Relativistic Multipoles and the Advance of the Perihelia

TL;DR

This paper investigates how different relativistic multipolar expansions affect the perihelion advance of test particles, revealing discrepancies at higher-order terms and providing an evolution equation for radial coordinate differences.

Contribution

It compares static solutions of Einstein's equations with identical Newtonian limits to analyze relativistic multipole effects on orbital precession.

Findings

Discrepancies appear at the fourth order in the expansion parameter.

Different multipole expansions lead to measurable differences in perihelion advance.

An evolution equation for radial coordinate differences is derived.

Abstract

In order to shed some light in the meaning of the relativistic multipolar expansions we consider different static solutions of the axially symmetric vacuum Einstein equations that in the non relativistic limit have same Newtonian moments. The motion of test particles orbiting around different deformed attraction centers with the same Newtonian limit is studied paying special attention to the advance of the perihelion. We find discrepancies in the fourth order of the dimensionless parameter (mass of the attraction center)/(semilatus rectum). An evolution equation for the difference of the radial coordinate due to the use of different general relativistic multipole expansions is presented.