Tracing the geometry around a massive, axisymmetric body to measure, through gravitational waves, its mass moments and electromagnetic moments

The geometry around a rotating massive body, which carries charge and electrical currents, could be described by its multipole moments (mass moments, mass-current moments, electric moments, and magnetic moments). When a small body is orbiting around such a massive body, it will move on geodesics, at least for a time interval that is short with respect to the characteristic time of the binary due to gravitational radiation. By monitoring the waves emitted by the small body we are actually tracing the geometry of the central object, and hence, in principle, we can infer all its multipole moments. This paper is a generalization of previous similar results by Ryan. Ryan explored the mass and mass-current moments of a stationary, axially symmetric, and reflection symmetric, with respect to its equatorial plane, metric, by analyzing the gravitational waves emitted from a test body which is orbiting around the central body in nearly circular equatorial orbits. In our study we suppose that the gravitating source is endowed with intense electromagnetic field as well. Because of 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 $l=0,1,2,\dots$. Four measurable quantities, the energy emitted by gravitational waves per logarithmic interval of frequency, the precession of the periastron, the precession of the orbital plane, 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. If any of these quantities are measured with sufficiently high accuracy, the lowest moments, including the electromagnetic ones, could be inferred and thus we could get valuable information about the internal structure of the compact massive body. The fact that the electromagnetic moments of spacetime can be measured demonstrates that one can obtain information about the electromagnetic field purely from gravitational-wave analysis. Additionally, these measurements could be used as a test of the no-hair theorem for black holes.