On the uniqueness of harmonic coordinates

TL;DR

This paper investigates the non-uniqueness of harmonic coordinate solutions in stationary asymptotically flat spacetimes with matter, highlighting dependence on smoothness conditions and boundary choices, with illustrative examples.

Contribution

It demonstrates the non-uniqueness of harmonic coordinates solutions in certain spacetimes and provides explicit examples, clarifying the influence of boundary and smoothness conditions.

Findings

Solutions depend on smoothness across matter boundaries

Explicit examples of static and stationary spacetimes in harmonic coordinates

Use of background metrics simplifies calculations

Abstract

Harmonic coordinate conditions in stationary asymptotically flat spacetimes with matter sources have more than one solution. The solutions depend on the degree of smoothness of the metric and its first derivatives, which we wish to impose across the material boundary, and on the conditions at infinity and at a suitable point inside the matter. This is illustrated in detail by simple fully solvable examples of static spherically symmetric spacetimes in global harmonic coordinates. Examples of stationary electrovacuum spacetimes described simply in harmonic coordinates are also given. They can represent the exterior fields of material discs. The use of an appropriate background metric considerably simplifies the calculations.