Comparing metrics at large: harmonic vs quo-harmonic coordinates

TL;DR

This paper compares harmonic and quo-harmonic coordinate systems for analyzing large-scale space-time structures, focusing on their properties, differences, and explicit coordinate determinations for key metrics like Schwarzschild, Curzon, and Kerr.

Contribution

It provides a detailed comparison of harmonic and quo-harmonic coordinates, including explicit calculations for important stationary metrics up to fifth order in their asymptotic expansions.

Findings

Harmonic and quo-harmonic coordinates differ at the linearized level.

Explicit coordinate systems are derived for Schwarzschild, Curzon, and Kerr metrics.

The study discusses implications for multipole definitions and matching interior and exterior solutions.

Abstract

To compare two space-times on large domains, and in particular the global structure of their manifolds, requires using identical frames of reference and associated coordinate conditions. In this paper we use and compare two classes of time-like congruences and corresponding adapted coordinates: the harmonic and quo-harmonic classes. Besides the intrinsic definition and some of their intrinsic properties and differences we consider with some detail their differences at the level of the linearized approximation of the field equations. The hard part of this paper is an explicit and general determination of the harmonic and quo-harmonic coordinates adapted to the stationary character of three well-know metrics, Schwarzschild's, Curzon's and Kerr's, to order five of their asymptotic expansions. It also contains some relevant remarks on such problems as defining the multipoles of vacuum solutions or matching interior and exterior solutions.