Time transfer and frequency shift to the order 1/c^4 in the field of an axisymmetric rotating body

TL;DR

This paper derives precise formulas for time transfer and frequency shifts caused by an axisymmetric rotating body within the post-Newtonian framework, including effects of mass, quadrupole, and spin moments up to order 1/c^4.

Contribution

It introduces a new method to calculate the influence of multipole moments on time transfer and frequency shift up to order 1/c^4 in the PPN formalism.

Findings

Explicit formulas for mass, quadrupole, and spin contributions.

Application to Earth's quadrupole and rotation effects in the ACES mission.

Complete expression of frequency shift for beta and gamma PPN parameters.

Abstract

Within the weak-field, post-Newtonian approximation of the metric theories of gravity, we determine the one-way time transfer up to the order 1/c^4, the unperturbed term being of order 1/c, and the frequency shift up to the order 1/c^4. We adapt the method of the world-function developed by Synge to the Nordtvedt-Will PPN formalism. We get an integral expression for the world-function up to the order 1/c^3 and we apply this result to the field of an isolated, axisymmetric rotating body. We give a new procedure enabling to calculate the influence of the mass and spin multipole moments of the body on the time transfer and the frequency shift up to the order 1/c^4. We obtain explicit formulas for the contributions of the mass, of the quadrupole moment and of the intrinsic angular momentum. In the case where the only PPN parameters different from zero are beta and gamma, we deduce from these results the complete expression of the frequency shift up to the order 1/c^4. We briefly discuss the influence of the quadrupole moment and of the rotation of the Earth on the frequency shifts in the ACES mission.