Measuring mass moments and electromagnetic moments of a massive, axisymmetric body, through gravitational waves

TL;DR

This paper develops a method to determine the multipole moments of a rotating, charged, axisymmetric body by analyzing gravitational waves emitted by a small orbiting test particle, linking wave measurements to the body's geometry.

Contribution

It introduces a power series approach to relate gravitational wave observables to the multipole moments of a rotating, charged body, enabling inference of its structure from wave data.

Findings

Derived power series expressions for gravitational wave quantities in terms of multipole moments.

Identified four measurable gravitational wave parameters linked to the body's multipole moments.

Provided a framework to infer the body's geometry from gravitational wave observations.

Abstract

The electrovacuum around a rotating massive body with electric charge density is described by its multipole moments (mass moments, mass-current moments, electric moments, and magnetic moments). A small uncharged test particle orbiting around such a body moves on geodesics if gravitational radiation is ignored. The waves emitted by the small body carry information about the geometry of the central object, and hence, in principle, we can infer all its multipole moments. Due to its axisymmetry the source is characterized now by four families of scalar multipole moments: its mass moments $M_l$, its mass-current moments $S_l$, its electrical moments $E_l$ and its magnetic moments $H_l$, where $l=0,1,2,...$. Four measurable quantities, the energy emitted by gravitational waves per logarithmic interval of frequency, the precession of the periastron (assuming almost circular orbits), the precession of the orbital plane (assuming almost equatorial orbits), and the number of cycles emitted per logarithmic interval of frequency, are presented as power series of the newtonian orbital velocity of the test body. The power series coefficients are simple polynomials of the various moments.