On the gravitational field of static and stationary axial symmetric bodies with multi-polar structure

TL;DR

This paper interprets multi-polar solutions of Einstein's equations as configurations of bars with linear densities related to Legendre polynomials, and uses inverse scattering to generate solutions for rotating bodies with complex structures.

Contribution

It introduces a physical interpretation of multi-polar gravitational fields as bar configurations and develops integral representations for solution generation using inverse scattering methods.

Findings

Multi-polar solutions correspond to bar models with densities linked to Legendre polynomials.

Integral representations of the gamma function are derived for these solutions.

New solutions for rotating bodies with multi-polar structures are obtained.

Abstract

We give a physical interpretation to the multi-polar Erez-Rozen-Quevedo solution of the Einstein Equations in terms of bars. We find that each multi-pole correspond to the Newtonian potential of a bar with linear density proportional to a Legendre Polynomial. We use this fact to find an integral representation of the $\gamma$ function. These integral representations are used in the context of the inverse scattering method to find solutions associated to one or more rotating bodies each one with their own multi-polar structure.