# A quadrupolar generalization of the Erez-Rosen coordinates

## TL;DR

This paper introduces a unique coordinate system for the M Q-solution that generalizes Erez-Rosen coordinates, enabling better analysis of relativistic multipole moments and correcting previous misconceptions about quadrupole bounds.

## Contribution

It derives a new coordinate system for the M Q-solution, generalizes Erez-Rosen coordinates, and explores extensions to solutions with multiple relativistic multipole moments.

## Key findings

- MSA coordinates are unique solutions to a specific PDE.
- The coordinates recover known asymptotic expansions and Erez-Rosen coordinates.
- The method allows correction of previous bounds on quadrupole moments.

## Abstract

The MSA system of coordinates [1] for the M Q-solution [2] is proved to be the unique solution of certain partial differential equation with boundary and asymptotic conditions. Such a differential equation is derived from the orthogonality condition between two surfaces which hold a functional relationship. The obtained expressions for the MSA system recover the asymptotic expansions previously calculated [1] for those coordinates, as well as the Erez-Rosen coordinates in the spherical case. It is also shown that the event horizon of the M Q-solution can be easily obtained from those coordinates leading to already known results. But in addition, it allows us to correct a mistaken conclusion related to some bound imposed to the value of the quadrupole moment [3]. Finally, it is explored the possibility of extending this method of generalizing the Erez-Rosen coordinates to the general case of solutions with any finite number of Relativistic Multipole Moments (RMM). It is discussed as well, the possibility of determining the Weyl moments of those solutions from their corresponding MSA coordinates, aiming to establish a relation between the uniqueness of the MSA coordinates and the solutions itself.