methods.

## Key findings

- Existence of a unique STCMC-foliation near infinity
- Definition of a new covariant total center of mass
- Center of mass transforms under Poincaré group and evolves like a point particle

## Abstract

We propose a new foliation of asymptotically Euclidean initial data sets by 2-spheres of constant spacetime mean curvature (STCMC). The leaves of the foliation have the STCMC-property regardless of the initial data set in which the foliation is constructed which asserts that there is a plethora of STCMC 2-spheres in a neighborhood of spatial infinity of any asymptotically flat spacetime. The STCMC-foliation can be understood as a covariant relativistic generalization of the CMC-foliation suggested by Huisken and Yau. We show that a unique STCMC-foliation exists near infinity of any asymptotically Euclidean initial data set with non-vanishing energy which allows for the definition of a new notion of total center of mass for isolated systems. This STCMC-center of mass transforms equivariantly under the asymptotic Poincar\'e group of the ambient spacetime and in particular evolves under the Einstein evolution equations like a point particle in Special Relativity. The new definition also remedies subtle deficiencies in the CMC-approach to defining the total center of mass suggested by Huisken and Yau which were described by Cederbaum and Nerz.