From geodesics of the multipole solutions to the perturbed Kepler problem

TL;DR

This paper derives relativistic corrections to Keplerian orbits using multipole solutions of Einstein's equations, explicitly calculating perihelion precession influenced by quadrupole and higher moments.

Contribution

It introduces a method to connect multipole moments of a relativistic source with classical orbital equations, extending the understanding of orbital precession in general relativity.

Findings

Relativistic corrections to Keplerian motion are expressed in terms of multipole moments.

Perihelion precession is quantified considering quadrupole and higher moments.

MSA coordinates generalize Schwarzschild coordinates for orbital analysis.

Abstract

A static and axisymmetric solution of the Einstein vacuum equations with a finite number of Relativistic Multipole Moments (RMM) is written in MSA coordinates up to certain order of approximation, and the structure of its metric components is explicitly shown. From the equation of equatorial geodesics we obtain the Binet equation for the orbits and it allows us to determine the gravitational potential that leads to the equivalent classical orbital equations of the perturbed Kepler problem. The relativistic corrections to Keplerian motion are provided by the different contributions of the RMM of the source starting from the Monopole (Schwarzschild correction). In particular, the perihelion precession of the orbit is calculated in terms of the quadrupole and 2$^4$-pole moments. Since the MSA coordinates generalize the Schwarzschild coordinates, the result obtained allows measurement of the relevance of the quadrupole moment in the first order correction to the perihelion frequency-shift.