Spherical harmonics representation of the gravitational phase shift

We investigate the evolution of the general relativistic phase of an electromagnetic (EM) wave as it propagates in a weak gravitational field. Treating the source as an isolated, weakly aspherical gravitating body, we represent its external gravitational field using Cartesian symmetric trace-free (STF) mass multipole moments. The light propagation equation is solvable along the trajectory of a light ray to all STF orders ℓ. We express the STF moments via spherical harmonic coefficients of various degree and order, Cℓk, Sℓk. Although we focus primarily on the quadrupole (ℓ = 2), octupole (ℓ = 3), and hexadecapole (ℓ = 4) cases, our approach is valid to all orders. The result is the gravitational phase shift expressed in terms of the spherical harmonics. These results are new, useful, and important as increasingly accurate clocks are used for many practical and scientific applications, including space-based time and frequency transfers, relativistic geodesy, and navigation. We also consider tidal contributions and contributions due to the Earth’s rotation. We estimate the characteristic numerical magnitudes of each term of the resulting overall gravitational phase shift.