Some remarks about the centre of mass of two particles in spaces of constant curvature

TL;DR

This paper examines different definitions of the centre of mass for two particles in 2D spaces of constant curvature, highlighting their equivalence only in flat space and differences in curved spaces.

Contribution

It compares three notions of centre of mass in curved spaces and shows their equivalence only in flat space, clarifying the concept in non-Euclidean geometries.

Findings

Different definitions coincide only in flat space.

In curved spaces, the notions of centre of mass differ.

The relativistic rule of lever aligns with other definitions only in zero curvature.

Abstract

The concept of centre of mass of two particles in 2D spaces of constant Gaussian curvature is discussed by recalling the notion of "relativistic rule of lever" introduced by Galperin [Comm. Math. Phys. 154 (1993), 63--84] and comparing it with two other definitions of centre of mass that arise naturally on the treatment of the 2-body problem in spaces of constant curvature: firstly as the collision point of particles that are initially at rest, and secondly as the centre of rotation of steady rotation solutions. It is shown that if the particles have distinct masses then these definitions are equivalent only if the curvature vanishes and instead lead to three different notions of centre of mass in the general case.