Faithful transformation of quasi-isotropic to Weyl-Papapetrou coordinates: A prerequisite to compare metrics

TL;DR

This paper clarifies the correct method to transform quasi-isotropic coordinates to Weyl-Papapetrou coordinates, essential for accurate comparison of numerical and analytical metrics around rotating stars, especially in non-vacuum regions.

Contribution

It provides a detailed procedure for faithful coordinate transformation from quasi-isotropic to Weyl-Papapetrou coordinates in non-vacuum regions, preventing misinterpretations.

Findings

Incorrect transformations lead to erroneous metric comparisons.

Proper transformation is crucial for accurate metric matching.

Highlights the importance of considering non-vacuum regions in coordinate changes.

Abstract

We demonstrate how one should transform correctly quasi-isotropic coordinates to Weyl-Papapetrou coordinates in order to compare the metric around a rotating star that has been constructed numerically in the former coordinates with an axially symmetric stationary metric that is given through an analytical form in the latter coordinates. Since a stationary metric associated with an isolated object that is built numerically partly refers to a non-vacuum solution (interior of the star) the transformation of its coordinates to Weyl-Papapetrou coordinates, which are usually used to describe vacuum axisymmetric and stationary solutions of Einstein equations, is not straightforward in the non-vacuum region. If this point is \textit{not} taken into consideration, one may end up to erroneous conclusions about how well a specific analytical metric matches the metric around the star, due to fallacious coordinate transformations.