Spherical harmonics representation of the gravitational phase shift

We investigate the general relativistic phase of an electromagnetic wave as it propagates in gravitational field Earth, which is modeled isolated, weakly aspherical gravitating body. introduce coordinate systems to describe light propagation Earth's vicinity along with relevant transformations, and discuss transformations between proper times. represent using Cartesian symmetric trace-free (STF) mass multipole moments. The equation solvable trajectory a ray all STF orders $\ensuremath{\ell}$. Although we focus primarily on quadrupole ($\ensuremath{\ell}=2$), octupole ($\ensuremath{\ell}=3$), hexadecapole ($\ensuremath{\ell}=4$) cases, our approach valid orders. express moments via spherical harmonic coefficients various degree order, ${C}_{\ensuremath{\ell}k},{S}_{\ensuremath{\ell}k}$. result shift expressed terms harmonics. These results are new. also consider contributions due tides rotation. estimate characteristic magnitudes each term resulting overall shift. assessed either large enough impact current-generation clocks or will become significant future-generation offer greater sensitivity. Our formulation useful for many practical scientific applications, including space-based time frequency transfers, geodesy navigation, well quantum communication links tests fundamental physics.