A quadrupolar generalization of the Erez-Rosen coordinates

Abstract

[EN]The MSA system of coordinates (Hernández-Pastora 2010 Class. Quantum Grav. 27 045006) for the MQ-solution (Hernández-Pastora and Martín 1994 Gen. Relativ. Gravit. 26 877) 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 (Hernández-Pastora 2010 Class. Quantum Grav. 27 045006) for those coordinates, as well as the Erez-Rosen coordinates in the spherical case. It is also shown that the event horizon of the MQ-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 (Hernández- Pastora and Herrera 2011 Class. Quantum Grav. 28 225026). 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.