Computing asymptotic invariants with the Ricci tensor on asymptotically flat and hyperbolic manifolds

TL;DR

This paper establishes a coordinate-free method to compute the mass and center of mass for asymptotically flat and hyperbolic manifolds using the Ricci tensor, simplifying previous approaches.

Contribution

It introduces a simple, coordinate-free proof linking classical and Ricci tensor-based definitions of mass and center of mass for asymptotically flat and hyperbolic manifolds.

Findings

Equivalence of classical and Ricci-based definitions of mass and center of mass

Coordinate-free proofs simplify understanding of asymptotic invariants

Extension of results to asymptotically hyperbolic manifolds

Abstract

We prove in a simple and coordinate-free way the equivalence bteween the classical definitions of the mass or the center of mass of an asymptotically flat manifold and their alternative definitions depending on the Ricci tensor and conformal Killing fields. This enables us to prove an analogous statement in the asymptotically hyperbolic case.