Coordinates are messy -- not only in General Relativity

TL;DR

The paper explores the complexities of defining physical quantities like the center of mass in both General Relativity and Newtonian Gravity, highlighting coordinate issues and providing explicit examples where standard conditions fail.

Contribution

It demonstrates that coordinate ambiguities affect the definition of physical quantities in Newtonian Gravity and General Relativity, and presents explicit examples challenging existing parity conditions.

Findings

Coordinate freedom complicates physical quantity definitions.

Explicit examples show failure of parity conditions.

Insights into the nature of the center of mass in relativistic settings.

Abstract

The coordinate freedom of General Relativity makes it challenging to find mathematically rigorous and physically sound definitions for physical quantities such as the center of mass of an isolated gravitating system. We will argue that a similar phenomenon occurs in Newtonian Gravity once one ahistorically drops the restriction that one should only work in Cartesian coordinates when studying Newtonian Gravity. This will also shed light on the nature of the challenge of defining the center of mass in General Relativity. Relatedly, we will give explicit examples of asymptotically Euclidean relativistic initial data sets which do not satisfy the Regge--Teitelboim parity conditions often used to achieve a satisfactory definition of center of mass. These originate in our joint work with Jan Metzger. This will require appealing to Bartnik's asymptotic harmonic coordinates.