On the existence of MSA coordinates

TL;DR

This paper introduces MSA coordinates for axially symmetric vacuum Einstein solutions, demonstrating their relation to symmetries and providing a method to transform Weyl solutions using these coordinates.

Contribution

It presents the construction of MSA coordinates at any multipole order and explores their connection to symmetries in the Einstein equations.

Findings

MSA coordinates can be explicitly calculated for any multipole order.

Existence of MSA coordinates is linked to symmetries of differential equations in GR.

The relationship between MSA coordinates and symmetries is established for specific cases.

Abstract

The static solutions of the axially symmetric vacuum Einstein equations with a finite number of Relativistic Multipole Moments are described by means of a function that can be written in the same analytic form as the Newtonian gravitational multipole potential. A family of so-called MSA (Multipole-Symmetry Adapted) coordinates are introduced and calculated at any multipole order to perform the transformation of the Weyl solutions. In analogy with a previous result obtained in Newtonian gravity, the existence of a symmetry of a certain system of differential equations leading to the determination of that kind of multipole solutions in General Relativity is explored. The relationship between the existence of this kind of coordinate and the symmetries mentioned is proved for some cases, and the characterization of the MSA system of coordinates by means of this relationship is discussed.