Approaches to the Monopole-Dynamic Dipole Vacuum Solution Concerning the Structure of its Ernst's Potential on the Symmetry Axis

TL;DR

This paper explores methods to analyze the Ernst's potential in vacuum stationary axisymmetric solutions, comparing exact and approximate solutions to understand their multipole structures using the FHP algorithm and Sibgatullin's method.

Contribution

It introduces an approximate solution for Ernst's potential as a power series and compares it with exact solutions to analyze multipole moments.

Findings

The approximate solution captures key multipole features.

Differences between exact and approximate solutions are characterized.

The FHP algorithm effectively computes relativistic multipole moments.

Abstract

The FHP algorithm allows to obtain the relativistic multipole moments of a vacuum stationary axisymmetric solution in terms of coefficients which appear in the expansion of its Ernst's potential on the symmetry axis. First of all, we will use this result in order to determine, at a certain approximation degree, the Ernst's potential on the symmetry axis of the metric whose only multipole moments are mass and angular momentum. By using Sibgatullin's method we analyse a series of exacts solutions with the afore mentioned multipole characteristic. Besides, we present an approximate solution whose Ernst's potential is introduced as a power series of a dimensionless parameter. The calculation of its multipole moments allows us to understand the existing differences between both approximations to the proposed pure multipole solution.