Relativistic Multipoles and the Advance of the Perihelia

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 the 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.